Behaviour of the Order of Tate–Shafarevich Groups for the Quadratic Twists of $$X_0(49)$$

Dąbrowski, Andrzej; Jędrzejak, Tomasz; Szymaszkiewicz, Lucjan

doi:10.1007/978-3-319-45032-2_4

Behaviour of the Order of Tate–Shafarevich Groups for the Quadratic Twists of $X_0(49)$

Andrzej Dąbrowski³,
Tomasz Jędrzejak³ &
Lucjan Szymaszkiewicz³

Conference paper
First Online: 16 January 2017

1340 Accesses

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 188))

Abstract

We present the results of our search for the orders of Tate–Shafarevich groups for the quadratic twists of $E=X_0(49)$.

Dedicated to John Coates on his seventieth birthday.

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Author information

Authors and Affiliations

Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451, Szczecin, Poland
Andrzej Dąbrowski, Tomasz Jędrzejak & Lucjan Szymaszkiewicz

Authors

Andrzej Dąbrowski
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Tomasz Jędrzejak
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Lucjan Szymaszkiewicz
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Corresponding author

Correspondence to Andrzej Dąbrowski .

Editor information

Editors and Affiliations

Mathematics Institute, University of Warwick, Coventry, United Kingdom
David Loeffler
Department of Mathematics, University College London, London, United Kingdom
Sarah Livia Zerbes

Appendices

Appendix: The Algorithm and the Implementation

The strategy is to use the construction described in the end of Sect. 2 to compute the coefficients $a_d$ for d satisfying condition (**) up to $32 \cdot 10^{9}$, and use Corollary 1.

We present our algorithm using computer algebra system PARI/GP [18].

The key point of the above implementation is using the qfrep function. Recall that qfrep ( q , B ) returns the vector whose i-th entry $(1 \leqslant i \leqslant B)$ is half the number of vectors v such that $q(v) = i$. Routine qfrep relies on qfminim function which enumerates, using the Fincke–Pohst algorithm, the vectors v for which $q(v) \leqslant B$.

We used the above PARI/GP script (with small modifications) to compute sha( B ) for $B=5\cdot 10^7$ (on standard desktop PC). We made further progress implementing qfrep function in C++ language. Our routine directly enumerates all vectors v for which $f_i(v)\leqslant B$ for $i=1,2,3,4$. Such a straightforward approach is more effective for forms $f_1, f_2, f_3, f_4$ than the sophisticated Fincke–Pohst algorithm used by PARI/GP. Moreover, we made some optimizations. Most important among them are:

time optimization—exploiting symmetries of $f_1, f_2, f_3, f_4$,
memory optimization—storing in memory only values of $a_d$ for $d\equiv 1\pmod 4$.

Enumerating numbers satisfying condition (**) (and factoring them) also takes some time. One can be speed it up by filtering out numbers which are not square-free. We did it by using a modified sieve of Eratosthenes. However, the real bottleneck is computing qfrep.

Let us add that our algorithm is quite easily parallelizable. It appears that performing computation in parallel will make possible to increase B substantially compared to our achievement.

Tables

For each positive integer $k\leqslant 1793$, the column headed $d_1$ gives the smallest integer d for which . One interesting observation is that all odd orders are realized by the integers $d_1$ satisfying the condition (*).

k	$d_1$	k	$d_1$	k	$d_1$	k	$d_1$
1	1	2	93	3	73	4	177
5	257	6	933	7	929	8	4337
9	2281	10	6073	11	3169	12	6609
13	5897	14	14177	15	12241	16	20497
17	10937	18	19713	19	5641	20	52257
21	18793	22	40769	23	31513	24	63473
25	26249	26	55617	27	23369	28	63849
29	62929	30	121881	31	49993	32	152769
33	65609	34	100857	35	62401	36	167073
37	98257	38	322921	39	96353	40	226913
41	133769	42	206273	43	151273	44	734001
45	110977	46	337681	47	129457	48	498129
49	253553	50	549817	51	152953	52	518137
53	152249	54	702353	55	291457	56	612529
57	247369	58	673817	59	368857	60	953313
61	365249	62	964793	63	626377	64	847793
65	290657	66	1319649	67	527729	68	1217049
69	536017	70	1091841	71	957361	72	2060353
73	637297	74	1501329	75	423097	76	1135649
77	1465469	78	1707729	79	955769	80	1827193
81	570113	82	2874369	83	682009	84	1234137
85	1101593	86	2827553	87	1899481	88	2229529
89	1885673	90	2341817	91	1323689	92	2799217
93	1381337	94	3018513	95	1242169	96	2904801
97	1917697	98	4294313	99	1790897	100	3567881
101	1625321	102	4518273	103	1866857	104	5884041
105	1781569	106	4184049	107	2915713	108	6165329
109	2182249	110	4479897	111	3647689	112	4909017
113	1465313	114	5427489	115	2761841	116	6469849
117	2687257	118	6350073	119	3393449	120	4884177
121	3524041	122	7486329	123	3485513	124	7240809
125	3613193	126	4935001	127	4229657	128	7353921
129	3486257	130	7753601	131	4459601	132	4110177
133	4693177	134	8832657	135	3247313	136	8314777
137	4296977	138	10538889	139	5507297	140	11180073
141	3688081	142	10114889	143	6025801	144	6302409
145	3653369	146	14245449	147	5294833	148	11250761
149	3106921	150	10362081	151	5946337	152	13688313
153	6790073	154	10240521	155	7491361	156	15464089
157	4103641	158	10692817	159	7016777	160	11928953
161	6718193	162	15799897	163	6645721	164	13927593
165	10108297	166	12971121	167	6120929	168	19275657
169	10688753	170	21326937	171	6078337	172	9310449
173	8860361	174	16962969	175	9539281	176	23387809
177	9032609	178	18940849	179	9425113	180	24630321
181	9843529	182	21259921	183	5634809	184	20181561
185	15130393	186	18589729	187	10534921	188	26802777

k	$d_1$	k	$d_1$	k	$d_1$	k	$d_1$
189	9128969	190	18017673	191	15028201	192	27317361
193	17238961	194	27932529	195	13267873	196	18475521
197	19233889	198	32768481	199	18412817	200	17722329
201	14262433	202	35158337	203	17371513	204	31450409
205	15861473	206	55430673	207	17282137	208	25378833
209	16847953	210	20905977	211	13179577	212	30805273
213	17764913	214	47121369	215	14565193	216	24423177
217	17119073	218	31973313	219	20660377	220	30225721
221	20663593	222	32356041	223	24736529	224	31522353
225	17100961	226	44057561	227	23429761	228	61238433
229	20035217	230	30310809	231	17521937	232	45713721
233	26153209	234	53720529	235	19521001	236	30965713
237	17479313	238	45580921	239	19624729	240	59076249
241	24796313	242	46666337	243	15196457	244	47964921
245	24126161	246	67880649	247	15737417	248	59498961
249	27527393	250	67310681	251	31900529	252	71401089
253	21488809	254	42480201	255	21141041	256	62559121
257	21436001	258	44968137	259	23661529	260	68143553
261	28188257	262	83482809	263	45616297	264	74407953
265	33502577	266	76802441	267	45721681	268	56817777
269	30511001	270	80564961	271	42257857	272	65262849
273	30407369	274	51619593	275	34562401	276	86165913
277	28530241	278	100543353	279	29771201	280	55275609
281	33775801	282	102490809	283	38382041	284	89933937
285	27594521	286	94586473	287	41793233	288	84939537
289	47313209	290	94270929	291	25854097	292	74695377
293	49080337	294	129084873	295	48796537	296	97992497
297	41571113	298	108653521	299	63138337	300	114844137
301	28987073	302	117318657	303	70938377	304	120142353
305	54726241	306	106517777	307	62983121	308	108212241
309	54211177	310	132195057	311	57847201	312	107424529
313	64804081	314	136571681	315	67153729	316	72567049
317	39044641	318	103714801	319	68778097	320	124306361
321	67515881	322	99630969	323	42683441	324	126667249
325	81725521	326	128132673	327	22476089	328	69421713
329	82804153	330	131429937	331	35634569	332	103211529
333	53915137	334	127330809	335	58071737	336	145127361
337	82549193	338	127179537	339	81256817	340	212734713
341	94701017	342	120707049	343	74528177	344	180657537
345	82682417	346	163723673	347	83092201	348	127062681
349	64266313	350	158078897	351	45363041	352	165215121
353	86228729	354	150696393	355	74605177	356	158006489
357	96923641	358	168255201	359	101236001	360	199147433
361	61967953	362	208401153	363	75284753	364	179223529
365	81361481	366	190068321	367	128132273	368	233660249
369	104815049	370	166966409	371	78813817	372	166934289
373	85236353	374	259338137	375	100124561	376	150853497
377	86855849	378	195302193	379	117023129	380	170908593

k	$d_1$	k	$d_1$	k	$d_1$	k	$d_1$
381	119282593	382	229290681	383	124208657	384	250172553
385	84921329	386	242076657	387	81585241	388	198355873
389	111613529	390	317098497	391	68312473	392	164772969
393	137121073	394	317478617	395	184317257	396	267280841
397	157495033	398	198455217	399	95602057	400	201599049
401	141095993	402	315033657	403	162887129	404	252130649
405	129060721	406	239496657	407	75453481	408	309730961
409	145253513	410	263668161	411	114535441	412	258161649
413	70924577	414	201012681	415	156669217	416	348700969
417	133510961	418	255577281	419	133337329	420	243488697
421	163103713	422	349629081	423	129344561	424	248961057
425	113424457	426	158340513	427	161716729	428	457284881
429	149221609	430	306979737	431	188287097	432	263466921
433	113174249	434	379096881	435	116677553	436	341102721
437	138979921	438	344625297	439	142217729	440	402660633
441	173153249	442	337717857	443	124106569	444	303693153
445	187920529	446	348974817	447	141120313	448	374142729
449	169920161	450	415195257	451	119896241	452	456636161
453	172998929	454	370842873	455	186067649	456	284126217
457	211471489	458	322781001	459	175500121	460	424009721
461	175321193	462	383282377	463	227914553	464	411552969
465	151632193	466	405886881	467	160286201	468	493254001
469	133377289	470	381535529	471	150114793	472	467868809
473	199481561	474	367048641	475	231417217	476	428448673
477	216300353	478	464970993	479	182712193	480	377366529
481	203950673	482	481370849	483	208906417	484	466546089
485	221357009	486	611992833	487	213593561	488	445591689
489	152807297	490	365071281	491	227448577	492	324398649
493	152292209	494	327557561	495	158411977	496	455620881
497	235272161	498	298850577	499	159619337	500	487175313
501	87873329	502	433020569	503	209599633	504	468120129
505	233102873	506	680179593	507	248435521	508	409342217
509	167807489	510	499217601	511	201167833	512	502073881
513	256670657	514	601398561	515	265990297	516	591746313
517	225790673	518	474463697	519	244563961	520	560288513
521	191110121	522	724247857	523	259344209	524	455548713
525	263635321	526	426422649	527	313004473	528	484397841
529	231144521	530	556803769	531	257557873	532	652226129
533	328620697	534	579162081	535	335578081	536	618274897
537	312412721	538	888968217	539	291056657	540	577412049
541	243334657	542	871280337	543	389624233	544	484212369
545	320529089	546	874368489	547	260262113	548	784207257
549	298280401	550	598627929	551	359759921	552	374213393
553	217628393	554	840530793	555	258609433	556	789059793
557	376545137	558	899077953	559	310206713	560	849373977
561	276498809	562	720256769	563	359961713	564	605796249
565	505198489	566	660711921	567	374576513	568	693060153
569	285955057	570	582042129	571	194086553	572	971076633

k	$d_1$	k	$d_1$	k	$d_1$	k	$d_1$
573	269512417	574	812238113	575	394565449	576	531793497
577	376637449	578	558769737	579	177118241	580	800651713
581	314235617	582	971046561	583	278549137	584	713190633
585	343632217	586	910562577	587	299414377	588	806580569
589	339441217	590	969882153	591	404910817	592	1180596561
593	445593769	594	746080449	595	366680009	596	588145881
597	499709489	598	1000317921	599	453223081	600	857027697
601	482163793	602	709173529	603	464499577	604	1085969329
605	583993777	606	611140737	607	477113129	608	767941113
609	487518937	610	1205784057	611	345190481	612	1042100481
613	433569953	614	759143361	615	295290449	616	1060039817
617	271819777	618	916876041	619	399513761	620	833937729
621	351775609	622	1014335921	623	461803457	624	1063477353
625	604768433	626	1195237257	627	355313929	628	1130731233
629	356217217	630	1088179233	631	523841737	632	1296431537
633	373251833	634	1046573089	635	533158321	636	1129661633
637	437185369	638	1001125833	639	690863353	640	1049542113
641	393691721	642	1219662481	643	442113409	644	866810121
645	473179657	646	918013177	647	422803841	648	925311633
649	628298753	650	908022641	651	416028409	652	668155161
653	639545801	654	788481633	655	456778633	656	1253196393
657	553788233	658	1011782193	659	559500833	660	1095831129
661	502398097	662	1556083761	663	540229913	664	1511104233
665	487247689	666	1530974993	667	419408113	668	1006678033
669	557895281	670	1502333529	671	748804241	672	954448401
673	431472649	674	1503531033	675	567004513	676	1390863849
677	486471529	678	845750049	679	563530553	680	1401556137
681	472466377	682	1131586201	683	774729497	684	1153585497
685	424046153	686	1292386929	687	384060241	688	969826881
689	677852257	690	1146387201	691	472867673	692	1284311337
693	590461073	694	1448799433	695	552263017	696	1430856177
697	487192513	698	1065709793	699	789946601	700	1272089361
701	617717609	702	1305583953	703	622046417	704	1815425929
705	510979577	706	886583769	707	571233809	708	1259122593
709	634656713	710	1451816193	711	769553489	712	1902558001
713	919520761	714	1671253809	715	555591601	716	1737287913
717	690268577	718	1691545857	719	758546561	720	1417870169
721	626618809	722	2027470713	723	645226937	724	1660673337
725	357461201	726	1159545633	727	803008321	728	1172959881
729	645947833	730	1532369217	731	786522889	732	1220017089
733	822906313	734	1812159633	735	723749137	736	1677838089
737	797421113	738	1334772609	739	533625049	740	2156546177
741	898149521	742	1908161193	743	777702721	744	1475258537
745	764284097	746	1932838161	747	774980849	748	1597045321
749	805117801	750	1649813817	751	912356273	752	1922828889
753	535552681	754	2049963393	755	618088673	756	1439459169
757	1012636441	758	1734170937	759	704129537	760	2013014697
761	673372577	762	2328013713	763	832572401	764	1695155961

k	$d_1$	k	$d_1$	k	$d_1$	k	$d_1$
765	920887169	766	1176556089	767	831201097	768	1843728657
769	786715537	770	2185812897	771	637134697	772	1860573801
773	623324777	774	2428781793	775	854043041	776	1309574841
777	413798201	778	2198450769	779	790908361	780	1722152721
781	1058446409	782	1562389249	783	673866737	784	1600255897
785	859210817	786	2201361121	787	845971769	788	2347753809
789	1009504417	790	2403252273	791	780662161	792	2485110489
793	991564633	794	1670427609	795	988737601	796	1304526953
797	851169289	798	2039945329	799	909129017	800	1739103809
801	936222409	802	1575219297	803	1063972649	804	2389858689
805	810098873	806	2374269969	807	838474873	808	1497563313
809	980245633	810	1902373617	811	1228354273	812	1950931833
813	636479009	814	1263787449	815	1314825769	816	2641549089
817	691199521	818	2555306121	819	839732129	820	2539277841
821	880782593	822	2059914081	823	1530012697	824	1619827449
825	1079910089	826	2651407089	827	877717273	828	2528252409
829	1123787593	830	2115598521	831	879991633	832	2757188417
833	910187017	834	2576804673	835	1162667929	836	2967468729
837	1194444817	838	1496186553	839	1092092489	840	2828807529
841	1168598161	842	1741340841	843	1217448433	844	2961741017
845	1055512121	846	2086860777	847	1176846961	848	2857144161
849	1145287009	850	3129250569	851	1359540241	852	2734116361
853	1270901369	854	2849667881	855	1316031401	856	4109398617
857	1192513873	858	2589301401	859	1747673497	860	2374378953
861	1665814057	862	2457866009	863	864297857	864	2597941641
865	1105681657	866	2126163873	867	1474091393	868	2422942097
869	813330241	870	2587316257	871	1215280433	872	2864765121
873	1055546617	874	2955604521	875	1588324849	876	2540162769
877	1166140897	878	1740197649	879	1071471217	880	3251124937
881	1121666929	882	2896790241	883	1071890161	884	2631081489
885	1370348857	886	2229607857	887	1314894769	888	2641504209
889	1094391521	890	3179094297	891	1421740409	892	2424853489
893	2003784313	894	2601422553	895	1324413313	896	3435753297
897	1457825417	898	3103542033	899	1602399217	900	4071247833
901	798447257	902	3530623273	903	1367841121	904	2494992417
905	1568185337	906	1507151409	907	1403055569	908	3180171369
909	720843289	910	3779810841	911	1697289089	912	2537937033
913	1444968089	914	3481002897	915	1492811897	916	3819843393
917	1201733153	918	2923319177	919	1520806369	920	3225574697
921	1219732793	922	3645265017	923	1480790041	924	2561150937
925	1303714201	926	2471502009	927	1672411249	928	3086409481
929	1134207353	930	3259442041	931	1342180313	932	3616278537
933	1823027209	934	3784692993	935	1818849121	936	3746518561
937	1919893561	938	4533867321	939	1944481873	940	2560471321
941	1902900761	942	2760382353	943	1998745937	944	4549819233
945	1727638009	946	2346199689	947	1791347633	948	3866224849
949	1776360337	950	2284988073	951	1859912113	952	3095542209
953	1968641401	954	4091400681	955	1242925193	956	4225706769

k	$d_1$	k	$d_1$	k	$d_1$	k	$d_1$
957	1892915281	958	4045337521	959	1343608489	960	3763601697
961	1814149921	962	4314724129	963	1530725993	964	3609639993
965	2095192361	966	2436844953	967	1351773289	968	3652908681
969	1829988737	970	3910470657	971	1267624417	972	3916603681
973	1608509857	974	3902482497	975	2267415617	976	4570911097
977	1857529337	978	4374788097	979	2397032977	980	5057369153
981	1670595169	982	4467133353	983	1846662313	984	4018295033
985	1762663633	986	3335288433	987	1284164017	988	4156050761
989	1285009081	990	4916862897	991	2476064449	992	4493035633
993	1271996969	994	4194691593	995	1940749649	996	4604658081
997	1711463473	998	4304953977	999	1545011873	1000	5834979417
1001	1847473897	1002	4000561161	1003	2587251313	1004	4444371537
1005	1970441177	1006	3405718369	1007	1382096873	1008	5178973193
1009	1676958233	1010	3411899313	1011	1764844217	1012	3933861297
1013	2665391921	1014	4085187033	1015	2155796977	1016	4976692977
1017	1727206529	1018	4202383529	1019	2238334649	1020	4709707473
1021	1906163689	1022	4994064329	1023	2099589449	1024	3672154313
1025	2398817921	1026	4792036593	1027	2531008121	1028	6935325609
1029	2054505073	1030	3141709761	1031	2776699313	1032	5715222753
1033	2134171513	1034	4244558761	1035	1927111289	1036	5127372177
1037	1838393969	1038	4227048849	1039	2140121833	1040	2998130601
1041	2288746241	1042	4985786569	1043	2278021241	1044	3622165017
1045	1723885337	1046	4033170177	1047	1948369009	1048	4866766521
1049	2181026153	1050	4926721809	1051	1899948697	1052	4513174257
1053	1780107257	1054	6008035881	1055	1981477217	1056	4505164217
1057	2128430041	1058	3772506593	1059	2822718281	1060	4274440017
1061	1704499681	1062	4783736721	1063	1541303833	1064	4850259177
1065	2098010809	1066	5239266033	1067	2637658841	1068	4392626921
1069	2149662329	1070	5038266297	1071	2158972121	1072	4546632633
1073	2086733353	1074	3829269273	1075	2998234721	1076	6655000377
1077	2648174233	1078	5672683977	1079	2252184889	1080	5032809849
1081	1771959913	1082	5685794529	1083	2053479553	1084	5816631921
1085	1511663233	1086	4288434441	1087	2787474721	1088	4952525601
1089	2244439553	1090	6708573953	1091	3044027353	1092	4837226241
1093	2746126073	1094	7094498513	1095	2877152249	1096	6873218697
1097	1772069249	1098	4869092089	1099	2817189961	1100	4743326977
1101	3022586257	1102	4954226601	1103	2498748409	1104	6842596993
1105	2487064193	1106	7970709993	1107	2353272241	1108	5216995617
1109	1926922913	1110	4832534281	1111	2792473897	1112	5454046977
1113	3116936497	1114	6769807377	1115	2517392177	1116	5703735801
1117	2671437473	1118	7489651409	1119	2477257033	1120	5843140473
1121	2207569633	1122	6967177233	1123	3071942201	1124	5210538337
1125	2860318481	1126	4725738361	1127	2158369337	1128	4380257841
1129	2955926249	1130	6026522217	1131	2854320649	1132	8687265337
1133	3132253633	1134	3941363721	1135	3007750849	1136	8538508113
1137	2660576873	1138	4614141057	1139	2714667961	1140	5295320529
1141	2520918889	1142	5026383321	1143	3100841593	1144	5145053313
1145	3892302041	1146	6648477441	1147	3307165321	1148	7713100497

k	$d_1$	k	$d_1$	k	$d_1$	k	$d_1$
1149	3459028681	1150	8423248593	1151	3855433433	1152	6303556081
1153	4081831337	1154	8181165073	1155	2370744913	1156	5505018033
1157	3265573513	1158	6115577073	1159	3420560993	1160	8620240729
1161	2074438369	1162	6685243161	1163	3091638289	1164	6461731329
1165	2557770689	1166	8386327569	1167	3909415849	1168	7717795649
1169	3346828609	1170	7157766009	1171	2546895473	1172	6856502937
1173	2793810361	1174	6047040057	1175	2157996097	1176	7838284001
1177	3455089697	1178	6647141153	1179	2842924577	1180	3590709753
1181	2476686001	1182	6121270401	1183	3243393473	1184	8151936081
1185	2750240993	1186	8253927273	1187	4210292057	1188	7049052497
1189	3727000777	1190	6917832177	1191	1880429633	1192	8525295417
1193	2945848097	1194	7813345929	1195	4223004449	1196	7625410321
1197	2935348321	1198	8133389889	1199	3453727913	1200	9033894193
1201	3900859073	1202	7929221217	1203	4421113777	1204	5713891737
1205	3245362609	1206	8728800369	1207	2985830417	1208	9863502889
1209	3775474961	1210	8046600153	1211	3447639473	1212	4400227329
1213	2407830617	1214	8811977777	1215	2882908073	1216	7791324753
1217	2283976033	1218	4647909473	1219	3339966881	1220	9564956097
1221	2923551601	1222	6811015929	1223	3830663849	1224	7028709697
1225	4063280401	1226	6696778449	1227	3844261441	1228	8576958993
1229	3535227961	1230	7620325329	1231	3092311897	1232	8908428153
1233	3776364689	1234	9188637753	1235	4696847321	1236	7724221737
1237	3856164217	1238	8480502209	1239	3861511657	1240	9144421801
1241	3395460017	1242	9294254817	1243	4180275881	1244	7201797177
1245	4078091713	1246	8112886233	1247	4499842297	1248	7881839529
1249	3775972633	1250	7019585673	1251	4105962433	1252	5848850001
1253	3991021849	1254	6065628801	1255	3564439913	1256	10740230169
1257	3463552753	1258	9561871929	1259	3185608033	1260	6169512273
1261	4293851609	1262	9078298377	1263	3156206177	1264	8582985849
1265	3897792881	1266	10075921769	1267	4139956313	1268	10380813601
1269	4570493801	1270	9356528801	1271	3890225513	1272	9442108113
1273	5438791777	1274	9367475217	1275	4870511089	1276	11355293009
1277	4570534433	1278	5457766953	1279	4582329937	1280	7966743801
1281	3303470393	1282	10603149393	1283	2858681489	1284	8329309041
1285	4838698793	1286	7925188569	1287	4214651369	1288	6365154713
1289	4383924977	1290	7700745369	1291	4783441193	1292	10899763737
1293	3183701393	1294	7947641353	1295	4216864601	1296	10375292481
1297	4884197377	1298	8881826633	1299	3896076089	1300	12504176953
1301	4842107689	1302	5756693721	1303	2723480233	1304	10029103001
1305	5075648857	1306	10231827369	1307	4881446321	1308	10362529857
1309	4663594841	1310	8630540241	1311	4361721577	1312	6785253561
1313	4473287153	1314	11339103273	1315	5213216977	1316	9030929009
1317	3765257761	1318	8397845481	1319	5921712193	1320	10518869337
1321	3553813609	1322	10411769897	1323	4880881361	1324	6586436201
1325	4041084001	1326	9637389353	1327	4263481897	1328	10794914769
1329	4020541969	1330	8861462817	1331	4655094673	1332	10320158377
1333	4225091729	1334	12687213841	1335	4849883897	1336	8635792713
1337	5056156097	1338	11247364209	1339	3533357441	1340	12338265921

k	$d_1$	k	$d_1$	k	$d_1$	k	$d_1$
1341	6188174921	1342	10179842689	1343	4931635369	1344	10717061457
1345	4992350177	1346	9914338641	1347	3640667057	1348	10149929057
1349	5178899449	1350	11908911953	1351	3790717769	1352	9688139169
1353	2687142617	1354	11319565489	1355	5446670857	1356	10184514657
1357	4413137921	1358	11811948801	1359	5491510897	1360	14981176697
1361	6845363297	1362	11520637489	1363	5726707801	1364	9844492793
1365	4074339769	1366	13189267281	1367	4590877273	1368	10263989121
1369	6090656801	1370	11577597441	1371	5360341369	1372	8969163897
1373	3729738193	1374	10030041849	1375	5790874417	1376	13672932177
1377	3354967201	1378	9721180921	1379	3593612993	1380	11572057281
1381	2975326049	1382	11990306353	1383	5409192193	1384	12071006817
1385	6181709449	1386	14069494849	1387	6109066921	1388	11024766993
1389	5274319777	1390	9642295441	1391	6249679649	1392	11173712089
1393	3920782057	1394	14970384881	1395	4291103633	1396	14873830449
1397	4970225489	1398	14282140161	1399	3113018473	1400	9907899873
1401	5602055801	1402	13224194153	1403	4388529953	1404	9080218569
1405	5874339553	1406	14392574233	1407	5313687361	1408	8070174249
1409	5340878497	1410	8445957249	1411	5799147161	1412	11307387921
1413	6256545641	1414	14603912313	1415	5182399633	1416	11291079633
1417	5761864793	1418	8611063521	1419	4283780297	1420	11965703817
1421	4919896369	1422	13148498001	1423	5418737449	1424	11819709609
1425	6452481929	1426	8849047089	1427	6464797601	1428	10724557281
1429	5850358097	1430	11668121193	1431	4450743881	1432	14117182377
1433	7530295477	1434	11005822041	1435	4781099249	1436	16464319257
1437	5625246553	1438	15200800833	1439	6232068121	1440	11543895273
1441	5582017241	1442	11467212681	1443	6162816449	1444	12587849569
1445	4505553209	1446	9179582801	1447	4759466801	1448	12803192537
1449	5190934153	1450	13376175873	1451	5026362641	1452	17479263937
1453	6196053953	1454	12803941329	1455	6038713553	1456	14943995697
1457	5826184769	1458	12952966161	1459	8336832073	1460	12575735841
1461	6697005209	1462	12862360281	1463	6953992769	1464	14337419209
1465	5551779049	1466	13519359177	1467	5740147537	1468	11113911169
1469	6633294233	1470	10581863009	1471	6089924993	1472	14453700969
1473	6641047097	1474	13050281721	1475	6676376873	1476	13635058089
1477	4879320689	1478	11540902641	1479	5218748209	1480	15361738681
1481	5142288889	1482	8955885801	1483	8016755057	1484	12068317977
1485	5782637041	1486	14976660297	1487	7880353297	1488	16452839881
1489	6484799473	1490	9905393521	1491	5707371641	1492	17674712841
1493	6689379689	1494	12127943433	1495	6879464849	1496	16182669297
1497	4306434697	1498	14691262881	1499	8152320377	1500	15122451873
1501	6323595697	1502	14457595441	1503	6009027289	1504	13185180849
1505	5685159793	1506	14022943169	1507	8097404209	1508	16948680681
1509	6547557281	1510	14908391969	1511	7678220273	1512	19449660561
1513	6075080353	1514	17372940729	1515	6823802473	1516	10667702049
1517	7286535961	1518	16155325929	1519	6453592841	1520	14756469729
1521	7044532081	1522	11995984617	1523	6981504113	1524	17180091273
1525	7773449641	1526	19556487329	1527	6239985569	1528	18678344721
1529	7319054153	1530	9633634761	1531	8359447409	1532	19243693401

k	$d_1$	k	$d_1$	k	$d_1$	k	$d_1$
1533	5383103641	1534	15336632257	1535	5435424889	1536	16518110361
1537	6493576057	1538	11799669513	1539	6261258617	1540	11453876897
1541	7007402449	1542	16811621457	1543	7119113321	1544	15084189201
1545	5032367849	1546	19853096433	1547	6206696401	1548	18095827017
1549	5585570233	1550	12228799353	1551	8651999977	1552	13545865481
1553	7061734417	1554	15351766761	1555	7881434881	1556	19088671361
1557	6358784129	1558	17521254633	1559	8866491401	1560	14777854617
1561	5406250657	1562	16055379201	1563	8111629553	1564	18455413729
1565	7557590969	1566	19354854017	1567	6376897561	1568	11162316201
1569	10325243801	1570	14174249937	1571	8346350809	1572	14805992793
1573	8639085097	1574	19876647729	1575	7310427353	1576	13383109041
1577	7692909481	1578	17923856217	1579	7980327521	1580	13791208929
1581	4445597953	1582	14074036089	1583	5656117361	1584	17916753801
1585	9527118401	1586	14217349353	1587	7082431481	1588	16185164297
1589	6205767769	1590	16566006801	1591	9414671033	1592	19394644593
1593	9484650257	1594	10596671121	1595	8488474417	1596	13021079817
1597	7823658209	1598	24921801969	1599	9490053809	1600	19530979913
1601	9941854033	1602	11917095513	1603	8980051961	1604	17863582801
1605	8408412017	1606	19317873369	1607	9030034649	1608	21150606513
1609	10040135537	1610	22641182337	1611	7404361369	1612	18619595009
1613	11724971569	1614	14844290169	1615	8054991409	1616	18427052337
1617	7654546177	1618	20158634841	1619	8074096649	1620	10101987681
1621	6833900393	1622	21247408473	1623	9205065929	1624	19442890041
1625	9320044529	1626	20777329497	1627	7474670633	1628	16407640929
1629	8062549801	1630	19745737113	1631	5708233177	1632	19947287721
1633	7857319273	1634	24913676649	1635	10897495537	1636	18635575281
1637	9765563569	1638	22715248593	1639	6518086921	1640	11710935681
1641	10877325889	1642	24782078073	1643	7997553217	1644	17307309497
1645	9688815713	1646	20375827873	1647	8266590337	1648	21286015017
1649	6771151313	1650	21418425473	1651	7759261313	1652	17058280017
1653	7055863681	1654	23824112489	1655	7797935281	1656	22941611697
1657	6898160657	1658	19080932649	1659	8573955281	1660	21767826777
1661	11803489417	1662	20306822649	1663	10557367441	1664	19351782129
1665	9643518041	1666	23188938609	1667	8053982201	1668	17951834217
1669	9542343233	1670	26324496353	1671	8320139033	1672	14965707817
1673	9223372409	1674	16030498793	1675	12397060721	1676	21302932753
1677	7388864993	1678	20675197713	1679	8424181121	1680	15850391313
1681	9979688393	1682	23169114809	1683	9821011049	1684	21532687521
1685	9032961017	1686	17768279433	1687	8223478961	1688	19766229081
1689	12157301161	1690	18187374489	1691	10753068737	1692	18319821657
1693	9909960329	1694	26639467017	1695	10258117313	1696	21471116241
1697	13692281329	1698	15424707057	1699	6648674609	1700	22105438041
1701	10328316337	1702	23340311481	1703	7672509353	1704	19865714313
1705	7345957961	1706	20083328049	1707	12110334193	1708	12603492609
1709	14665471217	1710	21153612153	1711	12214804297	1712	16901015217
1713	7883012129	1714	20265748617	1715	11526296753	1716	17900661153
1717	9022814057	1718	23173716129	1719	8558039537	1720	18317261649
1721	9755539897	1722	26642566113	1723	11295454553	1724	21852465793

k	$d_1$	k	$d_1$	k	$d_1$	k	$d_1$
1725	9820146593	1726	25820071817	1727	13386871657	1728	19234668121
1729	10200354113	1730	22356100321	1731	10496099857	1732	15563628777
1733	10280518337	1734	19458436089	1735	10344401849	1736	15334875609
1737	12478800689	1738	22729799577	1739	13185137881	1740	31556733609
1741	11977301593	1742	13991205561	1743	9726851417	1744	26735056217
1745	7959615929	1746	20732047473	1747	13544803529	1748	23934761337
1749	10109738497	1750	15138620841	1751	10531936889	1752	26307135033
1753	11742293489	1754	15293415993	1755	11740768681	1756	21444330777
1757	11408438473	1758	26855399481	1759	9561059017	1760	19279833721
1761	9953890057	1762	25903870689	1763	13747239089	1764	23244127729
1765	10512541913	1766	29607980417	1767	14566400561	1768	31222130153
1769	13852180417	1770	29675608953	1771	9052924193	1772	21842571921
1773	12653890049	1774	21060861537	1775	10345905937	1776	22913920929
1777	11299193849	1778	23773736001	1779	12959578369	1780	27545395593
1781	11502866521	1782	27768982497	1783	12903652081	1784	28861680633
1785	12271976993	1786	24459731697	1787	10862200793	1788	15961497577
1789	10612904713	1790	24324206537	1791	10638082801	1792	31848889713
1793	10500588257

Selected values $1795 \leqslant k \leqslant 2941$ are also considered.

k	$d_1$	k	$d_1$	k	$d_1$	k	$d_1$
1795	14302640609	1796	27358359081	1797	11675814881	1798	27273950169
1799	12288253057	1800	29409340097	1801	10747223609	1802	17116275873
1803	10761662033	1805	11795108209	1806	24360921033	1807	14006501017
1809	11315214497	1810	30452439257	1811	10525057129	1812	28348764753
1813	8922565193	1814	31795855249	1815	10367096249	1816	28742795673
1817	8638659049	1818	24484322193	1819	13819473449	1820	15894922737
1821	11682471617	1822	28719137049	1823	10534815857	1824	17666542041
1825	10081327513	1827	12077934553	1828	22366484121	1829	14616252913
1830	20798720121	1831	11063929649	1832	28289473809	1833	8002632881
1834	25611350433	1835	9212193857	1836	27665418993	1837	16898429161
1838	26484256857	1839	9016016953	1840	24896737497	1841	15427422041
1842	26136546297	1843	10786881689	1844	28223186289	1845	16636782473
1847	12676735273	1849	7747069097	1850	28252003081	1851	10879109849
1852	26361529041	1853	13897278121	1854	21989559201	1855	10695553697
1856	27284626329	1857	14003640817	1858	27657269313	1859	12177360529
1860	21356618529	1861	11888816113	1862	30328356601	1863	9906836593
1864	21976435497	1865	14472946801	1866	20748624513	1867	11552645537
1868	27402172977	1869	11303255617	1870	24913897737	1871	8443601753
1872	31756304409	1873	16026449393	1874	29842152657	1875	12307826081
1876	27713889569	1877	9654472721	1879	15210808849	1880	28870006929
1881	15669411673	1882	28934387553	1883	16555237537	1885	11660985689
1886	27292507377	1887	14953069561	1889	10128518657	1891	17239580153
1892	30427087497	1893	11212691801	1894	30501258393	1895	15299539457
1896	27590751057	1897	13077468809	1898	31827861641	1899	11483109641
1901	12606717017	1903	15584020681	1905	11813267161	1906	16367835009
1907	18412667969	1909	11836986241	1910	21635667673	1911	13272833377
1912	30472364553	1913	12531171217	1914	23353861521	1915	8202143393

k	$d_1$	k	$d_1$	k	$d_1$	k	$d_1$
1917	14755875137	1918	25708307601	1919	15066358481	1920	22043395081
1921	9023034793	1923	11690362561	1925	14786098913	1927	13989520417
1929	10757106673	1930	26996413497	1931	11823613913	1933	16867973321
1935	15576283369	1937	16795975817	1939	17334712729	1941	15871885753
1942	31072472337	1943	12023446697	1945	10890996689	1946	22747002657
1947	16774867913	1948	27535083801	1949	14984974273	1950	23125134057
1951	15365245153	1952	28463088977	1953	14557374529	1954	29968558449
1955	16991135209	1956	29772869289	1957	18647799593	1958	30016450209
1959	17536828609	1961	16387251809	1963	16939173761	1964	24017100321
1965	17519090473	1966	25475162953	1967	18063913321	1969	19189797049
1971	16094600017	1973	19185850561	1975	22692362201	1977	13792212017
1979	16926239609	1981	18654590113	1982	31863559753	1983	11258242889
1984	26625714369	1985	17771644441	1987	19770734129	1989	16941181249
1991	16964769281	1993	18166946593	1995	19893527657	1997	10175019889
1999	18426841921	2001	16120899073	2003	10683271289	2005	19279072217
2007	20008876177	2009	15135767321	2011	12772102033	2013	15387373321
2015	17258610449	2016	26923237881	2017	15488878849	2019	20066054209
2021	14898157433	2022	25753502769	2023	19001943929	2025	17314824481
2027	11829608209	2029	16940925433	2031	19334270129	2033	23685116761
2035	18311081017	2037	14836645081	2039	16373229137	2041	17298071521
2043	14748148673	2045	10982533169	2047	21082442393	2049	23866962377
2051	18769206193	2053	19125553481	2054	29534671497	2055	14107004761
2057	21482977393	2059	14756819873	2061	12912128689	2063	10837182401
2065	18331106561	2067	10481469337	2069	17415046001	2071	20282274193
2073	18819832793	2075	17545036193	2077	15578650289	2078	27729367257
2079	24057117089	2081	16148176609	2082	25882365777	2083	12490953649
2085	10832312753	2087	19455168137	2089	18270610081	2091	21856626049
2093	23238238169	2095	24068136049	2097	16522756057	2099	15962858257
2101	25476168961	2103	24847607321	2105	18015016057	2106	31923588929
2107	20803781729	2109	19673142289	2111	18392434289	2113	23220719273
2115	12636915289	2117	19315124009	2119	28994112073	2121	20050974761
2123	29088481481	2125	18655061473	2127	19574918641	2129	18895809337
2131	23112080113	2133	14523603209	2135	22249389529	2137	21773555593
2139	18025003393	2141	22630967561	2142	29046167529	2143	18693988129
2145	18877428953	2147	18136409801	2149	19695859753	2151	19861422377
2153	24196460777	2155	23555454961	2156	29713956801	2157	20859254233
2159	20632640297	2161	27332078161	2162	28975285761	2163	18060492217
2165	21744616193	2167	20857806913	2169	14622455329	2171	20886550177
2173	14434855249	2175	16846410769	2177	15977018641	2179	23282095961
2181	25607323801	2183	26851308977	2185	19995487417	2187	26886280081
2189	27014177657	2191	14464864153	2193	20535271249	2195	26841314281
2197	22297034849	2199	23926888417	2201	23201739361	2203	16987173097
2205	18142454081	2207	16398494993	2209	27148857433	2211	22374953593
2213	23897368169	2215	20780753537	2217	24137876041	2219	21973542001
2221	21305351633	2222	31449231969	2223	25203670673	2224	31563684681
2225	19548976033	2227	19151122577	2229	17388825793	2231	20015710609
2233	23774802841	2235	17229928721	2237	25054037273	2239	28215927697

k	$d_1$	k	$d_1$	k	$d_1$	k	$d_1$
2241	25364314097	2243	24693406769	2245	19663680161	2247	23166334049
2249	22198449089	2251	21782856049	2253	23058697577	2255	23490722401
2257	23182069777	2259	25375332953	2261	23321397889	2263	23134466353
2265	21099281657	2267	20500043689	2269	24136990729	2271	26367890993
2273	23874674297	2275	18425670833	2277	26640229873	2279	23580652217
2281	18777503057	2282	28804400601	2283	25006505753	2285	28135395473
2287	25885481009	2289	24003072217	2291	25217386561	2293	22590552449
2295	24478979353	2297	29279333513	2299	21622666817	2301	28206525689
2303	29479981529	2305	30915727681	2307	23347446401	2309	30173417497
2311	17259667001	2313	20830862281	2315	28875641633	2317	22175499113
2319	20768662297	2321	22111529257	2323	22985300057	2325	21970530497
2327	21542492929	2329	25253712697	2331	24055982809	2333	27344154281
2335	31659517921	2337	26380992137	2339	24173328793	2341	23702333329
2343	22874829473	2345	27503106937	2347	23997306689	2349	25080505033
2351	27005350529	2353	19594199089	2355	26290364593	2357	31187585617
2361	21535123417	2363	30890690297	2365	17997494969	2367	25941664313
2368	30521729001	2369	21152347649	2373	28640771921	2375	25445748593
2377	24642382537	2379	27949170833	2383	29346656233	2385	27353459369
2387	22092862993	2389	26541069889	2391	27922751849	2395	25679645297
2397	25567788161	2401	26806417097	2403	25471907233	2411	23201469721
2413	27471244057	2415	28729140457	2417	31918397593	2423	19416040537
2427	27250644433	2429	28702862873	2431	18452796697	2433	31076018153
2437	19044233393	2443	28724687897	2447	24087157561	2449	24340659377
2451	24310203641	2453	29613412849	2455	25877124769	2457	20105114921
2459	31572674153	2465	31474193953	2469	31751925329	2473	25877913169
2479	25781498417	2483	30687948241	2485	29489657473	2487	23214266969
2489	24403608241	2493	28496723993	2495	26242884937	2499	29141913769
2511	27983986649	2523	31630888169	2531	30568914073	2533	29836994353
2545	30815861849	2551	29163166121	2553	18839920273	2555	25973619241
2561	27573697457	2565	20167085041	2567	30338840489	2575	24854975473
2579	29969335969	2623	29674805977	2627	26057264561	2645	22700098081
2667	26463497129	2683	28569879721	2705	30513902753	2713	29668713889
2735	28004847841	2757	20013907409	2783	31014739937	2801	31532536313
2851	25306669001	2869	30730146737	2941	28715939033

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Dąbrowski, A., Jędrzejak, T., Szymaszkiewicz, L. (2016). Behaviour of the Order of Tate–Shafarevich Groups for the Quadratic Twists of $X_0(49)$ . In: Loeffler, D., Zerbes, S. (eds) Elliptic Curves, Modular Forms and Iwasawa Theory. JHC70 2015. Springer Proceedings in Mathematics & Statistics, vol 188. Springer, Cham. https://doi.org/10.1007/978-3-319-45032-2_4

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DOI: https://doi.org/10.1007/978-3-319-45032-2_4
Published: 16 January 2017
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-45031-5
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Behaviour of the Order of Tate–Shafarevich Groups for the Quadratic Twists of \(X_0(49)\)

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