curve #5969

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y² + xy + y = x³ − 76538901033634041235677069749357148x + 8177172578267831570049658150786264346130208149670006

a-invariants

[1, 0, 1, -76538901033634041235677069749357148, 8177172578267831570049658150786264346130208149670006]

c-invariants

[3673867249614433979312499347969143081, -7065077107623406482033705516700983364025248863268600021]

rank (lower bound)

≥ 24

torsion subgroup

trivial

conductor (N)

1815662763075819492679731665574431988241426419467083404483983040651142994921569249869810

discriminant (Δ)

-189828716701432934131785730945332228485587635587959637570795003883299453986144144939171150077944098687500

Faltings height

18.8680

naive height

252.5895

primes of bad reduction

2, 3, 5, 13, 19, 23, 79, 54711148930777111562285234059, 2464825716641823703871398562214754348798891574462747

regulator

3225389867147631730355414.5798089736484467711063422242917815463985962299334570944012118

submitted by

apkorte

submitted at

2026-10-27 05:20:23 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-27 05:21:26 UTC

Witness: 24 independent points log in to add more points →

(127501501587803315, -22160615421223438777126593)
(172060455443857550, -10083146169011353923098178)
(226191908839523396855/1849, -2018801382308449929090437164011/79507)
(77842976495795405, 51873388869220968746813877)
(-144527872649418208, -127358685120847847132411406)
(658812915774933320, 493659662631328678484765877)
(1575303139066157200/9, -322598647808508557812301681/27)
(238184322053847245, -58816875284474660444629323)
(265072853328584975, 80707900878346901943059127)
(173271463016152070, 10830250815043074368859627)
(157556334187039760, 5400194654962000396148562)
(-220995035684142085, 119577406477813075565947557)
(1265543988254036315/4, -1000211362368390270666387429/8)
(-282532199819164565, 85140477675995511948378677)
(1522983502905663095, 1850442530297171284147952727)
(206227546451739395, 34110984987826957275631677)
(16059850101772194235/9, 63628081225768512649559752229/27)
(175650153255836770, 12346372010302743234461927)
(633128521450633655/4, 42264935839056845468028141/8)
(193938323562051605, -25055316398119266976453983)
(908133447776955020, -829224376404076772981564823)
(216828573202181233398245/3024121, -290813243109644356460460540658069137/5258946419)
(2738493734750830037/16, 610477818882903552343778541/64)
(2433693202872601795/9, 2299465328844265453727003909/27)

Commentary

Specialization at T = 399/62 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -2315727727754537200T^8 - 385085454700810899360T^7 - 21602016129394505916480T^6 + 522753273151467123362400T^5 - 5286101242963131414221040T^4 + 36372271295130380430902640T^3 - 119866073107198500927623880T^2 - 44611766420397375005288280T - 17644996097302241306504187, A6 = 1527906580671423711878454400T^12 + 405036291873933234226307839360T^11 + 29619810420503480598512683442496T^10 + 83447431068127636496045716863680T^9 + 10401708370692526999699168797492240T^8 - 680407910276562698898410450598391200T^7 + 10353906760921511571152027566046979840T^6 - 64341197016007520540432295631610298720T^5 + 151935569584040641348413346807219047840T^4 - 98223779992621049253987068203224001200T^3 + 461589258984786807406908653536129872360T^2 + 245414512415050160861358750620214278280T + 41992607521879070487127955356093746390. It is a 2-neighbour of the fibration given in curve#4187 (itself a 2-neighbour of the fibration in curve#3900): its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-27 05:20:23 UTC · history