curve #1930

y² + xy + y = x³ − x² − 138214144554290187675134101249142x − 579879735751462082947294013830895109088949879291

a-invariants: [1, -1, 1, -138214144554290187675134101249142, -579879735751462082947294013830895109088949879291]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 105282506383638544223030541052846185291174420590490
discriminant (Δ): 23716292678395955644377149937563549663452112811340012809826362032637434820558412308740021250000000
Faltings height: 17.4274
naive height: 233.6327
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 29, 59, 71, 73, 103, 181, 251, 307, 563, 1663, 1913, 2089, 4507, 395027, 468967
regulator: 11468824455421.68339675014613101305977457937474985413082339617304
submitted by: Alvaro Lozano-Robledo
submitted at: 2026-09-25 19:17:10 UTC
last updated: 2026-09-25 19:17:10 UTC

Witness: 13 independent points log in to add more points →

(61609378834838375904575399847140567/4176810645302640601, -6560616577571446135978100564163196737930759188289841/8536251078851134926183904099)
(104996944666616006316856231730577532693/5915178375044958506041, 727832307346880006699399282358767330957968299228135114883/454937533056030713423391427495661)
(847484749701504442015207/57017601, -346980496689542909304710270408973149/430539905151)
(1238101957120178967100385988025514258570372034790748885427653/71468048652544471735635655892302948198937809, -901200419536043693713201445083587230972156047988823194466986743105817983070550819553161493/604182144820498867421249665309349907527304694797603569434062356377)
(26906667381071750696316431204383495792993/915540875409634997323329, 3989655597183156926179293065252274430282312106063122630074997/876025217355110757075621397151428767)
(287818100852041676324828015496136566408392510697524284900663222178842064077559369237382856988268080307081057510249538254299740954451603927647153/635921890537857774877610362842158685418064151262501279083759793794732345756257535177324723727052577454563054587825658239868849, 154358031791208579056679318482872282928241476664973499376018499480973963014738238600624217379054240652585182130331080714845365544621155670018856507482132041237016703947997606498583948325955464855496009939345464060387/507114072714811822054334858681275379172819780174232161090780149169404786594950616323803505515544930759251168231512310880275330667105573101473469019492108869643167548043024548282982274657257)
(1110217844410005247/81, 231286463859709611695667709/729)
(1528145819283341260532690136971808821668698560615315608239549371529795486200166831702407186509201806939266738507856806147495912134408897/112755622274505764490593958256427358388462472760933669966595821683033556497012109413078426974559573992070433554479085281, 7209433748577424947549399003502956016754965455303829935552225263996310339680339393813049063518158698955562413826887734930538859420237652619354344050348026386038801411291327815233349088793721562217818789/37862327624109302039282533935736040119623833246432033955196594941315728380915803068991943039927282541850220047364403950510123348346658014013839522699221248628436086472135909829679)
(3960570704806282136073291165107/291024458994721, 1212490682921238012726874259180682021590329879/4964714005728444221519)
(-21304147119935342101096295153060550293889625499072003976055044508602472313401704894058219532506133732607686832154823332521502074009884333846843813553907045463333850449087359810713135137/2708148572179932290820647289887753264216763456959826859116321206735661236791475854077852044499286583622790086885160054243531949728694041640527245432094116095549358482209, 639335969883639788531050390425148764430245307634190338094015616395322140525950693510220909911536432289656726495199445365539997370884669767759509512555735828766348130948378734678082921099718071307910566992177096564121320660937493567980217021076543196928100686139542111596614373/4456652067234490271569385998969967595583713950367529112492763868390960050025408115903591541152469754114849143056506299607422122677071292545036047437112661573770024807508071151877104260204179584284198323914567444918736990060128659417775271599260972943823)
(687319851007155044330342486274273839101134665733801137181824197/2966073850729800303389302536276592758920021241, 17995691512645544761678377238082199336962146352548695948465049041649292082525274569534101752301/161537344826828746128587442752375119458573253717762531255072526948211)
(31039836893948827669338418233426252300274708510925471222012659527741552351426034957313133942013690869791435291179139252712775745376493/265090662395065072273574000231353959810954567384557346947508110016389594072243840072232277902870773397935692135957369, 172028234099406522476330792495579380941565512920136345699420602774359141411931982326602860713127310921606171343104464220895181247769596433638736097387830904669133591381061092375227969374481800321464933/4316101462600607447937219506822403066250176444985854803994960172372924179231117298234992214067113669531031831094461819060754871086417494499900511843544038573492529646158452403)
(-7270739394626594558700053677391054951687/888324209396227932315289, 46382901088473871491393829136482365225353554218475714239617/837254030849748337209049141566587587)

Commentary

Rank exactly 13, torsion Z/2Z: 13 independent points; 2-isogeny Selmer bound 13, confirmed by a second 2-isogeny descent. Specialisation of Kihara's rank-8 family (S. Kihara, Proc. Japan Acad. 77A (2001) 11-12) with u = (s^2 - l)/(2s), (p,q,s) = (1,4,177/13).

last edited by Alvaro Lozano-Robledo at 2026-09-25 19:17:10 UTC · history