curve #3111

y² + xy + y = x³ − x² − 9978029353641888566131786978681328913618084650282x + 12131151121741358591772238579040028131778041440986508930372063198247403881

a-invariants: [1, -1, 1, -9978029353641888566131786978681328913618084650282, 12131151121741358591772238579040028131778041440986508930372063198247403881]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 7882602549544199351275160289497567761080531375360072630
discriminant (Δ): 3884216391571627352247148103129080677244623863456029601226759538509684066075959228151047326082283331910026298513999904959389078955441899776000000
Faltings height: 26.8338
naive height: 350.0870
primes of bad reduction: 2, 3, 5, 7, 11, 23, 29, 47, 73, 113, 173, 229, 251, 751, 1861, 2351, 3919, 4007, 4639, 42569, 134999, 321569
regulator: 12263635546002.1125555699708271264978039992516041862658627643
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:37:48 UTC
last updated: 2026-09-26 20:37:48 UTC

Witness: 11 independent points log in to add more points →

(1865677824830166637375041, -96582083595766139279175511279820921)
(2049943383615605961773841, -539599013498770943624190438434633321)
(5780219385295850823039409, -12148188885367916322897700529267837609)
(41039496537777927201075963801/22201, -181538486428174726794864822311201659852309/3307949)
(614281959890020138027070943459229369/70244611369, 453602676744624232478413437369051327921784236011309027/18617421063405653)
(187819734658600373493084091449/6889, 80873990474283726857907332214155439720267773/571787)
(175623214535052408563592261570258374748131181/316399363912466641, -2327372325650317584609007444815665903577625075173475775462261493019/177972705520255977354190439)
(463845130361627272273852795928401476441/238396163046889, -1059274998151893139387017635412553090518933141036217701827/3680856544325499051787)
(1598807711200230798134551871941999/871253289, -469553401199557324071326807567127566006429568973/25716783331413)
(1368131356348895679142638266309487209080724064512520169/719893662234973204214305661881, -109760026226888607626770703588500806121431710557359633077715428617754896576017301/610804918064109626749873495544774467997586221)
(1491619748735181883113906157570254579009/728197726849321, -10528187964646629859966068914846597656263051096891458102291/19650516878512959240619)

Commentary

2-isogenous image of D-z4old9101112-018, rational kernel x=-907749466601149462784559. Source: https://dujella.github.io/tors/z4old9101112.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:37:48 UTC · history