curve #3131

y² = x³ + x² − 109986054404072035905718914436733889x + 14037140715375786147567700636361626091030503635127071

a-invariants: [0, 1, 0, -109986054404072035905718914436733889, 14037140715375786147567700636361626091030503635127071]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 725195062080830777445016962580423655128128
discriminant (Δ): 29755690648100452225526027887626920749885367041422075731236756304863192205631185820219558357449795764224
Faltings height: 18.8526
naive height: 253.6706
primes of bad reduction: 2, 3, 11, 13, 29, 41, 47, 53, 61, 89, 193, 233, 257, 419, 509, 683, 3037, 14519, 22129
regulator: 37045299241.3464967059622169889809935468565471457906322268206
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:43 UTC
last updated: 2026-09-26 20:38:43 UTC

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

(282833611654981348917/1444, -162095409908071355030171542335/54872)
(1564244656360110333099/1225, -59959307884135903804547856868032/42875)
(1960618722959114340708285/315844, 2741459360360983225512801659236964043/177504328)
(11498584646133551250192381768942237/41474909763286609, -588479138380477376774947257776255411687389283945100/8446527001016665450765273)
(7187166168051981812387706/34586161, -2533212697157206573214182294398838257/203401212841)
(21578424639246648404981552297205882/74238869332163449, -1646391459037175390322535218214909771374499485091485/20227724192754929163621043)
(5355894392417124758385687915/13756681, -391965506315543101726749612068306713324224/51023529829)
(1010044240733075184289216341724212242068329/274625795236906728388225, -1011112625990006542457516111782877687521439398114444339744958208/143916964830653352754118171775666625)
(3104791397761565669248264573165136338903667314128059124563351016415381174458/16031910610981581591445839763999664152590635106471451798081, -1133986151981100864247500998038710363029211067262316237778115669614980171655742646753586101656382512423191295385/2029915333088671562483843803605153471562221042837965597899940564791293287865332867226271)
(749503108008605084181/3721, -1689798906465476276023816928172/226981)
(194929629094072269, -2109564027247788109727820)

Commentary

2-isogenous image of D-z4old9101112-014, rational kernel x=-378805909904245639. 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 20:38:43 UTC · history