curve #691

y² + xy + y = x³ − x² − 170700384026944526407042114523161014957x + 413672480022508285934178743417257094770023329091786960789

a-invariants: [1, -1, 1, -170700384026944526407042114523161014957, 413672480022508285934178743417257094770023329091786960789]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 272477462251985379286820733512680714450495871001031119894781628331624714896491284195957326271170
naive height: 275.7125 ★ record for rank ≥ 25
Faltings height: 21.0284 ★ record for rank ≥ 25
discriminant (Δ): 244408358275683636638671816734289148186540606903620569817329948906217619582004929679836204768521293229032073369600000 ★ record for rank ≥ 25
primes of bad reduction: 2, 5, 7, 13, 19, 43, 53, 59, 6650521999543582597820284157728903, 1355626871643268662547104228934136167711647486605987
regulator: 313336172249796547387045223.5830466859517581310574560774181882343732612873751640512574239
submitted by: FNacc
submitted at: 2026-09-10 02:08:55 UTC
last updated: 2026-09-10 02:08:55 UTC

(2333539875972734387, 5295602380421415127245162776)
(2431361028243932249, -3607110965222310192437772838)
(2233187909552714527, -6603305290521731591676766824)
(22599181096082416903/9, -25226156227929874876948117048/27)
(2425534977638519887, -3728630257046553197288042664)
(2514010933252338647, -647323965120163402770729224)
(-127134412399897897307/9, 67282047572880641718572398642/27)
(2316872610659823317, -5533369056476508640945357844)
(-11375247088445700433, 29724018108526195069914290776)
(1573534378649378047, -12205148293943229864184313624)
(-14135877523050558993, -1413603930352666456929092424)
(1060084045522050367, 15294019277415375572913764776)
(13960132232968075933/9, 333563888597802914327741604452/27)
(11626552370549162767, -809160133062991418728533224)
(107660914395643514983/9, 246683308349613695272716174952/27)
(8610650185976897163/4, 59966657533967103126120546433/8)
(167540544089577355207/81, 6075011888020530672288687726904/729)
(11873036895121820727, -7789141244448996733914866824)
(15378337469903858863, -37755232508475130119833034152)
(11717193449300755647, 4714580939652690145154625576)
(-2526933945886573649167/529, 407200102799048210308118121972532/12167)
(-320365069336459081193/121, 38737964951732650192855654292936/1331)
(-12012410170854264941, 27033844888151270638438555288)
(-673732993617873256193/81, 25856947855489241736417136074904/729)
(82939415766968813647, 746184763762427143501762015576)

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