curve #1993

y² + xy + y = x³ − x² − 202005056277292663263513760789382619682414593967562330105x + 770439162017176757138055457316491496826827238138704381947622393803259781488125439897

a-invariants: [1, -1, 1, -202005056277292663263513760789382619682414593967562330105, 770439162017176757138055457316491496826827238138704381947622393803259781488125439897]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 2409091324418488847528892674329981342840728238071834873259259850
discriminant (Δ): 271128676704855300311661589188149357102613242281876033228218091040713737478453512094760343251630095347678480785689680971460317390297980980541322614275894941530521600000000
Faltings height: 31.4086
naive height: 400.5573
primes of bad reduction: 2, 3, 5, 11, 13, 19, 43, 71, 89, 101, 113, 137, 197, 271, 397, 401, 563, 1087, 2311, 1576391, 2205527, 110976062006309
regulator: 19826492873191176254.89433759699753933562713329180496063416224436
submitted by: Steps Unbounded
submitted at: 2026-09-26 04:30:04 UTC
last updated: 2026-09-26 04:30:04 UTC

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

(241349050797723392459471957719/625, -13002504825536993731249778298130218994204448628/15625)
(622100673751537501665357406719/361, -4484376877230022196197352715621333579221547260/6859)
(72366418504725481107292207492991829/1957201, -18131578093188408711172647221776213657614140433151610/2738124199)
(-375665827895487194143325721, 919930727257481877579179124831313296734860)
(-1026489066569083200952712838104151/67081, -9167509567486197963079952246692341634167435180560/17373979)
(3531231290522043364603588279, 318033973764513293402697072335535999334860)
(-43998340706298335285554237815536727851/4510331281, -407827952953142533452937797969698314142963393261727810810/302909338500679)
(7804850931369738943732163897569/4761, -218820463985954979125373614909415503203413655760/328509)
(7078740719627843518861071881635423519/128391561, -18241832726368811592326724332849748577859725450214759740/1454804777691)
(20671961680639831055219397327843069877/6520078009, 211805189094943268073595485108798397587500308475163990000/526476738992723)
(2456334173678204910275565027827097390519/1778224561, 121732978743693990457666704608476477587844404227076982661740/74985951512809)
(139476596216343630566869560986306289996879/10141352333401, -24883734505023473610673278874600583650943339891010840299277860/32295633431979821149)
(108983958313986754119959985839283801129633/33237300721, 35978229777262589463229541978665951841172735329063083027728880/6059525531746231)

Commentary

New curve, 2-isogenous partner of curve#1918; the 13 witness points are transported through the rational isogeny and certified independent by the exact 2-descent character test.

last edited by Steps Unbounded at 2026-09-26 04:30:04 UTC · history