curve #3235

y² + xy = x³ − 5683314401210621492911632703706655785381168445x + 164890848036318916388498198309479413985191764686997097959363364445041

a-invariants: [1, 0, 0, -5683314401210621492911632703706655785381168445, 164890848036318916388498198309479413985191764686997097959363364445041]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 705477490224703868376944408577475677167161370706714635492150217886150043262
discriminant (Δ): 2925850661961454638992775323842355816326464424446627427660814665402796484380468765258243544371089059552882545240822179366381636945969152
Faltings height: 25.0088
naive height: 327.6752
primes of bad reduction: 2, 3, 7, 13, 17, 19, 29, 31, 37, 43, 53, 59, 71, 83, 131, 139, 167, 193, 389, 709, 977, 3359, 6221, 23813, 126859, 431099, 7562663, 274307221
regulator: 87983808990553633463.821412222595442674520326826582274025500090844108907387
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:56 UTC
last updated: 2026-09-26 20:43:56 UTC

Witness: 18 independent points log in to add more points →

(44641631922248734923714, 378977676133996348694765099006991)
(50921387038993434827874, 2743560436171297731192250474593615)
(-33686530052573969395230, -17835781763719108530351825468439089)
(42619799571493298954082, 292827379736122777599526074533199)
(5838847854036662207298, 11485029333192609885941978850009519)
(45248121238865553802986, 610091741742590528419332548432031)
(20432612029373596300830, 7569434220951625743173570363936139)
(-20379955995558953084190, -16500057012036585501568720282449201)
(37893141641450993359290, 1985598283050941174197688196727359)
(90893593185053575522434, 19981106231077783825926363388226031)
(37466689610256728180226, 2133036097413924302478261733769583)
(44597741180131259968938, 361829443357404721375545820385823)
(-53358687949900149432606, -17782713002210366978203433940336945)
(17692097214605877674682, 8359360374675174943088135065445439)
(83600785812042225095106, 16554601482925895999528983275399183)
(60360232761624133109747832, 468949438496772093487913290891082734449)
(44174705372674583511906, 186632952809786713542510794511183)
(12746151116772493790784930/361, 19690792123555484120569621125453221301/6859)

Commentary

Voznyy (2024). Source: https://dujella.github.io/tors/z2old141516171819.html (curve entry 8). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:43:57 UTC · history