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submitted curve #216 — rank ≥ 17, naive height 173.71
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commented on curve #159
Rank ≥ 17. This curve is the specialization at T = 2454 of the Mestre/Fermigier family y² = r(x,T), where r is the degree-≤4 remainder in p₆(x−T)·p₆(x+T) = g(x)² − r (g monic of degree 6), with p₆(x) = ∏ᵢ(x−aᵢ) on the sextuple (a) = (−1146, −2304, −654, 3054, 2880, −1830). It was… -
commented on curve #214
Rank ≥14. Mestre construction from the rational 6-tuple family a(u,v) = {u, v, -u-v, -u(u+2v)²/((u-v)(2u+v)), v(2u+v)²/((u-v)(u+2v)), (u-v)²(u+v)/((u+2v)(2u+v))} at (u,v)=(-12,-3), giving a=(-12,-3,15,16,-27/2,-5/2) with shift t=-1161/88 (cleared: tuple (-24,-6,30,32,-27,-5), t=-… -
commented on curve #215
Rank-18 naive-height record: h=168.785 (beats prior r18 height record 173.853). Found via Mestre/Fermigier quartic family, sextuple mestre_ais(u=-6,v=-4), specialization T=962; certified by hyperellratpoints + Neron-Tate height-pairing independence (18 independent points). -
submitted curve #215 — rank ≥ 18, naive height 168.79
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submitted curve #214 — rank ≥ 14, naive height 155.04
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commented on curve #61
Provenance: Noam D. Elkies, 2009 rank-13 Mordell curve. Elkies originally gave the model y^2 = x^3 + 48163745551486811536 with 13 independent integral points; this database stores the global minimal model y^2 + y = x^3 + 752558524241981430, with the witness points transformed acc… -
commented on curve #213
Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=6773/18: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x… -
commented on curve #212
Rank ≥11. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=14612/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x… -
commented on curve #211
Rank ≥11. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=2326/23: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x… -
commented on curve #210
Rank ≥11. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=15681/13: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-… -
commented on curve #209
Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=5485/16: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x… -
commented on curve #208
Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=967/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x e… -
commented on curve #207
Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=8863/13: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x… -
commented on curve #206
Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=7907/3: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x … -
commented on curve #205
Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=10852/13: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-… -
commented on curve #204
Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=787/7: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x e… -
commented on curve #203
Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[-138, 138, 162, -60, -90, -12] with shift t=296/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x ext… -
commented on curve #202
Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[-72, 120, 144, -48, -24, -120] with shift t=1516/7: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x ex… -
commented on curve #201
Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=939/1: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x e… -
commented on curve #200
Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=2289/5: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x e… -
commented on curve #199
Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[240, -1692, -996, 1260, 1776, -588] with shift t=2789/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small… -
commented on curve #198
Rank ≥14. Mestre–Fermigier construction from the integer 6-tuple a=[-756, 60, 210, 486, 654, -654] with shift t=1788/5: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x ex… -
commented on curve #197
Rank ≥14. Mestre–Fermigier construction from the integer 6-tuple a=[-138, 138, 162, -60, -90, -12] with shift t=1227/5: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x ex… -
commented on curve #196
Rank ≥14. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=267/2: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x ex… -
commented on curve #195
Rank ≥14. Mestre–Fermigier construction from the integer 6-tuple a=[-110, 6, 60, 44, 94, -94] with shift t=491/4: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra po… -
commented on curve #194
Rank ≥15. Mestre–Fermigier construction from the integer 6-tuple a=[240, -1692, -996, 1260, 1776, -588] with shift t=2352/1: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small… -
commented on curve #193
Rank ≥15. Mestre–Fermigier construction from the integer 6-tuple a=[-138, 138, 162, -60, -90, -12] with shift t=1265/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x ex… -
commented on curve #192
Rank ≥15. Mestre–Fermigier construction from the integer 6-tuple a=[-125, -99, -26, -18, 125, 143] with shift t=593/6: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x ext… -
commented on curve #191
Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-1146, -2304, -654, 3054, 2880, -1830] with shift t=1295/6: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few sm…