Record CM constructions of elliptic curves

Sep 2010

D = −1,000,000,013,079,299,   h(D) = 10,034,174

As described in Accelerating the CM method, the square root of the class polynomial for the Atkin invariant A71 was used to contruct the elliptic curve

y2 = x3 + x + c

over the prime field Fp, where p is a 10000-digit (probable) prime listed here, and the integer c is listed here. The trace of Frobenius for this curve is listed here.

May 2010

D = −10,000,006,055,889,179,   h(D) = 25,459,680

As described in Accelerating the CM method, a decomposition of the square root of the class polynomial for the Atkin invariant A71 was used to contruct the elliptic curve

y2 = x3 −3x + 15325252384887882227757421748102794318349518712709487389817905929239007568605

over the prime field Fp, where

p = 28948022309329048855892746252171992875431396939874100252456123922623314798263.

The trace of this curve is

t = −340282366920938463463374607431768304979.

This computation was performed on 12 computers working in parallel (3.0 GHz AMD Phenom II, 4 cores each), and took approximately 8 days.

March 2010

D = −506,112,046,263,599,   h(D) = 50,666,940

As described in Accelerating the CM method, a decomposition of the square root of the class polynomial for the Atkin invariant A71 was used to contruct the Edwards curve

x2 + y2 = 1 + 3499565016101407566774046926671095877424725326083135202080143113943636512545x2y2

over the prime field Fp, where

p = 28948022309329048855892746252171986268338819619472424415843054443714437912893.

The trace of this curve is

t = 340282366920938463463374607431768266146.

This computation was performed on 8 computers working in parallel (3.0 GHz AMD Phenom II, 4 cores each), and took approximately 6 days.

January 2010

D = −1,000,000,013,079,299,   h(D) = 10,034,174

As described in Class invariants by the CRT method, the square root of the class polynomial for the Atkin invariant A71 was used to contruct the elliptic curve

y2 = x3 −3x + 12229445650235697471539531853482081746072487194452039355467804333684298579047

over the prime field Fp, where

p = 28948022309329048855892746252171981646113288548904805961094058424256743169033.

The trace of this curve is

t = −340282366920938463463374607431768238979.

This computation was performed on 8 computers working in parallel (3.0 GHz AMD Phenom II, 4 cores each), and took approximately 6 days.

April 2009

D = −4,058,817,012,071,   h(D) = 5,000,000

The class polynomial for the Weber f invariant was used to contruct the elliptic curve

y2 = x3 −3x + 14958658426191810116189297981703822101772119993348226289290257122252980182781

over the prime field Fp, where

p = 57896044618658097711785492504343953926634992332820282019728792010007722821607.

The trace of this curve is

t = 445463228097262625385482521918971302688.

This computation was performed on 16 computers working in parallel (2.8 GHz AMD Athlon, 2 cores each), and took approximately 3 days.

October 2008

D = −102,197,306,669,747,   h(D) = 2,014,236

The class polynomial for the square of the Ramanujan invariant (as defined in Ramanujan and the modular j-invariant) was used to contruct the elliptic curve

y2 = x3−3x + 154344787563346744370152153588767287709323583533485442048

over the prime field Fp, where

p = 1317860422843322160610398725225958731902944552925978150597.

The trace of this curve is

t = −36302347346188540382304940685.

This computation was performed on 12 computers working in parallel (2.8 GHz AMD Athlon, 2 cores each), and took approximately 5 days.