**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

*y*^{2} = *x*^{3} + *x* + *c*

over the prime field F_{p}, 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

*y*^{2} = *x*^{3} −3*x* + 15325252384887882227757421748102794318349518712709487389817905929239007568605

over the prime field F_{p}, 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

*x*^{2} + *y*^{2} = 1 + 3499565016101407566774046926671095877424725326083135202080143113943636512545*x*^{2}*y*^{2}

over the prime field F_{p}, 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

*y*^{2} = *x*^{3} −3*x* + 12229445650235697471539531853482081746072487194452039355467804333684298579047

over the prime field F_{p}, 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

*y*^{2} = *x*^{3} −3*x* + 14958658426191810116189297981703822101772119993348226289290257122252980182781

over the prime field F_{p}, 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

*y*^{2} = *x*^{3}−3*x* + 154344787563346744370152153588767287709323583533485442048

over the prime field F_{p}, 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.