Prime Numbers

7.5 Counting points on elliptic curves 365
We observe that the polynomial calculations for the deeper discriminants
(i.e. possessed of higher class numbers) can be difficult. For example, there is
the precision issue when using floating-point arithmetic in Algorithm 7.5.8. It
is therefore worthwhile to contemplate means for establishing some explicit
curve parameters for small |D|, in this way obviating the need for class
polynomial calculations. To this end, we have compiled here a complete list
of curve parameter sets for all D with h(D) =1, 2:
D r s
−7 125 189
−8 125 98
−11 512 539
−19 512 513
−43 512000 512001
−67 85184000 85184001
−163 151931373056000 151931373056001
−15 1225 − 2080 √ 5 5929
−20 108250 + 29835 √ 5 174724
−24 1757 − 494 √ 2 1058
−35 −1126400 − 1589760 √ 5 2428447
−40 54175 − 1020 √ 5 51894
−51 75520 − 7936 √ 17 108241
−52 1778750 + 5125 √ 13 1797228
−88 181713125 − 44250 √ 2 181650546
−91 74752 − 36352 √ 13 205821
−115 269593600 − 89157120 √ 5 468954981
−123 1025058304000 − 1248832000 √ 41 1033054730449
−148 499833128054750 + 356500625 √ 37 499835296563372
−187 91878880000 − 1074017568000 √ 17 4520166756633
−232 1728371226151263375 − 11276414500 √ 29 1728371165425912854
−235 7574816832000 − 190341944320 √ 5 8000434358469
−267 3632253349307716000000 − 12320504793376000 √ 89
3632369580717474122449
−403 16416107434811840000 − 4799513373120384000 √ 13
33720998998872514077
−427 564510997315289728000 − 5784785611102784000 √ 61
609691617259594724421
Table 7.1 Explicit curve parameters of CM curves for class number 1 and 2
Algorithm 7.5.10 (Explicit CM curve parameters: Class numbers 1, 2).
Given prime p>3, this algorithm reports explicit CM curves y 2 = x 3 + ax + b
over Fp, with orders as specified in the [Option: Curve orders] step of Algorithm
7.5.9. The search herein is exhaustive over all discriminants D of class numbers