Here we briefly describe the supported curves

1. ED25519 - This is the Bernstein et al Edwards curve https://ed25519.cr.yp.to/ed25519-20110926.pdf
2. C25519 - The original Bernstein Montgomery curve https://cr.yp.to/ecdh/curve25519-20060209.pdf
3. NIST256 - The NIST standard 256-bit curve secp256r1 http://www.secg.org/SEC2-Ver-1.0.pdf
4. BRAINPOOL - The 256-bit Brainpool twisted curve P256r1 http://www.ecc-brainpool.org/download/Domain-parameters.pdf
5. ANNSI - French standard 256-bit curve https://safecurves.cr.yp.to/
6. HIFIVE - Our own suggested curve https://eprint.iacr.org/2015/991
7. GOLDILOCKS - New standard 448-bit curve https://eprint.iacr.org/2015/625
8. NIST384 - The NIST standard 384-bit curve secp384r1 http://www.secg.org/SEC2-Ver-1.0.pdf
9. C41417 - An efficient high-security 414-bit curve https://eprint.iacr.org/2014/526
10. NIST521 - The NIST standard 521-bit curve secp521r1 http://www.secg.org/SEC2-Ver-1.0.pdf
11-16. The Microsoft NUMS suggested standard curves - https://www.microsoft.com/en-us/research/publication/selecting-elliptic-curves-for-cryptography-an-efficiency-and-security-analysis/
17. SEC256K1 - The other NIST standard Bitcoin curve

Pairing-friendly curves

Note that efforts to standardise these curves seem to have stalled, probably due to new insights
into their true security. There seems to be a move from BN curves to BLS curves for 128-bit
security and above.

18. BN254 - The Nogami BN curve - https://eprint.iacr.org/2005/133
19. BN254CX - The CertiVox standard BN curve
20. BLS383 - Our own suggested new standard curve, a GT-Strong 383-bit BLS12 curve - https://eprint.iacr.org/2002/088
21. BLS381 - The zk-SNARK BLS12 standard curve - https://blog.z.cash/new-snark-curve/
22. FP256BN - An ISO standard 256-bit BN curve, as recommended for use by FIDO - https://tools.ietf.org/pdf/draft-kasamatsu-bncurves-02.pdf
23. FP512BN - An ISO standard 512-bit BN curve, as recommended for use by FIDO - https://tools.ietf.org/pdf/draft-kasamatsu-bncurves-02.pdf
24. BLS461 - A 461 bit BLS12 curve suggested by Barbulescu and Duquesne https://eprint.iacr.org/2017/334

25. BLS24 - An experimental (approx AES-192 bit security) BLS24 GT-Strong curve
26. BLS48 - An experimental (approx AES-256 bit security) BLS48 GT-Strong curve

Note that it is quite possible to add new curves, and some programs are supplied to assist in the process.
Let us know if you have a particular request - mike.scott@miracl.com