brainpoolP512r1 is the 512-bit Brainpool curve of RFC 5639, with parameters derived verifiably from the digits of π and e.
brainpoolP512r1 is a short Weierstrass curve y² = x³ + ax + b over a 512-bit prime field with a 512-bit prime-order subgroup and cofactor 1; its estimated security level is 256 bits. Status: secure, but regional or niche.
The Brainpool curves (RFC 5639) were generated in a verifiably pseudo-random way: the primes come from the digits of π and the curve coefficients from the digits of e, so anyone can recompute them (RFC 5639 Appendix A). brainpoolP512r1 is the 512-bit random curve; it has prime order (cofactor 1) and about 256 bits of security.
TLS supports it through RFC 7027 (TLS 1.2) and RFC 8734 (TLS 1.3, group brainpoolP512r1tls13 and signature scheme ecdsa_brainpoolP512r1tls13_sha512); COSE registers ESB512. German government PKI and electronic identity documents use the Brainpool curves, and NIST SP 800-186 allows them for interoperability.