brainpoolP512t1 is the counterpart of brainpoolP512r1 with a = −3, isomorphic to it and chosen for faster arithmetic.
brainpoolP512t1 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.
Each Brainpool 'r1' curve has a 't1' counterpart obtained through an isomorphism over GF(p) — the parameter Z of RFC 5639 §3 maps (x, y) to (Z²x, Z³y) — that turns a into −3, which allows faster point doubling. The two curves have isomorphic groups of the same prime order and the same security level (about 256 bits). See brainpoolP512r1.
TLS, JOSE and COSE register only the r1 curves; the t1 forms appear as X.509 OIDs (RFC 5639 §4.1) and in libraries such as OpenSSL and Bouncy Castle.