Which curve algorithm should be used for ES256?

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I am writing a PowerShell module for JWT tasks. So far the module can create RSA and HMAC signatures. I want to extend the module to include as well ECDSA signatures.
The signing and verifying of the data is done with OpenSSL. I used so far a key pair with secp256k1 and secp384r1 curve but the resulting signature was not verifiable in the debugging field of jwt.io.

Which curve algorithms should be used for ES256/384/512?