Oid of ecdsa key

Hi,
In my application there are three types of ecdsa keys which are initialized 
as:
 
AutoSeededRandomPool rng;
ECDSA<ECP, SHA256>::PrivateKey private_key;
private_key.Initialize(rng, oid);

with oid being one of 
oid = OID(CryptoPP::ASN1::secp256r1());
oid = OID(CryptoPP::ASN1::brainpoolP256r1());
oid = OID(CryptoPP::ASN1::brainpoolP384r1());

My problem is when I compare  mykey.GetGroupParameters().GetAlgorithmID() 
with CryptoPP::ASN1::xxxxxxr1() the two values are different. Any insights? 
How could I see the the curve of my key?

The key is of CryptoPP::ECDSA<CryptoPP::ECP, CryptoPP::SHA256>::PublicKey;

Thanks in advance!

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