Help determine points on P-256 lie on the actual curve

Question

The curve equation for P-256 is:

NIST P-256

y^2 = x^3-3x+41058363725152142129326129780047268409114441015993725554835256314039467401291

Below I am generating key data, including the secret key "d".

Can someone please help me:

1-Convert and plug in the 2 points below into the above equation to ensure the points lie on the curve.

2-Show me, using the curve equation above, how I can determine that x+y will land back on the curve as d, all values shown below?

Use jwcrypto to generate key data

key_data = jwk_crypto.JWK.generate(kty='EC', crv='P-256')

private_key_data = key_data.export(as_dict=True)

print(json.dumps(private_key_data, indent=2))

===>

{

'kty': 'EC',

'crv': 'P-256',

'x': 'vddm_d3R2dYNICxh2D93hjBZyEbpIWSW0fhPUXhHCgc',

'y': 'BjcTCjAp6oqDRbWDSCiCMxSKMDEAe4rAwYJqXHEbO98',

'd': 'B_nLv2KHBVUZEx6KSNULf3ZqZlYaVrpp8J7ncORbS_U'

}

Welcome to Cryptography.SE. First, your question is not totally clear. Second, coding-related issues even they are related to cryptography are off-topic here. You should ask in [so] with proper tags and include where you failed. We accept some coding to explain the problem on the Cryptographic side, not on the programmatic side. — kelalaka, Dec 26 '21 at 17:53
Note that with secret key $d$ one calculates the public key $[d]G$ where $G$ is generator of the curve group and this is called the scalar multiplication meaning add $G$ d-times. — kelalaka, Dec 26 '21 at 17:56
Are you asking similar to this? Proof that user public key corresponds the curve equation (secp256r1) — kelalaka, Dec 26 '21 at 18:27
The link you sent uses hex X Y inputs. How do I convert above to Hex first in order to feed to the equation - Or I would have assumed it needs to be converted to Integers first? — Steve237, Dec 26 '21 at 20:59
SageMath accepts hex values as integer inputs, which depends on the programming language that you use. — kelalaka, Dec 26 '21 at 21:02
Also can you let me know if the given X and Y is the first point, then where is the second point defined? And also how does the d: fit in the output at the end? — Steve237, Dec 26 '21 at 21:57
Those are Base64 encoded try here to get hex without a seperator and your point is a valid point and the public key of $d$ i.e $[d]G$ has coordinates as $(x,y)$. What is 2 points what is the second question. — kelalaka, Dec 27 '21 at 20:05