Using a public key's points (other than the generator point) to calculate the order of the group (SECP256k1)?

Question

Imagine if we were on a mission to try to calculate the order of the cyclic group $n$

n = 115792089237316195423570985008687907852837564279074904382605163141518161494337

Given the order of the finite field $p$

p = 115792089237316195423570985008687907853269984665640564039457584007908834671663

Generator point $G$:

G = (
55066263022277343669578718895168534326250603453777594175500187360389116729240,
32670510020758816978083085130507043184471273380659243275938904335757337482424
)

which has the private key of $1$.

Cofactor $h$ is $1$. The equation is $y^2 = x^3 + 7$

We can find the $n$ with Schoof-Elkies-Atkin algorithm

However, it's a bit confusing. Here's the solution in Sage. It presents one code example:

sage: p = 115792089237316195423570985008687907853269984665640564039457584007908834671663
sage: EllipticCurve(GF(p), [0,7]).order()
115792089237316195423570985008687907852837564279074904382605163141518161494337

What is [0, 7] specifically? These cannot be the coordinates of the G, since we already established that they are different.

And where is the $G$ even used in this calculation?

What if we used instead of $G$, another point on the elliptic curve, that we know for sure that it sits on the curve and is part of our cyclic group? Like this one:

(x,y) = (
44886295857190546091508615621464465421050773292389158775895365558788257183826,
79820197542983972470655013754473404410649480536210503962616926227235987362275
)

The private key for this point sits somewhere between $2^{129}$ and $2^{130}$. What would happen if we use this point, instead of $G$ for our Schoof-Elkies-Atkin to calculate the $n$ (order of the cyclic group)?

Answer 1

The $[a,b]$ in the SageMath code is for the short Weierstrass curve representation $y^2=x^3+ax+b.$ If the prime $p \neq 2,3$ all Weierstrass curves can be represented in the short form. Naturally, SageMath has a short form to define the curves with a short Weierstrass representation.

Curve order and point order may have different values if the curve order is not prime like the Curve448, Curve25519, etc.

Schoof-Elkies-Atkin calculates curve order which doesn't use the points.