An explicit equation for $X_1(13)$ and a computation using MAGMA

Question

By a general theory $X_1(13)$ is smooth over $\mathbb{Z}[1/13]$, and so is its Jacobian $J$.

And the hyperelliptic curve given by an affine model $y^2 = x^6 - 2x^5 + x^4 -2x^3 + 6x^2 -4x + 1$ is $X_1(13)$.

However, according to MAGMA, $J$ is bad at $2$.

What is wrong with my argument?

Here is my code:

P<x> := PolynomialRing(RationalField());
C := HyperellipticCurve(x^6 - 2 * x^5 + x^4 - 2 * x^3 + 6 * x^2 - 4 * x +1);
J := Jacobian(C);
BadPrimes(J);

Answer 1

To get a model with good reduction at $2$, take $y = 2Y + x^3 + x^2 + 1$, subtract $(x^3+x^2+1)^2$ from both sides, and divide by $4$ to get $$ Y^2 + (x^3+x^2+1) \, Y = -x^5-x^3+x^2-x. $$ (A similar tactic of un-completing the square is well-known for elliptic curves.)