<s>[INST] An explicit equation for $X_1(13)$ and a computation using MAGMA 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);

 [/INST] 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.)
</s>