Minimal twists of elliptic curves over $\mathbb{Q}$ (awaiting review)

The minimal quadratic twist of an elliptic curve $E$ defined over $\mathbb{Q}$ is defined as follows.

• First consider the finite set of all quadratic twists of $E$ which have minimal conductor. If this set contains just one curve, it is the minimal quadratic twist.
• Otherwise, sort the curves with minimal conductor into isogeny classes, and restrict attention to the curves whose class comes first in the LMFDB labelling; equivalently, sort the curves by the sequence of coefficients $(a_n)$ of their $L$-function and restrict to the curve or curves with the first such sequence.
• If $E$ does not have Complex Multiplication (CM), then the minimal isogeny class contains a unique curve with the same $j$-invariant as $E$, and this curve is the minimal quadratic twist of $E$.
• If $E$ does have CM, then the minimal isogeny class contains exactly two curves with $j$-invariant $j(E)$. In all but one case these two curves have distinct minimal discriminants, with the same sign, and we define the minimal quadratic twist to be the curve whose minimal discriminant has smallest absolute value.
• The exception is for elliptic curves with $j=66^3$, which have CM by the imaginary quadratic order with discriminant $-16$. The minimal conductor is $32$, and curves 32.a1 and 32.a2 (which are quadratic twists of each other by $-1$) both have minimal discriminant $2^9$. The minimal quadratic twist for $j=66^3$ is defined to be 32.a1.

All elliptic curves $E$ over $\mathbb{Q}$ with $j$-invariant $1728$ are quartic twists of each other. The smallest conductor of such a curve is $32$. Both the curves 32.a3 and 32.a4 have $j$-invariant $1728$, and they have minimal discriminants $-2^{12}$ and $2^6$ respectively.
We define the minimal quartic twist (or just minimal twist) of every elliptic curve with $j=1728$ to be the curve 32.a3, which has smaller discriminant, and equation $Y^2=X^3-X$.

All elliptic curves $E$ over $\mathbb{Q}$ with $j$-invariant $0$ are sextic twists of each other. The smallest conductor of such a curve is $27$. Both the curves 27.a3 and 27.a4 have $j$-invariant $0$, and they have minimal discriminants $-3^9$ and $-3^3$ respectively.
We define the minimal sextic twist (or just minimal twist) of every elliptic curve with $j=0$ to be the curve 27.a4, which has smaller discriminant, and equation $Y^2+Y=X^3$.

The minimal twist of an elliptic curve $E$ is its minimal quadratic twist, unless $j(E)=0$ or $1728$, in which cases the minimal twist is its minimal sextic or quartic twist respectively. The minimal quadratic twist depends only on the $j$-invariant unless $j=0$ or $1728$; in each of these cases, there are infinitely many different minimal quadratic twists, though only one minimal twist.