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Notation. Let $p$ be a prime number, $K$ a finite extension of $\mathbb{Q}_p$ and $E|K$ an elliptic curve which has good reduction. The discriminant $d_{E|K}$ of $E|K$ is an element of the multiplicative group $\mathfrak{o}^\times/\mathfrak{o}^{\times12}$, where $\mathfrak{o}$ is the ring of integers of $K$.

Question. Does the order of $d_{E|K}$ as an element of $\mathfrak{o}^\times/\mathfrak{o}^{\times12}$ show up somewhere ? Is it related to some other invariant of $E|K$ ?

Background. $E$ can be defined over $K$ by a minimal cubic

$f=y^2+a_1xy+a_3y-x^3-a_2x^2-a_4x-a_6=0$;f=y^2+a_1xy+a_3y-x^3-a_2x^2-a_4x-a_6=0,\ \ (a_i\in\mathfrak{o})$; its discriminant$d_f$is in$\mathfrak{o}^\times$(because$E$has good reduction). If we replace$f$by another minimal cubic$g$defining$E$, then$d_f$gets replaced by$d_g=u^{12}d_f$for some$u\in\mathfrak{o}^\times$. So the class$d_{E|K}$of$d_f$in$\mathfrak{o}^\times/\mathfrak{o}^{\times12}$depends only on$E|K$, not on the choice of a minimal cubic defining$E$. It can be shown that every class in the finite group$\mathfrak{o}^\times/\mathfrak{o}^{\times12}$is the discriminant of some good-reduction elliptic curve. Addendum. As Qing Liu remarks, one may ask, given an elliptic curve$E$over a finite extension $k|\mathbb{F}_p$, whether the order of the discriminant$d_{E|k}\in k^\times/k^{\times12}$shows up somewhere. When$p\neq2,3$, the two questions are equivalent. 3 deleted 1 characters in body Notation. Let$p$be a prime number,$K$a finite extension of$\mathbb{Q}_p$and$E|K$an elliptic curve which has good reduction. The discriminant$d_{E|K}$of$E|K$is an element of the multiplicative group$\mathfrak{o}^\times/\mathfrak{o}^{\times12}$, where$\mathfrak{o}$is the ring of integers of$K$. Question. Does the order of$d_{E|K}$as an element of$\mathfrak{o}^\times/\mathfrak{o}^{\times12}$show up somewhere ? Is it related to some other invariant of$E|K$? Background.$E$can be defined over$K$by a minimal cubic$f=y^2+a_1xy+a_3y-x^3-a_2x^2-a_4x-a_6=0$whose f=y^2+a_1xy+a_3y-x^3-a_2x^2-a_4x-a_6=0$;

its discriminant $d_f$ is in $\mathfrak{o}^\times$ (because $E$ has good reduction). If we replace $f$ by another minimal cubic $g$ defining $E$, then $d_f$ gets replaced by $d_g=u^{12}d_f$ for some $u\in\mathfrak{o}^\times$. So the class $d_{E|K}$ of $d_f$ in $\mathfrak{o}^\times/\mathfrak{o}^{\times12}$ depends only on $E|K$, not on the choice of a minimal cubic defining $E$. It can be shown that every class in the finite group $\mathfrak{o}^\times/\mathfrak{o}^{\times12}$ is the discriminant of some good-reduction elliptic curve.

Addendum. As Qing Liu remarks, one may ask, given an elliptic curve $E$ over a finite extension $k|\mathbb{F}_p$, whether the order of the discriminant $d_{E|k}\in k^\times/k^{\times12}$ shows up somewhere. When $p\neq2,3$, the two questions are equivalent.

Notation. Let $p$ be a prime number, $K$ a finite extension of $\mathbb{Q}_p$ and $E|K$ an elliptic curve which has good reduction. The discriminant $d_{E|K}$ of $E|K$ is an element of the multiplicative group $\mathfrak{o}^\times/\mathfrak{o}^{\times12}$, where $\mathfrak{o}$ is the ring of integers of $K$.

Question. Does the order of $d_{E|K}$ as an element of $\mathfrak{o}^\times/\mathfrak{o}^{\times12}$ show up somewhere ? Is it related to some other invariant of $E|K$ ?

Background. $E$ can be defined over $K$ by a minimal cubic

$f=y^2+a_1xy+a_3y-x^3-a_2x^2-a_4x-a_6=0$

whose discriminant $d_f$ is in $\mathfrak{o}^\times$ (because $E$ has good reduction). If we replace $f$ by another minimal cubic $g$ defining $E$, then $d_f$ gets replaced by $d_g=u^{12}d_f$ for some $u\in\mathfrak{o}^\times$. So the class $d_{E|K}$ of $d_f$ in $\mathfrak{o}^\times/\mathfrak{o}^{\times12}$ depends only on $E|K$, not on the choice of a minimal cubic defining $E$. It can be shown that every class in the finite group $\mathfrak{o}^\times/\mathfrak{o}^{\times12}$ is the discriminant of some good-reduction elliptic curve.

Addendum. As Qing Liu remarks, one may ask, given an elliptic curve $E$ over a finite extension $k|\mathbb{F}_p$, whether the order of the discriminant $d_{E|k}\in k^\times/k^{\times12}$ shows up somewhere. When $p\neq2,3$, the two questions are equivalent.

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