# Properties

 Label 1.25.af Base Field $\F_{5^{2}}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{5^{2}}$ Dimension: $1$ L-polynomial: $1 - 5 x + 25 x^{2}$ Frobenius angles: $\pm0.333333333333$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ Jacobians: 2

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is supersingular. $p$-rank: $0$ Slopes: $[1/2, 1/2]$

## Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21 651 15876 391251 9762501 244109376 6103437501 152588281251 3814701171876 95367441406251

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 21 651 15876 391251 9762501 244109376 6103437501 152588281251 3814701171876 95367441406251

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{5^{2}}$
 The base change of $A$ to $\F_{5^{6}}$ is the simple isogeny class 1.15625.jq and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $5$ and $\infty$.
All geometric endomorphisms are defined over $\F_{5^{6}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.25.f $2$ 1.625.z 1.25.k $3$ (not in LMFDB) 1.25.ak $6$ (not in LMFDB)