# Properties

 Label 2.5.ac_k Base Field $\F_{5}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{5}$ Dimension: $2$ L-polynomial: $( 1 - 2 x + 5 x^{2} )( 1 + 5 x^{2} )$ Frobenius angles: $\pm0.352416382350$, $\pm0.5$ Angle rank: $1$ (numerical) Jacobians: 4

This isogeny class is not simple.

## Newton polygon $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

This isogeny class contains the Jacobians of 4 curves, and hence is principally polarizable:

• $y^2=3x^6+2x^4+4x^3+2x^2+3$
• $y^2=3x^6+2x^5+4x^3+2x+3$
• $y^2=x^6+4x^5+2x^3+4x+1$
• $y^2=x^5+2x^4+x^3+2x^2+x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 24 1152 18648 368640 9515544 244363392 6099140568 152510791680 3819384768024 95423854465152

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 42 148 590 3044 15642 78068 390430 1955524 9771402

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ac $\times$ 1.5.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{2}}$ is 1.25.g $\times$ 1.25.k. The endomorphism algebra for each factor is: 1.25.g : $$\Q(\sqrt{-1})$$. 1.25.k : the quaternion algebra over $$\Q$$ ramified at $5$ and $\infty$.
All geometric endomorphisms are defined over $\F_{5^{2}}$.

## 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 2.5.c_k $2$ 2.25.q_eg 2.5.ae_k $4$ 2.625.abk_ve 2.5.e_k $4$ 2.625.abk_ve