# Properties

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

# Learn more about

## Invariants

 Base field: $\F_{5}$ Dimension: $2$ L-polynomial: $( 1 - 4 x + 5 x^{2} )( 1 + 5 x^{2} )$ Frobenius angles: $\pm0.147583617650$, $\pm0.5$ 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 12 720 15372 368640 10009452 251793360 6147109932 152510791680 3817505860812 95433118203600

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 30 122 590 3202 16110 78682 390430 1954562 9772350

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae $\times$ 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.ag $\times$ 1.25.k. The endomorphism algebra for each factor is: 1.25.ag : $$\Q(\sqrt{-1})$$. 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.e_k $2$ 2.25.e_ak 2.5.ac_k $4$ 2.625.abk_ve 2.5.c_k $4$ 2.625.abk_ve