# Properties

 Label 2.5.ae_n Base Field $\F_{5}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{5}$ Dimension: $2$ L-polynomial: $( 1 - 3 x + 5 x^{2} )( 1 - x + 5 x^{2} )$ Frobenius angles: $\pm0.265942140215$, $\pm0.428216853436$ Angle rank: $2$ (numerical) Jacobians: 3

This isogeny class is not simple.

## Newton polygon

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

## Point counts

This isogeny class contains the Jacobians of 3 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 15 945 20160 401625 9628575 243855360 6107451015 152251619625 3808015968960 95358161970225

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 36 158 644 3082 15606 78178 389764 1949702 9764676

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ad $\times$ 1.5.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{5}$.

## 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.ac_h $2$ 2.25.k_ch 2.5.c_h $2$ 2.25.k_ch 2.5.e_n $2$ 2.25.k_ch