# Properties

 Label 2.5.ad_h 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: $1 - 3 x + 7 x^{2} - 15 x^{3} + 25 x^{4}$ Frobenius angles: $\pm0.177952114464$, $\pm0.556618995437$ Angle rank: $2$ (numerical) Number field: 4.0.48069.2 Galois group: $D_{4}$ Jacobians: 4

This isogeny class is simple and geometrically 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 4 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 15 765 14715 386325 10381200 249904845 6093381435 152632757925 3818442864735 95307887059200

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 31 117 619 3318 15991 77997 390739 1955043 9759526

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The endomorphism algebra of this simple isogeny class is 4.0.48069.2.
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.d_h $2$ 2.25.f_j