# Properties

 Label 2.5.ab_ab 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: $1 - x - x^{2} - 5 x^{3} + 25 x^{4}$ Frobenius angles: $\pm0.169335136474$, $\pm0.720319309201$ Angle rank: $2$ (numerical) Number field: 4.0.1525.1 Galois group: $D_{4}$ Jacobians: 3

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 3 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 19 589 13471 438805 9968464 245966989 6183697231 152444806245 3813996125059 95381612955904

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 23 107 699 3190 15743 79147 390259 1952765 9767078

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The endomorphism algebra of this simple isogeny class is 4.0.1525.1.
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.b_ab $2$ 2.25.ad_bp