# Properties

 Label 2.5.ab_b 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: $\pm0.209104108027$, $\pm0.692387135011$ Angle rank: $2$ (numerical) Number field: 4.0.138269.1 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21 693 14049 444213 10160976 243398925 6147375801 152276660613 3805540590789 95368156950528

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 27 113 707 3250 15579 78685 389827 1948433 9765702

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The endomorphism algebra of this simple isogeny class is 4.0.138269.1.
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_b $2$ 2.25.b_bp