# Properties

 Label 2.5.ac_l 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: $( 1 - x + 5 x^{2} )^{2}$ Frobenius angles: $\pm0.428216853436$, $\pm0.428216853436$ Angle rank: $1$ (numerical) Jacobians: 1

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 1 curves, and hence is principally polarizable:

• $y^2=4x^6+x^5+x^4+x^3+x^2+x+4$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 25 1225 19600 354025 9150625 245862400 6191329225 152814537225 3804918384400 95290298805625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 44 154 564 2924 15734 79244 391204 1948114 9757724

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-19})$$$)$
All geometric endomorphisms are defined over $\F_{5}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.5.a_j $2$ 2.25.s_fb 2.5.c_l $2$ 2.25.s_fb 2.5.b_ae $3$ 2.125.bc_re
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.5.a_j $2$ 2.25.s_fb 2.5.c_l $2$ 2.25.s_fb 2.5.b_ae $3$ 2.125.bc_re 2.5.a_aj $4$ 2.625.ack_dhb 2.5.ab_ae $6$ (not in LMFDB)