# Properties

 Label 2.5.ab_ae 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: $1 - x - 4 x^{2} - 5 x^{3} + 25 x^{4}$ Frobenius angles: $\pm0.0948835201023$, $\pm0.761550186769$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Galois group: $C_2^2$ Jacobians: 1

This isogeny class is simple but not 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 1 curves, and hence is principally polarizable:

• $y^2=4x^5+2x^4+x^3+4x^2+4x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 16 448 12544 410368 9456976 245862400 6147422416 152474692608 3824496519424 95406021473728

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 17 98 657 3025 15734 78685 390337 1958138 9769577

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{-19})$$.
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{3}}$ is 1.125.ao 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-19})$$$)$
All geometric endomorphisms are defined over $\F_{5^{3}}$.

## 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.b_ae $2$ 2.25.aj_ce 2.5.c_l $3$ 2.125.abc_re 2.5.ac_l $6$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.5.b_ae $2$ 2.25.aj_ce 2.5.c_l $3$ 2.125.abc_re 2.5.ac_l $6$ (not in LMFDB) 2.5.a_j $6$ (not in LMFDB) 2.5.a_aj $12$ (not in LMFDB)