# Properties

 Label 2.5.ae_i 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: $1 - 4 x + 8 x^{2} - 20 x^{3} + 25 x^{4}$ Frobenius angles: $\pm0.0320471084245$, $\pm0.532047108424$ Angle rank: $1$ (numerical) Number field: $$\Q(i, \sqrt{6})$$ 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:

• $y^2=3x^6+4x^5+x^4+x^2+x+3$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 10 580 12490 336400 9629050 244129540 6042262810 151912857600 3815627066890 95367440008900

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 26 98 534 3082 15626 77338 388894 1953602 9765626

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The endomorphism algebra of this simple isogeny class is $$\Q(i, \sqrt{6})$$.
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{4}}$ is 1.625.abu 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-6})$$$)$
All geometric endomorphisms are defined over $\F_{5^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{5^{2}}$  The base change of $A$ to $\F_{5^{2}}$ is the simple isogeny class 2.25.a_abu and its endomorphism algebra is $$\Q(i, \sqrt{6})$$.

## 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.e_i $2$ 2.25.a_abu 2.5.a_ac $8$ (not in LMFDB) 2.5.a_c $8$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.5.e_i $2$ 2.25.a_abu 2.5.a_ac $8$ (not in LMFDB) 2.5.a_c $8$ (not in LMFDB) 2.5.ag_r $24$ (not in LMFDB) 2.5.g_r $24$ (not in LMFDB)