# Properties

 Label 2.5.ae_o 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: $( 1 - 2 x + 5 x^{2} )^{2}$ Frobenius angles: $\pm0.352416382350$, $\pm0.352416382350$ 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:

• $y^2=3x^6+3x^4+3x^2+3$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 16 1024 21904 409600 9265936 236913664 6094612624 153413222400 3824074114576 95358193583104

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 38 170 654 2962 15158 78010 392734 1957922 9764678

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ac 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$
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_g $2$ 2.25.m_di 2.5.e_o $2$ 2.25.m_di 2.5.c_ab $3$ 2.125.bs_bcg
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.5.a_g $2$ 2.25.m_di 2.5.e_o $2$ 2.25.m_di 2.5.c_ab $3$ 2.125.bs_bcg 2.5.ai_ba $4$ 2.625.bc_cdq 2.5.ag_s $4$ 2.625.bc_cdq 2.5.ac_c $4$ 2.625.bc_cdq 2.5.a_ag $4$ 2.625.bc_cdq 2.5.c_c $4$ 2.625.bc_cdq 2.5.g_s $4$ 2.625.bc_cdq 2.5.i_ba $4$ 2.625.bc_cdq 2.5.ac_ab $6$ (not in LMFDB) 2.5.a_ai $8$ (not in LMFDB) 2.5.a_i $8$ (not in LMFDB) 2.5.ae_l $12$ (not in LMFDB) 2.5.e_l $12$ (not in LMFDB)