# Properties

 Label 2.5.ae_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: $1 - 4 x + 11 x^{2} - 20 x^{3} + 25 x^{4}$ Frobenius angles: $\pm0.185749715683$, $\pm0.480916950984$ Angle rank: $1$ (numerical) Number field: $$\Q(\zeta_{12})$$ Galois group: $C_2^2$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 13 793 16900 381433 10005853 251539600 6147184693 152176891113 3809093856100 95362793192953

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 32 134 612 3202 16094 78682 389572 1950254 9765152

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\zeta_{12})$$.
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{3}}$ is 1.125.e 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$
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.e_l $2$ 2.25.g_l 2.5.i_ba $3$ 2.125.i_kg 2.5.ac_ab $4$ 2.625.ao_aqn
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.5.e_l $2$ 2.25.g_l 2.5.i_ba $3$ 2.125.i_kg 2.5.ac_ab $4$ 2.625.ao_aqn 2.5.c_ab $4$ 2.625.ao_aqn 2.5.ai_ba $6$ (not in LMFDB) 2.5.a_ag $6$ (not in LMFDB) 2.5.ag_s $12$ (not in LMFDB) 2.5.ae_o $12$ (not in LMFDB) 2.5.ac_c $12$ (not in LMFDB) 2.5.a_g $12$ (not in LMFDB) 2.5.c_c $12$ (not in LMFDB) 2.5.e_o $12$ (not in LMFDB) 2.5.g_s $12$ (not in LMFDB) 2.5.a_ai $24$ (not in LMFDB) 2.5.a_i $24$ (not in LMFDB)