# Properties

 Label 2.3.a_a Base Field $\F_{3}$ Dimension $2$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $1 + 9 x^{4}$ Frobenius angles: $\pm0.250000000000$, $\pm0.750000000000$ Angle rank: $0$ (numerical) Number field: $$\Q(i, \sqrt{6})$$ Galois group: $C_2^2$ Jacobians: 2

This isogeny class is simple but not geometrically simple.

## Newton polygon

This isogeny class is supersingular. $p$-rank: $0$ Slopes: $[1/2, 1/2, 1/2, 1/2]$

## Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

• $y^2=x^5+2$
• $y^2=2x^5+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 10 100 730 10000 59050 532900 4782970 40960000 387420490 3486902500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 10 28 118 244 730 2188 6238 19684 59050

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is $$\Q(i, \sqrt{6})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{4}}$ is 1.81.s 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$

## 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 2.3.a_ag $8$ (not in LMFDB) 2.3.a_g $8$ (not in LMFDB) 2.3.ag_p $24$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.3.a_ag $8$ (not in LMFDB) 2.3.a_g $8$ (not in LMFDB) 2.3.ag_p $24$ (not in LMFDB) 2.3.ad_g $24$ (not in LMFDB) 2.3.a_ad $24$ (not in LMFDB) 2.3.a_d $24$ (not in LMFDB) 2.3.d_g $24$ (not in LMFDB) 2.3.g_p $24$ (not in LMFDB)