# Properties

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

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## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $( 1 - 3 x^{2} )^{2}$ Frobenius angles: $0$, $0$, $1$, $1$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{3})$$ Galois group: $C_2$ Jacobians: 0

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 is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4 16 676 4096 58564 456976 4778596 40960000 387381124 3429742096

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 -2 28 46 244 622 2188 6238 19684 58078

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is the quaternion algebra over $$\Q(\sqrt{3})$$ ramified at both real infinite places.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{2}}$ is 1.9.ag 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 $\F_{3^{2}}$.

## 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_d $3$ 2.27.a_acc 2.3.a_g $4$ 2.81.abk_ss 2.3.a_a $8$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.3.a_d $3$ 2.27.a_acc 2.3.a_g $4$ 2.81.abk_ss 2.3.a_a $8$ (not in LMFDB) 2.3.ag_p $12$ (not in LMFDB) 2.3.ad_g $12$ (not in LMFDB) 2.3.a_ad $12$ (not in LMFDB) 2.3.d_g $12$ (not in LMFDB) 2.3.g_p $12$ (not in LMFDB)