# Properties

 Label 2.3.ac_b Base Field $\F_{3}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $1 - 2 x + x^{2} - 6 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.0292466093486$, $\pm0.637420057318$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ 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:

• $y^2=2x^6+2x^5+2x^4+2x^3+x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 3 57 324 5529 58323 467856 4600011 43264425 380016036 3458495577

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 8 8 68 242 638 2102 6596 19304 58568

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-2}, \sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{3}}$ is 1.27.ak 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
All geometric endomorphisms are defined over $\F_{3^{3}}$.

## 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 2.3.c_b $2$ 2.9.ac_af 2.3.e_k $3$ 2.27.au_fy 2.3.ae_k $6$ 2.729.ado_fhm
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.3.c_b $2$ 2.9.ac_af 2.3.e_k $3$ 2.27.au_fy 2.3.ae_k $6$ 2.729.ado_fhm 2.3.a_c $6$ 2.729.ado_fhm 2.3.a_ac $12$ (not in LMFDB) 2.3.ae_i $24$ (not in LMFDB) 2.3.e_i $24$ (not in LMFDB)