# Properties

 Label 2.3.ad_g 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: $( 1 - 3 x + 3 x^{2} )( 1 + 3 x^{2} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.5$ Angle rank: $0$ (numerical) Jacobians: 1

This isogeny class is not 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 1 curves, and hence is principally polarizable:

• $y^2=2x^6+2x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4 112 784 5824 66124 614656 4964572 42515200 387459856 3501133552

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 1 13 28 73 271 838 2269 6481 19684 59293

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 1.3.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 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 $\F_{3^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad $\times$ 1.9.g. The endomorphism algebra for each factor is: 1.9.ad : $$\Q(\sqrt{-3})$$. 1.9.g : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$

## 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.ag_p $3$ 2.27.a_cc 2.3.a_ad $3$ 2.27.a_cc 2.3.a_g $3$ 2.27.a_cc 2.3.d_g $3$ 2.27.a_cc 2.3.g_p $3$ 2.27.a_cc
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.3.ag_p $3$ 2.27.a_cc 2.3.a_ad $3$ 2.27.a_cc 2.3.a_g $3$ 2.27.a_cc 2.3.d_g $3$ 2.27.a_cc 2.3.g_p $3$ 2.27.a_cc 2.3.a_ad $6$ 2.729.ee_gmg 2.3.a_ag $12$ (not in LMFDB) 2.3.a_d $12$ (not in LMFDB) 2.3.a_a $24$ (not in LMFDB)