Properties

 Label 2.3.a_d 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: $1 + 3 x^{2} + 9 x^{4}$ Frobenius angles: $\pm0.333333333333$, $\pm0.666666666667$ Angle rank: $0$ (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 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:

• $y^2=x^6+x^3+2x+1$
• $y^2=2x^6+2x^3+x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 13 169 676 8281 59293 456976 4785157 44129449 387381124 3515659849

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 16 28 100 244 622 2188 6724 19684 59536

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\zeta_{12})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.acc 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.d 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is the simple isogeny class 2.27.a_acc and its endomorphism algebra is the quaternion algebra over $$\Q(\sqrt{3})$$ ramified at both real infinite places.

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 $3$ 2.27.a_acc 2.3.ag_p $4$ 2.81.s_jj 2.3.a_ad $4$ 2.81.s_jj
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.3.a_ag $3$ 2.27.a_acc 2.3.ag_p $4$ 2.81.s_jj 2.3.a_ad $4$ 2.81.s_jj 2.3.g_p $4$ 2.81.s_jj 2.3.ad_g $12$ (not in LMFDB) 2.3.a_ad $12$ (not in LMFDB) 2.3.a_g $12$ (not in LMFDB) 2.3.d_g $12$ (not in LMFDB) 2.3.a_a $24$ (not in LMFDB)