Properties

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

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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:

Point counts of the abelian variety

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

Point counts of the curve

$r$ 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

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)