Properties

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

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $1 - 2 x^{2} + 9 x^{4}$
Frobenius angles:  $\pm0.195913276015$, $\pm0.804086723985$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\zeta_{8})\)
Galois group:  $C_2^2$
Jacobians:  2

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 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 8 64 776 9216 58568 602176 4785992 42614784 387417224 3430210624

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 6 28 110 244 822 2188 6494 19684 58086

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\).
Endomorphism algebra over $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.ac 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
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.
TwistExtension DegreeCommon base change
2.3.ae_k$4$2.81.bc_nu
2.3.a_c$4$2.81.bc_nu
2.3.e_k$4$2.81.bc_nu
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.3.ae_k$4$2.81.bc_nu
2.3.a_c$4$2.81.bc_nu
2.3.e_k$4$2.81.bc_nu
2.3.ae_i$8$(not in LMFDB)
2.3.e_i$8$(not in LMFDB)
2.3.ac_b$12$(not in LMFDB)
2.3.c_b$12$(not in LMFDB)