Properties

Label 1.343.a
Base Field $\F_{7^{3}}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{7^{3}}$
Dimension:  $1$
L-polynomial:  $1 + 343 x^{2}$
Frobenius angles:  $\pm0.5$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-7}) \)
Galois group:  $C_2$
Jacobians:  2

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

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
$A(\F_{q^r})$ 344 118336 40353608 13841051904 4747561509944 1628413678617664 558545864083284008 191581231352883840000 65712362363534280139544 22539340290701753210883136

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 344 118336 40353608 13841051904 4747561509944 1628413678617664 558545864083284008 191581231352883840000 65712362363534280139544 22539340290701753210883136

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{3}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-7}) \).
Endomorphism algebra over $\overline{\F}_{7^{3}}$
The base change of $A$ to $\F_{7^{6}}$ is the simple isogeny class 1.117649.bak and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $7$ and $\infty$.
All geometric endomorphisms are defined over $\F_{7^{6}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{7^{3}}$.

SubfieldPrimitive Model
$\F_{7}$1.7.a

Twists

This isogeny class has no twists.