Properties

Label 1.331.a
Base Field $\F_{331}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{331}$
Dimension:  $1$
L-polynomial:  $1 + 331 x^{2}$
Frobenius angles:  $\pm0.5$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-331}) \)
Galois group:  $C_2$
Jacobians:  12

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 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 332 110224 36264692 12003393600 3973195810652 1315127885854864 435307306210734212 144086718331745798400 47692703775754250891372 15786284949782603436665104

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 332 110224 36264692 12003393600 3973195810652 1315127885854864 435307306210734212 144086718331745798400 47692703775754250891372 15786284949782603436665104

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{331}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-331}) \).
Endomorphism algebra over $\overline{\F}_{331}$
The base change of $A$ to $\F_{331^{2}}$ is the simple isogeny class 1.109561.zm and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $331$ and $\infty$.
All geometric endomorphisms are defined over $\F_{331^{2}}$.

Base change

This is a primitive isogeny class.

Twists

This isogeny class has no twists.