Properties

Label 1.23.a
Base field $\F_{23}$
Dimension $1$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{23}$
Dimension:  $1$
L-polynomial:  $1 + 23 x^{2}$
Frobenius angles:  $\pm0.5$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-23}) \)
Galois group:  $C_2$
Jacobians:  6

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 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $24$ $576$ $12168$ $278784$ $6436344$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $24$ $576$ $12168$ $278784$ $6436344$ $148060224$ $3404825448$ $78310425600$ $1801152661464$ $41426524086336$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{23}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-23}) \).
Endomorphism algebra over $\overline{\F}_{23}$
The base change of $A$ to $\F_{23^{2}}$ is the simple isogeny class 1.529.bu and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $23$ and $\infty$.
All geometric endomorphisms are defined over $\F_{23^{2}}$.

Base change

This is a primitive isogeny class.

Twists

This isogeny class has no twists.