Properties

Label 1.11.a
Base field $\F_{11}$
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:  $\F_{11}$
Dimension:  $1$
L-polynomial:  $1 + 11 x^{2}$
Frobenius angles:  $\pm0.5$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-11}) \)
Galois group:  $C_2$
Jacobians:  4

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 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 12 144 1332 14400 161052 1774224 19487172 214329600 2357947692 25937746704

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 12 144 1332 14400 161052 1774224 19487172 214329600 2357947692 25937746704

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-11}) \).
Endomorphism algebra over $\overline{\F}_{11}$
The base change of $A$ to $\F_{11^{2}}$ is the simple isogeny class 1.121.w and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $11$ and $\infty$.
All geometric endomorphisms are defined over $\F_{11^{2}}$.

Base change

This is a primitive isogeny class.

Twists

This isogeny class has no twists.