Properties

Label 1.11.ad
Base field $\F_{11}$
Dimension $1$
$p$-rank $1$
Ordinary Yes
Supersingular No
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:  $1 - 3 x + 11 x^{2}$
Frobenius angles:  $\pm0.350615407277$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-35}) \)
Galois group:  $C_2$
Jacobians:  2

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

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
$A(\F_{q^r})$ 9 135 1404 14715 160479 1769040 19485909 214382835 2358033444 25937418375

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 9 135 1404 14715 160479 1769040 19485909 214382835 2358033444 25937418375

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-35}) \).
All geometric endomorphisms are defined over $\F_{11}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.11.d$2$1.121.n