Properties

Label 2.11.ag_s
Base field $\F_{11}$
Dimension $2$
$p$-rank $2$
Ordinary Yes
Supersingular No
Simple Yes
Geometrically simple No
Primitive Yes
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{11}$
Dimension:  $2$
L-polynomial:  $1 - 6 x + 18 x^{2} - 66 x^{3} + 121 x^{4}$
Frobenius angles:  $\pm0.0290991158233$, $\pm0.529099115823$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(i, \sqrt{13})\)
Galois group:  $C_2^2$
Jacobians:  2

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

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
$A(\F_{q^r})$ 68 14416 1655732 207821056 25878546548 3138426760144 379509165371012 45940387292319744 5559929409777049892 672749994945773415376

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 122 1242 14190 160686 1771562 19474818 214315294 2357952822 25937424602

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{13})\).
Endomorphism algebra over $\overline{\F}_{11}$
The base change of $A$ to $\F_{11^{4}}$ is 1.14641.ais 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$
All geometric endomorphisms are defined over $\F_{11^{4}}$.
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.11.g_s$2$2.121.a_ais
2.11.a_ae$8$(not in LMFDB)
2.11.a_e$8$(not in LMFDB)