Properties

Label 2.11.ai_bg
Base Field $\F_{11}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{11}$
Dimension:  $2$
L-polynomial:  $1 - 8 x + 32 x^{2} - 88 x^{3} + 121 x^{4}$
Frobenius angles:  $\pm0.0750991438595$, $\pm0.424900856141$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(i, \sqrt{6})\)
Galois group:  $C_2^2$
Jacobians:  3

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 3 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 58 14500 1760938 210250000 25769191258 3138431750500 379935240802378 45953639424000000 5559842595347832058 672749994932236862500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 122 1324 14358 160004 1771562 19496684 214377118 2357916004 25937424602

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{6})\).
Endomorphism algebra over $\overline{\F}_{11}$
The base change of $A$ to $\F_{11^{4}}$ is 1.14641.afm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$
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 are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.11.i_bg$2$2.121.a_afm
2.11.i_bg$4$(not in LMFDB)
2.11.a_ak$8$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.11.i_bg$2$2.121.a_afm
2.11.i_bg$4$(not in LMFDB)
2.11.a_ak$8$(not in LMFDB)
2.11.a_k$8$(not in LMFDB)
2.11.ag_x$24$(not in LMFDB)
2.11.g_x$24$(not in LMFDB)