Properties

Label 2.11.aj_bm
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:  $1 - 9 x + 38 x^{2} - 99 x^{3} + 121 x^{4}$
Frobenius angles:  $\pm0.0468428922585$, $\pm0.380176225592$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{17})\)
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
$A(\F_{q^r})$ 52 13936 1769872 211214016 25719331612 3132446896384 379744766941228 45953547054914304 5559917317647236752 672742381710223208176

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 117 1332 14425 159693 1768182 19486911 214376689 2357947692 25937131077

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{17})\).
Endomorphism algebra over $\overline{\F}_{11}$
The base change of $A$ to $\F_{11^{6}}$ is 1.1771561.acna 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$
All geometric endomorphisms are defined over $\F_{11^{6}}$.
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.a_f$3$(not in LMFDB)
2.11.j_bm$3$(not in LMFDB)
2.11.a_f$6$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.11.a_f$3$(not in LMFDB)
2.11.j_bm$3$(not in LMFDB)
2.11.a_f$6$(not in LMFDB)
2.11.a_af$12$(not in LMFDB)