Properties

Label 2.11.ag_z
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:  $1 - 6 x + 25 x^{2} - 66 x^{3} + 121 x^{4}$
Frobenius angles:  $\pm0.193114434330$, $\pm0.473552232337$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-2}, \sqrt{-3})\)
Galois group:  $C_2^2$
Jacobians:  11

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 11 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 75 16425 1822500 213672825 26013826875 3146721210000 379921651716675 45943907113816425 5559605875406722500 672747467936585135625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 136 1368 14596 161526 1776238 19495986 214331716 2357815608 25937327176

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{-3})\).
Endomorphism algebra over $\overline{\F}_{11}$
The base change of $A$ to $\F_{11^{3}}$ is 1.1331.s 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
All geometric endomorphisms are defined over $\F_{11^{3}}$.

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.g_z$2$2.121.o_cx
2.11.m_cg$3$(not in LMFDB)
2.11.am_cg$6$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.11.g_z$2$2.121.o_cx
2.11.m_cg$3$(not in LMFDB)
2.11.am_cg$6$(not in LMFDB)
2.11.a_ao$6$(not in LMFDB)
2.11.a_o$12$(not in LMFDB)
2.11.ae_i$24$(not in LMFDB)
2.11.e_i$24$(not in LMFDB)