Properties

Label 2.11.ai_bm
Base field $\F_{11}$
Dimension $2$
$p$-rank $2$
Ordinary Yes
Supersingular No
Simple No
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:  $( 1 - 4 x + 11 x^{2} )^{2}$
Frobenius angles:  $\pm0.293962833700$, $\pm0.293962833700$
Angle rank:  $1$ (numerical)
Jacobians:  3

This isogeny class is not 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
$A(\F_{q^r})$ 64 16384 1960000 220463104 25962232384 3131484160000 379411494766144 45944091111849984 5560119669688360000 672766404448046301184

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 134 1468 15054 161204 1767638 19469804 214332574 2358033508 25938057254

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
All geometric endomorphisms are defined over $\F_{11}$.

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_g$2$2.121.m_ks
2.11.i_bm$2$2.121.m_ks
2.11.e_f$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.11.a_g$2$2.121.m_ks
2.11.i_bm$2$2.121.m_ks
2.11.e_f$3$(not in LMFDB)
2.11.a_ag$4$(not in LMFDB)
2.11.ae_f$6$(not in LMFDB)