Properties

Label 2.11.af_o
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

Learn more about

Invariants

Base field:  $\F_{11}$
Dimension:  $2$
L-polynomial:  $1 - 5 x + 14 x^{2} - 55 x^{3} + 121 x^{4}$
Frobenius angles:  $\pm0.105104110453$, $\pm0.561562556214$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{-19})\)
Galois group:  $C_2^2$
Jacobians:  7

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 7 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 76 14896 1669264 210986944 26054551876 3142195845376 379697887884916 45955090747125504 5560368732876601744 672755273810511468976

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 7 125 1252 14409 161777 1773686 19484507 214383889 2358139132 25937628125

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-19})\).
Endomorphism algebra over $\overline{\F}_{11}$
The base change of $A$ to $\F_{11^{3}}$ is 1.1331.abo 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$
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.f_o$2$2.121.d_aei
2.11.k_bv$3$(not in LMFDB)
2.11.ak_bv$6$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.11.f_o$2$2.121.d_aei
2.11.k_bv$3$(not in LMFDB)
2.11.ak_bv$6$(not in LMFDB)
2.11.a_ad$6$(not in LMFDB)
2.11.f_o$6$(not in LMFDB)
2.11.a_d$12$(not in LMFDB)