Properties

Label 1.121.s
Base field $\F_{11^{2}}$
Dimension $1$
$p$-rank $1$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive no
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{11^{2}}$
Dimension:  $1$
L-polynomial:  $1 + 18 x + 121 x^{2}$
Frobenius angles:  $\pm0.805017770958$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-10}) \)
Galois group:  $C_2$
Jacobians:  $6$
Isomorphism classes:  6

This isogeny class is simple and geometrically simple, not primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $140$ $14560$ $1770860$ $214381440$ $25937103500$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $140$ $14560$ $1770860$ $214381440$ $25937103500$ $3138431427040$ $379749817530860$ $45949729783426560$ $5559917316877190540$ $672749994881328364000$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11^{2}}$.

Endomorphism algebra over $\F_{11^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-10}) \).

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{11^{2}}$.

SubfieldPrimitive Model
$\F_{11}$1.11.ac
$\F_{11}$1.11.c

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
1.121.as$2$(not in LMFDB)