Properties

Label 1.121.ad
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

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{11^{2}}$
Dimension:  $1$
L-polynomial:  $1 - 3 x + 121 x^{2}$
Frobenius angles:  $\pm0.456458445761$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-19}) \)
Galois group:  $C_2$
Jacobians:  $5$
Isomorphism classes:  5

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $119$ $14875$ $1772624$ $214333875$ $25937221079$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $119$ $14875$ $1772624$ $214333875$ $25937221079$ $3138430792000$ $379749865455359$ $45949729666939875$ $5559917309045808464$ $672749994943013246875$

Jacobians and polarizations

This isogeny class contains the Jacobians of 5 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 \(\Q(\sqrt{-19}) \).

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.af
$\F_{11}$1.11.f

Twists

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

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