Properties

Label 2.49.av_hr
Base field $\F_{7^{2}}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{7^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 21 x + 199 x^{2} - 1029 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.0816995674884$, $\pm0.321155419757$
Angle rank:  $2$ (numerical)
Number field:  4.0.2427237.1
Galois group:  $D_{4}$
Jacobians:  $16$
Isomorphism classes:  16

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1551$ $5662701$ $13863422727$ $33235938086373$ $79786477354344816$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $29$ $2359$ $117839$ $5765323$ $282454754$ $13841050831$ $678222270083$ $33232938408019$ $1628413728252977$ $79792267216439614$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{2}}$.

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.2427237.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.49.v_hr$2$(not in LMFDB)