Properties

Label 2.49.av_hr
Base Field $\F_{7^{2}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
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

This isogeny class is simple and 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 16 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1551 5662701 13863422727 33235938086373 79786477354344816 191577959731011790629 459985992090562697140623 1104427934737568361596936997 2651731058111204212095865353903 6366805834224365854473506703641856

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

Decomposition and endomorphism algebra

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

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)