Properties

Label 2.49.av_hz
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:  $1 - 21 x + 207 x^{2} - 1029 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.188420471035$, $\pm0.266233865926$
Angle rank:  $2$ (numerical)
Number field:  4.0.164725.1
Galois group:  $D_{4}$
Jacobians:  6

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 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1559 5704381 13923131231 33279866443909 79806409265639024 191583237120040145461 459986122096890084274199 1104427313669916848806373349 2651730727886119677866682577511 6366805740786850551048744059704576

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 29 2375 118343 5772939 282525314 13841432111 678222461771 33232919719699 1628413525463537 79792266045429950

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.164725.1.
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_hz$2$(not in LMFDB)