Properties

Label 2.49.au_he
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:  $1 - 20 x + 186 x^{2} - 980 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.0883562958487$, $\pm0.345388837514$
Angle rank:  $2$ (numerical)
Number field:  4.0.81216.1
Galois group:  $D_{4}$
Jacobians:  28

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 28 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1588 5697744 13867090900 33230701630464 79784329442455828 191578366201333410000 459986657310244533256948 1104428187943134012864282624 2651731074872682735296972058100 6366805809512704005975291774565584

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 30 2374 117870 5764414 282447150 13841080198 678223250910 33232946027134 1628413738546110 79792266906739654

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.81216.1.
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.u_he$2$(not in LMFDB)