Properties

Label 2.41.at_gh
Base Field $\F_{41}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{41}$
Dimension:  $2$
L-polynomial:  $1 - 19 x + 163 x^{2} - 779 x^{3} + 1681 x^{4}$
Frobenius angles:  $\pm0.0648514433265$, $\pm0.331738269742$
Angle rank:  $2$ (numerical)
Number field:  4.0.1123949.1
Galois group:  $D_{4}$
Jacobians:  7

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 7 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1047 2767221 4756563927 7983324663381 13420159017819312 22562612330832089469 37929111304691614287543 63759054559127439389730309 107178945880691649347053549407 180167786043534625378694696453376

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 23 1647 69017 2825195 115834618 4749919407 194753678821 7984928190259 327381979946747 13422659540145102

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is 4.0.1123949.1.
All geometric endomorphisms are defined over $\F_{41}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.41.t_gh$2$(not in LMFDB)