Properties

Label 3.13.ap_ed_ash
Base Field $\F_{13}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{13}$
Dimension:  $3$
L-polynomial:  $1 - 15 x + 107 x^{2} - 475 x^{3} + 1391 x^{4} - 2535 x^{5} + 2197 x^{6}$
Frobenius angles:  $\pm0.0356147865305$, $\pm0.203805463241$, $\pm0.408190245538$
Angle rank:  $3$ (numerical)
Number field:  6.0.279340175.1
Galois group:  $S_4\times C_2$

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 671 4509791 10676155523 23168460479879 51044453464796891 112437272100305768759 247090793820651893465408 542787389009674883450894375 1192488803215307020031633525831 2619961820501932501511744178896831

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 159 2213 28403 370269 4826031 62755188 815710611 10604103749 137856712139

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 6.0.279340175.1.
All geometric endomorphisms are defined over $\F_{13}$.

Base change

This is a primitive isogeny class.

Twists

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