Properties

Label 3.13.ap_ed_ash
Base field $\F_{13}$
Dimension $3$
$p$-rank $3$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
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, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $671$ $4509791$ $10676155523$ $23168460479879$ $51044453464796891$

Point counts of the (virtual) curve

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

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13}$.

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 6.0.279340175.1.

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)