Properties

Label 3.13.ap_ee_aso
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:  $1 - 15 x + 108 x^{2} - 482 x^{3} + 1404 x^{4} - 2535 x^{5} + 2197 x^{6}$
Frobenius angles:  $\pm0.0430990397297$, $\pm0.222225562473$, $\pm0.395227696733$
Angle rank:  $3$ (numerical)
Number field:  6.0.365587668.1
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
$A(\F_{q^r})$ 678 4571076 10791466326 23251344887424 51045199500731808 112395971660466168036 247063647712058721521382 542787046237881197925468672 1192498013715060193302368871702 2619967843446249868930769792664576

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 161 2237 28505 370274 4824257 62748293 815710097 10604185655 137857029056

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 6.0.365587668.1.
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_ee_so$2$(not in LMFDB)