Properties

Label 3.13.aq_eq_aux
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:  $1 - 16 x + 120 x^{2} - 543 x^{3} + 1560 x^{4} - 2704 x^{5} + 2197 x^{6}$
Frobenius angles:  $\pm0.0186782147920$, $\pm0.208870362755$, $\pm0.359149016839$
Angle rank:  $3$ (numerical)
Number field:  6.0.9709203.1
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
$A(\F_{q^r})$ 615 4391715 10769211495 23323805483475 51056734370604075 112340505835142908155 247017510590676700158255 542782457877041823738771075 1192509017310997013503638712320 2619966322644293656283832526776075

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 154 2233 28594 370358 4821877 62736574 815703202 10604283508 137856949034

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 6.0.9709203.1.
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.q_eq_ux$2$(not in LMFDB)