# Properties

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

## Invariants

 Base field: $\F_{13}$ Dimension: $3$ L-polynomial: $1 - 15 x + 110 x^{2} - 496 x^{3} + 1430 x^{4} - 2535 x^{5} + 2197 x^{6}$ Frobenius angles: $\pm0.0520504550069$, $\pm0.271344466885$, $\pm0.356632440861$ Angle rank: $3$ (numerical) Number field: 6.0.5611187.1 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.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 692 4694528 11023593908 23408287410176 51013925488225472 112265907265155318272 246978771567299420628692 542792262946231369189203968 1192545125878570895730884198324 2620002402814583034388045374291968

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 165 2285 28697 370044 4818669 62726733 815717937 10604604599 137858847500

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The endomorphism algebra of this simple isogeny class is 6.0.5611187.1.
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.
 Twist Extension Degree Common base change 3.13.p_eg_tc $2$ (not in LMFDB)