Properties

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

Learn more about

Invariants

Base field:  $\F_{13}$
Dimension:  $3$
L-polynomial:  $1 - 15 x + 109 x^{2} - 487 x^{3} + 1417 x^{4} - 2535 x^{5} + 2197 x^{6}$
Frobenius angles:  $\pm0.103362670615$, $\pm0.212954520112$, $\pm0.386876328178$
Angle rank:  $3$ (numerical)
Number field:  6.0.319611111.1
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
$A(\F_{q^r})$ 687 4644807 10945226979 23430789293991 51194738741758707 112504491215063671023 247144610265913121230848 542846936277244377955333959 1192534846672217710084252249239 2619983787442601875420928888080287

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 163 2267 28723 371359 4828915 62768852 815800099 10604513195 137857868003

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 6.0.319611111.1.
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_ef_st$2$(not in LMFDB)