Properties

Label 3.3.ae_k_av
Base Field $\F_{3}$
Dimension $3$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

Learn more about

Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $1 - 4 x + 10 x^{2} - 21 x^{3} + 30 x^{4} - 36 x^{5} + 27 x^{6}$
Frobenius angles:  $\pm0.0145064862012$, $\pm0.383559653096$, $\pm0.564732805964$
Angle rank:  $2$ (numerical)
Number field:  6.0.309123.1
Galois group:  $D_{6}$
Jacobians:  0

This isogeny class is simple and geometrically simple.

Newton polygon

$p$-rank:  $2$
Slopes:  $[0, 0, 1/2, 1/2, 1, 1]$

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
$A(\F_{q^r})$ 7 903 14539 358491 12358717 354475359 9850388611 283090663443 7647092485312 202171542731823

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 14 21 50 210 665 2058 6578 19740 57974

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is 6.0.309123.1.
All geometric endomorphisms are defined over $\F_{3}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.e_k_v$2$3.9.e_ai_acx
3.3.c_ac_aj$3$(not in LMFDB)
3.3.c_e_d$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.e_k_v$2$3.9.e_ai_acx
3.3.c_ac_aj$3$(not in LMFDB)
3.3.c_e_d$3$(not in LMFDB)
3.3.ac_ac_j$6$(not in LMFDB)
3.3.ac_e_ad$6$(not in LMFDB)