Properties

Label 3.3.ae_k_av
Base field $\F_{3}$
Dimension $3$
$p$-rank $2$
Ordinary no
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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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:  $0$
Isomorphism classes:  1

This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $7$ $903$ $14539$ $358491$ $12358717$

Point counts of the (virtual) curve

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

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3}$.

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is 6.0.309123.1.

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)