Properties

Label 3.4.ag_r_abj
Base field $\F_{2^{2}}$
Dimension $3$
$p$-rank $3$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $3$
L-polynomial:  $1 - 6 x + 17 x^{2} - 35 x^{3} + 68 x^{4} - 96 x^{5} + 64 x^{6}$
Frobenius angles:  $\pm0.125725122630$, $\pm0.183009650264$, $\pm0.584457690022$
Angle rank:  $3$ (numerical)
Number field:  6.0.400967.1
Galois group:  $A_4\times C_2$
Isomorphism classes:  1

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $13$ $3731$ $213109$ $17285723$ $1240653973$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $15$ $50$ $263$ $1174$ $4308$ $16540$ $66375$ $263327$ $1046420$

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 $\F_{2^{2}}$.

Endomorphism algebra over $\F_{2^{2}}$
The endomorphism algebra of this simple isogeny class is 6.0.400967.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
3.4.g_r_bj$2$3.16.ac_f_cl