Properties

Label 3.2.ae_j_ao
Base field $\F_{2}$
Dimension $3$
$p$-rank $2$
Ordinary No
Supersingular No
Simple No
Geometrically simple No
Primitive Yes
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{2}$
Dimension:  $3$
L-polynomial:  $( 1 - 2 x + 2 x^{2} )( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$
Frobenius angles:  $\pm0.174442860055$, $\pm0.250000000000$, $\pm0.546783656212$
Angle rank:  $2$ (numerical)
Jacobians:  0

This isogeny class is not 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
$A(\F_{q^r})$ 2 140 806 5600 72242 394940 1813198 14716800 134765618 1036672700

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 7 11 23 59 91 111 223 515 987

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ac $\times$ 2.2.ac_d and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 2.16.ac_b. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
3.2.a_b_ac$2$3.4.c_f_q
3.2.a_b_c$2$3.4.c_f_q
3.2.e_j_o$2$3.4.c_f_q
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.2.a_b_ac$2$3.4.c_f_q
3.2.a_b_c$2$3.4.c_f_q
3.2.e_j_o$2$3.4.c_f_q
3.2.ac_f_ai$8$(not in LMFDB)
3.2.c_f_i$8$(not in LMFDB)