Properties

Label 2.4.a_ae
Base Field $\F_{2^2}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^2}$
Dimension:  $2$
Weil polynomial:  $1 - 4 x^{2} + 16 x^{4}$
Frobenius angles:  $\pm0.166666666667$, $\pm0.833333333333$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\zeta_{12})\)
Galois group:  $C_2^2$

This isogeny class is simple.

Newton polygon

This isogeny class is supersingular.

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

Point counts

This isogeny class contains a Jacobian, and hence is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 13 169 4225 74529 1047553 17850625 268419073 4328718849 68720001025 1097367287809

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 9 65 289 1025 4353 16385 66049 262145 1046529

Decomposition

This is a simple isogeny class.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^2}$.

SubfieldPrimitive Model
$\F_{2}$2.2.ac_c
$\F_{2}$2.2.c_c