Properties

Label 2.2.a_ae
Base Field $\F_{2}$
Dimension $2$
$p$-rank $0$
Principally polarizable
Does not contain a Jacobian

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Invariants

Base field:  $\F_{2}$
Dimension:  $2$
Weil polynomial:  $( 1 - 2 x^{2} )^{2}$
Frobenius angles:  $0.0$, $0.0$, $1.0$, $1.0$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{2}) \)
Galois group:  $C_2$

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 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
$A(\F_{q^r})$ 1 1 49 81 961 2401 16129 50625 261121 923521

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 -3 9 1 33 33 129 193 513 897

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.

Additional information

This isogeny class appears as a sporadic example in the classification of abelian varieties with one rational point.