Properties

Label 2.3.a_ag
Base Field $\F_{3}$
Dimension $2$
$p$-rank $0$
Principally polarizable
Does not contain a Jacobian

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
Weil polynomial:  $( 1 - 3 x^{2} )^{2}$
Frobenius angles:  $0.0$, $0.0$, $1.0$, $1.0$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{3}) \)
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
$A(\F_{q^r})$ 4 16 676 4096 58564 456976 4778596 40960000 387381124 3429742096

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 -2 28 46 244 622 2188 6238 19684 58078

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.