Properties

Label 3.2.ae_j_ap
Base Field $\F_{2}$
Dimension $3$
$p$-rank $3$
Does not contain a Jacobian

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Invariants

Base field:  $\F_{2}$
Dimension:  $3$
Weil polynomial:  $1 - 4 x + 9 x^{2} - 15 x^{3} + 18 x^{4} - 16 x^{5} + 8 x^{6}$
Frobenius angles:  $\pm0.0435981566527$, $\pm0.329312442367$, $\pm0.527830414776$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\zeta_{7})\)
Galois group:  $C_6$

This isogeny class is simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class does not contain a Jacobian, and it is unknown whether it is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1 71 421 2059 25621 209237 1560896 15222187 151446751 1147846421

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 7 8 7 24 52 90 231 575 1092

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.