Properties

Label 2.19.ak_by
Base Field $\F_{19}$
Dimension $2$
$p$-rank $2$
Principally polarizable
Contains a Jacobian

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Invariants

Base field:  $\F_{19}$
Dimension:  $2$
Weil polynomial:  $1 - 10 x + 50 x^{2} - 190 x^{3} + 361 x^{4}$
Frobenius angles:  $\pm0.0511346671405$, $\pm0.448865332859$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(i, \sqrt{13})\)
Galois group:  $V_4$

This isogeny class is simple.

Newton polygon

This isogeny class is ordinary.

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

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
$A(\F_{q^r})$ 212 129744 46568132 16833505536 6120057514052 2213314951942224 799031830790578772 288438918993308160000 104126940843523446384692 37589973457558189176538704

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 10 362 6790 129166 2471650 47045882 893899870 16983416158 322686428890 6131066257802

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.