Properties

Label 1.19.f
Base Field $\F_{19}$
Dimension $1$
$p$-rank $1$
Principally polarizable
Contains a Jacobian

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Invariants

Base field:  $\F_{19}$
Dimension:  $1$
Weil polynomial:  $1 + 5 x + 19 x^{2}$
Frobenius angles:  $\pm0.694430027533$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-51}) \)
Galois group:  $C_2$

This isogeny class is simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

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
$A(\F_{q^r})$ 25 375 6700 130875 2476375 47034000 893925925 16983517875 322686894100 6131071134375

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 25 375 6700 130875 2476375 47034000 893925925 16983517875 322686894100 6131071134375

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.