Properties

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

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Invariants

Base field:  $\F_{19}$
Dimension:  $2$
L-polynomial:  $1 - 12 x + 69 x^{2} - 228 x^{3} + 361 x^{4}$
Frobenius angles:  $\pm0.106312411237$, $\pm0.357895350441$
Angle rank:  $2$ (numerical)
Number field:  4.0.12625.1
Galois group:  $D_{4}$
Jacobians:  8

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains the Jacobians of 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 191 128161 47539136 16981973305 6125692347071 2213008391520256 799044670009668791 288449384824034140905 104127995238661094476736 37589995770672827115745441

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 8 356 6932 130308 2473928 47039366 893914232 16984032388 322689696428 6131069897156

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is 4.0.12625.1.
All geometric endomorphisms are defined over $\F_{19}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.19.m_cr$2$(not in LMFDB)