Properties

Label 2.19.an_cz
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:  $1 - 13 x + 77 x^{2} - 247 x^{3} + 361 x^{4}$
Frobenius angles:  $\pm0.0986133210333$, $\pm0.318874605641$
Angle rank:  $2$ (numerical)
Number field:  4.0.64389.1
Galois group:  $D_{4}$
Jacobians:  2

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 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 179 125121 47489237 17014078701 6128624886224 2212876729285569 798996913370243249 288445608159578024949 104128030656023260494527 37590028970091756735864576

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 7 347 6925 130555 2475112 47036567 893860807 16983810019 322689806185 6131075312102

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is 4.0.64389.1.
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.n_cz$2$(not in LMFDB)