Properties

Label 2.19.ak_cj
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:  $1 - 10 x + 61 x^{2} - 190 x^{3} + 361 x^{4}$
Frobenius angles:  $\pm0.236824788216$, $\pm0.365068605593$
Angle rank:  $2$ (numerical)
Number field:  4.0.140864.1
Galois group:  $D_{4}$
Jacobians:  4

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 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 223 138929 48859300 17088405929 6130940329303 2212940301530000 798990535013385343 288441516833365815689 104127243159049773789700 37589947403608916796295809

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 10 384 7120 131124 2476050 47037918 893853670 16983569124 322687365760 6131062008304

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is 4.0.140864.1.
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.k_cj$2$(not in LMFDB)