Properties

Label 2.19.ak_cf
Base Field $\F_{19}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{19}$
Dimension:  $2$
L-polynomial:  $1 - 10 x + 57 x^{2} - 190 x^{3} + 361 x^{4}$
Frobenius angles:  $\pm0.173854422602$, $\pm0.405491683409$
Angle rank:  $2$ (numerical)
Number field:  4.0.820800.3
Galois group:  $D_{4}$
Jacobians:  8

This isogeny class is simple and geometrically simple.

Newton polygon

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

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
$A(\F_{q^r})$ 219 135561 48021444 17002738425 6130443137979 2213819494209936 799094906199094899 288446091332880808425 104127110457816217373604 37589922863120783333154441

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 10 376 7000 130468 2475850 47056606 893970430 16983838468 322686954520 6131058005656

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is 4.0.820800.3.
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_cf$2$(not in LMFDB)