Properties

Label 2.19.ak_ck
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:  $( 1 - 6 x + 19 x^{2} )( 1 - 4 x + 19 x^{2} )$
Frobenius angles:  $\pm0.258380448083$, $\pm0.348268167089$
Angle rank:  $2$ (numerical)
Jacobians:  8

This isogeny class is not 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
$A(\F_{q^r})$ 224 139776 49069664 17108582400 6130448043104 2212593545949696 798952242414789344 288440549357971046400 104127493568053717359584 37589983010397427949154816

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 10 386 7150 131278 2475850 47030546 893810830 16983512158 322688141770 6131067815906

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The isogeny class factors as 1.19.ag $\times$ 1.19.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ac_o$2$(not in LMFDB)
2.19.c_o$2$(not in LMFDB)
2.19.k_ck$2$(not in LMFDB)