Properties

Label 2.19.ak_ci
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:  $1 - 10 x + 60 x^{2} - 190 x^{3} + 361 x^{4}$
Frobenius angles:  $\pm0.219146719950$, $\pm0.377690954553$
Angle rank:  $2$ (numerical)
Number field:  4.0.288576.1
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
$A(\F_{q^r})$ 222 138084 48649302 17067734736 6131186130102 2213236847696004 799024323493479102 288442668910231680000 104127099218085493490382 37589925218957464314310404

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 10 382 7090 130966 2476150 47044222 893891470 16983636958 322686919690 6131058389902

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is 4.0.288576.1.
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_ci$2$(not in LMFDB)