Properties

Label 2.19.al_cm
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:  $1 - 11 x + 64 x^{2} - 209 x^{3} + 361 x^{4}$
Frobenius angles:  $\pm0.165808925970$, $\pm0.370946582130$
Angle rank:  $2$ (numerical)
Number field:  4.0.349112.1
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
$A(\F_{q^r})$ 206 133076 48115832 17034792608 6130763962866 2213470309936064 799069449922614818 288448112752124296832 104127537296738062291928 37589940421461346132563156

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 9 369 7014 130713 2475979 47049186 893941953 16983957489 322688277282 6131060869489

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is 4.0.349112.1.
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.l_cm$2$(not in LMFDB)