Properties

Label 2.19.aj_bt
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:  $1 - 9 x + 45 x^{2} - 171 x^{3} + 361 x^{4}$
Frobenius angles:  $\pm0.116535216599$, $\pm0.468549984315$
Angle rank:  $2$ (numerical)
Number field:  4.0.2058997.1
Galois group:  $D_{4}$
Jacobians:  7

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 7 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 227 133249 46870733 16887045517 6129376351472 2214228406289809 799085520922156121 288442626062387377653 104127427900407754641071 37590016523920554069232384

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 11 371 6833 129579 2475416 47065295 893959931 16983634435 322687938269 6131073282086

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is 4.0.2058997.1.
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.j_bt$2$(not in LMFDB)