Properties

Label 2.19.aj_bt
Base field $\F_{19}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
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:  $7$
Isomorphism classes:  7

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $227$ $133249$ $46870733$ $16887045517$ $6129376351472$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{19}$.

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is 4.0.2058997.1.

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)