Properties

Label 2.19.ak_cj
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:  $1 - 10 x + 61 x^{2} - 190 x^{3} + 361 x^{4}$
Frobenius angles:  $\pm0.236824788216$, $\pm0.365068605593$
Angle rank:  $2$ (numerical)
Number field:  4.0.140864.1
Galois group:  $D_{4}$
Jacobians:  $4$
Isomorphism classes:  4

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $223$ $138929$ $48859300$ $17088405929$ $6130940329303$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $10$ $384$ $7120$ $131124$ $2476050$ $47037918$ $893853670$ $16983569124$ $322687365760$ $6131062008304$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 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 4.0.140864.1.

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.k_cj$2$(not in LMFDB)