Properties

Label 2.19.ak_cf
Base field $\F_{19}$
Dimension $2$
$p$-rank $1$
Ordinary no
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:  $1 - 10 x + 57 x^{2} - 190 x^{3} + 361 x^{4}$
Frobenius angles:  $\pm0.173854422602$, $\pm0.405491683409$
Angle rank:  $2$ (numerical)
Number field:  4.0.820800.3
Galois group:  $D_{4}$
Jacobians:  $8$
Isomorphism classes:  8

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $219$ $135561$ $48021444$ $17002738425$ $6130443137979$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $10$ $376$ $7000$ $130468$ $2475850$ $47056606$ $893970430$ $16983838468$ $322686954520$ $6131058005656$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 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 4.0.820800.3.

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_cf$2$(not in LMFDB)