Abelian variety isogeny class 2.19.an_db over $\F_{19}$

• Label: 2.19.an_db
• 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

Invariants

Base field:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$1 - 13 x + 79 x^{2} - 247 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.161616782251$, $\pm0.288204697885$
Angle rank:	$2$ (numerical)
Number field:	4.0.21125.1
Galois group:	$C_4$
Jacobians:	$2$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$181$	$126881$	$48039391$	$17109268445$	$6138926902096$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$7$	$351$	$7003$	$131283$	$2479272$	$47050491$	$893866813$	$16983571443$	$322688189557$	$6131069106406$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=18x^6+2x^5+16x^4+2x^3+11x^2+13x+3$
• $y^2=15x^6+5x^5+18x^4+10x^3+11x^2+15$

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 4.0.21125.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.19.n_db	$2$	(not in LMFDB)