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

• Label: 2.19.an_cy
• 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

Invariants

Base field:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$1 - 13 x + 76 x^{2} - 247 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.0603553363735$, $\pm0.329969159439$
Angle rank:	$2$ (numerical)
Number field:	4.0.43928.1
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$178$	$124244$	$47214856$	$16965766688$	$6122994315838$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$7$	$345$	$6886$	$130185$	$2472837$	$47026434$	$893823679$	$16983659793$	$322688982466$	$6131070755305$

Jacobians and polarizations

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

• $y^2=10x^5+16x^4+16x^3+15x^2+5x+8$
• $y^2=3x^6+7x^5+18x^4+17x^3+10x^2+17x+3$

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 4.0.43928.1.

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_cy	$2$	(not in LMFDB)