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

• Label: 2.19.aj_bp
• 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:	$1 - 9 x + 41 x^{2} - 171 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.0387402455333$, $\pm0.487338161656$
Angle rank:	$2$ (numerical)
Number field:	4.0.404685.1
Galois group:	$D_{4}$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$223$	$130009$	$46130449$	$16808213565$	$6123139391248$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$363$	$6725$	$128971$	$2472896$	$47048991$	$893848799$	$16983168931$	$322686779645$	$6131069107878$

Jacobians and polarizations

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

• $y^2=6x^6+7x^5+12x^4+11x^3+9x^2+13x+14$
• $y^2=2x^6+7x^5+9x^4+7x^3+2x^2+14x+15$

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 4.0.404685.1.

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.j_bp	$2$	(not in LMFDB)