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

• Label: 2.19.al_ci
• 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:	$1 - 11 x + 60 x^{2} - 209 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.0899168499442$, $\pm0.402539378619$
Angle rank:	$2$ (numerical)
Number field:	4.0.444312.2
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$202$	$129684$	$47196088$	$16911831072$	$6122057290822$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$361$	$6882$	$129769$	$2472459$	$47043970$	$893945649$	$16983929425$	$322688208246$	$6131066027161$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

• $y^2=8x^6+5x^4+4x^3+15x^2+x+8$
• $y^2=12x^6+9x^5+5x^4+x^3+x^2+17x+12$
• $y^2=14x^6+5x^5+18x^4+11x^3+18x^2+3x+6$
• $y^2=16x^6+9x^5+10x^4+12x^3+9x^2+8x+8$

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 4.0.444312.2.

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.l_ci	$2$	(not in LMFDB)