Abelian variety isogeny class 2.19.al_ch over F19\F_{19}

• Label: 2.19.al_ch
• Base field: $\F_{19}$
• Dimension: $2$
• pp-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:	$1 - 11 x + 59 x^{2} - 209 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.0641455582635$, $\pm0.408994580500$
Angle rank:	$2$ (numerical)
Number field:	4.0.291597.2
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$201$	$128841$	$46967067$	$16879845933$	$6119201069616$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$359$	$6849$	$129523$	$2471304$	$47039051$	$893916543$	$16983744979$	$322687359387$	$6131063631014$

Jacobians and polarizations

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

• $y^2=2 x^6+5 x^5+5 x^4+10 x^3+10 x^2+5 x+15$
• $y^2=3 x^6+4 x^5+5 x^4+4 x^3+18 x^2+15 x+12$

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 4.0.291597.2.

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_ch	$2$	(not in LMFDB)