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

• Label: 2.19.al_cl
• 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:	$1 - 11 x + 63 x^{2} - 209 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.148084750558$, $\pm0.380020549717$
Angle rank:	$2$ (numerical)
Number field:	4.0.443205.2
Galois group:	$D_{4}$
Jacobians:	$8$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$205$	$132225$	$47885335$	$17004796125$	$6128994732400$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$367$	$6981$	$130483$	$2475264$	$47050027$	$893959971$	$16984039603$	$322688549499$	$6131062547302$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

• $y^2=13x^6+17x^5+5x^4+12x^3+18x^2+7x+14$
• $y^2=2x^6+4x^5+14x^4+11x^3+12x^2+16$
• $y^2=3x^6+6x^5+18x^4+x^3+12x+12$
• $y^2=8x^6+16x^5+9x^4+8x^3+18x^2+8x+12$
• $y^2=12x^6+15x^5+2x^4+5x^2+7x+18$
• $y^2=15x^6+16x^5+5x^4+9x^3+7x^2+7x+18$
• $y^2=18x^6+11x^5+x^4+8x^3+13x+8$
• $y^2=7x^6+16x^5+6x^4+3x^3+2x^2+3x+3$

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 4.0.443205.2.

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_cl	$2$	(not in LMFDB)