Abelian variety isogeny class 1.19.ah over $\F_{19}$

• Label: 1.19.ah
• Base field: $\F_{19}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{19}$
Dimension:	$1$
L-polynomial:	$1 - 7 x + 19 x^{2}$
Frobenius angles:	$\pm0.203259864187$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}) \)
Galois group:	$C_2$
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:	$1$
Slopes:	$[0, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$13$	$351$	$6916$	$130923$	$2479243$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$13$	$351$	$6916$	$130923$	$2479243$	$47056464$	$893886097$	$16983462483$	$322686721084$	$6131061331551$

Jacobians and polarizations

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

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 \(\Q(\sqrt{-3}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
1.19.h	$2$	1.361.al
1.19.ab	$3$	(not in LMFDB)
1.19.i	$3$	(not in LMFDB)