Abelian variety isogeny class 1.361.abm over $\F_{19^{2}}$

• Label: 1.361.abm
• Base field: $\F_{19^{2}}$
• Dimension: $1$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{19^{2}}$
Dimension:	$1$
L-polynomial:	$( 1 - 19 x )^{2}$
	$1 - 38 x + 361 x^{2}$
Frobenius angles:	$0$, $0$
Angle rank:	$0$ (numerical)
Number field:	\(\Q\)
Galois group:	Trivial
Jacobians:	$2$

This isogeny class is simple and geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$324$	$129600$	$47032164$	$16983302400$	$6131061305604$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$324$	$129600$	$47032164$	$16983302400$	$6131061305604$	$2213314824974400$	$799006683995140644$	$288441413533654041600$	$104127350297265866137284$	$37589973457533696060840000$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{19^{2}}$.

Endomorphism algebra over $\F_{19^{2}}$

The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q\) ramified at $19$ and $\infty$.

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
1.361.bm	$2$	(not in LMFDB)
1.361.a	$4$	(not in LMFDB)