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

• Label: 1.361.j
• Base field: $\F_{19^{2}}$
• 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:	$\F_{19^{2}}$
Dimension:	$1$
L-polynomial:	$1 + 9 x + 361 x^{2}$
Frobenius angles:	$\pm0.576112408269$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-1363}) \)
Galois group:	$C_2$
Jacobians:	$6$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$371$	$130963$	$47036864$	$16983412803$	$6131070865451$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$371$	$130963$	$47036864$	$16983412803$	$6131070865451$	$2213314931833600$	$799006684004615891$	$288441413579016536643$	$104127350298450638043584$	$37589973457536989896564003$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 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 \(\Q(\sqrt{-1363}) \).

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.aj	$2$	(not in LMFDB)