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

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

Invariants

Base field:	$\F_{19^{2}}$
Dimension:	$1$
L-polynomial:	$1 - 26 x + 361 x^{2}$
Frobenius angles:	$\pm0.260146938293$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}) \)
Galois group:	$C_2$
Jacobians:	$16$

This isogeny class is simple and geometrically simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$336$	$130368$	$47056464$	$16983821568$	$6131069159376$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$336$	$130368$	$47056464$	$16983821568$	$6131069159376$	$2213314901179200$	$799006684270354896$	$288441413534752601088$	$104127350297602681851984$	$37589973457549801194193728$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 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 \(\Q(\sqrt{-3}) \).

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{19^{2}}$.

Subfield	Primitive Model
$\F_{19}$	1.19.ai
$\F_{19}$	1.19.i

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
1.361.ba	$2$	(not in LMFDB)
1.361.al	$3$	(not in LMFDB)
1.361.bl	$3$	(not in LMFDB)