Abelian variety isogeny class 1.349.abj over $\F_{349}$

• Label: 1.349.abj
• Base field: $\F_{349}$
• 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:	$\F_{349}$
Dimension:	$1$
L-polynomial:	$1 - 35 x + 349 x^{2}$
Frobenius angles:	$\pm0.113814939953$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-19}) \)
Galois group:	$C_2$
Jacobians:	$5$
Isomorphism classes:	5

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$315$	$121275$	$42502320$	$14835449475$	$5177584756575$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$315$	$121275$	$42502320$	$14835449475$	$5177584756575$	$1806976784289600$	$630634882866992955$	$220091573704046242275$	$76811959213316115006960$	$26807373765263833783531875$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which 0 are hyperelliptic):

• $y^2=x^3+319 x+319$
• $y^2=x^3+331 x+331$
• $y^2=x^3+30 x+30$
• $y^2=x^3+219 x+219$
• $y^2=x^3+280 x+211$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{349}$.

Endomorphism algebra over $\F_{349}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-19}) \).

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
1.349.bj	$2$	(not in LMFDB)