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

• Label: 1.349.g
• 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:	$1 + 6 x + 349 x^{2}$
Frobenius angles:	$\pm0.551338499210$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-85}) \)
Galois group:	$C_2$
Jacobians:	$12$
Isomorphism classes:	12

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$356$	$122464$	$42502484$	$14835288960$	$5177587061636$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$356$	$122464$	$42502484$	$14835288960$	$5177587061636$	$1806976786306144$	$630634880156055284$	$220091573667325386240$	$76811959213313798472356$	$26807373765254003364529504$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which 0 are hyperelliptic):

• $y^2=x^3+285 x+221$
• $y^2=x^3+13 x+13$
• $y^2=x^3+116 x+232$
• $y^2=x^3+167 x+167$
• $y^2=x^3+25 x+25$
• $y^2=x^3+161 x+322$
• $y^2=x^3+283 x+283$
• $y^2=x^3+305 x+305$
• $y^2=x^3+216 x+216$
• $y^2=x^3+321 x+293$
• $y^2=x^3+276 x+203$
• $y^2=x^3+312 x+275$

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 \(\Q(\sqrt{-85}) \).

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.ag	$2$	(not in LMFDB)