Abelian variety isogeny class 1.311.ag over $\F_{311}$

• Label: 1.311.ag
• Base field: $\F_{311}$
• 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:	$\F_{311}$
Dimension:	$1$
L-polynomial:	$1 - 6 x + 311 x^{2}$
Frobenius angles:	$\pm0.445586276780$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-302}) \)
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$306$	$97308$	$30085614$	$9354801888$	$2909387449026$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$306$	$97308$	$30085614$	$9354801888$	$2909387449026$	$904820328207900$	$281399113358689086$	$87515123943652990848$	$27217203547320728159634$	$8464550303318504032520028$

Jacobians and polarizations

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

• $y^2=x^3+86 x+86$
• $y^2=x^3+113 x+113$
• $y^2=x^3+119 x+119$
• $y^2=x^3+167 x+167$
• $y^2=x^3+97 x+134$
• $y^2=x^3+18 x+198$
• $y^2=x^3+301 x+201$
• $y^2=x^3+210 x+210$
• $y^2=x^3+70 x+148$
• $y^2=x^3+264 x+264$
• $y^2=x^3+260 x+260$
• $y^2=x^3+62 x+60$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{311}$.

Endomorphism algebra over $\F_{311}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-302}) \).

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
1.311.g	$2$	(not in LMFDB)