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

• Label: 1.311.as
• 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:	$1 - 18 x + 311 x^{2}$
Frobenius angles:	$\pm0.329518380815$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-230}) \)
Galois group:	$C_2$
Jacobians:	$20$
Isomorphism classes:	20

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$294$	$97020$	$30091194$	$9355056480$	$2909388496854$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$294$	$97020$	$30091194$	$9355056480$	$2909388496854$	$904820237008380$	$281399111765557674$	$87515123955190081920$	$27217203547978544086854$	$8464550303322799314205500$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which 0 are hyperelliptic):

• $y^2=x^3+211 x+211$
• $y^2=x^3+298 x+168$
• $y^2=x^3+246 x+246$
• $y^2=x^3+184 x+184$
• $y^2=x^3+38 x+38$
• $y^2=x^3+264 x+105$
• $y^2=x^3+274 x+215$
• $y^2=x^3+188 x+188$
• $y^2=x^3+5 x+5$
• $y^2=x^3+71 x+159$
• $y^2=x^3+139 x+285$
• $y^2=x^3+300 x+300$
• $y^2=x^3+199 x+199$
• $y^2=x^3+92 x+92$
• $y^2=x^3+210 x+133$
• $y^2=x^3+239 x+141$
• $y^2=x^3+305 x+305$
• $y^2=x^3+177 x+177$
• $y^2=x^3+250 x+250$
• $y^2=x^3+218 x+221$

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 \(\Q(\sqrt{-230}) \).

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.s	$2$	(not in LMFDB)