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

• Label: 1.311.ap
• 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:	$1 - 15 x + 311 x^{2}$
Frobenius angles:	$\pm0.360173519252$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-1019}) \)
Galois group:	$C_2$
Jacobians:	$13$
Isomorphism classes:	13

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$297$	$97119$	$30090852$	$9354987675$	$2909387257227$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$297$	$97119$	$30090852$	$9354987675$	$2909387257227$	$904820244389424$	$281399112441812277$	$87515123964855189075$	$27217203547889923134732$	$8464550303317480046237079$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 13 curves (of which 0 are hyperelliptic):

• $y^2=x^3+187 x+187$
• $y^2=x^3+105 x+105$
• $y^2=x^3+7 x+7$
• $y^2=x^3+34 x+63$
• $y^2=x^3+8 x+8$
• $y^2=x^3+12 x+132$
• $y^2=x^3+191 x+235$
• $y^2=x^3+117 x+117$
• $y^2=x^3+158 x+158$
• $y^2=x^3+309 x+309$
• $y^2=x^3+271 x+182$
• $y^2=x^3+50 x+239$
• $y^2=x^3+220 x+220$

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 \(\Q(\sqrt{-1019}) \).

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.p	$2$	(not in LMFDB)