Abelian variety isogeny class 1.131.al over $\F_{131}$

• Label: 1.131.al
• Base field: $\F_{131}$
• 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:	$\F_{131}$
Dimension:	$1$
L-polynomial:	$1 - 11 x + 131 x^{2}$
Frobenius angles:	$\pm0.340441692746$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-403}) \)
Galois group:	$C_2$
Jacobians:	$2$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$121$	$17303$	$2251084$	$294514363$	$38579256551$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$121$	$17303$	$2251084$	$294514363$	$38579256551$	$5053908688400$	$662062603422341$	$86730203849463603$	$11361656661045528244$	$1488377021754439004903$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 0 are hyperelliptic):

• $y^2=x^3+66 x+1$
• $y^2=x^3+113 x+113$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{131}$.

Endomorphism algebra over $\F_{131}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-403}) \).

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
1.131.l	$2$	(not in LMFDB)