Abelian variety isogeny class 1.59.ai over $\F_{59}$

• Label: 1.59.ai
• Base field: $\F_{59}$
• 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:	$\F_{59}$
Dimension:	$1$
L-polynomial:	$1 - 8 x + 59 x^{2}$
Frobenius angles:	$\pm0.325650265238$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-43}) \)
Galois group:	$C_2$
Jacobians:	$4$
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$52$	$3536$	$206284$	$12121408$	$714903332$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$52$	$3536$	$206284$	$12121408$	$714903332$	$42180127184$	$2488649470268$	$146830445468928$	$8662996000430356$	$511116754290832976$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which 0 are hyperelliptic):

• $y^2=x^3+11 x+11$
• $y^2=x^3+16 x+16$
• $y^2=x^3+54 x+49$
• $y^2=x^3+13 x+13$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59}$.

Endomorphism algebra over $\F_{59}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-43}) \).

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
1.59.i	$2$	(not in LMFDB)