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

• Label: 1.59.ag
• 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:	$1 - 6 x + 59 x^{2}$
Frobenius angles:	$\pm0.372279067924$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}) \)
Galois group:	$C_2$
Jacobians:	$7$
Isomorphism classes:	7

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$54$	$3564$	$206226$	$12117600$	$714875814$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$3564$	$206226$	$12117600$	$714875814$	$42180228684$	$2488652515746$	$146830461782400$	$8662995902898774$	$511116752379597804$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which 0 are hyperelliptic):

• $y^2=x^3+33 x+7$
• $y^2=x^3+57 x+57$
• $y^2=x^3+30 x+30$
• $y^2=x^3+14 x+28$
• $y^2=x^3+9 x+9$
• $y^2=x^3+35 x+11$
• $y^2=x^3+18 x+36$

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 \(\Q(\sqrt{-2}) \).

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.g	$2$	(not in LMFDB)