Abelian variety isogeny class 1.61.b over $\F_{61}$

• Label: 1.61.b
• Base field: $\F_{61}$
• 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:	$\F_{61}$
Dimension:	$1$
L-polynomial:	$1 + x + 61 x^{2}$
Frobenius angles:	$\pm0.520391647268$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}) \)
Galois group:	$C_2$
Jacobians:	$5$
Isomorphism classes:	5
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$63$	$3843$	$226800$	$13838643$	$844614603$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$63$	$3843$	$226800$	$13838643$	$844614603$	$51520795200$	$3142741298823$	$191707288863363$	$11694146210737200$	$713342913017148603$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which 0 are hyperelliptic):

• $y^2=x^3+26 x+26$
• $y^2=x^3+38 x+15$
• $y^2=x^3+37 x+13$
• $y^2=x^3+5$
• $y^2=x^3+53 x+45$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61}$.

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
1.61.ab	$2$	(not in LMFDB)
1.61.ao	$3$	(not in LMFDB)
1.61.n	$3$	(not in LMFDB)