Abelian variety isogeny class 1.67.af over $\F_{67}$

• Label: 1.67.af
• Base field: $\F_{67}$
• 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:	$\F_{67}$
Dimension:	$1$
L-polynomial:	$1 - 5 x + 67 x^{2}$
Frobenius angles:	$\pm0.401201937998$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}) \)
Galois group:	$C_2$
Jacobians:	$5$
Isomorphism classes:	5

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$63$	$4599$	$301644$	$20148219$	$1350051633$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$63$	$4599$	$301644$	$20148219$	$1350051633$	$90458209296$	$6060715663779$	$406067709431475$	$27206534283752628$	$1822837801853436039$

Jacobians and polarizations

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

• $y^2=x^3+44 x+44$
• $y^2=x^3+53 x+39$
• $y^2=x^3+63 x+63$
• $y^2=x^3+6$
• $y^2=x^3+25 x+50$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{67}$.

Endomorphism algebra over $\F_{67}$

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
1.67.f	$2$	(not in LMFDB)
1.67.al	$3$	(not in LMFDB)
1.67.q	$3$	(not in LMFDB)