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

• Label: 1.67.ae
• 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:	$1 - 4 x + 67 x^{2}$
Frobenius angles:	$\pm0.421429069538$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-7}) \)
Galois group:	$C_2$
Jacobians:	$10$
Isomorphism classes:	10

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$64$	$4608$	$301504$	$20146176$	$1350055744$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$64$	$4608$	$301504$	$20146176$	$1350055744$	$90458436096$	$6060716468416$	$406067693395968$	$27206534133825088$	$1822837802440647168$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which 0 are hyperelliptic):

• $y^2=x^3+61 x+55$
• $y^2=x^3+52 x+37$
• $y^2=x^3+14 x+14$
• $y^2=x^3+30 x+60$
• $y^2=x^3+42 x+42$
• $y^2=x^3+22 x+44$
• $y^2=x^3+19 x+38$
• $y^2=x^3+5 x+5$
• $y^2=x^3+17 x+17$
• $y^2=x^3+33 x+33$

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 \(\Q(\sqrt{-7}) \).

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
1.67.e	$2$	(not in LMFDB)