Abelian variety isogeny class 1.71.ak over $\F_{71}$

• Label: 1.71.ak
• Base field: $\F_{71}$
• 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:	$\F_{71}$
Dimension:	$1$
L-polynomial:	$1 - 10 x + 71 x^{2}$
Frobenius angles:	$\pm0.297788873486$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-46}) \)
Galois group:	$C_2$
Jacobians:	$4$
Isomorphism classes:	4
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$62$	$5084$	$359042$	$25420000$	$1804232302$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$62$	$5084$	$359042$	$25420000$	$1804232302$	$128099722844$	$9095114338162$	$645753512880000$	$45848500948027742$	$3255243554609637404$

Jacobians and polarizations

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

• $y^2=x^3+58 x+51$
• $y^2=x^3+38 x+38$
• $y^2=x^3+54 x+23$
• $y^2=x^3+56 x+56$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{71}$.

Endomorphism algebra over $\F_{71}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-46}) \).

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
1.71.k	$2$	(not in LMFDB)