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

• Label: 1.71.g
• 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:	$1 + 6 x + 71 x^{2}$
Frobenius angles:	$\pm0.615871442562$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-62}) \)
Galois group:	$C_2$
Jacobians:	$8$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$78$	$5148$	$356850$	$25410528$	$1804311678$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$78$	$5148$	$356850$	$25410528$	$1804311678$	$128099871900$	$9095116785378$	$645753580737408$	$45848500660983150$	$3255243547840769628$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which 0 are hyperelliptic):

• $y^2=x^3+27 x+27$
• $y^2=x^3+41 x+41$
• $y^2=x^3+14 x+27$
• $y^2=x^3+50 x+66$
• $y^2=x^3+48 x+48$
• $y^2=x^3+29 x+61$
• $y^2=x^3+20 x+20$
• $y^2=x^3+20 x+69$

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 \(\Q(\sqrt{-62}) \).

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.ag	$2$	(not in LMFDB)