Abelian variety isogeny class 2.71.abd_nj over $\F_{71}$

• Label: 2.71.abd_nj
• Base field: $\F_{71}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$1 - 29 x + 347 x^{2} - 2059 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.0270940053421$, $\pm0.242092067010$
Angle rank:	$2$ (numerical)
Number field:	4.0.122157.1
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2

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:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3301$	$24681577$	$127965163075$	$645766929509725$	$3255224922566550256$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$43$	$4895$	$357535$	$25412211$	$1804219028$	$128099765747$	$9095111703413$	$645753441934291$	$45848500047415045$	$3255243547744294430$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=51x^6+13x^5+14x^4+24x^3+29x^2+2x+13$
• $y^2=5x^6+15x^5+6x^4+22x^3+54x^2+65x+63$

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 4.0.122157.1.

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
2.71.bd_nj	$2$	(not in LMFDB)