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

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

Invariants

Base field:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$( 1 - 16 x + 71 x^{2} )^{2}$
	$1 - 32 x + 398 x^{2} - 2272 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.101666819831$, $\pm0.101666819831$
Angle rank:	$1$ (numerical)
Jacobians:	$3$
Cyclic group of points:	no
Non-cyclic primes:	$2, 7$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3136$	$24285184$	$127608986176$	$645605491277824$	$3255251579835443776$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$4814$	$356536$	$25405854$	$1804233800$	$128100768878$	$9095127601880$	$645753615909694$	$45848501544584296$	$3255243558216906254$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

• $y^2=60 x^5+29 x^4+59 x^3+29 x^2+60 x$
• $y^2=26 x^6+20 x^4+20 x^2+26$
• $y^2=3 x^6+64 x^4+64 x^2+3$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.aq^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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
2.71.a_aek	$2$	(not in LMFDB)
2.71.bg_pi	$2$	(not in LMFDB)
2.71.q_hd	$3$	(not in LMFDB)