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

• Label: 2.71.abb_mm
• 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: no

Invariants

Base field:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$( 1 - 14 x + 71 x^{2} )( 1 - 13 x + 71 x^{2} )$
	$1 - 27 x + 324 x^{2} - 1917 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.187913521440$, $\pm0.219552767034$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	4

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3422$	$25014820$	$128391044600$	$646173424667680$	$3255540182092083962$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$45$	$4961$	$358722$	$25428201$	$1804393755$	$128101331738$	$9095122766565$	$645753494062321$	$45848500049355822$	$3255243544709626601$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.ao $\times$ 1.71.an and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.71.ao : \(\Q(\sqrt{-22}) \).
• 1.71.an : \(\Q(\sqrt{-115}) \).

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.ab_abo	$2$	(not in LMFDB)
2.71.b_abo	$2$	(not in LMFDB)
2.71.bb_mm	$2$	(not in LMFDB)