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

• Label: 2.71.abd_no
• 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:	$( 1 - 15 x + 71 x^{2} )( 1 - 14 x + 71 x^{2} )$
	$1 - 29 x + 352 x^{2} - 2059 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.150643965450$, $\pm0.187913521440$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3306$	$24735492$	$128121715800$	$646016844564000$	$3255503550934065006$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$43$	$4905$	$357970$	$25422041$	$1804373453$	$128101626522$	$9095129407043$	$645753571345681$	$45848500663821790$	$3255243547585445505$

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.ap $\times$ 1.71.ao 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.ap : \(\Q(\sqrt{-59}) \).
• 1.71.ao : \(\Q(\sqrt{-22}) \).

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_acq	$2$	(not in LMFDB)
2.71.b_acq	$2$	(not in LMFDB)
2.71.bd_no	$2$	(not in LMFDB)