Abelian variety isogeny class 2.83.abf_pq over $\F_{83}$

• Label: 2.83.abf_pq
• Base field: $\F_{83}$
• 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:	$\F_{83}$
Dimension:	$2$
L-polynomial:	$( 1 - 16 x + 83 x^{2} )( 1 - 15 x + 83 x^{2} )$
	$1 - 31 x + 406 x^{2} - 2573 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.158801688027$, $\pm0.192168636682$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	12

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4692$	$46450800$	$327083280048$	$2253050532216000$	$15516926384833852572$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$53$	$6741$	$572036$	$47474297$	$3939265363$	$326942518374$	$27136065644041$	$2252292283704433$	$186940254887381948$	$15516041177414264661$

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 $\F_{83}$.

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.aq $\times$ 1.83.ap and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.83.aq : \(\Q(\sqrt{-19}) \).
• 1.83.ap : \(\Q(\sqrt{-107}) \).

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
2.83.ab_acw	$2$	(not in LMFDB)
2.83.b_acw	$2$	(not in LMFDB)
2.83.bf_pq	$2$	(not in LMFDB)