Abelian variety isogeny class 2.625.adq_fcs over $\F_{5^{4}}$

• Label: 2.625.adq_fcs
• Base field: $\F_{5^{4}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{5^{4}}$
Dimension:	$2$
L-polynomial:	$( 1 - 25 x )^{2}( 1 - 44 x + 625 x^{2} )$
	$1 - 94 x + 3450 x^{2} - 58750 x^{3} + 390625 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.157542425424$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$335232$	$151833277440$	$59596360679544192$	$23282992558198463201280$	$9094946618618951519263393152$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$532$	$388690$	$244106692$	$152587420030$	$95367427455652$	$59604644768186770$	$37252902983988092212$	$23283064365290456982910$	$14551915228361168897272372$	$9094947017729046870963322450$

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 $\F_{5^{4}}$.

Endomorphism algebra over $\F_{5^{4}}$

The isogeny class factors as 1.625.aby $\times$ 1.625.abs and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.625.aby : the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.
• 1.625.abs : \(\Q(\sqrt{-141}) \).

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
2.625.ag_abko	$2$	(not in LMFDB)
2.625.g_abko	$2$	(not in LMFDB)
2.625.dq_fcs	$2$	(not in LMFDB)
2.625.at_fu	$3$	(not in LMFDB)