Abelian variety isogeny class 2.25.as_fb over $\F_{5^{2}}$

• Label: 2.25.as_fb
• Base field: $\F_{5^{2}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 9 x + 25 x^{2} )^{2}$
	$1 - 18 x + 131 x^{2} - 450 x^{3} + 625 x^{4}$
Frobenius angles:	$\pm0.143566293129$, $\pm0.143566293129$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

This isogeny class is not simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$289$	$354025$	$242487184$	$152814537225$	$95444634758929$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$564$	$15518$	$391204$	$9773528$	$244197294$	$6103828088$	$152589286084$	$3814702013198$	$95367439482324$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

• $y^2=2ax^6+2ax^5+2ax^4+2ax^3+2ax^2+2ax+2a$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{2}}$.

Endomorphism algebra over $\F_{5^{2}}$

The isogeny class factors as 1.25.aj^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{5^{2}}$.

Subfield	Primitive Model
$\F_{5}$	2.5.a_aj

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.25.a_abf	$2$	2.625.ack_dhb
2.25.s_fb	$2$	2.625.ack_dhb
2.25.j_ce	$3$	(not in LMFDB)