Abelian variety isogeny class 2.5.b_ac over $\F_{5}$

• Label: 2.5.b_ac
• Base field: $\F_{5}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$2$
L-polynomial:	$( 1 - 3 x + 5 x^{2} )( 1 + 4 x + 5 x^{2} )$
	$1 + x - 2 x^{2} + 5 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.265942140215$, $\pm0.852416382350$
Angle rank:	$2$ (numerical)
Jacobians:	$2$
Cyclic group of points:	yes

This isogeny class is not simple, 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$30$	$540$	$18720$	$432000$	$9708150$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$7$	$21$	$148$	$689$	$3107$	$15786$	$77063$	$390529$	$1950772$	$9769101$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=4 x^6+x^5+x^4+x+4$
• $y^2=2 x^6+2 x^5+2 x^4+3 x^2+1$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ad $\times$ 1.5.e and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.5.ad : \(\Q(\sqrt{-11}) \).
• 1.5.e : \(\Q(\sqrt{-1}) \).

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
2.5.ah_w	$2$	2.25.af_bs
2.5.ab_ac	$2$	2.25.af_bs
2.5.h_w	$2$	2.25.af_bs