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

• Label: 2.5.ab_i
• 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:	$( 1 - 2 x + 5 x^{2} )( 1 + x + 5 x^{2} )$
	$1 - x + 8 x^{2} - 5 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.352416382350$, $\pm0.571783146564$
Angle rank:	$2$ (numerical)
Jacobians:	$3$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$28$	$1120$	$16576$	$380800$	$9822988$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$41$	$134$	$609$	$3145$	$15446$	$77509$	$391969$	$1958030$	$9761201$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ac $\times$ 1.5.b 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.ac : \(\Q(\sqrt{-1}) \).
• 1.5.b : \(\Q(\sqrt{-19}) \).

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.ad_m	$2$	2.25.p_ea
2.5.b_i	$2$	2.25.p_ea
2.5.d_m	$2$	2.25.p_ea