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

• Label: 2.5.a_j
• 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:	$( 1 - x + 5 x^{2} )( 1 + x + 5 x^{2} )$
	$1 + 9 x^{2} + 25 x^{4}$
Frobenius angles:	$\pm0.428216853436$, $\pm0.571783146564$
Angle rank:	$1$ (numerical)
Jacobians:	$2$
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$35$	$1225$	$15680$	$354025$	$9761675$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$44$	$126$	$564$	$3126$	$15734$	$78126$	$391204$	$1953126$	$9757724$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

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

Endomorphism algebra over $\overline{\F}_{5}$

The base change of $A$ to $\F_{5^{2}}$ is 1.25.j^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$

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.ac_l	$2$	2.25.s_fb
2.5.c_l	$2$	2.25.s_fb
2.5.a_aj	$4$	2.625.ack_dhb