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

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

Invariants

Base field:	$\F_{5}$
Dimension:	$2$
L-polynomial:	$1 + 4 x + 9 x^{2} + 20 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.483189752728$, $\pm0.896114405083$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-26 +2 \sqrt{5}})\)
Galois group:	$D_{4}$
Jacobians:	$2$
Cyclic group of points:	yes

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$59$	$649$	$17936$	$353705$	$9659539$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$28$	$142$	$564$	$3090$	$15958$	$77962$	$390564$	$1949110$	$9777228$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-26 +2 \sqrt{5}})\).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.5.ae_j	$2$	2.25.c_abd