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

• Label: 2.25.aq_ej
• Base field: $\F_{5^{2}}$
• 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:	$\F_{5^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 9 x + 25 x^{2} )( 1 - 7 x + 25 x^{2} )$
	$1 - 16 x + 113 x^{2} - 400 x^{3} + 625 x^{4}$
Frobenius angles:	$\pm0.143566293129$, $\pm0.253183311107$
Angle rank:	$2$ (numerical)
Jacobians:	$6$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$323$	$373065$	$246162176$	$153189815625$	$95446998996563$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$596$	$15754$	$392164$	$9773770$	$244167086$	$6103553914$	$152587809604$	$3814697136730$	$95367437511476$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=4x^6+(4a+4)x^5+3ax^4+3x^3+3ax^2+(4a+4)x+4$
• $y^2=2ax^6+(3a+1)x^5+(3a+1)x^4+3x^3+(3a+1)x^2+(3a+1)x+2a$
• $y^2=4ax^6+4ax^5+(a+4)x^4+(4a+3)x^3+(a+4)x^2+4ax+4a$
• $y^2=(2a+3)x^6+(4a+3)x^5+(2a+3)x^4+(2a+1)x^3+(2a+3)x^2+(4a+3)x+2a+3$
• $y^2=(a+4)x^6+(a+3)x^5+3x^4+(2a+3)x^3+3x^2+(a+3)x+a+4$
• $y^2=(2a+3)x^6+2ax^5+(3a+3)x^4+3ax^3+(3a+3)x^2+2ax+2a+3$

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 $\times$ 1.25.ah and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.25.aj : \(\Q(\sqrt{-19}) \).
• 1.25.ah : \(\Q(\sqrt{-51}) \).

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
2.25.ac_an	$2$	2.625.abe_bux
2.25.c_an	$2$	2.625.abe_bux
2.25.q_ej	$2$	2.625.abe_bux