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

• Label: 2.25.aq_eh
• Base field: $\F_{5^{2}}$
• 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:	$\F_{5^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 16 x + 111 x^{2} - 400 x^{3} + 625 x^{4}$
Frobenius angles:	$\pm0.0738526172967$, $\pm0.284366381360$
Angle rank:	$2$ (numerical)
Number field:	4.0.46224.1
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$321$	$370113$	$244628964$	$152738602953$	$95357916129201$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$592$	$15658$	$391012$	$9764650$	$244116214$	$6103367770$	$152587603012$	$3814699921018$	$95367462167152$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5^{2}}$

The endomorphism algebra of this simple isogeny class is 4.0.46224.1.

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.q_eh	$2$	2.625.abi_bdr