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

• Label: 2.25.am_cw
• 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:	$1 - 12 x + 74 x^{2} - 300 x^{3} + 625 x^{4}$
Frobenius angles:	$\pm0.104680351346$, $\pm0.418388652673$
Angle rank:	$2$ (numerical)
Number field:	4.0.35136.1
Galois group:	$D_{4}$
Jacobians:	$14$
Isomorphism classes:	18

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$388$	$392656$	$244710436$	$152212313088$	$95313424978468$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$630$	$15662$	$389662$	$9760094$	$244153878$	$6103772318$	$152588933182$	$3814698214190$	$95367434599350$

Jacobians and polarizations

This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.35136.1.

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.m_cw	$2$	2.625.e_asg