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

• Label: 2.25.am_cy
• 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:	$1 - 12 x + 76 x^{2} - 300 x^{3} + 625 x^{4}$
Frobenius angles:	$\pm0.131218376688$, $\pm0.408414038432$
Angle rank:	$2$ (numerical)
Number field:	4.0.2361600.2
Galois group:	$D_{4}$
Jacobians:	$16$
Isomorphism classes:	32

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$390$	$395460$	$245843910$	$152427684240$	$95335675119750$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$634$	$15734$	$390214$	$9762374$	$244160314$	$6103807934$	$152589183934$	$3814698508814$	$95367423654874$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.2361600.2.

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_cy	$2$	2.625.i_ags