Abelian variety isogeny class 3.5.c_m_s over $\F_{5}$

• Label: 3.5.c_m_s
• Base field: $\F_{5}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$3$
L-polynomial:	$1 + 2 x + 12 x^{2} + 18 x^{3} + 60 x^{4} + 50 x^{5} + 125 x^{6}$
Frobenius angles:	$\pm0.402917635499$, $\pm0.537763293788$, $\pm0.716593418271$
Angle rank:	$3$ (numerical)
Number field:	6.0.27060976.1
Galois group:	$S_4\times C_2$
Jacobians:	$9$
Isomorphism classes:	12
Cyclic group of points:	no
Non-cyclic primes:	$2$

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:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$268$	$34304$	$1802032$	$237795328$	$29767489228$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$46$	$116$	$610$	$3048$	$15604$	$78744$	$390018$	$1954052$	$9767166$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which 2 are hyperelliptic):

• $y^2=x^8+x^5+3 x^4+4 x^3+3 x^2+4 x+3$
• $y^2=x^8+x^6+2 x^3+4 x+3$
• $3 x^3 y+4 x^3 z+x^2 y^2+x^2 y z+2 x y^2 z+x z^3+y^4=0$
• $4 x^4+2 x^3 z+4 x^2 y^2+3 x^2 y z+x^2 z^2+x z^3+y^3 z=0$
• $4 x^3 y+x^3 z+2 x^2 y^2+2 x^2 y z+x y z^2+x z^3+y^3 z=0$
• $x^4+2 x^3 y+4 x^3 z+x^2 y^2+4 x^2 z^2+x y z^2+x z^3+y^3 z=0$
• $4 x^4+3 x^3 y+x^3 z+3 x^2 y^2+x^2 y z+2 x^2 z^2+x y^3+x z^3+y^2 z^2=0$
• $4 x^4+2 x^3 z+4 x^2 y^2+4 x^2 y z+2 x^2 z^2+x y^3+x z^3+y^2 z^2=0$
• $3 x^4+2 x^3 y+3 x^2 y^2+x^2 y z+x y^3+x z^3+y^4+y^2 z^2=0$

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 6.0.27060976.1.

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
3.5.ac_m_as	$2$	3.25.u_hk_bsw