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

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

Invariants

Base field:	$\F_{5}$
Dimension:	$3$
L-polynomial:	$1 + 2 x^{3} + 125 x^{6}$
Frobenius angles:	$\pm0.176169533239$, $\pm0.490497133427$, $\pm0.842836199906$
Angle rank:	$1$ (numerical)
Number field:	6.0.21717639.1
Galois group:	$D_{6}$
Jacobians:	$31$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$128$	$15872$	$2097152$	$244111360$	$30517729408$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$26$	$132$	$626$	$3126$	$16364$	$78126$	$390626$	$1950900$	$9765626$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 31 curves (of which 15 are hyperelliptic):

• $y^2=x^7+x^3+2 x^2+4 x+3$
• $y^2=x^7+x^4+4 x^3+4 x^2+2 x+4$
• $y^2=x^7+2 x^4+4 x+2$
• $y^2=x^7+x^5+x^2+x+4$
• $y^2=x^7+2 x^5+2 x^2+x$
• $y^2=x^7+2 x^5+2 x^4+x^3+2 x^2+3 x+3$
• $y^2=x^7+3 x^5+x^4+4 x^3+4 x^2+x+4$
• $y^2=x^7+3 x^5+2 x^4+x^3+1$
• $y^2=x^8+x^4+x^3+3 x^2+2 x+1$
• $y^2=x^8+x^4+x^3+4 x^2+2 x+3$
• $y^2=x^8+x^5+2 x^4+2 x^3+4 x+2$
• $y^2=x^8+x^5+3 x^4+4 x^2+4 x+3$
• $y^2=x^8+x^5+3 x^4+3 x^3+3 x^2+3 x+4$
• $y^2=x^8+x^6+x^4+x^3+2 x^2+2 x+1$
• $y^2=x^8+x^6+x^5+4 x^4+4 x^3+4 x+1$
• $3 x^4+3 x^3 y+2 x^3 z+x^2 y^2+x^2 y z+x z^3+y^4=0$
• $3 x^4+2 x^3 y+2 x^3 z+2 x^2 y^2+x y^2 z+x z^3+y^4=0$
• $3 x^4+x^3 y+4 x^3 z+2 x^2 y^2+2 x y^2 z+x z^3+y^4=0$
• $2 x^4+3 x^3 y+x^3 z+2 x^2 y^2+x^2 y z+2 x y^2 z+x z^3+y^4=0$
• $x^4+3 x^3 y+2 x^3 z+2 x^2 y^2+x^2 z^2+x z^3+y^3 z=0$
• and

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{3}}$.

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is 6.0.21717639.1.

Endomorphism algebra over $\overline{\F}_{5}$

The base change of $A$ to $\F_{5^{3}}$ is 1.125.c^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-31}) \)$)$

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.a_a_ac	$2$	3.25.a_a_jm
3.5.a_a_ac	$6$	(not in LMFDB)