Abelian variety isogeny class 3.17.f_k_e over $\F_{17}$

• Label: 3.17.f_k_e
• Base field: $\F_{17}$
• 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:	$\F_{17}$
Dimension:	$3$
L-polynomial:	$1 + 5 x + 10 x^{2} + 4 x^{3} + 170 x^{4} + 1445 x^{5} + 4913 x^{6}$
Frobenius angles:	$\pm0.233616406589$, $\pm0.644093291910$, $\pm0.865081944886$
Angle rank:	$3$ (numerical)
Number field:	6.0.349684823903256.1
Galois group:	$S_4\times C_2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6548$	$23834720$	$118279202132$	$588060222422400$	$2867769979134330688$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$23$	$285$	$4901$	$84297$	$1422508$	$24139725$	$410283169$	$6975914417$	$118586131547$	$2015993274300$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 61 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

• $y^2=x^7+16x^6+14x^5+8x^4+7x^3+2x^2+12x+12$
• $y^2=x^7+11x^6+11x^5+3x^4+15x^3+11x^2+x$
• $y^2=x^8+14x^7+13x^6+x^5+14x^4+13x^3+2x^2+3x+6$
• $y^2=x^8+14x^7+4x^6+5x^5+9x^3+15x^2+3x$
• $y^2=x^7+12x^6+10x^5+11x^4+5x^3+12x^2+13x+15$
• $y^2=x^8+11x^7+10x^6+13x^5+9x^4+4x^3+5x^2+15x+3$
• $y^2=x^8+10x^7+14x^6+4x^5+8x^4+14x^3+6x^2+2x+7$
• $y^2=x^8+12x^7+13x^6+12x^5+6x^4+3x^3+13x^2+10x+13$
• $y^2=x^8+12x^7+13x^6+14x^5+16x^4+6x^3+8x^2+9x+3$
• $y^2=x^8+14x^7+14x^6+8x^4+2x^3+x^2+13x+1$
• $y^2=3x^7+16x^6+14x^5+8x^4+10x^3+13x^2+9x+6$
• $y^2=3x^8+3x^7+13x^6+16x^5+16x^3+x^2+4x+7$
• $y^2=3x^8+x^7+16x^5+5x^4+10x^3+6x^2+12x+8$
• $y^2=x^8+5x^7+5x^6+2x^5+8x^4+14x^3+12x^2+13x+5$
• $y^2=x^8+9x^7+4x^5+13x^4+11x^3+11x^2+15x+14$
• $y^2=x^8+x^7+x^6+14x^5+2x^4+x^3+6x^2+8x+9$
• $y^2=x^8+15x^7+2x^6+15x^5+12x^4+11x^3+x^2+10x+3$
• $y^2=x^8+9x^7+16x^5+3x^4+5x^2+6x+11$
• $y^2=x^8+8x^7+x^6+8x^5+15x^4+9x^3+12x^2+6x+8$
• $y^2=x^8+8x^7+x^6+14x^5+4x^4+x^3+10x^2+7x+7$
• and

• $y^2=x^8+9x^7+5x^6+10x^5+15x^4+4x^3+15x^2+6x+12$
• $y^2=x^7+3x^6+10x^5+7x^4+5x^3+14x^2+6x+8$
• $y^2=3x^7+13x^6+15x^5+7x^4+15x^3+12x^2+11x+13$
• $y^2=x^8+3x^7+9x^6+12x^5+10x^4+4x^3+x^2+2x+13$
• $y^2=x^8+11x^7+15x^6+6x^5+15x^4+6x^3+13x^2+11x+6$
• $y^2=x^8+16x^7+2x^6+4x^5+8x^4+11x^3+13x^2+9x+2$
• $y^2=3x^8+15x^7+8x^6+9x^5+12x^3+12x^2+6x+4$
• $y^2=3x^8+8x^7+x^6+13x^5+13x^4+9x^3+4x^2+12x+12$
• $y^2=3x^8+16x^7+16x^6+14x^5+5x^4+14x^3+13x^2+12x+13$
• $y^2=x^8+12x^7+5x^6+7x^5+14x^4+8x^3+5x^2+5x+7$
• $y^2=3x^8+5x^7+14x^6+6x^5+8x^4+2x^3+14x^2+6x+1$
• $y^2=x^8+10x^7+15x^6+5x^5+2x^4+2x^2+9x+4$
• $y^2=x^8+14x^7+3x^6+6x^5+12x^4+6x^3+4x^2+9x+16$
• $y^2=3x^8+5x^7+10x^6+8x^5+15x^4+6x^3+15x^2+12x+7$
• $y^2=3x^8+9x^7+10x^6+12x^5+8x^4+9x^3+10x^2+15x+15$
• $y^2=3x^8+2x^7+14x^6+13x^5+13x^4+8x^3+4x^2+12x+12$
• $y^2=x^8+15x^7+9x^6+2x^5+5x^4+5x^3+16x^2+2x+14$
• $y^2=3x^8+2x^6+x^5+4x^4+10x^3+4x^2+10x+16$
• $y^2=3x^8+4x^7+13x^6+6x^5+3x^4+9x^3+x+1$
• $y^2=x^8+14x^7+10x^6+3x^5+14x^4+16x^3+7x^2+9x+13$
• $y^2=x^8+4x^7+x^6+11x^5+2x^4+10x^3+9x^2+16$
• $y^2=x^8+15x^7+5x^6+15x^5+4x^4+2x^3+16x^2+10x+14$
• $y^2=3x^8+7x^7+8x^5+11x^4+9x^3+8x^2+9x$
• $y^2=3x^8+5x^7+6x^6+3x^5+10x^4+10x^3+13x^2+16x+15$
• $y^2=3x^8+2x^7+9x^6+4x^5+2x^4+6x^3+12x+14$
• $y^2=x^8+13x^7+13x^6+16x^5+14x^4+15x^3+5x^2+6x+15$
• $y^2=3x^8+5x^7+6x^6+4x^5+8x^4+10x^3+13x^2+5x+2$
• $y^2=x^8+15x^7+7x^6+16x^5+5x^4+2x^3+12x^2+2x+10$
• $y^2=3x^8+14x^7+5x^5+8x^4+10x^3+3x^2+9x+16$
• $y^2=3x^8+5x^7+16x^5+3x^4+6x^3+9x^2+14x+1$
• $y^2=x^8+6x^7+13x^6+5x^5+8x^4+7x^3+x^2+15x+11$
• $y^2=3x^8+6x^7+6x^5+13x^4+10x^3+15x^2+x+1$
• $y^2=3x^8+14x^7+5x^6+7x^4+x^3+13x^2+2x+7$
• $y^2=3x^8+8x^7+6x^6+13x^5+15x^4+13x^3+9x^2+11x+11$
• $y^2=3x^8+2x^7+4x^6+5x^5+16x^4+13x^3+5x^2+12x+8$
• $y^2=x^8+9x^7+x^6+3x^5+7x^3+11x^2+7x+5$
• $y^2=3x^8+14x^7+15x^6+11x^5+4x^4+4x^3+10x^2+x+2$
• $y^2=x^8+7x^7+15x^6+13x^4+2x^3+16x^2+7x+4$
• $y^2=x^8+5x^7+16x^6+12x^5+13x^4+6x^3+6x^2+x+6$
• $y^2=3x^8+11x^7+14x^6+9x^5+16x^4+16x^3+15x^2+7x+10$
• $y^2=3x^8+11x^7+3x^5+3x^3+x^2+15x+14$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17}$.

Endomorphism algebra over $\F_{17}$

The endomorphism algebra of this simple isogeny class is 6.0.349684823903256.1.

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
3.17.af_k_ae	$2$	(not in LMFDB)