Abelian variety isogeny class 2.19.ak_cg over $\F_{19}$

• Label: 2.19.ak_cg
• Base field: $\F_{19}$
• 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:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$1 - 10 x + 58 x^{2} - 190 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.188320984213$, $\pm0.397309458316$
Angle rank:	$2$ (numerical)
Number field:	4.0.40400.1
Galois group:	$D_{4}$
Jacobians:	$18$
Isomorphism classes:	24

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$220$	$136400$	$48230380$	$17024902400$	$6130937675500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$378$	$7030$	$130638$	$2476050$	$47053578$	$893950270$	$16983778398$	$322686776890$	$6131056700698$

Jacobians and polarizations

This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=15x^6+5x^5+5x^4+8x^3+8x^2+10x+8$
• $y^2=11x^6+7x^5+6x^4+9x^3+11x^2+16x$
• $y^2=10x^6+6x^5+13x^4+7x^3+14x^2+10x+18$
• $y^2=2x^6+15x^5+8x^4+2x^3+14x^2+7x+8$
• $y^2=13x^6+18x^5+8x^4+7x^3+9x^2+x+14$
• $y^2=14x^6+13x^5+x^4+8x^3+5x^2+6x+3$
• $y^2=3x^6+3x^5+17x^4+12x^3+13x^2+4x$
• $y^2=13x^6+10x^5+17x^4+2x^3+6x+3$
• $y^2=4x^5+9x^4+14x^3+6x^2+5x+2$
• $y^2=16x^6+6x^5+6x^4+8x^3+10x^2+11x+12$
• $y^2=8x^6+15x^5+9x^4+3x^3+12x^2+4x+14$
• $y^2=10x^6+9x^5+14x^4+5x^3+3x^2+15x+14$
• $y^2=14x^6+13x^5+11x^4+11x^3+8x^2+16x+10$
• $y^2=8x^6+7x^5+16x^4+13x^3+10x^2+8x+6$
• $y^2=x^6+5x^5+17x^4+16x^3+8x+2$
• $y^2=10x^6+4x^5+3x^4+11x^3+18x^2+6x+8$
• $y^2=11x^6+5x^5+17x^4+3x^3+15x^2+7x+13$
• $y^2=15x^6+3x^5+2x^4+3x^3+15x^2+3x+8$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{19}$.

Endomorphism algebra over $\F_{19}$

The endomorphism algebra of this simple isogeny class is 4.0.40400.1.

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
2.19.k_cg	$2$	(not in LMFDB)