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

• Label: 2.19.ad_bk
• 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:	$1 - 3 x + 36 x^{2} - 57 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.366038693329$, $\pm0.520517985920$
Angle rank:	$2$ (numerical)
Number field:	4.0.1384888.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	12
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$338$	$154804$	$47918936$	$16903358368$	$6126826949438$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$17$	$425$	$6986$	$129705$	$2474387$	$47047394$	$893857289$	$16983590353$	$322689004526$	$6131067825305$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

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

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 4.0.1384888.1.

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.d_bk	$2$	(not in LMFDB)