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

• Label: 2.19.ac_u
• 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:	$1 - 2 x + 20 x^{2} - 38 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.289273908625$, $\pm0.625899319086$
Angle rank:	$2$ (numerical)
Number field:	4.0.1963840.6
Galois group:	$D_{4}$
Jacobians:	$20$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$342$	$144324$	$47022606$	$17067756240$	$6139536045102$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$18$	$398$	$6858$	$130966$	$2479518$	$47026766$	$893790342$	$16983679966$	$322687599282$	$6131067530078$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

• $y^2=2 x^6+17 x^5+12 x^4+9 x^3+13 x^2+6 x+8$
• $y^2=5 x^6+7 x^5+4 x^4+16 x^3+7 x^2+9 x+3$
• $y^2=12 x^6+8 x^5+17 x^4+10 x^3+10 x^2+11 x+18$
• $y^2=2 x^6+6 x^5+18 x^4+5 x^3+3 x^2+4 x+16$
• $y^2=15 x^6+x^5+17 x^4+5 x^3+6 x^2+18 x+5$
• $y^2=8 x^5+2 x^4+12 x^3+9 x+4$
• $y^2=x^6+12 x^5+11 x^4+7 x^3+15 x^2+11 x+6$
• $y^2=4 x^6+16 x^5+7 x^4+5 x^3+2 x^2+5 x+15$
• $y^2=4 x^6+9 x^5+4 x^3+17 x^2+7 x+13$
• $y^2=15 x^6+8 x^4+17 x^3+4 x^2+18 x+5$
• $y^2=9 x^6+13 x^5+3 x^4+6 x^2+8 x+10$
• $y^2=9 x^6+3 x^5+17 x^4+3 x^2+2 x+17$
• $y^2=14 x^6+8 x^5+18 x^4+2 x^3+2 x^2+18 x+6$
• $y^2=7 x^6+11 x^5+11 x^4+12 x^3+18 x^2+8 x+18$
• $y^2=18 x^6+2 x^5+x^4+6 x^3+4 x^2+11 x+1$
• $y^2=8 x^6+3 x^5+17 x^4+2 x^2+10 x+17$
• $y^2=16 x^6+14 x^5+13 x^4+7 x^3+3 x^2+5 x+14$
• $y^2=9 x^6+5 x^5+6 x^3+8 x^2+14 x+2$
• $y^2=7 x^6+16 x^5+9 x^4+14 x^3+12 x^2+10 x+11$
• $y^2=2 x^6+4 x^5+2 x^4+11 x^3+7 x+3$

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 4.0.1963840.6.

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.c_u	$2$	(not in LMFDB)