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

• Label: 2.19.ag_bs
• 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:	$1 - 6 x + 44 x^{2} - 114 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.317362850973$, $\pm0.453539019738$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-64 -6 \sqrt{3}})\)
Galois group:	$D_{4}$
Jacobians:	$10$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$286$	$150436$	$48676342$	$16967375568$	$6123778398166$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$414$	$7094$	$130198$	$2473154$	$47041566$	$893876858$	$16983492574$	$322687621262$	$6131071040814$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-64 -6 \sqrt{3}})\).

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.g_bs	$2$	(not in LMFDB)