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

• Label: 2.19.am_cw
• Base field: $\F_{19}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$( 1 - 6 x + 19 x^{2} )^{2}$
	$1 - 12 x + 74 x^{2} - 228 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.258380448083$, $\pm0.258380448083$
Angle rank:	$1$ (numerical)
Jacobians:	$4$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$196$	$132496$	$48804196$	$17171481600$	$6140553384196$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$366$	$7112$	$131758$	$2479928$	$47041566$	$893773112$	$16983053278$	$322686513128$	$6131068835406$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{19}$

The isogeny class factors as 1.19.ag^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.19.a_c	$2$	(not in LMFDB)
2.19.m_cw	$2$	(not in LMFDB)
2.19.g_r	$3$	(not in LMFDB)