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

• Label: 2.19.m_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:	$\pm0.741619551917$, $\pm0.741619551917$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$676$	$132496$	$45346756$	$17171481600$	$6121596362596$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$366$	$6608$	$131758$	$2472272$	$47041566$	$893970368$	$16983053278$	$322688882432$	$6131068835406$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{19}$

The isogeny class factors as 1.19.g^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.am_cw	$2$	(not in LMFDB)
2.19.a_c	$2$	(not in LMFDB)
2.19.ag_r	$3$	(not in LMFDB)