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

• Label: 2.19.af_y
• 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:	$( 1 - 7 x + 19 x^{2} )( 1 + 2 x + 19 x^{2} )$
	$1 - 5 x + 24 x^{2} - 95 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.203259864187$, $\pm0.573681533379$
Angle rank:	$2$ (numerical)
Jacobians:	$16$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$286$	$138996$	$46710664$	$17005326624$	$6145998770626$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$15$	$385$	$6810$	$130489$	$2482125$	$47058946$	$893826375$	$16983534769$	$322687711230$	$6131057977825$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{19}$

The isogeny class factors as 1.19.ah $\times$ 1.19.c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.19.ah : \(\Q(\sqrt{-3}) \).
• 1.19.c : \(\Q(\sqrt{-2}) \).

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.aj_ca	$2$	(not in LMFDB)
2.19.f_y	$2$	(not in LMFDB)
2.19.j_ca	$2$	(not in LMFDB)
2.19.b_bk	$3$	(not in LMFDB)
2.19.k_cc	$3$	(not in LMFDB)