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

• Label: 2.19.al_co
• 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:	$( 1 - 7 x + 19 x^{2} )( 1 - 4 x + 19 x^{2} )$
	$1 - 11 x + 66 x^{2} - 209 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.203259864187$, $\pm0.348268167089$
Angle rank:	$2$ (numerical)
Jacobians:	$6$
Isomorphism classes:	32

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$208$	$134784$	$48577984$	$17093306880$	$6133488510448$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$373$	$7080$	$131161$	$2477079$	$47043286$	$893874501$	$16983666481$	$322687757400$	$6131061600853$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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.ae 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.ae : \(\Q(\sqrt{-15}) \).

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.ad_k	$2$	(not in LMFDB)
2.19.d_k	$2$	(not in LMFDB)
2.19.l_co	$2$	(not in LMFDB)
2.19.af_bq	$3$	(not in LMFDB)
2.19.e_g	$3$	(not in LMFDB)