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

• Label: 2.19.an_dc
• 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: no

Invariants

Base field:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$( 1 - 7 x + 19 x^{2} )( 1 - 6 x + 19 x^{2} )$
	$1 - 13 x + 80 x^{2} - 247 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.203259864187$, $\pm0.258380448083$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	4

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$182$	$127764$	$48315176$	$17156149920$	$6143598863402$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$7$	$353$	$7042$	$131641$	$2481157$	$47054306$	$893836783$	$16983207601$	$322686128758$	$6131062620353$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

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.ag 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.ag : \(\Q(\sqrt{-10}) \).

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.ab_ae	$2$	(not in LMFDB)
2.19.b_ae	$2$	(not in LMFDB)
2.19.n_dc	$2$	(not in LMFDB)
2.19.ah_bs	$3$	(not in LMFDB)
2.19.c_ak	$3$	(not in LMFDB)