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

• Label: 2.19.al_ck
• 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:	$( 1 - 8 x + 19 x^{2} )( 1 - 3 x + 19 x^{2} )$
	$1 - 11 x + 62 x^{2} - 209 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.130073469147$, $\pm0.388176076177$
Angle rank:	$2$ (numerical)
Jacobians:	$4$
Isomorphism classes:	18

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$204$	$131376$	$47655216$	$16974304704$	$6126953978724$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$365$	$6948$	$130249$	$2474439$	$47049446$	$893966901$	$16984068049$	$322688697372$	$6131064544925$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{19}$

The isogeny class factors as 1.19.ai $\times$ 1.19.ad 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.ai : \(\Q(\sqrt{-3}) \).
• 1.19.ad : \(\Q(\sqrt{-67}) \).

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.af_o	$2$	(not in LMFDB)
2.19.f_o	$2$	(not in LMFDB)
2.19.l_ck	$2$	(not in LMFDB)
2.19.ac_bj	$3$	(not in LMFDB)
2.19.e_r	$3$	(not in LMFDB)