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

• Label: 2.19.ab_bl
• Base field: $\F_{19}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$1 - x + 37 x^{2} - 19 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.440576737334$, $\pm0.522585002744$
Angle rank:	$2$ (numerical)
Number field:	4.0.138725.1
Galois group:	$D_{4}$
Jacobians:	$4$

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$379$	$158801$	$47420101$	$16824807149$	$6127512765424$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$19$	$435$	$6913$	$129099$	$2474664$	$47064351$	$893900971$	$16983323139$	$322687245637$	$6131068570150$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{19}$

The endomorphism algebra of this simple isogeny class is 4.0.138725.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.19.b_bl	$2$	(not in LMFDB)