Abelian variety isogeny class 2.47.au_hl over $\F_{47}$

• Label: 2.47.au_hl
• Base field: $\F_{47}$
• 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:	$\F_{47}$
Dimension:	$2$
L-polynomial:	$( 1 - 11 x + 47 x^{2} )( 1 - 9 x + 47 x^{2} )$
	$1 - 20 x + 193 x^{2} - 940 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.203632126579$, $\pm0.272081565027$
Angle rank:	$2$ (numerical)
Jacobians:	$9$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1443$	$4852809$	$10858448016$	$23850047011401$	$52609017853268643$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$28$	$2196$	$104584$	$4887620$	$229388108$	$10779290622$	$506622090884$	$23811274517764$	$1119130404498328$	$52599132073407636$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):

• $y^2=35 x^6+x^5+27 x^4+41 x^3+16 x^2+42 x+46$
• $y^2=22 x^6+28 x^5+19 x^4+9 x^3+11 x^2+6 x+31$
• $y^2=35 x^6+4 x^5+3 x^4+x^3+3 x^2+4 x+35$
• $y^2=43 x^6+9 x^5+3 x^4+40 x^3+16 x^2+21 x+39$
• $y^2=27 x^6+27 x^5+29 x^4+9 x^3+46 x^2+4 x+42$
• $y^2=30 x^6+46 x^5+3 x^4+15 x^3+18 x^2+11 x+41$
• $y^2=21 x^6+9 x^5+18 x^4+37 x^2+8 x+14$
• $y^2=29 x^6+40 x^5+23 x^4+44 x^3+39 x^2+41 x+15$
• $y^2=23 x^6+36 x^5+32 x^4+38 x^3+14 x^2+37 x+46$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{47}$.

Endomorphism algebra over $\F_{47}$

The isogeny class factors as 1.47.al $\times$ 1.47.aj and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.47.al : \(\Q(\sqrt{-67}) \).
• 1.47.aj : \(\Q(\sqrt{-107}) \).

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
2.47.ac_af	$2$	(not in LMFDB)
2.47.c_af	$2$	(not in LMFDB)
2.47.u_hl	$2$	(not in LMFDB)