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

• Label: 2.47.aq_gb
• 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:	$( 1 - 9 x + 47 x^{2} )( 1 - 7 x + 47 x^{2} )$
	$1 - 16 x + 157 x^{2} - 752 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.272081565027$, $\pm0.329450163008$
Angle rank:	$2$ (numerical)
Jacobians:	$27$
Isomorphism classes:	36
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1599$	$5012865$	$10902698352$	$23843717436825$	$52598982070352679$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$2268$	$105008$	$4886324$	$229344352$	$10778924286$	$506620946080$	$23811282400036$	$1119130528877456$	$52599132805095468$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 27 curves (of which all are hyperelliptic):

• $y^2=31 x^6+43 x^5+32 x^4+24 x^3+2 x^2+11 x+38$
• $y^2=5 x^6+3 x^5+15 x^4+44 x^3+35 x^2+32 x+20$
• $y^2=20 x^6+18 x^5+41 x^4+29 x^3+35 x^2+25 x+19$
• $y^2=22 x^6+4 x^5+27 x^4+5 x^3+25 x^2+18 x+29$
• $y^2=7 x^6+4 x^5+39 x^4+16 x^3+46 x^2+3 x+8$
• $y^2=42 x^6+18 x^5+13 x^4+46 x^3+45 x^2+16 x+28$
• $y^2=16 x^6+4 x^5+42 x^4+41 x^3+21 x^2+x+2$
• $y^2=18 x^6+11 x^5+6 x^4+45 x^3+12 x^2+44 x+3$
• $y^2=25 x^6+37 x^5+14 x^4+45 x^3+x^2+34 x+4$
• $y^2=23 x^6+17 x^5+44 x^4+14 x^3+22 x^2+16 x+44$
• $y^2=34 x^6+2 x^5+44 x^4+27 x^3+44 x^2+2 x+34$
• $y^2=8 x^6+36 x^5+4 x^4+2 x^3+25 x^2+8 x+32$
• $y^2=29 x^6+31 x^5+11 x^4+8 x^3+35 x^2+40 x+15$
• $y^2=10 x^6+27 x^5+11 x^4+44 x^3+13 x^2+28 x+39$
• $y^2=28 x^6+22 x^5+28 x^4+37 x^3+28 x^2+22 x+28$
• $y^2=45 x^6+13 x^5+15 x^4+22 x^3+20 x^2+44 x+44$
• $y^2=45 x^6+42 x^5+30 x^4+6 x^3+30 x^2+42 x+45$
• $y^2=21 x^6+30 x^5+30 x^4+36 x^3+10 x^2+19 x+6$
• $y^2=46 x^6+21 x^5+8 x^4+27 x^3+27 x^2+2 x+45$
• $y^2=38 x^6+5 x^5+43 x^4+10 x^3+19 x^2+10 x+15$
• $y^2=38 x^6+7 x^5+21 x^4+22 x^3+36 x^2+34 x+44$
• $y^2=5 x^6+8 x^5+15 x^4+8 x^3+13 x^2+34 x+38$
• $y^2=33 x^6+19 x^5+4 x^4+46 x^3+4 x^2+19 x+33$
• $y^2=11 x^6+12 x^5+23 x^4+26 x^3+20 x^2+24 x+13$
• $y^2=30 x^6+6 x^5+45 x^4+11 x^3+15 x^2+32 x+22$
• $y^2=38 x^6+33 x^5+30 x^4+20 x^3+29 x^2+10 x+23$
• $y^2=30 x^6+26 x^5+15 x^4+33 x^3+20 x^2+41 x+45$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{47}$

The isogeny class factors as 1.47.aj $\times$ 1.47.ah 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.aj : \(\Q(\sqrt{-107}) \).
• 1.47.ah : \(\Q(\sqrt{-139}) \).

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_bf	$2$	(not in LMFDB)
2.47.c_bf	$2$	(not in LMFDB)
2.47.q_gb	$2$	(not in LMFDB)