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

• Label: 2.47.q_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:	$( 1 + 7 x + 47 x^{2} )( 1 + 9 x + 47 x^{2} )$
	$1 + 16 x + 157 x^{2} + 752 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.670549836992$, $\pm0.727918434973$
Angle rank:	$2$ (numerical)
Jacobians:	$27$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3135$	$5012865$	$10656843120$	$23843717436825$	$52599282971003175$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$64$	$2268$	$102640$	$4886324$	$229345664$	$10778924286$	$506625294848$	$23811282400036$	$1119130417328080$	$52599132805095468$

Jacobians and polarizations

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

• $y^2=25 x^6+18 x^5+44 x^4+33 x^3+38 x^2+21 x+17$
• $y^2=25 x^6+15 x^5+28 x^4+32 x^3+34 x^2+19 x+6$
• $y^2=4 x^6+13 x^5+27 x^4+34 x^3+7 x^2+5 x+32$
• $y^2=42 x^6+29 x^5+43 x^4+x^3+5 x^2+13 x+34$
• $y^2=35 x^6+20 x^5+7 x^4+33 x^3+42 x^2+15 x+40$
• $y^2=46 x^6+13 x^5+12 x^4+28 x^3+9 x^2+22 x+15$
• $y^2=22 x^6+29 x^5+46 x^4+27 x^3+23 x^2+19 x+38$
• $y^2=13 x^6+21 x^5+20 x^4+9 x^3+40 x^2+37 x+10$
• $y^2=5 x^6+45 x^5+31 x^4+9 x^3+19 x^2+35 x+29$
• $y^2=14 x^6+41 x^5+37 x^4+31 x^3+42 x^2+22 x+37$
• $y^2=35 x^6+38 x^5+37 x^4+43 x^3+37 x^2+38 x+35$
• $y^2=11 x^6+26 x^5+29 x^4+38 x^3+5 x^2+11 x+44$
• $y^2=4 x^6+14 x^5+8 x^4+40 x^3+34 x^2+12 x+28$
• $y^2=2 x^6+43 x^5+21 x^4+37 x^3+12 x^2+15 x+36$
• $y^2=15 x^6+42 x^5+15 x^4+45 x^3+15 x^2+42 x+15$
• $y^2=9 x^6+12 x^5+3 x^4+42 x^3+4 x^2+37 x+37$
• $y^2=9 x^6+46 x^5+6 x^4+20 x^3+6 x^2+46 x+9$
• $y^2=23 x^6+6 x^5+6 x^4+26 x^3+2 x^2+32 x+20$
• $y^2=28 x^6+23 x^5+11 x^4+43 x^3+43 x^2+38 x+9$
• $y^2=17 x^6+x^5+18 x^4+2 x^3+32 x^2+2 x+3$
• $y^2=2 x^6+35 x^5+11 x^4+16 x^3+39 x^2+29 x+32$
• $y^2=25 x^6+40 x^5+28 x^4+40 x^3+18 x^2+29 x+2$
• $y^2=24 x^6+x^5+20 x^4+42 x^3+20 x^2+x+24$
• $y^2=21 x^6+40 x^5+14 x^4+24 x^3+4 x^2+33 x+12$
• $y^2=9 x^6+30 x^5+37 x^4+8 x^3+28 x^2+19 x+16$
• $y^2=2 x^6+24 x^5+9 x^4+6 x^3+4 x^2+3 x+21$
• $y^2=6 x^6+24 x^5+3 x^4+16 x^3+4 x^2+27 x+9$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{47}$

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

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.aq_gb	$2$	(not in LMFDB)
2.47.ac_bf	$2$	(not in LMFDB)
2.47.c_bf	$2$	(not in LMFDB)