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

• Label: 2.47.q_gc
• 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:	$( 1 + 8 x + 47 x^{2} )^{2}$
	$1 + 16 x + 158 x^{2} + 752 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.698301488982$, $\pm0.698301488982$
Angle rank:	$1$ (numerical)
Jacobians:	$39$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3136$	$5017600$	$10651891264$	$23845642240000$	$52599503316705856$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$64$	$2270$	$102592$	$4886718$	$229346624$	$10778871710$	$506625793472$	$23811281427838$	$1119130389342784$	$52599133151904350$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 39 curves (of which all are hyperelliptic):

• $y^2=19 x^6+13 x^5+5 x^4+42 x^3+20 x^2+29 x$
• $y^2=10 x^6+23 x^5+12 x^4+11 x^3+12 x^2+23 x+10$
• $y^2=x^6+15 x^5+4 x^4+20 x^3+4 x^2+15 x+1$
• $y^2=17 x^6+17 x^4+17 x^2+17$
• $y^2=46 x^6+41 x^5+12 x^4+28 x^3+12 x^2+41 x+46$
• $y^2=x^6+30 x^5+x^4+5 x^3+3 x^2+35 x+27$
• $y^2=41 x^6+23 x^5+29 x^4+5 x^3+43 x^2+11 x+45$
• $y^2=8 x^6+39 x^5+46 x^4+22 x^3+46 x^2+39 x+8$
• $y^2=16 x^6+17 x^5+27 x^4+2 x^3+24 x^2+32 x+28$
• $y^2=44 x^6+46 x^5+31 x^4+31 x^2+46 x+44$
• $y^2=37 x^6+x^5+29 x^4+10 x^3+44 x^2+17 x+3$
• $y^2=43 x^6+36 x^5+7 x^4+4 x^3+7 x^2+36 x+43$
• $y^2=24 x^6+22 x^5+20 x^4+38 x^3+20 x^2+22 x+24$
• $y^2=39 x^6+13 x^5+22 x^4+37 x^3+22 x^2+13 x+39$
• $y^2=33 x^6+14 x^5+38 x^4+29 x^3+43 x^2+37 x+41$
• $y^2=x^6+35 x^4+35 x^2+1$
• $y^2=32 x^6+43 x^5+39 x^4+39 x^2+43 x+32$
• $y^2=45 x^6+7 x^5+25 x^4+32 x^3+37 x^2+3 x+46$
• $y^2=7 x^6+13 x^5+32 x^4+40 x^3+4 x^2+45 x+8$
• $y^2=x^6+36 x^5+30 x^4+29 x^3+30 x^2+36 x+1$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{47}$

The isogeny class factors as 1.47.i^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-31}) \)$)$

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
2.47.aq_gc	$2$	(not in LMFDB)
2.47.a_be	$2$	(not in LMFDB)
2.47.ai_r	$3$	(not in LMFDB)