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

• Label: 2.47.am_fa
• 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:	$( 1 - 6 x + 47 x^{2} )^{2}$
	$1 - 12 x + 130 x^{2} - 564 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.355830380849$, $\pm0.355830380849$
Angle rank:	$1$ (numerical)
Jacobians:	$48$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1764$	$5143824$	$10910638116$	$23821583901696$	$52588452181045284$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$2326$	$105084$	$4881790$	$229298436$	$10778836822$	$506623038300$	$23811303958654$	$1119130580745828$	$52599132068734486$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 48 curves (of which all are hyperelliptic):

• $y^2=31 x^6+40 x^5+2 x^4+24 x^3+10 x^2+9 x+44$
• $y^2=26 x^6+x^5+21 x^4+5 x^3+21 x^2+x+26$
• $y^2=25 x^6+14 x^5+9 x^4+37 x^3+9 x^2+14 x+25$
• $y^2=22 x^6+30 x^5+6 x^4+19 x^3+43 x^2+17 x+8$
• $y^2=29 x^6+46 x^5+44 x^4+19 x^3+20 x^2+13 x+38$
• $y^2=4 x^6+27 x^5+2 x^4+15 x^3+23 x^2+17 x+35$
• $y^2=29 x^6+22 x^5+17 x^4+41 x^3+21 x^2+23 x+23$
• $y^2=41 x^6+41 x^5+44 x^4+41 x^3+43 x^2+5 x+38$
• $y^2=27 x^6+16 x^5+28 x^4+22 x^3+6 x^2+18 x+17$
• $y^2=43 x^6+19 x^5+12 x^4+31 x^3+12 x^2+19 x+43$
• $y^2=40 x^6+33 x^5+23 x^4+26 x^3+23 x^2+33 x+40$
• $y^2=31 x^6+4 x^5+6 x^4+7 x^3+3 x^2+x+45$
• $y^2=43 x^6+18 x^4+33 x^3+2 x^2+39$
• $y^2=37 x^6+43 x^5+33 x^3+38 x^2+26 x$
• $y^2=28 x^6+26 x^5+26 x^4+10 x^3+38 x^2+23 x+7$
• $y^2=x^6+x^5+25 x^4+9 x^3+25 x^2+x+1$
• $y^2=26 x^6+46 x^5+39 x^4+39 x^2+46 x+26$
• $y^2=33 x^6+2 x^5+12 x^4+23 x^3+12 x^2+2 x+33$
• $y^2=33 x^6+21 x^5+33 x^4+46 x^3+41 x^2+24 x+20$
• $y^2=23 x^6+32 x^5+45 x^4+40 x^3+45 x^2+32 x+23$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{47}$

The isogeny class factors as 1.47.ag^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-38}) \)$)$

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.a_cg	$2$	(not in LMFDB)
2.47.m_fa	$2$	(not in LMFDB)
2.47.g_al	$3$	(not in LMFDB)