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

• Label: 2.47.ae_dt
• 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:	$( 1 - 3 x + 47 x^{2} )( 1 - x + 47 x^{2} )$
	$1 - 4 x + 97 x^{2} - 188 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.429786591898$, $\pm0.476764235367$
Angle rank:	$2$ (numerical)
Jacobians:	$30$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2115$	$5285385$	$10835128080$	$23776965095625$	$52590453654291075$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$44$	$2388$	$104360$	$4872644$	$229307164$	$10779454206$	$506625239668$	$23811280399876$	$1119130370998520$	$52599132305549268$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):

• $y^2=11 x^6+27 x^5+5 x^4+46 x^3+31 x^2+2 x+43$
• $y^2=20 x^6+41 x^5+20 x^4+42 x^3+20 x^2+41 x+20$
• $y^2=x^6+21 x^5+35 x^4+7 x^3+39 x^2+25 x+9$
• $y^2=38 x^6+36 x^5+30 x^4+43 x^3+45 x^2+34 x+46$
• $y^2=31 x^6+25 x^5+32 x^4+5 x^3+34 x^2+24 x+10$
• $y^2=15 x^6+10 x^5+9 x^4+19 x^3+42 x^2+35 x+10$
• $y^2=12 x^6+41 x^5+37 x^4+4 x^3+24 x^2+35 x+27$
• $y^2=44 x^6+27 x^5+x^4+2 x^3+14 x^2+28 x+40$
• $y^2=34 x^6+33 x^5+10 x^4+20 x^3+11 x^2+31 x+32$
• $y^2=30 x^6+26 x^5+44 x^4+39 x^3+35 x^2+40 x+40$
• $y^2=4 x^6+41 x^5+36 x^4+43 x^3+36 x^2+41 x+4$
• $y^2=2 x^6+7 x^5+22 x^4+13 x^3+10 x^2+3 x+1$
• $y^2=29 x^6+23 x^5+14 x^4+27 x^3+17 x^2+26 x+22$
• $y^2=4 x^6+10 x^5+42 x^4+20 x^3+32 x^2+43 x+14$
• $y^2=40 x^6+45 x^5+32 x^4+30 x^3+36 x^2+43 x+43$
• $y^2=12 x^6+44 x^5+22 x^4+12 x^3+40 x^2+13 x+8$
• $y^2=41 x^6+43 x^5+33 x^4+36 x^3+35 x^2+22 x+35$
• $y^2=36 x^6+3 x^5+22 x^4+16 x^3+38 x^2+14 x+21$
• $y^2=37 x^6+10 x^5+18 x^4+42 x^3+3 x^2+29 x+3$
• $y^2=45 x^6+4 x^5+2 x^4+36 x^3+36 x^2+27 x+39$
• $y^2=24 x^6+17 x^5+x^4+14 x^3+37 x^2+8 x+17$
• $y^2=26 x^6+26 x^5+19 x^4+3 x^3+43 x^2+13 x+22$
• $y^2=22 x^6+44 x^4+44 x^3+15 x^2+23$
• $y^2=10 x^6+16 x^5+26 x^4+36 x^3+35 x^2+34 x+39$
• $y^2=9 x^6+30 x^5+32 x^4+3 x^3+27 x^2+22 x+37$
• $y^2=33 x^6+9 x^5+42 x^4+6 x^3+2 x^2+24 x+40$
• $y^2=29 x^6+42 x^5+37 x^4+3 x^3+37 x^2+42 x+29$
• $y^2=26 x^6+12 x^5+9 x^4+31 x^3+7 x^2+9 x+23$
• $y^2=40 x^6+42 x^5+38 x^4+17 x^3+19 x^2+34 x+5$
• $y^2=11 x^6+27 x^5+46 x^4+45 x^3+20 x^2+37 x+31$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{47}$

The isogeny class factors as 1.47.ad $\times$ 1.47.ab 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.ad : \(\Q(\sqrt{-179}) \).
• 1.47.ab : \(\Q(\sqrt{-187}) \).

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_dn	$2$	(not in LMFDB)
2.47.c_dn	$2$	(not in LMFDB)
2.47.e_dt	$2$	(not in LMFDB)