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

• Label: 2.47.g_dz
• 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:	$( 1 + 3 x + 47 x^{2} )^{2}$
	$1 + 6 x + 103 x^{2} + 282 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.570213408102$, $\pm0.570213408102$
Angle rank:	$1$ (numerical)
Jacobians:	$50$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2601$	$5267025$	$10697351184$	$23783909765625$	$52611533192505681$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$2380$	$103032$	$4874068$	$229399074$	$10779316990$	$506620274382$	$23811290421988$	$1119130595587944$	$52599131691643900$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 50 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{47}$

The isogeny class factors as 1.47.d^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-179}) \)$)$

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.ag_dz	$2$	(not in LMFDB)
2.47.a_dh	$2$	(not in LMFDB)
2.47.ad_abm	$3$	(not in LMFDB)