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

• Label: 2.47.au_ha
• Base field: $\F_{47}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{47}$
Dimension:	$2$
L-polynomial:	$1 - 20 x + 182 x^{2} - 940 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.0605356534996$, $\pm0.341840273347$
Angle rank:	$2$ (numerical)
Number field:	4.0.35136.1
Galois group:	$D_{4}$
Jacobians:	$14$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1432$	$4800064$	$10789456984$	$23804400587776$	$52590853665327832$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$28$	$2174$	$103924$	$4878270$	$229308908$	$10778923838$	$506622312644$	$23811292742014$	$1119130547473468$	$52599132501193214$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{47}$

The endomorphism algebra of this simple isogeny class is 4.0.35136.1.

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.u_ha	$2$	(not in LMFDB)