Properties

Label 6.2.9251107100235625.1
Degree 66
Signature [2,2][2, 2]
Discriminant 9.251×10159.251\times 10^{15}
Root discriminant 458.18458.18
Ramified primes 5,43,5715,43,571
Class number 22 (GRH)
Class group [2] (GRH)
Galois group S6S_6 (as 6T16)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 101*x^4 - 938*x^3 - 3752*x^2 - 12252*x + 9233)
 
gp: K = bnfinit(y^6 - 3*y^5 - 101*y^4 - 938*y^3 - 3752*y^2 - 12252*y + 9233, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^6 - 3*x^5 - 101*x^4 - 938*x^3 - 3752*x^2 - 12252*x + 9233);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^6 - 3*x^5 - 101*x^4 - 938*x^3 - 3752*x^2 - 12252*x + 9233)
 

x63x5101x4938x33752x212252x+9233 x^{6} - 3x^{5} - 101x^{4} - 938x^{3} - 3752x^{2} - 12252x + 9233 Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  66
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  [2,2][2, 2]
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   92511071002356259251107100235625 =544335713\medspace = 5^{4}\cdot 43^{3}\cdot 571^{3} Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  458.18458.18
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  52/3431/25711/2458.17594556432655^{2/3}43^{1/2}571^{1/2}\approx 458.1759455643265
Ramified primes:   55, 4343, 571571 Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  Q(24553)\Q(\sqrt{24553})
Aut(K/Q)\Aut(K/\Q):   C1C_1
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over Q\Q.
This is not a CM field.

Integral basis (with respect to field generator aa)

11, aa, a2a^{2}, a3a^{3}, a4a^{4}, 12363313a5+8334792363313a4+9773662363313a3160145787771a2+10996722363313a+2148012363313\frac{1}{2363313}a^{5}+\frac{833479}{2363313}a^{4}+\frac{977366}{2363313}a^{3}-\frac{160145}{787771}a^{2}+\frac{1099672}{2363313}a+\frac{214801}{2363313} Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  11
Inessential primes:  None

Class group and class number

Ideal class group:  C2C_{2}, which has order 22 (assuming GRH)
sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 
Narrow class group:  C2×C2C_{2}\times C_{2}, which has order 44 (assuming GRH)

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  33
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   1 -1  (order 22) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   8949013399132882363313a573 ⁣ ⁣002363313a4+31 ⁣ ⁣972363313a358 ⁣ ⁣38787771a2+77 ⁣ ⁣982363313a42 ⁣ ⁣982363313\frac{894901339913288}{2363313}a^{5}-\frac{73\!\cdots\!00}{2363313}a^{4}+\frac{31\!\cdots\!97}{2363313}a^{3}-\frac{58\!\cdots\!38}{787771}a^{2}+\frac{77\!\cdots\!98}{2363313}a-\frac{42\!\cdots\!98}{2363313}, 40160183076277787771a595691990696214787771a441 ⁣ ⁣07787771a340 ⁣ ⁣33787771a217 ⁣ ⁣06787771a60 ⁣ ⁣80787771\frac{40160183076277}{787771}a^{5}-\frac{95691990696214}{787771}a^{4}-\frac{41\!\cdots\!07}{787771}a^{3}-\frac{40\!\cdots\!33}{787771}a^{2}-\frac{17\!\cdots\!06}{787771}a-\frac{60\!\cdots\!80}{787771}, 31 ⁣ ⁣002363313a520 ⁣ ⁣942363313a4+21 ⁣ ⁣812363313a315 ⁣ ⁣96787771a2+62 ⁣ ⁣372363313a38 ⁣ ⁣792363313\frac{31\!\cdots\!00}{2363313}a^{5}-\frac{20\!\cdots\!94}{2363313}a^{4}+\frac{21\!\cdots\!81}{2363313}a^{3}-\frac{15\!\cdots\!96}{787771}a^{2}+\frac{62\!\cdots\!37}{2363313}a-\frac{38\!\cdots\!79}{2363313} Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  536421.287716 536421.287716 (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

lims1(s1)ζK(s)=(2r1(2π)r2RhwD(22(2π)2536421.287716229251107100235625(0.880701787902 \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 536421.287716 \cdot 2}{2\cdot\sqrt{9251107100235625}}\cr\approx \mathstrut & 0.880701787902 \end{aligned} (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 101*x^4 - 938*x^3 - 3752*x^2 - 12252*x + 9233)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^6 - 3*x^5 - 101*x^4 - 938*x^3 - 3752*x^2 - 12252*x + 9233, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^6 - 3*x^5 - 101*x^4 - 938*x^3 - 3752*x^2 - 12252*x + 9233);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^6 - 3*x^5 - 101*x^4 - 938*x^3 - 3752*x^2 - 12252*x + 9233);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

S6S_6 (as 6T16):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 720
The 11 conjugacy class representatives for S6S_6
Character table for S6S_6

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and Q\Q.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling algebras

Twin sextic algebra: 6.2.613825.1
Degree 6 sibling: 6.2.613825.1
Degree 10 sibling: deg 10
Degree 12 siblings: deg 12, deg 12
Degree 15 siblings: deg 15, deg 15
Degree 20 siblings: deg 20, deg 20, deg 20
Degree 30 siblings: deg 30, deg 30, deg 30, deg 30, deg 30, deg 30
Degree 36 sibling: data not computed
Degree 40 siblings: deg 40, deg 40, some data not computed
Degree 45 sibling: data not computed
Minimal sibling: 6.2.613825.1

Frobenius cycle types

pp 22 33 55 77 1111 1313 1717 1919 2323 2929 3131 3737 4141 4343 4747 5353 5959
Cycle type 5,1{\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.1.0.1}{1} } 4,2{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} } R 3,13{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{3} 5,1{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.1.0.1}{1} } 4,2{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} } 4,12{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2} 4,2{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} } 5,1{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} } 3,2,1{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} } 22,12{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2} 2,14{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4} 3,2,1{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} } R 3,2,1{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} } 23{\href{/padicField/53.2.0.1}{2} }^{3} 4,2{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

ppLabelPolynomial ee ff cc Galois group Slope content
55 Copy content Toggle raw display 5.2.3.4a1.1x6+12x5+54x4+112x3+108x2+53x+8x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 53 x + 8332244S3×C3S_3\times C_3[ ]36[\ ]_{3}^{6}
4343 Copy content Toggle raw display 43.1.2.1a1.2x2+129x^{2} + 129221111C2C_2[ ]2[\ ]_{2}
43.2.2.2a1.1x4+84x3+1770x2+295x+9x^{4} + 84 x^{3} + 1770 x^{2} + 295 x + 9222222C4C_4[ ]22[\ ]_{2}^{2}
571571 Copy content Toggle raw display Deg 66223333

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)