Properties

Label 10.2.1395650309.1
Degree 1010
Signature [2,4][2, 4]
Discriminant 13956503091395650309
Root discriminant 8.218.21
Ramified primes 17,103,347,229717,103,347,2297
Class number 11
Class group trivial
Galois group S10S_{10} (as 10T45)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + x^8 - 4*x^7 + 2*x^6 - 2*x^5 + 3*x^4 - 1)
 
Copy content gp:K = bnfinit(y^10 - y^9 + y^8 - 4*y^7 + 2*y^6 - 2*y^5 + 3*y^4 - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - x^9 + x^8 - 4*x^7 + 2*x^6 - 2*x^5 + 3*x^4 - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - x^9 + x^8 - 4*x^7 + 2*x^6 - 2*x^5 + 3*x^4 - 1)
 

x10x9+x84x7+2x62x5+3x41 x^{10} - x^{9} + x^{8} - 4x^{7} + 2x^{6} - 2x^{5} + 3x^{4} - 1 Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  1010
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  [2,4][2, 4]
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   13956503091395650309 =171033472297\medspace = 17\cdot 103\cdot 347\cdot 2297 Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  8.218.21
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  171/21031/23471/222971/237358.40345892741617^{1/2}103^{1/2}347^{1/2}2297^{1/2}\approx 37358.403458927416
Ramified primes:   1717, 103103, 347347, 22972297 Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  Q(1395650309\Q(\sqrt{1395650309})
Aut(K/Q)\Aut(K/\Q):   C1C_1
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over Q\Q.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator aa)

11, aa, a2a^{2}, a3a^{3}, a4a^{4}, a5a^{5}, a6a^{6}, a7a^{7}, a8a^{8}, a9a^{9} Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Yes
Index:  11
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order 11
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order 11
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  55
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   1 -1  (order 22) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   aa, a9a8+a73a6+a5a4+a2a+1a^{9}-a^{8}+a^{7}-3a^{6}+a^{5}-a^{4}+a^{2}-a+1, a92a8+2a74a6+4a53a4+2a3+1a^{9}-2a^{8}+2a^{7}-4a^{6}+4a^{5}-3a^{4}+2a^{3}+1, a92a8+2a74a6+5a53a4+2a32a2a+1a^{9}-2a^{8}+2a^{7}-4a^{6}+5a^{5}-3a^{4}+2a^{3}-2a^{2}-a+1, a8a72a5+a4+a3a2+a1a^{8}-a^{7}-2a^{5}+a^{4}+a^{3}-a^{2}+a-1 Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  2.343092286753046 2.343092286753046
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

lims1(s1)ζK(s)=(2r1(2π)r2RhwD(22(2π)42.343092286753046121395650309(0.195501707764210 \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)^{4}\cdot 2.343092286753046 \cdot 1}{2\cdot\sqrt{1395650309}}\cr\approx \mathstrut & 0.195501707764210 \end{aligned}

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + x^8 - 4*x^7 + 2*x^6 - 2*x^5 + 3*x^4 - 1) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^10 - x^9 + x^8 - 4*x^7 + 2*x^6 - 2*x^5 + 3*x^4 - 1, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - x^9 + x^8 - 4*x^7 + 2*x^6 - 2*x^5 + 3*x^4 - 1); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - x^9 + x^8 - 4*x^7 + 2*x^6 - 2*x^5 + 3*x^4 - 1); 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

S10S_{10} (as 10T45):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 3628800
The 42 conjugacy class representatives for S10S_{10}
Character table for S10S_{10}

Intermediate fields

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

Sibling fields

Degree 20 sibling: data not computed
Degree 45 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

pp 22 33 55 77 1111 1313 1717 1919 2323 2929 3131 3737 4141 4343 4747 5353 5959
Cycle type 10{\href{/padicField/2.10.0.1}{10} } 10{\href{/padicField/3.10.0.1}{10} } 6,4{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} } 7,3{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.3.0.1}{3} } 6,4{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} } 4,3,2,1{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} } R 5,4,1{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} } 10{\href{/padicField/23.10.0.1}{10} } 10{\href{/padicField/29.10.0.1}{10} } 3,23,1{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} } 7,2,1{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} } 5,3,12{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2} 6,4{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} } 8,12{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2} 5,3,12{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2} 5,3,2{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\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.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=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
1717 Copy content Toggle raw display 17.1.2.1a1.1x2+17x^{2} + 17221111C2C_2[ ]2[\ ]_{2}
17.8.1.0a1.1x8+11x4+12x3+6x+3x^{8} + 11 x^{4} + 12 x^{3} + 6 x + 3118800C8C_8[ ]8[\ ]^{8}
103103 Copy content Toggle raw display Q103\Q_{103}x+98x + 98111100Trivial[ ][\ ]
Q103\Q_{103}x+98x + 98111100Trivial[ ][\ ]
103.1.2.1a1.2x2+515x^{2} + 515221111C2C_2[ ]2[\ ]_{2}
103.6.1.0a1.1x6+96x3+9x2+30x+5x^{6} + 96 x^{3} + 9 x^{2} + 30 x + 5116600C6C_6[ ]6[\ ]^{6}
347347 Copy content Toggle raw display Q347\Q_{347}xx111100Trivial[ ][\ ]
Q347\Q_{347}xx111100Trivial[ ][\ ]
Q347\Q_{347}xx111100Trivial[ ][\ ]
Q347\Q_{347}xx111100Trivial[ ][\ ]
Deg 22221111C2C_2[ ]2[\ ]_{2}
Deg 44114400C4C_4[ ]4[\ ]^{4}
22972297 Copy content Toggle raw display Q2297\Q_{2297}xx111100Trivial[ ][\ ]
Q2297\Q_{2297}xx111100Trivial[ ][\ ]
Deg 22221111C2C_2[ ]2[\ ]_{2}
Deg 66116600C6C_6[ ]6[\ ]^{6}

Spectrum of ring of integers

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