Properties

Label 12.0.12950250637492224.2
Degree 1212
Signature [0,6][0, 6]
Discriminant 1.295×10161.295\times 10^{16}
Root discriminant 22.0122.01
Ramified primes 2,3,72,3,7
Class number 44
Class group [4]
Galois group C6×S3C_6\times S_3 (as 12T18)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9)
 
Copy content gp:K = bnfinit(y^12 - 4*y^11 + 22*y^10 - 48*y^9 + 108*y^8 - 132*y^7 + 170*y^6 - 140*y^5 + 107*y^4 - 12*y^3 - 12*y^2 + 12*y + 9, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9)
 

x124x11+22x1048x9+108x8132x7+170x6140x5+107x4++9 x^{12} - 4 x^{11} + 22 x^{10} - 48 x^{9} + 108 x^{8} - 132 x^{7} + 170 x^{6} - 140 x^{5} + 107 x^{4} + \cdots + 9 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:  1212
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:  [0,6][0, 6]
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:   1295025063749222412950250637492224 =2243876\medspace = 2^{24}\cdot 3^{8}\cdot 7^{6} 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:  22.0122.01
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:  2234/371/245.790004298278022^{2}3^{4/3}7^{1/2}\approx 45.79000429827802
Ramified primes:   22, 33, 77 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\Q
Aut(K/Q)\Aut(K/\Q):   C6C_6
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.
Maximal CM subfield:  Q(2,7)\Q(\sqrt{-2}, \sqrt{7})

Integral basis (with respect to field generator aa)

11, aa, a2a^{2}, a3a^{3}, a4a^{4}, a5a^{5}, a6a^{6}, 13a713a6+13a413a3\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}, 13a813a6+13a513a3\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}, 13a913a3\frac{1}{3}a^{9}-\frac{1}{3}a^{3}, 127a10127a9427a8127a7127a61327a5+1027a413a3+19a219a13\frac{1}{27}a^{10}-\frac{1}{27}a^{9}-\frac{4}{27}a^{8}-\frac{1}{27}a^{7}-\frac{1}{27}a^{6}-\frac{13}{27}a^{5}+\frac{10}{27}a^{4}-\frac{1}{3}a^{3}+\frac{1}{9}a^{2}-\frac{1}{9}a-\frac{1}{3}, 12577231a11+8842859077a101356202577231a9+1021872577231a8+605652577231a7+6016872577231a6395084859077a5+653048627a4+175183859077a395981286359a2+357608859077a6595286359\frac{1}{2577231}a^{11}+\frac{8842}{859077}a^{10}-\frac{135620}{2577231}a^{9}+\frac{102187}{2577231}a^{8}+\frac{60565}{2577231}a^{7}+\frac{601687}{2577231}a^{6}-\frac{395084}{859077}a^{5}+\frac{6530}{48627}a^{4}+\frac{175183}{859077}a^{3}-\frac{95981}{286359}a^{2}+\frac{357608}{859077}a-\frac{6595}{286359} 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:  No
Index:  Not computed
Inessential primes:  33

Class group and class number

Ideal class group:  C4C_{4}, which has order 44
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:  C4C_{4}, which has order 44
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:   37892859077a117543182577231a10+35975772577231a9117795952577231a8+249616422577231a7453886312577231a6+540931362577231a5124937648627a4+195478195453a311588239859077a2+3264223859077a+185548286359\frac{37892}{859077}a^{11}-\frac{754318}{2577231}a^{10}+\frac{3597577}{2577231}a^{9}-\frac{11779595}{2577231}a^{8}+\frac{24961642}{2577231}a^{7}-\frac{45388631}{2577231}a^{6}+\frac{54093136}{2577231}a^{5}-\frac{1249376}{48627}a^{4}+\frac{1954781}{95453}a^{3}-\frac{11588239}{859077}a^{2}+\frac{3264223}{859077}a+\frac{185548}{286359}, 2182512577231a113756892577231a10+1085486859077a9377681859077a8+969622286359a7+2511215859077a6+103727752577231a5+27105448627a4+1215766859077a3+4984738859077a2+3258124859077a+256801286359\frac{218251}{2577231}a^{11}-\frac{375689}{2577231}a^{10}+\frac{1085486}{859077}a^{9}-\frac{377681}{859077}a^{8}+\frac{969622}{286359}a^{7}+\frac{2511215}{859077}a^{6}+\frac{10372775}{2577231}a^{5}+\frac{271054}{48627}a^{4}+\frac{1215766}{859077}a^{3}+\frac{4984738}{859077}a^{2}+\frac{3258124}{859077}a+\frac{256801}{286359}, 2425582577231a11321980859077a10+49286272577231a9104105412577231a8+189219052577231a7212050062577231a6+1628735286359a529368148627a41632973859077a3+1554550286359a23187016859077a354995286359\frac{242558}{2577231}a^{11}-\frac{321980}{859077}a^{10}+\frac{4928627}{2577231}a^{9}-\frac{10410541}{2577231}a^{8}+\frac{18921905}{2577231}a^{7}-\frac{21205006}{2577231}a^{6}+\frac{1628735}{286359}a^{5}-\frac{293681}{48627}a^{4}-\frac{1632973}{859077}a^{3}+\frac{1554550}{286359}a^{2}-\frac{3187016}{859077}a-\frac{354995}{286359}, 22544859077a112236092577231a10+14496312577231a925496062577231a8+74028252577231a758595032577231a6+117295902577231a51866848627a4+816754286359a3+637349859077a2+626740859077a+18829286359\frac{22544}{859077}a^{11}-\frac{223609}{2577231}a^{10}+\frac{1449631}{2577231}a^{9}-\frac{2549606}{2577231}a^{8}+\frac{7402825}{2577231}a^{7}-\frac{5859503}{2577231}a^{6}+\frac{11729590}{2577231}a^{5}-\frac{18668}{48627}a^{4}+\frac{816754}{286359}a^{3}+\frac{637349}{859077}a^{2}+\frac{626740}{859077}a+\frac{18829}{286359}, 32965286359a113693295453a10+687032286359a91278257286359a8+3466585286359a73766847286359a6+224049195453a5913245403a4+5825918286359a357648795453a2+57948595453a+20873095453\frac{32965}{286359}a^{11}-\frac{36932}{95453}a^{10}+\frac{687032}{286359}a^{9}-\frac{1278257}{286359}a^{8}+\frac{3466585}{286359}a^{7}-\frac{3766847}{286359}a^{6}+\frac{2240491}{95453}a^{5}-\frac{91324}{5403}a^{4}+\frac{5825918}{286359}a^{3}-\frac{576487}{95453}a^{2}+\frac{579485}{95453}a+\frac{208730}{95453} 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:  3833.8726435419103 3833.8726435419103
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(20(2π)63833.87264354191034212950250637492224(4.14579478575610 \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^{0}\cdot(2\pi)^{6}\cdot 3833.8726435419103 \cdot 4}{2\cdot\sqrt{12950250637492224}}\cr\approx \mathstrut & 4.14579478575610 \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^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9) 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^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9, 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^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9); 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^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9); 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

C6×S3C_6\times S_3 (as 12T18):

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 solvable group of order 36
The 18 conjugacy class representatives for C6×S3C_6\times S_3
Character table for C6×S3C_6\times S_3

Intermediate fields

Q(7)\Q(\sqrt{7}) , Q(14)\Q(\sqrt{-14}) , Q(2)\Q(\sqrt{-2}) , Q(2,7)\Q(\sqrt{-2}, \sqrt{7}), 6.0.14224896.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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

Galois closure: deg 36
Degree 18 siblings: 18.6.12237489813936932340847607808.2, deg 18
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 R R 6,23{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3} R 62{\href{/padicField/11.6.0.1}{6} }^{2} 62{\href{/padicField/13.6.0.1}{6} }^{2} 26{\href{/padicField/17.2.0.1}{2} }^{6} 34{\href{/padicField/19.3.0.1}{3} }^{4} 6,23{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3} 62{\href{/padicField/29.6.0.1}{6} }^{2} 62{\href{/padicField/31.6.0.1}{6} }^{2} 26{\href{/padicField/37.2.0.1}{2} }^{6} 62{\href{/padicField/41.6.0.1}{6} }^{2} 62{\href{/padicField/43.6.0.1}{6} }^{2} 62{\href{/padicField/47.6.0.1}{6} }^{2} 26{\href{/padicField/53.2.0.1}{2} }^{6} 34{\href{/padicField/59.3.0.1}{3} }^{4}

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
22 Copy content Toggle raw display 2.3.4.24b1.1x12+4x10+4x9+6x8+12x7+12x6+12x5+17x4+16x3+8x2+12x+9x^{12} + 4 x^{10} + 4 x^{9} + 6 x^{8} + 12 x^{7} + 12 x^{6} + 12 x^{5} + 17 x^{4} + 16 x^{3} + 8 x^{2} + 12 x + 944332424C6×C2C_6\times C_2[2,3]3[2, 3]^{3}
33 Copy content Toggle raw display Q3\Q_{3}x+1x + 1111100Trivial[ ][\ ]
Q3\Q_{3}x+1x + 1111100Trivial[ ][\ ]
Q3\Q_{3}x+1x + 1111100Trivial[ ][\ ]
Q3\Q_{3}x+1x + 1111100Trivial[ ][\ ]
Q3\Q_{3}x+1x + 1111100Trivial[ ][\ ]
Q3\Q_{3}x+1x + 1111100Trivial[ ][\ ]
3.1.3.4a2.1x3+6x2+3x^{3} + 6 x^{2} + 3331144C3C_3[2][2]
3.1.3.4a2.1x3+6x2+3x^{3} + 6 x^{2} + 3331144C3C_3[2][2]
77 Copy content Toggle raw display 7.6.2.6a1.2x12+2x10+10x9+9x8+22x7+39x6+52x5+82x4+78x3+60x2+36x+16x^{12} + 2 x^{10} + 10 x^{9} + 9 x^{8} + 22 x^{7} + 39 x^{6} + 52 x^{5} + 82 x^{4} + 78 x^{3} + 60 x^{2} + 36 x + 16226666C6×C2C_6\times C_2[ ]26[\ ]_{2}^{6}

Artin representations

Label Dimension Conductor Artin stem field GG Ind χ(c)\chi(c)
* 1.1.1t1.a.a11 11 Q\Q C1C_1 11 11
* 1.28.2t1.a.a11 227 2^{2} \cdot 7 Q(7)\Q(\sqrt{7}) C2C_2 (as 2T1) 11 11
* 1.56.2t1.b.a11 237 2^{3} \cdot 7 Q(14)\Q(\sqrt{-14}) C2C_2 (as 2T1) 11 1-1
* 1.8.2t1.b.a11 23 2^{3} Q(2)\Q(\sqrt{-2}) C2C_2 (as 2T1) 11 1-1
1.9.3t1.a.a11 32 3^{2} Q(ζ9)+\Q(\zeta_{9})^+ C3C_3 (as 3T1) 00 11
1.9.3t1.a.b11 32 3^{2} Q(ζ9)+\Q(\zeta_{9})^+ C3C_3 (as 3T1) 00 11
1.252.6t1.i.a11 22327 2^{2} \cdot 3^{2} \cdot 7 6.6.144027072.1 C6C_6 (as 6T1) 00 11
1.504.6t1.f.a11 23327 2^{3} \cdot 3^{2} \cdot 7 6.0.1152216576.2 C6C_6 (as 6T1) 00 1-1
1.504.6t1.f.b11 23327 2^{3} \cdot 3^{2} \cdot 7 6.0.1152216576.2 C6C_6 (as 6T1) 00 1-1
1.72.6t1.a.a11 2332 2^{3} \cdot 3^{2} 6.0.3359232.1 C6C_6 (as 6T1) 00 1-1
1.252.6t1.i.b11 22327 2^{2} \cdot 3^{2} \cdot 7 6.6.144027072.1 C6C_6 (as 6T1) 00 11
1.72.6t1.a.b11 2332 2^{3} \cdot 3^{2} 6.0.3359232.1 C6C_6 (as 6T1) 00 1-1
2.4536.3t2.a.a22 23347 2^{3} \cdot 3^{4} \cdot 7 3.1.4536.1 S3S_3 (as 3T2) 11 00
2.18144.6t3.g.a22 25347 2^{5} \cdot 3^{4} \cdot 7 6.0.658409472.1 D6D_{6} (as 6T3) 11 00
* 2.504.6t5.b.a22 23327 2^{3} \cdot 3^{2} \cdot 7 6.0.14224896.1 S3×C3S_3\times C_3 (as 6T5) 00 00
* 2.2016.12t18.c.a22 25327 2^{5} \cdot 3^{2} \cdot 7 12.0.12950250637492224.2 C6×S3C_6\times S_3 (as 12T18) 00 00
* 2.2016.12t18.c.b22 25327 2^{5} \cdot 3^{2} \cdot 7 12.0.12950250637492224.2 C6×S3C_6\times S_3 (as 12T18) 00 00
* 2.504.6t5.b.b22 23327 2^{3} \cdot 3^{2} \cdot 7 6.0.14224896.1 S3×C3S_3\times C_3 (as 6T5) 00 00

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers

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