Properties

Label 12.12.629...792.1
Degree 1212
Signature [12,0][12, 0]
Discriminant 6.296×10236.296\times 10^{23}
Root discriminant 96.2296.22
Ramified primes 2,3,7,132,3,7,13
Class number 44 (GRH)
Class group [2, 2] (GRH)
Galois group C12C_{12} (as 12T1)

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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 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573)
 
Copy content gp:K = bnfinit(y^12 - 273*y^10 + 28665*y^8 - 1444716*y^6 + 35395542*y^4 - 371653191*y^2 + 1114959573, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573)
 

x12273x10+28665x81444716x6+35395542x4371653191x2+1114959573 x^{12} - 273x^{10} + 28665x^{8} - 1444716x^{6} + 35395542x^{4} - 371653191x^{2} + 1114959573 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:  [12,0][12, 0]
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:   629582418765353043873792629582418765353043873792 =21236761311\medspace = 2^{12}\cdot 3^{6}\cdot 7^{6}\cdot 13^{11} 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:  96.2296.22
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:  231/271/21311/1296.217569592725082\cdot 3^{1/2}7^{1/2}13^{11/12}\approx 96.21756959272508
Ramified primes:   22, 33, 77, 1313 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(13)\Q(\sqrt{13})
Aut(K/Q)\Aut(K/\Q) == Gal(K/Q)\Gal(K/\Q):   C12C_{12}
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over Q\Q.
Conductor:  1092=2237131092=2^{2}\cdot 3\cdot 7\cdot 13
Dirichlet character group:    {\lbraceχ1092(1,)\chi_{1092}(1,·), χ1092(839,)\chi_{1092}(839,·), χ1092(841,)\chi_{1092}(841,·), χ1092(587,)\chi_{1092}(587,·), χ1092(589,)\chi_{1092}(589,·), χ1092(1007,)\chi_{1092}(1007,·), χ1092(337,)\chi_{1092}(337,·), χ1092(83,)\chi_{1092}(83,·), χ1092(757,)\chi_{1092}(757,·), χ1092(673,)\chi_{1092}(673,·), χ1092(167,)\chi_{1092}(167,·), χ1092(671,)\chi_{1092}(671,·)}\rbrace
This is not a CM field.

Integral basis (with respect to field generator aa)

11, aa, 121a2\frac{1}{21}a^{2}, 121a3\frac{1}{21}a^{3}, 1441a4\frac{1}{441}a^{4}, 1441a5\frac{1}{441}a^{5}, 19261a6\frac{1}{9261}a^{6}, 19261a7\frac{1}{9261}a^{7}, 1194481a8\frac{1}{194481}a^{8}, 1194481a9\frac{1}{194481}a^{9}, 14084101a10\frac{1}{4084101}a^{10}, 14084101a11\frac{1}{4084101}a^{11} 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:  Not computed
Index:  11
Inessential primes:  None

Class group and class number

Ideal class group:  C2×C2C_{2}\times C_{2}, which has order 44 (assuming GRH)
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:  C2×C2×C2×C2C_{2}\times C_{2}\times C_{2}\times C_{2}, which has order 1616 (assuming GRH)
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:  1111
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:   19261a62147a4+37a22\frac{1}{9261}a^{6}-\frac{2}{147}a^{4}+\frac{3}{7}a^{2}-2, 14084101a1011194481a8+449261a61163a4+5521a211\frac{1}{4084101}a^{10}-\frac{11}{194481}a^{8}+\frac{44}{9261}a^{6}-\frac{11}{63}a^{4}+\frac{55}{21}a^{2}-11, 1194481a811029a6+349a42921a2+6\frac{1}{194481}a^{8}-\frac{1}{1029}a^{6}+\frac{3}{49}a^{4}-\frac{29}{21}a^{2}+6, 1194481a889261a6+20441a41621a2+3\frac{1}{194481}a^{8}-\frac{8}{9261}a^{6}+\frac{20}{441}a^{4}-\frac{16}{21}a^{2}+3, 14084101a1011194481a8+449261a61163a4+5521a210\frac{1}{4084101}a^{10}-\frac{11}{194481}a^{8}+\frac{44}{9261}a^{6}-\frac{11}{63}a^{4}+\frac{55}{21}a^{2}-10, 14084101a111453789a1013194481a9+1121609a8+619261a74079261a6124441a5+751441a4+9721a356221a211a+82\frac{1}{4084101}a^{11}-\frac{1}{453789}a^{10}-\frac{13}{194481}a^{9}+\frac{11}{21609}a^{8}+\frac{61}{9261}a^{7}-\frac{407}{9261}a^{6}-\frac{124}{441}a^{5}+\frac{751}{441}a^{4}+\frac{97}{21}a^{3}-\frac{562}{21}a^{2}-11a+82, 11361367a1132194481a9+227783a8+2147a72147a674147a5+5263a4+233a3523a226a+64\frac{1}{1361367}a^{11}-\frac{32}{194481}a^{9}+\frac{2}{27783}a^{8}+\frac{2}{147}a^{7}-\frac{2}{147}a^{6}-\frac{74}{147}a^{5}+\frac{52}{63}a^{4}+\frac{23}{3}a^{3}-\frac{52}{3}a^{2}-26a+64, 14084101a104194481a9+121609a8+51323a71259261a6100441a5+386441a4+10021a339121a219a+69\frac{1}{4084101}a^{10}-\frac{4}{194481}a^{9}+\frac{1}{21609}a^{8}+\frac{5}{1323}a^{7}-\frac{125}{9261}a^{6}-\frac{100}{441}a^{5}+\frac{386}{441}a^{4}+\frac{100}{21}a^{3}-\frac{391}{21}a^{2}-19a+69, 14084101a11+84084101a1010194481a980194481a8+51323a7+401323a617147a5409441a4+43a3+757a25a35\frac{1}{4084101}a^{11}+\frac{8}{4084101}a^{10}-\frac{10}{194481}a^{9}-\frac{80}{194481}a^{8}+\frac{5}{1323}a^{7}+\frac{40}{1323}a^{6}-\frac{17}{147}a^{5}-\frac{409}{441}a^{4}+\frac{4}{3}a^{3}+\frac{75}{7}a^{2}-5a-35, 24084101a11+114084101a1022194481a9110194481a8+899261a7+3769261a655147a5496441a4+14521a3+627a251a+35\frac{2}{4084101}a^{11}+\frac{11}{4084101}a^{10}-\frac{22}{194481}a^{9}-\frac{110}{194481}a^{8}+\frac{89}{9261}a^{7}+\frac{376}{9261}a^{6}-\frac{55}{147}a^{5}-\frac{496}{441}a^{4}+\frac{145}{21}a^{3}+\frac{62}{7}a^{2}-51a+35, 14084101a11114084101a1010194481a9+4364827a8+319261a75489261a620441a5+1007441a44121a31063a2+35a+178\frac{1}{4084101}a^{11}-\frac{11}{4084101}a^{10}-\frac{10}{194481}a^{9}+\frac{43}{64827}a^{8}+\frac{31}{9261}a^{7}-\frac{548}{9261}a^{6}-\frac{20}{441}a^{5}+\frac{1007}{441}a^{4}-\frac{41}{21}a^{3}-\frac{106}{3}a^{2}+35a+178 Copy content Toggle raw display (assuming GRH)
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:  13423777.933 13423777.933 (assuming GRH)
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(212(2π)013423777.93342629582418765353043873792(0.13859207812 \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^{12}\cdot(2\pi)^{0}\cdot 13423777.933 \cdot 4}{2\cdot\sqrt{629582418765353043873792}}\cr\approx \mathstrut & 0.13859207812 \end{aligned} (assuming GRH)

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 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573) 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 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573, 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 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573); 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 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573); 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

C12C_{12} (as 12T1):

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 cyclic group of order 12
The 12 conjugacy class representatives for C12C_{12}
Character table for C12C_{12}

Intermediate fields

Q(13)\Q(\sqrt{13}) , 3.3.169.1, 4.4.15502032.1, Q(ζ13)+\Q(\zeta_{13})^+

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]
 

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 43{\href{/padicField/5.4.0.1}{4} }^{3} R 12{\href{/padicField/11.12.0.1}{12} } R 62{\href{/padicField/17.6.0.1}{6} }^{2} 12{\href{/padicField/19.12.0.1}{12} } 62{\href{/padicField/23.6.0.1}{6} }^{2} 62{\href{/padicField/29.6.0.1}{6} }^{2} 43{\href{/padicField/31.4.0.1}{4} }^{3} 12{\href{/padicField/37.12.0.1}{12} } 12{\href{/padicField/41.12.0.1}{12} } 34{\href{/padicField/43.3.0.1}{3} }^{4} 43{\href{/padicField/47.4.0.1}{4} }^{3} 26{\href{/padicField/53.2.0.1}{2} }^{6} 12{\href{/padicField/59.12.0.1}{12} }

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 2.6.2.12a1.2x12+2x10+2x9+x8+4x7+5x6+2x5+6x4+8x3+x2+4x+5x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 5 x^{6} + 2 x^{5} + 6 x^{4} + 8 x^{3} + x^{2} + 4 x + 522661212C12C_{12}[2]6[2]^{6}
33 Copy content Toggle raw display 3.3.2.3a1.2x6+4x4+2x3+4x2+4x+4x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4223333C6C_6[ ]23[\ ]_{2}^{3}
3.3.2.3a1.2x6+4x4+2x3+4x2+4x+4x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4223333C6C_6[ ]23[\ ]_{2}^{3}
77 Copy content Toggle raw display 7.6.2.6a1.1x12+2x10+10x9+9x8+22x7+39x6+52x5+82x4+78x3+60x2+43x+9x^{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} + 43 x + 9226666C12C_{12}[ ]26[\ ]_{2}^{6}
1313 Copy content Toggle raw display 13.1.12.11a1.12x12+156x^{12} + 1561212111111C12C_{12}[ ]12[\ ]_{12}

Spectrum of ring of integers

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