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Properties

Label 12.0.162...664.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.623\times 10^{45}$
Root discriminant \(5854.94\)
Ramified primes $2,3,19,181$
Class number $2592$ (GRH)
Class group [6, 6, 72] (GRH)
Galois group $S_3\times D_6$ (as 12T37)

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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 - 6*x^11 + 45*x^10 + 6346*x^9 - 28632*x^8 + 56838*x^7 + 20608493*x^6 - 62855850*x^5 - 485618601*x^4 + 33838345412*x^3 - 45847562742*x^2 - 446214581256*x + 25518486110788)
 
Copy content gp:K = bnfinit(y^12 - 6*y^11 + 45*y^10 + 6346*y^9 - 28632*y^8 + 56838*y^7 + 20608493*y^6 - 62855850*y^5 - 485618601*y^4 + 33838345412*y^3 - 45847562742*y^2 - 446214581256*y + 25518486110788, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 45*x^10 + 6346*x^9 - 28632*x^8 + 56838*x^7 + 20608493*x^6 - 62855850*x^5 - 485618601*x^4 + 33838345412*x^3 - 45847562742*x^2 - 446214581256*x + 25518486110788);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^11 + 45*x^10 + 6346*x^9 - 28632*x^8 + 56838*x^7 + 20608493*x^6 - 62855850*x^5 - 485618601*x^4 + 33838345412*x^3 - 45847562742*x^2 - 446214581256*x + 25518486110788)
 

\( x^{12} - 6 x^{11} + 45 x^{10} + 6346 x^{9} - 28632 x^{8} + 56838 x^{7} + 20608493 x^{6} + \cdots + 25518486110788 \) 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:  $12$
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:  $[0, 6]$
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:   \(1622827545666780088361541738287738688918257664\) \(\medspace = 2^{18}\cdot 3^{20}\cdot 19^{6}\cdot 181^{10}\) 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:  \(5854.94\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}3^{11/6}19^{1/2}181^{5/6}\approx 7031.417842506584$
Ramified primes:   \(2\), \(3\), \(19\), \(181\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{-19}, \sqrt{-362})\)

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{81}a^{6}-\frac{1}{27}a^{5}-\frac{1}{9}a^{4}-\frac{40}{81}a^{3}+\frac{1}{9}a^{2}+\frac{11}{27}a-\frac{19}{81}$, $\frac{1}{81}a^{7}+\frac{1}{9}a^{5}-\frac{13}{81}a^{4}-\frac{10}{27}a^{3}+\frac{2}{27}a^{2}-\frac{1}{81}a-\frac{10}{27}$, $\frac{1}{1134}a^{8}-\frac{2}{567}a^{7}-\frac{1}{567}a^{6}-\frac{8}{567}a^{5}-\frac{1}{81}a^{4}-\frac{230}{567}a^{3}-\frac{313}{1134}a^{2}+\frac{278}{567}a+\frac{97}{567}$, $\frac{1}{1134}a^{9}-\frac{2}{567}a^{7}+\frac{2}{567}a^{6}-\frac{2}{63}a^{5}+\frac{92}{567}a^{4}-\frac{97}{378}a^{3}-\frac{20}{63}a^{2}-\frac{37}{567}a-\frac{88}{567}$, $\frac{1}{23814}a^{10}-\frac{5}{23814}a^{9}-\frac{1}{3402}a^{8}+\frac{53}{11907}a^{7}-\frac{20}{3969}a^{6}+\frac{1760}{11907}a^{5}+\frac{19}{378}a^{4}-\frac{3251}{23814}a^{3}+\frac{5479}{23814}a^{2}-\frac{4951}{11907}a+\frac{2788}{11907}$, $\frac{1}{21\cdots 98}a^{11}-\frac{21\cdots 31}{10\cdots 99}a^{10}+\frac{86\cdots 89}{21\cdots 98}a^{9}+\frac{30\cdots 65}{21\cdots 98}a^{8}-\frac{10\cdots 40}{51\cdots 19}a^{7}+\frac{30\cdots 45}{10\cdots 99}a^{6}+\frac{47\cdots 71}{71\cdots 66}a^{5}-\frac{93\cdots 30}{10\cdots 99}a^{4}-\frac{46\cdots 07}{21\cdots 98}a^{3}+\frac{26\cdots 07}{21\cdots 98}a^{2}+\frac{22\cdots 31}{21\cdots 51}a-\frac{14\cdots 18}{35\cdots 33}$ 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:  $2$

Class group and class number

Ideal class group:  $C_{6}\times C_{6}\times C_{72}$, which has order $2592$ (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:  $C_{6}\times C_{6}\times C_{72}$, which has order $2592$ (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:  $5$
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:   \( -1 \)  (order $2$) 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:   $\frac{10\cdots 94}{46\cdots 87}a^{11}-\frac{10\cdots 27}{41\cdots 83}a^{10}+\frac{51\cdots 17}{41\cdots 83}a^{9}-\frac{31\cdots 00}{12\cdots 49}a^{8}-\frac{20\cdots 82}{12\cdots 49}a^{7}+\frac{21\cdots 38}{37\cdots 47}a^{6}-\frac{70\cdots 62}{12\cdots 49}a^{5}-\frac{50\cdots 95}{12\cdots 49}a^{4}+\frac{36\cdots 53}{37\cdots 47}a^{3}+\frac{10\cdots 62}{12\cdots 49}a^{2}-\frac{20\cdots 96}{46\cdots 87}a-\frac{92\cdots 61}{37\cdots 47}$, $\frac{18\cdots 96}{35\cdots 33}a^{11}-\frac{90\cdots 22}{35\cdots 33}a^{10}-\frac{15\cdots 58}{35\cdots 33}a^{9}+\frac{30\cdots 36}{11\cdots 11}a^{8}-\frac{49\cdots 64}{35\cdots 33}a^{7}-\frac{11\cdots 40}{35\cdots 33}a^{6}+\frac{24\cdots 28}{35\cdots 33}a^{5}-\frac{36\cdots 64}{11\cdots 11}a^{4}-\frac{72\cdots 14}{56\cdots 91}a^{3}+\frac{11\cdots 14}{35\cdots 33}a^{2}-\frac{21\cdots 20}{35\cdots 33}a-\frac{51\cdots 95}{35\cdots 33}$, $\frac{15\cdots 78}{51\cdots 19}a^{11}+\frac{13\cdots 95}{73\cdots 17}a^{10}+\frac{15\cdots 93}{51\cdots 19}a^{9}-\frac{37\cdots 01}{24\cdots 39}a^{8}+\frac{73\cdots 02}{73\cdots 17}a^{7}+\frac{10\cdots 56}{51\cdots 19}a^{6}-\frac{21\cdots 94}{51\cdots 19}a^{5}+\frac{39\cdots 61}{17\cdots 73}a^{4}+\frac{14\cdots 81}{17\cdots 73}a^{3}-\frac{91\cdots 47}{51\cdots 19}a^{2}+\frac{20\cdots 34}{51\cdots 19}a+\frac{47\cdots 19}{51\cdots 19}$, $\frac{29\cdots 17}{10\cdots 38}a^{11}-\frac{20\cdots 07}{23\cdots 22}a^{10}-\frac{38\cdots 93}{35\cdots 33}a^{9}-\frac{29\cdots 25}{10\cdots 38}a^{8}-\frac{73\cdots 68}{11\cdots 11}a^{7}-\frac{23\cdots 14}{35\cdots 33}a^{6}-\frac{80\cdots 23}{79\cdots 74}a^{5}-\frac{66\cdots 17}{37\cdots 94}a^{4}-\frac{54\cdots 54}{35\cdots 33}a^{3}-\frac{82\cdots 43}{71\cdots 66}a^{2}-\frac{58\cdots 57}{39\cdots 37}a-\frac{32\cdots 22}{39\cdots 37}$, $\frac{59\cdots 21}{35\cdots 33}a^{11}-\frac{98\cdots 62}{39\cdots 37}a^{10}+\frac{79\cdots 87}{35\cdots 33}a^{9}+\frac{38\cdots 83}{35\cdots 33}a^{8}-\frac{22\cdots 60}{11\cdots 11}a^{7}+\frac{68\cdots 76}{35\cdots 33}a^{6}+\frac{79\cdots 77}{35\cdots 33}a^{5}-\frac{53\cdots 41}{11\cdots 11}a^{4}+\frac{16\cdots 27}{35\cdots 33}a^{3}+\frac{17\cdots 01}{11\cdots 11}a^{2}-\frac{40\cdots 01}{11\cdots 11}a+\frac{44\cdots 67}{11\cdots 11}$ 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:  \( 1054956121597384.5 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1054956121597384.5 \cdot 2592}{2\cdot\sqrt{1622827545666780088361541738287738688918257664}}\cr\approx \mathstrut & 2.08824967681451 \end{aligned}\] (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 - 6*x^11 + 45*x^10 + 6346*x^9 - 28632*x^8 + 56838*x^7 + 20608493*x^6 - 62855850*x^5 - 485618601*x^4 + 33838345412*x^3 - 45847562742*x^2 - 446214581256*x + 25518486110788) 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 - 6*x^11 + 45*x^10 + 6346*x^9 - 28632*x^8 + 56838*x^7 + 20608493*x^6 - 62855850*x^5 - 485618601*x^4 + 33838345412*x^3 - 45847562742*x^2 - 446214581256*x + 25518486110788, 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 - 6*x^11 + 45*x^10 + 6346*x^9 - 28632*x^8 + 56838*x^7 + 20608493*x^6 - 62855850*x^5 - 485618601*x^4 + 33838345412*x^3 - 45847562742*x^2 - 446214581256*x + 25518486110788); 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 = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^11 + 45*x^10 + 6346*x^9 - 28632*x^8 + 56838*x^7 + 20608493*x^6 - 62855850*x^5 - 485618601*x^4 + 33838345412*x^3 - 45847562742*x^2 - 446214581256*x + 25518486110788); 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

$S_3\times D_6$ (as 12T37):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 72
The 18 conjugacy class representatives for $S_3\times D_6$
Character table for $S_3\times D_6$

Intermediate fields

\(\Q(\sqrt{-362}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{6878}) \), \(\Q(\sqrt{-19}, \sqrt{-362})\), 6.0.5873208011345484288.2

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.6.0.1}{6} }^{2}$ ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{6}$ ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ R ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.2.0.1}{2} }^{6}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.2.6a1.1$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$$2$$2$$6$$C_2^2$$$[3]^{2}$$
2.2.2.6a1.1$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$$2$$2$$6$$C_2^2$$$[3]^{2}$$
2.2.2.6a1.1$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$$2$$2$$6$$C_2^2$$$[3]^{2}$$
\(3\) Copy content Toggle raw display 3.2.3.10a1.1$x^{6} + 6 x^{5} + 18 x^{4} + 32 x^{3} + 36 x^{2} + 24 x + 11$$3$$2$$10$$D_{6}$$$[\frac{5}{2}]_{2}^{2}$$
3.2.3.10a1.1$x^{6} + 6 x^{5} + 18 x^{4} + 32 x^{3} + 36 x^{2} + 24 x + 11$$3$$2$$10$$D_{6}$$$[\frac{5}{2}]_{2}^{2}$$
\(19\) Copy content Toggle raw display 19.2.2.2a1.2$x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
19.2.2.2a1.2$x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
19.2.2.2a1.2$x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(181\) Copy content Toggle raw display 181.2.6.10a1.4$x^{12} + 1062 x^{11} + 469947 x^{10} + 110915280 x^{9} + 14726353155 x^{8} + 1043025098382 x^{7} + 30808510677349 x^{6} + 2086050196764 x^{5} + 58905412620 x^{4} + 887322240 x^{3} + 7519152 x^{2} + 42672 x + 27757$$6$$2$$10$$C_6\times C_2$$$[\ ]_{6}^{2}$$

Spectrum of ring of integers

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