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Properties

Label 12.0.131...136.76
Degree $12$
Signature $[0, 6]$
Discriminant $1.316\times 10^{17}$
Root discriminant \(26.71\)
Ramified primes $2,3$
Class number $3$
Class group [3]
Galois group $S_3^2$ (as 12T16)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 21*x^10 - 26*x^9 - 18*x^8 + 90*x^7 + 57*x^6 - 522*x^5 + 882*x^4 - 542*x^3 + 489*x^2 - 1050*x + 625)
 
Copy content gp:K = bnfinit(y^12 - 6*y^11 + 21*y^10 - 26*y^9 - 18*y^8 + 90*y^7 + 57*y^6 - 522*y^5 + 882*y^4 - 542*y^3 + 489*y^2 - 1050*y + 625, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 21*x^10 - 26*x^9 - 18*x^8 + 90*x^7 + 57*x^6 - 522*x^5 + 882*x^4 - 542*x^3 + 489*x^2 - 1050*x + 625);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^11 + 21*x^10 - 26*x^9 - 18*x^8 + 90*x^7 + 57*x^6 - 522*x^5 + 882*x^4 - 542*x^3 + 489*x^2 - 1050*x + 625)
 

\( x^{12} - 6 x^{11} + 21 x^{10} - 26 x^{9} - 18 x^{8} + 90 x^{7} + 57 x^{6} - 522 x^{5} + 882 x^{4} + \cdots + 625 \) 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:   \(131621703842267136\) \(\medspace = 2^{22}\cdot 3^{22}\) 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:  \(26.71\)
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:  $2^{11/6}3^{13/6}\approx 38.51687498161186$
Ramified primes:   \(2\), \(3\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $S_3$
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:  \(\Q(\sqrt{-2}, \sqrt{-3})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{18}a^{6}-\frac{1}{6}a^{5}+\frac{1}{3}a^{4}+\frac{5}{18}a^{3}+\frac{1}{3}a^{2}-\frac{1}{6}a+\frac{7}{18}$, $\frac{1}{18}a^{7}-\frac{1}{6}a^{5}+\frac{5}{18}a^{4}+\frac{1}{6}a^{3}-\frac{1}{6}a^{2}-\frac{1}{9}a+\frac{1}{6}$, $\frac{1}{18}a^{8}-\frac{2}{9}a^{5}+\frac{1}{6}a^{4}-\frac{1}{3}a^{3}-\frac{1}{9}a^{2}-\frac{1}{3}a+\frac{1}{6}$, $\frac{1}{90}a^{9}+\frac{1}{45}a^{8}-\frac{1}{90}a^{7}+\frac{4}{9}a^{5}-\frac{7}{18}a^{4}-\frac{11}{30}a^{3}-\frac{1}{90}a^{2}-\frac{37}{90}a-\frac{1}{18}$, $\frac{1}{180}a^{10}-\frac{1}{180}a^{9}+\frac{1}{60}a^{8}-\frac{1}{90}a^{7}-\frac{1}{36}a^{6}-\frac{1}{4}a^{5}-\frac{13}{180}a^{4}+\frac{17}{45}a^{3}-\frac{13}{60}a^{2}-\frac{79}{180}a-\frac{1}{12}$, $\frac{1}{12348900}a^{11}+\frac{1431}{1372100}a^{10}-\frac{9559}{12348900}a^{9}-\frac{148093}{6174450}a^{8}-\frac{66293}{12348900}a^{7}-\frac{65477}{2469780}a^{6}-\frac{2921443}{12348900}a^{5}-\frac{750838}{3087225}a^{4}-\frac{373103}{12348900}a^{3}+\frac{1915013}{12348900}a^{2}-\frac{3030161}{12348900}a-\frac{398}{123489}$ 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:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  $C_{3}$, which has order $3$
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:  $C_{3}$, which has order $3$
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:   \( \frac{99308}{3087225} a^{11} - \frac{495518}{3087225} a^{10} + \frac{1580128}{3087225} a^{9} - \frac{320221}{1029075} a^{8} - \frac{944548}{1029075} a^{7} + \frac{408958}{205815} a^{6} + \frac{3999952}{1029075} a^{5} - \frac{13288022}{1029075} a^{4} + \frac{15575692}{1029075} a^{3} - \frac{4495546}{3087225} a^{2} + \frac{41301712}{3087225} a - \frac{2364986}{123489} \)  (order $6$) 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:   $\frac{357733}{6174450}a^{11}-\frac{592601}{2058150}a^{10}+\frac{315116}{343025}a^{9}-\frac{3385043}{6174450}a^{8}-\frac{5088362}{3087225}a^{7}+\frac{2215997}{617445}a^{6}+\frac{21660553}{3087225}a^{5}-\frac{71443318}{3087225}a^{4}+\frac{168128671}{6174450}a^{3}-\frac{8186978}{3087225}a^{2}+\frac{147964727}{6174450}a-\frac{8061337}{246978}$, $\frac{8282}{3087225}a^{11}-\frac{34133}{2058150}a^{10}+\frac{348089}{6174450}a^{9}-\frac{177887}{3087225}a^{8}-\frac{82253}{686050}a^{7}+\frac{247826}{617445}a^{6}+\frac{783973}{6174450}a^{5}-\frac{1270657}{686050}a^{4}+\frac{8711594}{3087225}a^{3}+\frac{262807}{1029075}a^{2}-\frac{3801409}{1029075}a+\frac{234194}{123489}$, $\frac{1084927}{6174450}a^{11}-\frac{1785569}{2058150}a^{10}+\frac{17077487}{6174450}a^{9}-\frac{5005181}{3087225}a^{8}-\frac{5051366}{1029075}a^{7}+\frac{6535838}{617445}a^{6}+\frac{131704039}{6174450}a^{5}-\frac{142222103}{2058150}a^{4}+\frac{502202779}{6174450}a^{3}-\frac{8635409}{1029075}a^{2}+\frac{26460031}{343025}a-\frac{12680356}{123489}$, $\frac{99541}{2058150}a^{11}-\frac{735694}{3087225}a^{10}+\frac{2356664}{3087225}a^{9}-\frac{1398754}{3087225}a^{8}-\frac{1353289}{1029075}a^{7}+\frac{1201043}{411630}a^{6}+\frac{17969368}{3087225}a^{5}-\frac{13039809}{686050}a^{4}+\frac{15366389}{686050}a^{3}-\frac{19001711}{6174450}a^{2}+\frac{131342707}{6174450}a-\frac{3719695}{123489}$, $\frac{24823}{686050}a^{11}-\frac{1132627}{6174450}a^{10}+\frac{1812496}{3087225}a^{9}-\frac{1212071}{3087225}a^{8}-\frac{1042186}{1029075}a^{7}+\frac{458461}{205815}a^{6}+\frac{12916937}{3087225}a^{5}-\frac{30321383}{2058150}a^{4}+\frac{35680663}{2058150}a^{3}-\frac{9461582}{3087225}a^{2}+\frac{99038543}{6174450}a-\frac{2779454}{123489}$ 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:  \( 7696.04332288 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 7696.04332288 \cdot 3}{6\cdot\sqrt{131621703842267136}}\cr\approx \mathstrut & 0.652608857678 \end{aligned}\]

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 - 6*x^11 + 21*x^10 - 26*x^9 - 18*x^8 + 90*x^7 + 57*x^6 - 522*x^5 + 882*x^4 - 542*x^3 + 489*x^2 - 1050*x + 625) 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 + 21*x^10 - 26*x^9 - 18*x^8 + 90*x^7 + 57*x^6 - 522*x^5 + 882*x^4 - 542*x^3 + 489*x^2 - 1050*x + 625, 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 + 21*x^10 - 26*x^9 - 18*x^8 + 90*x^7 + 57*x^6 - 522*x^5 + 882*x^4 - 542*x^3 + 489*x^2 - 1050*x + 625); 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 - 6*x^11 + 21*x^10 - 26*x^9 - 18*x^8 + 90*x^7 + 57*x^6 - 522*x^5 + 882*x^4 - 542*x^3 + 489*x^2 - 1050*x + 625); 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

$S_3^2$ (as 12T16):

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 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), 6.2.362797056.3 x3

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 6 sibling: 6.2.362797056.3
Degree 9 sibling: 9.1.9521245937664.11
Degree 18 siblings: deg 18, deg 18, deg 18
Minimal sibling: 6.2.362797056.3

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 ${\href{/padicField/5.2.0.1}{2} }^{6}$ ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.6.22a1.1$x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 141 x^{6} + 126 x^{5} + 90 x^{4} + 50 x^{3} + 21 x^{2} + 6 x + 3$$6$$2$$22$$D_6$$$[3]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.1.6.11a1.11$x^{6} + 9 x^{3} + 9 x^{2} + 9 x + 3$$6$$1$$11$$S_3\times C_3$$$[2, \frac{5}{2}]_{2}$$
3.1.6.11a1.11$x^{6} + 9 x^{3} + 9 x^{2} + 9 x + 3$$6$$1$$11$$S_3\times C_3$$$[2, \frac{5}{2}]_{2}$$

Spectrum of ring of integers

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