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Properties

Label 12.0.228...000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2.288\times 10^{20}$
Root discriminant \(49.73\)
Ramified primes $2,3,5,29$
Class number $2$
Class group [2]
Galois group $C_3:S_3^3:C_4$ (as 12T245)

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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 + 12*x^10 - 12*x^9 + 54*x^8 - 108*x^7 + 192*x^6 - 324*x^5 + 585*x^4 - 532*x^3 + 756*x^2 - 624*x + 656)
 
Copy content gp:K = bnfinit(y^12 + 12*y^10 - 12*y^9 + 54*y^8 - 108*y^7 + 192*y^6 - 324*y^5 + 585*y^4 - 532*y^3 + 756*y^2 - 624*y + 656, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 12*x^10 - 12*x^9 + 54*x^8 - 108*x^7 + 192*x^6 - 324*x^5 + 585*x^4 - 532*x^3 + 756*x^2 - 624*x + 656);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 + 12*x^10 - 12*x^9 + 54*x^8 - 108*x^7 + 192*x^6 - 324*x^5 + 585*x^4 - 532*x^3 + 756*x^2 - 624*x + 656)
 

\( x^{12} + 12 x^{10} - 12 x^{9} + 54 x^{8} - 108 x^{7} + 192 x^{6} - 324 x^{5} + 585 x^{4} - 532 x^{3} + \cdots + 656 \) 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:   \(228834243072000000000\) \(\medspace = 2^{18}\cdot 3^{12}\cdot 5^{9}\cdot 29^{2}\) 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:  \(49.73\)
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:  $2^{3/2}3^{25/18}5^{3/4}29^{1/2}\approx 234.22859052096263$
Ramified primes:   \(2\), \(3\), \(5\), \(29\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_1$
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 $\Q$.
This is not a CM field.
Maximal CM subfield:  4.0.8000.2

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{7}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{4}-\frac{1}{2}a$, $\frac{1}{232}a^{9}+\frac{9}{232}a^{7}+\frac{11}{116}a^{6}+\frac{27}{232}a^{5}+\frac{2}{29}a^{4}+\frac{47}{232}a^{3}+\frac{41}{116}a^{2}+\frac{15}{58}a+\frac{1}{29}$, $\frac{1}{464}a^{10}+\frac{9}{464}a^{8}+\frac{11}{232}a^{7}+\frac{27}{464}a^{6}-\frac{25}{116}a^{5}+\frac{47}{464}a^{4}-\frac{17}{232}a^{3}-\frac{43}{116}a^{2}+\frac{1}{58}a$, $\frac{1}{464}a^{11}-\frac{1}{464}a^{9}+\frac{11}{232}a^{8}-\frac{5}{464}a^{7}+\frac{7}{116}a^{6}+\frac{9}{464}a^{5}+\frac{19}{232}a^{4}-\frac{1}{116}a^{3}-\frac{1}{2}a^{2}-\frac{17}{58}a-\frac{5}{29}$ 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:  $C_{2}$, which has order $2$
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:  $C_{2}$, which has order $2$
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:   $\frac{1}{116}a^{9}+\frac{9}{116}a^{7}-\frac{7}{116}a^{6}+\frac{27}{116}a^{5}-\frac{21}{58}a^{4}+\frac{47}{116}a^{3}-\frac{63}{116}a^{2}+\frac{15}{29}a+\frac{2}{29}$, $\frac{1}{464}a^{11}+\frac{11}{464}a^{10}+\frac{21}{464}a^{9}+\frac{121}{464}a^{8}+\frac{1}{16}a^{7}+\frac{345}{464}a^{6}-\frac{497}{464}a^{5}+\frac{443}{464}a^{4}-\frac{513}{232}a^{3}+\frac{279}{58}a^{2}+\frac{65}{29}a+\frac{209}{29}$, $\frac{3}{464}a^{11}+\frac{7}{464}a^{10}-\frac{33}{464}a^{9}+\frac{113}{464}a^{8}-\frac{155}{464}a^{7}+\frac{589}{464}a^{6}-\frac{539}{464}a^{5}+\frac{1379}{464}a^{4}-\frac{611}{232}a^{3}+\frac{139}{29}a^{2}-\frac{48}{29}a+\frac{142}{29}$, $\frac{15}{464}a^{11}-\frac{7}{58}a^{10}+\frac{215}{464}a^{9}-\frac{13}{8}a^{8}+\frac{1691}{464}a^{7}-\frac{245}{29}a^{6}+\frac{7769}{464}a^{5}-\frac{6325}{232}a^{4}+\frac{9459}{232}a^{3}-\frac{6275}{116}a^{2}+\frac{2719}{58}a-\frac{685}{29}$, $\frac{273}{232}a^{11}+\frac{4223}{464}a^{10}+\frac{865}{116}a^{9}+\frac{25253}{464}a^{8}-\frac{20811}{232}a^{7}+\frac{63973}{464}a^{6}-\frac{22471}{58}a^{5}+\frac{385347}{464}a^{4}-\frac{19593}{29}a^{3}+\frac{130243}{116}a^{2}-\frac{66625}{58}a+\frac{38829}{29}$ 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:  \( 70306.32457498257 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 70306.32457498257 \cdot 2}{2\cdot\sqrt{228834243072000000000}}\cr\approx \mathstrut & 0.285965142410099 \end{aligned}\]

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 + 12*x^10 - 12*x^9 + 54*x^8 - 108*x^7 + 192*x^6 - 324*x^5 + 585*x^4 - 532*x^3 + 756*x^2 - 624*x + 656) 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 + 12*x^10 - 12*x^9 + 54*x^8 - 108*x^7 + 192*x^6 - 324*x^5 + 585*x^4 - 532*x^3 + 756*x^2 - 624*x + 656, 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 + 12*x^10 - 12*x^9 + 54*x^8 - 108*x^7 + 192*x^6 - 324*x^5 + 585*x^4 - 532*x^3 + 756*x^2 - 624*x + 656); 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 + 12*x^10 - 12*x^9 + 54*x^8 - 108*x^7 + 192*x^6 - 324*x^5 + 585*x^4 - 532*x^3 + 756*x^2 - 624*x + 656); 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

$C_3:S_3^3:C_4$ (as 12T245):

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 2592
The 30 conjugacy class representatives for $C_3:S_3^3:C_4$
Character table for $C_3:S_3^3:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.8000.2

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

Sibling fields

Degree 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: 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 R ${\href{/padicField/7.12.0.1}{12} }$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ R ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{5}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.4.0.1}{4} }^{3}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.2.6a1.6$x^{4} + 2 x^{3} + 7 x^{2} + 14 x + 7$$2$$2$$6$$C_4$$$[3]^{2}$$
2.4.2.12a1.9$x^{8} + 2 x^{5} + 6 x^{4} + x^{2} + 6 x + 7$$2$$4$$12$$C_4\times C_2$$$[3]^{4}$$
\(3\) Copy content Toggle raw display 3.4.3.12a14.1$x^{12} + 6 x^{11} + 12 x^{10} + 8 x^{9} + 6 x^{8} + 30 x^{7} + 42 x^{6} + 12 x^{5} + 15 x^{4} + 42 x^{3} + 12 x^{2} + 17$$3$$4$$12$12T41$$[\frac{3}{2}, \frac{3}{2}]_{2}^{4}$$
\(5\) Copy content Toggle raw display 5.3.4.9a1.1$x^{12} + 12 x^{10} + 12 x^{9} + 54 x^{8} + 108 x^{7} + 162 x^{6} + 324 x^{5} + 405 x^{4} + 432 x^{3} + 491 x^{2} + 324 x + 81$$4$$3$$9$$C_{12}$$$[\ ]_{4}^{3}$$
\(29\) Copy content Toggle raw display $\Q_{29}$$x + 27$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{29}$$x + 27$$1$$1$$0$Trivial$$[\ ]$$
29.1.2.1a1.1$x^{2} + 29$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.1.2.1a1.1$x^{2} + 29$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.3.1.0a1.1$x^{3} + 2 x + 27$$1$$3$$0$$C_3$$$[\ ]^{3}$$
29.3.1.0a1.1$x^{3} + 2 x + 27$$1$$3$$0$$C_3$$$[\ ]^{3}$$

Spectrum of ring of integers

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