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Properties

Label 12.8.3531579275390625.1
Degree $12$
Signature $[8, 2]$
Discriminant $3.532\times 10^{15}$
Root discriminant \(19.75\)
Ramified primes $3,5,29,31,89$
Class number $1$
Class group trivial
Galois group $S_3\wr C_4$ (as 12T264)

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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 - 7*x^10 - 4*x^9 + 14*x^8 + 21*x^7 - 2*x^6 - 28*x^5 - 20*x^4 + 4*x^3 + 14*x^2 + 7*x + 1)
 
Copy content gp:K = bnfinit(y^12 - 7*y^10 - 4*y^9 + 14*y^8 + 21*y^7 - 2*y^6 - 28*y^5 - 20*y^4 + 4*y^3 + 14*y^2 + 7*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 7*x^10 - 4*x^9 + 14*x^8 + 21*x^7 - 2*x^6 - 28*x^5 - 20*x^4 + 4*x^3 + 14*x^2 + 7*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 7*x^10 - 4*x^9 + 14*x^8 + 21*x^7 - 2*x^6 - 28*x^5 - 20*x^4 + 4*x^3 + 14*x^2 + 7*x + 1)
 

\( x^{12} - 7x^{10} - 4x^{9} + 14x^{8} + 21x^{7} - 2x^{6} - 28x^{5} - 20x^{4} + 4x^{3} + 14x^{2} + 7x + 1 \) 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:  $[8, 2]$
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:   \(3531579275390625\) \(\medspace = 3^{6}\cdot 5^{9}\cdot 29\cdot 31^{2}\cdot 89\) 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:  \(19.75\)
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:  $3^{1/2}5^{3/4}29^{1/2}31^{1/2}89^{1/2}\approx 1638.1851312400668$
Ramified primes:   \(3\), \(5\), \(29\), \(31\), \(89\) 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:  \(\Q(\sqrt{12905}) \)
$\Aut(K/\Q)$:   $C_1$
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$ 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:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$
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:  $9$
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:   $2a^{11}-a^{10}-13a^{9}-a^{8}+25a^{7}+25a^{6}-11a^{5}-39a^{4}-17a^{3}+9a^{2}+16a+4$, $a$, $a^{11}-a^{10}-6a^{9}+2a^{8}+12a^{7}+9a^{6}-11a^{5}-17a^{4}-3a^{3}+7a^{2}+7a+1$, $a^{11}-8a^{9}-4a^{8}+21a^{7}+24a^{6}-16a^{5}-42a^{4}-15a^{3}+19a^{2}+20a+5$, $a^{11}-a^{10}-6a^{9}+3a^{8}+11a^{7}+4a^{6}-9a^{5}-11a^{4}+3a^{3}+4a^{2}+2a$, $a^{11}-a^{10}-6a^{9}+2a^{8}+11a^{7}+10a^{6}-6a^{5}-20a^{4}-9a^{3}+5a^{2}+11a+3$, $a^{11}-a^{10}-7a^{9}+3a^{8}+17a^{7}+7a^{6}-17a^{5}-23a^{4}-a^{3}+12a^{2}+10a+3$, $a^{11}-7a^{9}-5a^{8}+15a^{7}+27a^{6}-5a^{5}-39a^{4}-23a^{3}+12a^{2}+20a+6$, $a^{11}-7a^{9}-3a^{8}+14a^{7}+14a^{6}-5a^{5}-15a^{4}-6a^{3}+3a^{2}-1$ 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:  \( 2512.55345376 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{2}\cdot 2512.55345376 \cdot 1}{2\cdot\sqrt{3531579275390625}}\cr\approx \mathstrut & 0.213648838729 \end{aligned}\]

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 - 7*x^10 - 4*x^9 + 14*x^8 + 21*x^7 - 2*x^6 - 28*x^5 - 20*x^4 + 4*x^3 + 14*x^2 + 7*x + 1) 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 - 7*x^10 - 4*x^9 + 14*x^8 + 21*x^7 - 2*x^6 - 28*x^5 - 20*x^4 + 4*x^3 + 14*x^2 + 7*x + 1, 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 - 7*x^10 - 4*x^9 + 14*x^8 + 21*x^7 - 2*x^6 - 28*x^5 - 20*x^4 + 4*x^3 + 14*x^2 + 7*x + 1); 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 - 7*x^10 - 4*x^9 + 14*x^8 + 21*x^7 - 2*x^6 - 28*x^5 - 20*x^4 + 4*x^3 + 14*x^2 + 7*x + 1); 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

$S_3\wr C_4$ (as 12T264):

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 5184
The 36 conjugacy class representatives for $S_3\wr C_4$
Character table for $S_3\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\)

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 sibling: data not computed
Degree 18 sibling: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.0.10729184571822861436640625.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ R R ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ R R ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(3\) Copy content Toggle raw display 3.2.2.2a1.1$x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_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$$[\ ]$$
$\Q_{29}$$x + 27$$1$$1$$0$Trivial$$[\ ]$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.3.1.0a1.1$x^{3} + 2 x + 27$$1$$3$$0$$C_3$$$[\ ]^{3}$$
\(31\) Copy content Toggle raw display $\Q_{31}$$x + 28$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{31}$$x + 28$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{31}$$x + 28$$1$$1$$0$Trivial$$[\ ]$$
31.1.2.1a1.2$x^{2} + 93$$2$$1$$1$$C_2$$$[\ ]_{2}$$
31.1.2.1a1.1$x^{2} + 31$$2$$1$$1$$C_2$$$[\ ]_{2}$$
31.2.1.0a1.1$x^{2} + 29 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
31.3.1.0a1.1$x^{3} + x + 28$$1$$3$$0$$C_3$$$[\ ]^{3}$$
\(89\) Copy content Toggle raw display $\Q_{89}$$x + 86$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{89}$$x + 86$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{89}$$x + 86$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{89}$$x + 86$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{89}$$x + 86$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{89}$$x + 86$$1$$1$$0$Trivial$$[\ ]$$
89.2.1.0a1.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
89.2.1.0a1.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
89.1.2.1a1.2$x^{2} + 267$$2$$1$$1$$C_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)