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Properties

Label 12.0.6775409390765625.3
Degree $12$
Signature $[0, 6]$
Discriminant $6.775\times 10^{15}$
Root discriminant \(20.86\)
Ramified primes $3,5,29$
Class number $4$
Class group [2, 2]
Galois group $D_6$ (as 12T3)

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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 + 29*x^10 - 90*x^9 + 225*x^8 - 426*x^7 + 667*x^6 - 822*x^5 + 1020*x^4 - 1020*x^3 + 1379*x^2 - 957*x + 1189)
 
Copy content gp:K = bnfinit(y^12 - 6*y^11 + 29*y^10 - 90*y^9 + 225*y^8 - 426*y^7 + 667*y^6 - 822*y^5 + 1020*y^4 - 1020*y^3 + 1379*y^2 - 957*y + 1189, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 29*x^10 - 90*x^9 + 225*x^8 - 426*x^7 + 667*x^6 - 822*x^5 + 1020*x^4 - 1020*x^3 + 1379*x^2 - 957*x + 1189);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^11 + 29*x^10 - 90*x^9 + 225*x^8 - 426*x^7 + 667*x^6 - 822*x^5 + 1020*x^4 - 1020*x^3 + 1379*x^2 - 957*x + 1189)
 

\( x^{12} - 6 x^{11} + 29 x^{10} - 90 x^{9} + 225 x^{8} - 426 x^{7} + 667 x^{6} - 822 x^{5} + 1020 x^{4} + \cdots + 1189 \) 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:   \(6775409390765625\) \(\medspace = 3^{6}\cdot 5^{6}\cdot 29^{6}\) 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:  \(20.86\)
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:  $3^{1/2}5^{1/2}29^{1/2}\approx 20.85665361461421$
Ramified primes:   \(3\), \(5\), \(29\) 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:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $D_6$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{5}, \sqrt{-87})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{249527}a^{10}-\frac{5}{249527}a^{9}-\frac{100812}{249527}a^{8}-\frac{95776}{249527}a^{7}-\frac{97172}{249527}a^{6}-\frac{121870}{249527}a^{5}-\frac{111647}{249527}a^{4}+\frac{65155}{249527}a^{3}-\frac{56119}{249527}a^{2}+\frac{19191}{249527}a+\frac{62748}{249527}$, $\frac{1}{498305419}a^{11}+\frac{993}{498305419}a^{10}+\frac{52045341}{498305419}a^{9}-\frac{85734532}{498305419}a^{8}-\frac{227681403}{498305419}a^{7}-\frac{36713992}{498305419}a^{6}-\frac{123983650}{498305419}a^{5}+\frac{193563443}{498305419}a^{4}+\frac{194722611}{498305419}a^{3}-\frac{120116010}{498305419}a^{2}-\frac{206856096}{498305419}a+\frac{243030525}{498305419}$ 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:  $C_{2}\times C_{2}$, which has order $4$
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:  $C_{2}\times C_{2}$, which has order $4$
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:   $\frac{1936}{872689}a^{11}-\frac{10648}{872689}a^{10}+\frac{46826}{872689}a^{9}-\frac{130857}{872689}a^{8}+\frac{267326}{872689}a^{7}-\frac{399511}{872689}a^{6}+\frac{413575}{872689}a^{5}-\frac{287253}{872689}a^{4}+\frac{496099}{872689}a^{3}-\frac{585385}{872689}a^{2}+\frac{48749}{45931}a+\frac{67175}{872689}$, $\frac{175100}{498305419}a^{11}-\frac{1160753}{498305419}a^{10}+\frac{8270855}{498305419}a^{9}-\frac{26929579}{498305419}a^{8}+\frac{77711925}{498305419}a^{7}-\frac{134485292}{498305419}a^{6}+\frac{198607131}{498305419}a^{5}-\frac{185869455}{498305419}a^{4}+\frac{186442882}{498305419}a^{3}-\frac{140811288}{498305419}a^{2}+\frac{301014028}{498305419}a-\frac{89315146}{498305419}$, $\frac{678620}{498305419}a^{11}-\frac{4075894}{498305419}a^{10}+\frac{19411375}{498305419}a^{9}-\frac{56547335}{498305419}a^{8}+\frac{131627998}{498305419}a^{7}-\frac{222541228}{498305419}a^{6}+\frac{281798416}{498305419}a^{5}-\frac{287249240}{498305419}a^{4}+\frac{12045030}{21665453}a^{3}-\frac{318461904}{498305419}a^{2}+\frac{738776827}{498305419}a-\frac{156874968}{498305419}$, $\frac{179274}{498305419}a^{11}-\frac{1183710}{498305419}a^{10}+\frac{5200200}{498305419}a^{9}-\frac{12939454}{498305419}a^{8}+\frac{30699681}{498305419}a^{7}-\frac{35390387}{498305419}a^{6}+\frac{81006922}{498305419}a^{5}-\frac{106847785}{498305419}a^{4}+\frac{249901617}{498305419}a^{3}-\frac{523987503}{498305419}a^{2}+\frac{192748342}{498305419}a-\frac{634984096}{498305419}$, $\frac{1361414}{498305419}a^{11}-\frac{8174745}{498305419}a^{10}+\frac{35752095}{498305419}a^{9}-\frac{99104545}{498305419}a^{8}+\frac{216243752}{498305419}a^{7}-\frac{15042937}{21665453}a^{6}+\frac{445996623}{498305419}a^{5}-\frac{21542470}{21665453}a^{4}+\frac{32501585}{26226601}a^{3}-\frac{521794604}{498305419}a^{2}+\frac{870982549}{498305419}a-\frac{361113467}{498305419}$ 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:  \( 217.33692993 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 217.33692993 \cdot 4}{2\cdot\sqrt{6775409390765625}}\cr\approx \mathstrut & 0.32491889151 \end{aligned}\]

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 + 29*x^10 - 90*x^9 + 225*x^8 - 426*x^7 + 667*x^6 - 822*x^5 + 1020*x^4 - 1020*x^3 + 1379*x^2 - 957*x + 1189) 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 + 29*x^10 - 90*x^9 + 225*x^8 - 426*x^7 + 667*x^6 - 822*x^5 + 1020*x^4 - 1020*x^3 + 1379*x^2 - 957*x + 1189, 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 + 29*x^10 - 90*x^9 + 225*x^8 - 426*x^7 + 667*x^6 - 822*x^5 + 1020*x^4 - 1020*x^3 + 1379*x^2 - 957*x + 1189); 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 + 29*x^10 - 90*x^9 + 225*x^8 - 426*x^7 + 667*x^6 - 822*x^5 + 1020*x^4 - 1020*x^3 + 1379*x^2 - 957*x + 1189); 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

$D_6$ (as 12T3):

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 12
The 6 conjugacy class representatives for $D_6$
Character table for $D_6$

Intermediate fields

\(\Q(\sqrt{-87}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-435}) \), 3.1.87.1 x3, \(\Q(\sqrt{5}, \sqrt{-87})\), 6.0.658503.1, 6.2.946125.1 x3, 6.0.82312875.1 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

Degree 6 siblings: 6.2.946125.1, 6.0.82312875.1
Minimal sibling: 6.2.946125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}$ R R ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ R ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.1.0.1}{1} }^{12}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.2.0.1}{2} }^{6}$

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
\(3\) Copy content Toggle raw display 3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(5\) Copy content Toggle raw display 5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(29\) Copy content Toggle raw display 29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.1.2.1a1.2$x^{2} + 58$$2$$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)