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Properties

Label 12.6.168147445940224.2
Degree $12$
Signature $[6, 3]$
Discriminant $-1.681\times 10^{14}$
Root discriminant \(15.33\)
Ramified primes $2,37$
Class number $1$
Class group trivial
Galois group $(C_6\times C_2):C_2$ (as 12T15)

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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 - 4*x^11 - x^10 + 20*x^9 - 18*x^8 - 6*x^7 - 25*x^6 + 78*x^5 - 48*x^4 - 10*x^3 + 21*x^2 - 8*x + 1)
 
Copy content gp:K = bnfinit(y^12 - 4*y^11 - y^10 + 20*y^9 - 18*y^8 - 6*y^7 - 25*y^6 + 78*y^5 - 48*y^4 - 10*y^3 + 21*y^2 - 8*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - x^10 + 20*x^9 - 18*x^8 - 6*x^7 - 25*x^6 + 78*x^5 - 48*x^4 - 10*x^3 + 21*x^2 - 8*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 - x^10 + 20*x^9 - 18*x^8 - 6*x^7 - 25*x^6 + 78*x^5 - 48*x^4 - 10*x^3 + 21*x^2 - 8*x + 1)
 

\( x^{12} - 4 x^{11} - x^{10} + 20 x^{9} - 18 x^{8} - 6 x^{7} - 25 x^{6} + 78 x^{5} - 48 x^{4} - 10 x^{3} + \cdots + 1 \) 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:  $[6, 3]$
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:   \(-168147445940224\) \(\medspace = -\,2^{16}\cdot 37^{6}\) 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:  \(15.33\)
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:  $2^{5/3}37^{1/2}\approx 19.31156727893628$
Ramified primes:   \(2\), \(37\) 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:  \(\Q(\sqrt{-1}) \)
$\Aut(K/\Q)$:   $C_6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{21}a^{10}+\frac{2}{21}a^{9}-\frac{5}{21}a^{8}+\frac{1}{3}a^{7}-\frac{1}{21}a^{6}+\frac{3}{7}a^{5}-\frac{4}{21}a^{4}+\frac{1}{21}a^{3}+\frac{8}{21}a^{2}+\frac{1}{21}a-\frac{10}{21}$, $\frac{1}{147}a^{11}-\frac{1}{147}a^{10}-\frac{4}{147}a^{9}-\frac{41}{147}a^{8}+\frac{2}{49}a^{7}-\frac{37}{147}a^{6}+\frac{20}{49}a^{5}+\frac{62}{147}a^{4}-\frac{58}{147}a^{3}-\frac{37}{147}a^{2}+\frac{8}{147}a+\frac{16}{147}$ 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:  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:  Trivial group, which has order $1$
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:  $8$
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:   $\frac{277}{147}a^{11}-\frac{405}{49}a^{10}+\frac{5}{147}a^{9}+\frac{6122}{147}a^{8}-\frac{6619}{147}a^{7}-\frac{768}{49}a^{6}-\frac{5591}{147}a^{5}+\frac{25826}{147}a^{4}-\frac{5619}{49}a^{3}-\frac{5650}{147}a^{2}+\frac{2533}{49}a-\frac{1721}{147}$, $\frac{554}{147}a^{11}-\frac{2129}{147}a^{10}-\frac{907}{147}a^{9}+\frac{11033}{147}a^{8}-\frac{8240}{147}a^{7}-\frac{1718}{49}a^{6}-\frac{14255}{147}a^{5}+\frac{41432}{147}a^{4}-\frac{19448}{147}a^{3}-\frac{10705}{147}a^{2}+\frac{10501}{147}a-\frac{697}{49}$, $a^{11}-4a^{10}-a^{9}+20a^{8}-18a^{7}-6a^{6}-25a^{5}+78a^{4}-48a^{3}-10a^{2}+21a-7$, $\frac{467}{147}a^{11}-\frac{592}{49}a^{10}-\frac{860}{147}a^{9}+\frac{9301}{147}a^{8}-\frac{6410}{147}a^{7}-\frac{1583}{49}a^{6}-\frac{12475}{147}a^{5}+\frac{34288}{147}a^{4}-\frac{4761}{49}a^{3}-\frac{9572}{147}a^{2}+\frac{2720}{49}a-\frac{1537}{147}$, $\frac{905}{147}a^{11}-\frac{3152}{147}a^{10}-\frac{859}{49}a^{9}+\frac{16907}{147}a^{8}-\frac{7408}{147}a^{7}-\frac{10021}{147}a^{6}-\frac{27565}{147}a^{5}+\frac{18939}{49}a^{4}-\frac{12695}{147}a^{3}-\frac{5998}{49}a^{2}+\frac{9844}{147}a-\frac{1319}{147}$, $\frac{370}{147}a^{11}-\frac{401}{49}a^{10}-\frac{428}{49}a^{9}+\frac{6488}{147}a^{8}-\frac{583}{49}a^{7}-\frac{3841}{147}a^{6}-\frac{12100}{147}a^{5}+\frac{6732}{49}a^{4}-\frac{718}{49}a^{3}-\frac{2097}{49}a^{2}+\frac{905}{49}a-\frac{205}{147}$, $\frac{156}{49}a^{11}-\frac{1672}{147}a^{10}-\frac{1193}{147}a^{9}+\frac{8882}{147}a^{8}-\frac{4591}{147}a^{7}-\frac{4940}{147}a^{6}-\frac{4591}{49}a^{5}+\frac{30745}{147}a^{4}-\frac{8944}{147}a^{3}-\frac{9161}{147}a^{2}+\frac{6068}{147}a-\frac{1052}{147}$, $\frac{124}{21}a^{11}-\frac{484}{21}a^{10}-\frac{60}{7}a^{9}+\frac{2491}{21}a^{8}-\frac{1958}{21}a^{7}-\frac{155}{3}a^{6}-\frac{452}{3}a^{5}+450a^{4}-\frac{4759}{21}a^{3}-\frac{723}{7}a^{2}+\frac{2459}{21}a-\frac{583}{21}$ 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:  \( 399.890130029 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 399.890130029 \cdot 1}{2\cdot\sqrt{168147445940224}}\cr\approx \mathstrut & 0.244785185742 \end{aligned}\]

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 - 4*x^11 - x^10 + 20*x^9 - 18*x^8 - 6*x^7 - 25*x^6 + 78*x^5 - 48*x^4 - 10*x^3 + 21*x^2 - 8*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 - 4*x^11 - x^10 + 20*x^9 - 18*x^8 - 6*x^7 - 25*x^6 + 78*x^5 - 48*x^4 - 10*x^3 + 21*x^2 - 8*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 - 4*x^11 - x^10 + 20*x^9 - 18*x^8 - 6*x^7 - 25*x^6 + 78*x^5 - 48*x^4 - 10*x^3 + 21*x^2 - 8*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 = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 - x^10 + 20*x^9 - 18*x^8 - 6*x^7 - 25*x^6 + 78*x^5 - 48*x^4 - 10*x^3 + 21*x^2 - 8*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

$C_3:D_4$ (as 12T15):

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 24
The 9 conjugacy class representatives for $(C_6\times C_2):C_2$
Character table for $(C_6\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.148.1 x3, 4.2.21904.1, 6.6.810448.1

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

Galois closure: deg 24
Degree 12 sibling: 12.0.72712409055232.2
Minimal sibling: 12.0.72712409055232.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ ${\href{/padicField/5.2.0.1}{2} }^{6}$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{6}$ ${\href{/padicField/17.2.0.1}{2} }^{6}$ ${\href{/padicField/19.4.0.1}{4} }^{3}$ ${\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.4.0.1}{4} }^{3}$ R ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{3}$

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 2.2.6.16a2.8$x^{12} + 8 x^{11} + 33 x^{10} + 90 x^{9} + 180 x^{8} + 278 x^{7} + 339 x^{6} + 330 x^{5} + 254 x^{4} + 152 x^{3} + 67 x^{2} + 20 x + 9$$6$$2$$16$$(C_6\times C_2):C_2$$$[2, 2]_{3}^{2}$$
\(37\) Copy content Toggle raw display 37.2.2.2a1.2$x^{4} + 66 x^{3} + 1093 x^{2} + 132 x + 41$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
37.2.2.2a1.2$x^{4} + 66 x^{3} + 1093 x^{2} + 132 x + 41$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
37.2.2.2a1.2$x^{4} + 66 x^{3} + 1093 x^{2} + 132 x + 41$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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