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Properties

Label 12.0.36099543110...3968.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{33}\cdot 3^{6}\cdot 7^{8}$
Root discriminant $42.64$
Ramified primes $2, 3, 7$
Class number $194$ (GRH)
Class group $[194]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53857, -30344, 38696, -10924, 10378, -5164, 3402, -1060, 487, -96, 34, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 34*x^10 - 96*x^9 + 487*x^8 - 1060*x^7 + 3402*x^6 - 5164*x^5 + 10378*x^4 - 10924*x^3 + 38696*x^2 - 30344*x + 53857)
 
gp: K = bnfinit(x^12 - 4*x^11 + 34*x^10 - 96*x^9 + 487*x^8 - 1060*x^7 + 3402*x^6 - 5164*x^5 + 10378*x^4 - 10924*x^3 + 38696*x^2 - 30344*x + 53857, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 34 x^{10} - 96 x^{9} + 487 x^{8} - 1060 x^{7} + 3402 x^{6} - 5164 x^{5} + 10378 x^{4} - 10924 x^{3} + 38696 x^{2} - 30344 x + 53857 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(36099543110378323968=2^{33}\cdot 3^{6}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.64$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(336=2^{4}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{336}(1,·)$, $\chi_{336}(197,·)$, $\chi_{336}(193,·)$, $\chi_{336}(169,·)$, $\chi_{336}(29,·)$, $\chi_{336}(289,·)$, $\chi_{336}(221,·)$, $\chi_{336}(149,·)$, $\chi_{336}(25,·)$, $\chi_{336}(121,·)$, $\chi_{336}(317,·)$, $\chi_{336}(53,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{20379747} a^{10} + \frac{2080679}{20379747} a^{9} - \frac{1317103}{20379747} a^{8} - \frac{488922}{6793249} a^{7} - \frac{709}{30463} a^{6} - \frac{624140}{6793249} a^{5} + \frac{4974806}{20379747} a^{4} + \frac{2834665}{20379747} a^{3} + \frac{3156852}{6793249} a^{2} + \frac{2251951}{6793249} a - \frac{6086281}{20379747}$, $\frac{1}{2753986317047283} a^{11} + \frac{14709590}{917995439015761} a^{10} + \frac{94527342907752}{917995439015761} a^{9} + \frac{138289553802995}{2753986317047283} a^{8} - \frac{100494336749768}{917995439015761} a^{7} + \frac{29041166148736}{917995439015761} a^{6} + \frac{162803203708201}{2753986317047283} a^{5} - \frac{53406270951329}{917995439015761} a^{4} + \frac{323752620055344}{917995439015761} a^{3} + \frac{327302348674954}{2753986317047283} a^{2} + \frac{230444022118316}{917995439015761} a - \frac{368477815253402}{917995439015761}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{194}$, which has order $194$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 279.150027194 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), 4.0.18432.2, 6.6.1229312.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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 ${\href{/LocalNumberField/5.12.0.1}{12} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.12.33.374$x^{12} + 28 x^{10} - 6 x^{8} + 40 x^{6} - 56 x^{4} - 32 x^{2} - 56$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
$3$3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$