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Properties

Label 12.0.84346657391...0000.8
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 5^{6}\cdot 7^{10}$
Root discriminant $55.44$
Ramified primes $2, 3, 5, 7$
Class number $2800$ (GRH)
Class group $[20, 140]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5366971, -2040774, 2459025, -880834, 553824, -143382, 60185, -11094, 3372, -410, 93, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 93*x^10 - 410*x^9 + 3372*x^8 - 11094*x^7 + 60185*x^6 - 143382*x^5 + 553824*x^4 - 880834*x^3 + 2459025*x^2 - 2040774*x + 5366971)
 
gp: K = bnfinit(x^12 - 6*x^11 + 93*x^10 - 410*x^9 + 3372*x^8 - 11094*x^7 + 60185*x^6 - 143382*x^5 + 553824*x^4 - 880834*x^3 + 2459025*x^2 - 2040774*x + 5366971, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 93 x^{10} - 410 x^{9} + 3372 x^{8} - 11094 x^{7} + 60185 x^{6} - 143382 x^{5} + 553824 x^{4} - 880834 x^{3} + 2459025 x^{2} - 2040774 x + 5366971 \)

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:  \(843466573910016000000=2^{18}\cdot 3^{6}\cdot 5^{6}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.44$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(131,·)$, $\chi_{840}(289,·)$, $\chi_{840}(361,·)$, $\chi_{840}(299,·)$, $\chi_{840}(529,·)$, $\chi_{840}(419,·)$, $\chi_{840}(251,·)$, $\chi_{840}(59,·)$, $\chi_{840}(169,·)$, $\chi_{840}(121,·)$, $\chi_{840}(731,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{6} + \frac{1}{3} a^{3} - \frac{1}{6}$, $\frac{1}{6} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{36} a^{8} + \frac{1}{18} a^{7} + \frac{1}{18} a^{6} + \frac{1}{18} a^{5} + \frac{1}{36} a^{4} - \frac{1}{18} a^{3} + \frac{2}{9} a^{2} - \frac{2}{9} a - \frac{17}{36}$, $\frac{1}{36} a^{9} - \frac{1}{18} a^{7} - \frac{1}{18} a^{6} - \frac{1}{12} a^{5} + \frac{1}{18} a^{4} + \frac{1}{6} a^{2} + \frac{11}{36} a + \frac{1}{9}$, $\frac{1}{3398074884} a^{10} - \frac{5}{3398074884} a^{9} + \frac{4125433}{849518721} a^{8} - \frac{33003449}{1699037442} a^{7} + \frac{249277759}{3398074884} a^{6} + \frac{49536659}{3398074884} a^{5} + \frac{38206766}{849518721} a^{4} + \frac{87086020}{283172907} a^{3} + \frac{1055699105}{3398074884} a^{2} + \frac{487184347}{1132691628} a + \frac{161990189}{1699037442}$, $\frac{1}{480531963571092} a^{11} + \frac{23567}{160177321190364} a^{10} - \frac{688342104382}{120132990892773} a^{9} + \frac{104412951241}{240265981785546} a^{8} + \frac{1539783025997}{160177321190364} a^{7} + \frac{7390858777781}{480531963571092} a^{6} - \frac{8705151980374}{120132990892773} a^{5} - \frac{956744632015}{80088660595182} a^{4} - \frac{12906151149361}{160177321190364} a^{3} + \frac{66669607521755}{160177321190364} a^{2} - \frac{11764263247551}{26696220198394} a + \frac{45192314307949}{240265981785546}$

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

Class group and class number

$C_{20}\times C_{140}$, which has order $2800$ (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:  \( 104.88200347693757 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-210}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{5}, \sqrt{-42})\), 6.0.29042496000.9, 6.0.232339968.1, 6.6.300125.1

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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$