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Properties

Label 12.0.18669417722...000.14
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{18}\cdot 5^{6}\cdot 7^{6}$
Root discriminant $86.95$
Ramified primes $2, 3, 5, 7$
Class number $13936$ (GRH)
Class group $[4, 3484]$ (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![119894041, -33807978, 38478141, -9646066, 5597562, -958098, 367625, -42342, 11814, -830, 177, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 177*x^10 - 830*x^9 + 11814*x^8 - 42342*x^7 + 367625*x^6 - 958098*x^5 + 5597562*x^4 - 9646066*x^3 + 38478141*x^2 - 33807978*x + 119894041)
 
gp: K = bnfinit(x^12 - 6*x^11 + 177*x^10 - 830*x^9 + 11814*x^8 - 42342*x^7 + 367625*x^6 - 958098*x^5 + 5597562*x^4 - 9646066*x^3 + 38478141*x^2 - 33807978*x + 119894041, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 177 x^{10} - 830 x^{9} + 11814 x^{8} - 42342 x^{7} + 367625 x^{6} - 958098 x^{5} + 5597562 x^{4} - 9646066 x^{3} + 38478141 x^{2} - 33807978 x + 119894041 \)

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:  \(186694177220038656000000=2^{18}\cdot 3^{18}\cdot 5^{6}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.95$
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:  \(2520=2^{3}\cdot 3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{2520}(1,·)$, $\chi_{2520}(1091,·)$, $\chi_{2520}(1681,·)$, $\chi_{2520}(841,·)$, $\chi_{2520}(1931,·)$, $\chi_{2520}(419,·)$, $\chi_{2520}(1849,·)$, $\chi_{2520}(1009,·)$, $\chi_{2520}(2099,·)$, $\chi_{2520}(1259,·)$, $\chi_{2520}(169,·)$, $\chi_{2520}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{14} a^{4} - \frac{1}{7} a^{3} - \frac{1}{14} a^{2} + \frac{1}{7} a + \frac{1}{14}$, $\frac{1}{14} a^{5} - \frac{5}{14} a^{3} + \frac{5}{14} a + \frac{1}{7}$, $\frac{1}{14} a^{6} + \frac{2}{7} a^{3} - \frac{1}{7} a + \frac{5}{14}$, $\frac{1}{14} a^{7} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2} - \frac{3}{14} a - \frac{2}{7}$, $\frac{1}{196} a^{8} - \frac{1}{49} a^{7} + \frac{1}{98} a^{6} - \frac{3}{98} a^{5} - \frac{5}{196} a^{4} + \frac{31}{98} a^{3} + \frac{1}{98} a^{2} - \frac{33}{98} a - \frac{27}{196}$, $\frac{1}{196} a^{9} + \frac{1}{98} a^{6} - \frac{1}{196} a^{5} - \frac{43}{98} a^{3} + \frac{3}{49} a^{2} - \frac{81}{196} a + \frac{23}{98}$, $\frac{1}{604122442756} a^{10} - \frac{5}{604122442756} a^{9} + \frac{30158871}{21575801527} a^{8} - \frac{1688896761}{302061221378} a^{7} + \frac{13581018307}{604122442756} a^{6} + \frac{14230825439}{604122442756} a^{5} + \frac{5476018001}{302061221378} a^{4} - \frac{48350387587}{151030610689} a^{3} + \frac{2875561667}{604122442756} a^{2} + \frac{197447057125}{604122442756} a - \frac{29681509464}{151030610689}$, $\frac{1}{408684603667334708} a^{11} + \frac{338241}{408684603667334708} a^{10} - \frac{166245963462287}{102171150916833677} a^{9} - \frac{892192011447643}{408684603667334708} a^{8} - \frac{14120497968953211}{408684603667334708} a^{7} + \frac{13115589158846649}{408684603667334708} a^{6} + \frac{297543529544299}{29191757404809622} a^{5} - \frac{12125719417252345}{408684603667334708} a^{4} - \frac{141070244552660757}{408684603667334708} a^{3} + \frac{133590666344531407}{408684603667334708} a^{2} - \frac{96461527423941485}{204342301833667354} a - \frac{143208805787802009}{408684603667334708}$

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

Class group and class number

$C_{4}\times C_{3484}$, which has order $13936$ (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:  \( 201.0008347866989 \) (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

\(\Q(\sqrt{-42}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-210}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{5}, \sqrt{-42})\), 6.0.3456649728.1, 6.6.820125.1, 6.0.432081216000.29

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$$[3]^{6}$
$3$3.12.18.82$x^{12} - 9 x^{9} + 9 x^{8} - 9 x^{5} - 9 x^{4} - 9 x^{3} + 9$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$