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Properties

Label 12.0.12163740463...2096.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{16}\cdot 47^{6}$
Root discriminant $83.90$
Ramified primes $2, 3, 47$
Class number $10980$ (GRH)
Class group $[3, 3660]$ (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![14609528, -3480240, 3879852, -829016, 490488, -95424, 39033, -6486, 1995, -260, 63, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 63*x^10 - 260*x^9 + 1995*x^8 - 6486*x^7 + 39033*x^6 - 95424*x^5 + 490488*x^4 - 829016*x^3 + 3879852*x^2 - 3480240*x + 14609528)
 
gp: K = bnfinit(x^12 - 6*x^11 + 63*x^10 - 260*x^9 + 1995*x^8 - 6486*x^7 + 39033*x^6 - 95424*x^5 + 490488*x^4 - 829016*x^3 + 3879852*x^2 - 3480240*x + 14609528, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 63 x^{10} - 260 x^{9} + 1995 x^{8} - 6486 x^{7} + 39033 x^{6} - 95424 x^{5} + 490488 x^{4} - 829016 x^{3} + 3879852 x^{2} - 3480240 x + 14609528 \)

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:  \(121637404636973946372096=2^{18}\cdot 3^{16}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.90$
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:  $2, 3, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3384=2^{3}\cdot 3^{2}\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{3384}(1,·)$, $\chi_{3384}(2821,·)$, $\chi_{3384}(2257,·)$, $\chi_{3384}(1033,·)$, $\chi_{3384}(1693,·)$, $\chi_{3384}(2161,·)$, $\chi_{3384}(565,·)$, $\chi_{3384}(1129,·)$, $\chi_{3384}(2725,·)$, $\chi_{3384}(3289,·)$, $\chi_{3384}(1597,·)$, $\chi_{3384}(469,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{68} a^{9} + \frac{1}{17} a^{8} - \frac{2}{17} a^{7} - \frac{1}{34} a^{6} + \frac{7}{68} a^{5} - \frac{3}{34} a^{4} - \frac{5}{17} a^{3} + \frac{1}{17} a^{2} + \frac{5}{17} a$, $\frac{1}{105214447652} a^{10} - \frac{5}{105214447652} a^{9} + \frac{2691286921}{26303611913} a^{8} + \frac{2386658280}{26303611913} a^{7} + \frac{22779182087}{105214447652} a^{6} - \frac{22839926463}{105214447652} a^{5} + \frac{210197011}{3094542578} a^{4} - \frac{578946707}{1384400627} a^{3} - \frac{23393275093}{52607223826} a^{2} - \frac{10456420633}{26303611913} a + \frac{295901993}{1547271289}$, $\frac{1}{1469530190355484} a^{11} + \frac{3489}{734765095177742} a^{10} + \frac{8489811776489}{1469530190355484} a^{9} + \frac{18487127849963}{734765095177742} a^{8} + \frac{182664184997331}{1469530190355484} a^{7} - \frac{63077565440976}{367382547588871} a^{6} + \frac{335825330699347}{1469530190355484} a^{5} + \frac{59150149225479}{734765095177742} a^{4} - \frac{38270476538528}{367382547588871} a^{3} + \frac{55909323346551}{734765095177742} a^{2} + \frac{62182221201306}{367382547588871} a - \frac{2620401117262}{21610738093463}$

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

Class group and class number

$C_{3}\times C_{3660}$, which has order $10980$ (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:  \( 481.70037561485367 \) (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

\(\Q(\sqrt{-47}) \), \(\Q(\sqrt{-94}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{2}, \sqrt{-47})\), 6.0.681182703.2, 6.0.348765543936.8, 6.6.3359232.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
$47$47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$