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Properties

Label 12.0.21088444099...336.15
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{18}\cdot 7^{10}\cdot 19^{6}$
Root discriminant $229.26$
Ramified primes $2, 3, 7, 19$
Class number $1601664$ (GRH)
Class group $[2, 2, 2, 200208]$ (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![11307344896, 0, 675452112, 0, 25800888, 0, 594845, 0, 10851, 0, 111, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 111*x^10 + 10851*x^8 + 594845*x^6 + 25800888*x^4 + 675452112*x^2 + 11307344896)
 
gp: K = bnfinit(x^12 + 111*x^10 + 10851*x^8 + 594845*x^6 + 25800888*x^4 + 675452112*x^2 + 11307344896, 1)
 

Normalized defining polynomial

\( x^{12} + 111 x^{10} + 10851 x^{8} + 594845 x^{6} + 25800888 x^{4} + 675452112 x^{2} + 11307344896 \)

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:  \(21088444099773325470807822336=2^{12}\cdot 3^{18}\cdot 7^{10}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $229.26$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(1,·)$, $\chi_{4788}(2887,·)$, $\chi_{4788}(1063,·)$, $\chi_{4788}(3533,·)$, $\chi_{4788}(4561,·)$, $\chi_{4788}(1331,·)$, $\chi_{4788}(3649,·)$, $\chi_{4788}(2393,·)$, $\chi_{4788}(1787,·)$, $\chi_{4788}(2621,·)$, $\chi_{4788}(4295,·)$, $\chi_{4788}(607,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{56} a^{6} + \frac{3}{56} a^{4} - \frac{9}{28} a^{2} + \frac{1}{7}$, $\frac{1}{224} a^{7} + \frac{5}{112} a^{5} + \frac{45}{224} a^{3} + \frac{9}{56} a$, $\frac{1}{1792} a^{8} + \frac{3}{896} a^{6} - \frac{23}{1792} a^{4} + \frac{125}{448} a^{2} - \frac{1}{7}$, $\frac{1}{1792} a^{9} - \frac{1}{896} a^{7} - \frac{103}{1792} a^{5} + \frac{5}{64} a^{3} - \frac{17}{56} a$, $\frac{1}{14514849676544} a^{10} - \frac{1045631091}{7257424838272} a^{8} - \frac{114654247391}{14514849676544} a^{6} - \frac{13069353069}{259193744224} a^{4} - \frac{347599395051}{907178104784} a^{2} - \frac{3359112687}{8099804507}$, $\frac{1}{96465690950311424} a^{11} + \frac{11475331724237}{96465690950311424} a^{9} - \frac{27799786052317}{96465690950311424} a^{7} - \frac{1632996388217997}{96465690950311424} a^{5} - \frac{497479124220433}{24116422737577856} a^{3} + \frac{194746742260649}{1507276421098616} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{200208}$, which has order $1601664$ (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:  \( 38984.4787659612 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-133}) \), \(\Q(\sqrt{-399}) \), 3.3.3969.1, \(\Q(\sqrt{3}, \sqrt{-133})\), 6.6.3024568512.1, 6.0.48406202655552.5, 6.0.2269040749479.8

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
$3$3.12.18.51$x^{12} + 27 x^{11} + 15 x^{10} + 36 x^{9} - 36 x^{8} - 18 x^{7} + 21 x^{6} - 18 x^{4} + 27 x^{2} - 27 x + 36$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
$7$7.12.10.2$x^{12} + 35 x^{6} + 441$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$19$19.12.6.1$x^{12} + 41154 x^{6} - 2476099 x^{2} + 423412929$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$