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Properties

Label 12.0.17321270440...4656.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{18}\cdot 127^{10}$
Root discriminant $588.68$
Ramified primes $2, 3, 127$
Class number $16343040$ (GRH)
Class group $[2, 8, 8, 8, 15960]$ (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![252015625, 0, 33659206875, 0, 1472174475, 0, 24967184, 0, 202311, 0, 762, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 762*x^10 + 202311*x^8 + 24967184*x^6 + 1472174475*x^4 + 33659206875*x^2 + 252015625)
 
gp: K = bnfinit(x^12 + 762*x^10 + 202311*x^8 + 24967184*x^6 + 1472174475*x^4 + 33659206875*x^2 + 252015625, 1)
 

Normalized defining polynomial

\( x^{12} + 762 x^{10} + 202311 x^{8} + 24967184 x^{6} + 1472174475 x^{4} + 33659206875 x^{2} + 252015625 \)

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:  \(1732127044066296934075975910854656=2^{12}\cdot 3^{18}\cdot 127^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $588.68$
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:  $2, 3, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4572=2^{2}\cdot 3^{2}\cdot 127\)
Dirichlet character group:    $\lbrace$$\chi_{4572}(1,·)$, $\chi_{4572}(4571,·)$, $\chi_{4572}(1505,·)$, $\chi_{4572}(4171,·)$, $\chi_{4572}(781,·)$, $\chi_{4572}(2287,·)$, $\chi_{4572}(401,·)$, $\chi_{4572}(2285,·)$, $\chi_{4572}(3067,·)$, $\chi_{4572}(1885,·)$, $\chi_{4572}(3791,·)$, $\chi_{4572}(2687,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{127} a^{6}$, $\frac{1}{635} a^{7} + \frac{1}{5} a^{5} - \frac{2}{5} a^{3} + \frac{2}{5} a$, $\frac{1}{3175} a^{8} + \frac{12}{3175} a^{6} - \frac{7}{25} a^{4} - \frac{8}{25} a^{2}$, $\frac{1}{15875} a^{9} + \frac{12}{15875} a^{7} - \frac{32}{125} a^{5} - \frac{33}{125} a^{3} + \frac{2}{5} a$, $\frac{1}{1258230195625} a^{10} + \frac{154361362}{1258230195625} a^{8} - \frac{813022989}{1258230195625} a^{6} - \frac{4237480108}{9907324375} a^{4} + \frac{8049282}{79258595} a^{2} + \frac{5255285}{15851719}$, $\frac{1}{798976174221875} a^{11} - \frac{142323744}{6291150978125} a^{9} - \frac{4147195282}{6291150978125} a^{7} + \frac{2144463030342}{6291150978125} a^{5} + \frac{963652076}{1981464875} a^{3} - \frac{5825004}{79258595} a$

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

Class group and class number

$C_{2}\times C_{8}\times C_{8}\times C_{8}\times C_{15960}$, which has order $16343040$ (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:  \( -\frac{243}{4461684625} a^{11} - \frac{1262}{35131375} a^{9} - \frac{257697}{35131375} a^{7} - \frac{851607}{1405255} a^{5} - \frac{4872018}{276625} a^{3} + \frac{126396}{11065} a \) (order $4$)
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:  \( 120153.21081985293 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-381}) \), \(\Q(\sqrt{381}) \), 3.3.1306449.2, \(\Q(i, \sqrt{381})\), 6.0.109235775334464.1, 6.0.41618830402430784.1, 6.6.650294225037981.2

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.12.18.51$x^{12} + 27 x^{11} + 15 x^{10} + 36 x^{9} - 36 x^{8} - 18 x^{7} + 21 x^{6} - 18 x^{4} + 27 x^{2} - 27 x + 36$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
$127$127.12.10.1$x^{12} - 1270 x^{6} + 11758041$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$