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Properties

Label 12.0.16201367308...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{16}\cdot 5^{9}\cdot 19^{6}$
Root discriminant $126.12$
Ramified primes $2, 3, 5, 19$
Class number $163072$ (GRH)
Class group $[2, 2, 2, 2, 2, 14, 364]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7952847101, 67270842, 1053927204, 2436664, 59716095, 30672, 1824563, 36, 30834, -4, 273, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 273*x^10 - 4*x^9 + 30834*x^8 + 36*x^7 + 1824563*x^6 + 30672*x^5 + 59716095*x^4 + 2436664*x^3 + 1053927204*x^2 + 67270842*x + 7952847101)
 
gp: K = bnfinit(x^12 + 273*x^10 - 4*x^9 + 30834*x^8 + 36*x^7 + 1824563*x^6 + 30672*x^5 + 59716095*x^4 + 2436664*x^3 + 1053927204*x^2 + 67270842*x + 7952847101, 1)
 

Normalized defining polynomial

\( x^{12} + 273 x^{10} - 4 x^{9} + 30834 x^{8} + 36 x^{7} + 1824563 x^{6} + 30672 x^{5} + 59716095 x^{4} + 2436664 x^{3} + 1053927204 x^{2} + 67270842 x + 7952847101 \)

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:  \(16201367308849608000000000=2^{12}\cdot 3^{16}\cdot 5^{9}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $126.12$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3420=2^{2}\cdot 3^{2}\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{3420}(1,·)$, $\chi_{3420}(229,·)$, $\chi_{3420}(1063,·)$, $\chi_{3420}(2281,·)$, $\chi_{3420}(2887,·)$, $\chi_{3420}(2509,·)$, $\chi_{3420}(3343,·)$, $\chi_{3420}(1747,·)$, $\chi_{3420}(1141,·)$, $\chi_{3420}(1369,·)$, $\chi_{3420}(2203,·)$, $\chi_{3420}(607,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{19} a^{6} - \frac{6}{19} a^{4} - \frac{2}{19} a^{3} + \frac{9}{19} a^{2} + \frac{6}{19} a + \frac{1}{19}$, $\frac{1}{19} a^{7} - \frac{6}{19} a^{5} - \frac{2}{19} a^{4} + \frac{9}{19} a^{3} + \frac{6}{19} a^{2} + \frac{1}{19} a$, $\frac{1}{19} a^{8} - \frac{2}{19} a^{5} - \frac{8}{19} a^{4} - \frac{6}{19} a^{3} - \frac{2}{19} a^{2} - \frac{2}{19} a + \frac{6}{19}$, $\frac{1}{19} a^{9} - \frac{8}{19} a^{5} + \frac{1}{19} a^{4} - \frac{6}{19} a^{3} - \frac{3}{19} a^{2} - \frac{1}{19} a + \frac{2}{19}$, $\frac{1}{453372501419} a^{10} + \frac{8151649774}{453372501419} a^{9} + \frac{1683438992}{453372501419} a^{8} - \frac{4570686895}{453372501419} a^{7} - \frac{3782691505}{453372501419} a^{6} + \frac{132064476604}{453372501419} a^{5} - \frac{63602297557}{453372501419} a^{4} - \frac{218404924956}{453372501419} a^{3} - \frac{222784698754}{453372501419} a^{2} - \frac{49068543025}{453372501419} a - \frac{337818242}{6385528189}$, $\frac{1}{13729673886249803051200879} a^{11} - \frac{4340504158875}{13729673886249803051200879} a^{10} + \frac{141353593498403586702350}{13729673886249803051200879} a^{9} - \frac{109230161675210540025424}{13729673886249803051200879} a^{8} + \frac{145924406587456147225066}{13729673886249803051200879} a^{7} + \frac{23910217239152760834250}{13729673886249803051200879} a^{6} - \frac{5511810908240619151275171}{13729673886249803051200879} a^{5} - \frac{5351343092744302039527959}{13729673886249803051200879} a^{4} + \frac{6726286438466887202984081}{13729673886249803051200879} a^{3} - \frac{6822650848209841936126002}{13729673886249803051200879} a^{2} - \frac{1237288915928416823258973}{13729673886249803051200879} a - \frac{61444081573127290050924}{193375688538729620439449}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{14}\times C_{364}$, which has order $163072$ (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:  \( 201.000834787 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.0.722000.3, 6.6.820125.1

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 ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
$3$3.12.16.14$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$$3$$4$$16$$C_{12}$$[2]^{4}$
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$