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Properties

Label 12.0.13405059158...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{18}\cdot 5^{9}\cdot 11^{6}$
Root discriminant $57.62$
Ramified primes $3, 5, 11$
Class number $1832$ (GRH)
Class group $[2, 2, 458]$ (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![1522711, -332247, 599634, -166064, 152430, -15957, 25088, -441, 2079, -1, 78, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 78*x^10 - x^9 + 2079*x^8 - 441*x^7 + 25088*x^6 - 15957*x^5 + 152430*x^4 - 166064*x^3 + 599634*x^2 - 332247*x + 1522711)
 
gp: K = bnfinit(x^12 + 78*x^10 - x^9 + 2079*x^8 - 441*x^7 + 25088*x^6 - 15957*x^5 + 152430*x^4 - 166064*x^3 + 599634*x^2 - 332247*x + 1522711, 1)
 

Normalized defining polynomial

\( x^{12} + 78 x^{10} - x^{9} + 2079 x^{8} - 441 x^{7} + 25088 x^{6} - 15957 x^{5} + 152430 x^{4} - 166064 x^{3} + 599634 x^{2} - 332247 x + 1522711 \)

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:  \(1340505915846345703125=3^{18}\cdot 5^{9}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.62$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(495=3^{2}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{495}(32,·)$, $\chi_{495}(1,·)$, $\chi_{495}(98,·)$, $\chi_{495}(197,·)$, $\chi_{495}(166,·)$, $\chi_{495}(263,·)$, $\chi_{495}(364,·)$, $\chi_{495}(362,·)$, $\chi_{495}(331,·)$, $\chi_{495}(428,·)$, $\chi_{495}(34,·)$, $\chi_{495}(199,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{1121} a^{9} - \frac{1}{19} a^{8} - \frac{112}{1121} a^{7} + \frac{48}{1121} a^{6} - \frac{300}{1121} a^{5} + \frac{6}{59} a^{4} - \frac{166}{1121} a^{3} - \frac{397}{1121} a^{2} + \frac{520}{1121} a - \frac{238}{1121}$, $\frac{1}{1121} a^{10} - \frac{230}{1121} a^{8} + \frac{166}{1121} a^{7} + \frac{290}{1121} a^{6} + \frac{350}{1121} a^{5} - \frac{166}{1121} a^{4} - \frac{102}{1121} a^{3} - \frac{483}{1121} a^{2} + \frac{175}{1121} a + \frac{9}{19}$, $\frac{1}{201073062197124817997729} a^{11} - \frac{65153875453503940652}{201073062197124817997729} a^{10} + \frac{64038218963154033408}{201073062197124817997729} a^{9} + \frac{15034371448827415333896}{201073062197124817997729} a^{8} + \frac{5891158252097171855760}{201073062197124817997729} a^{7} - \frac{7884518988036513476302}{201073062197124817997729} a^{6} - \frac{29290456575187385809411}{201073062197124817997729} a^{5} - \frac{35927789629001254485933}{201073062197124817997729} a^{4} + \frac{81270137646511655132927}{201073062197124817997729} a^{3} + \frac{1283335726316838392499}{10582792747217095684091} a^{2} - \frac{36641189050423823894950}{201073062197124817997729} a - \frac{71385620663272783854977}{201073062197124817997729}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{458}$, which has order $1832$ (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.0.136125.2, 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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }$ R ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$3$3.12.18.74$x^{12} - 6 x^{11} - 3 x^{10} - 12 x^{9} + 9 x^{8} + 9 x^{7} + 12 x^{6} - 9 x^{3} - 9$$6$$2$$18$$C_{12}$$[2]_{2}^{2}$
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$