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Properties

Label 12.0.23145177442...0853.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 11^{6}\cdot 13^{11}$
Root discriminant $60.31$
Ramified primes $3, 11, 13$
Class number $1936$ (GRH)
Class group $[2, 22, 44]$ (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![7219369, -3811497, 3811497, -829609, 829609, -84137, 84137, -4265, 4265, -105, 105, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 105*x^10 - 105*x^9 + 4265*x^8 - 4265*x^7 + 84137*x^6 - 84137*x^5 + 829609*x^4 - 829609*x^3 + 3811497*x^2 - 3811497*x + 7219369)
 
gp: K = bnfinit(x^12 - x^11 + 105*x^10 - 105*x^9 + 4265*x^8 - 4265*x^7 + 84137*x^6 - 84137*x^5 + 829609*x^4 - 829609*x^3 + 3811497*x^2 - 3811497*x + 7219369, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 105 x^{10} - 105 x^{9} + 4265 x^{8} - 4265 x^{7} + 84137 x^{6} - 84137 x^{5} + 829609 x^{4} - 829609 x^{3} + 3811497 x^{2} - 3811497 x + 7219369 \)

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:  \(2314517744209204100853=3^{6}\cdot 11^{6}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.31$
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:  $3, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(429=3\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{429}(32,·)$, $\chi_{429}(1,·)$, $\chi_{429}(98,·)$, $\chi_{429}(100,·)$, $\chi_{429}(197,·)$, $\chi_{429}(166,·)$, $\chi_{429}(199,·)$, $\chi_{429}(298,·)$, $\chi_{429}(395,·)$, $\chi_{429}(164,·)$, $\chi_{429}(362,·)$, $\chi_{429}(133,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1282969} a^{7} + \frac{416248}{1282969} a^{6} + \frac{56}{1282969} a^{5} - \frac{547600}{1282969} a^{4} + \frac{896}{1282969} a^{3} - \frac{156355}{1282969} a^{2} + \frac{3584}{1282969} a + \frac{292244}{1282969}$, $\frac{1}{1282969} a^{8} + \frac{64}{1282969} a^{6} + \frac{518923}{1282969} a^{5} + \frac{1280}{1282969} a^{4} + \frac{229416}{1282969} a^{3} + \frac{8192}{1282969} a^{2} + \frac{552359}{1282969} a + \frac{8192}{1282969}$, $\frac{1}{1282969} a^{9} - \frac{461569}{1282969} a^{6} - \frac{2304}{1282969} a^{5} + \frac{635653}{1282969} a^{4} - \frac{49152}{1282969} a^{3} + \frac{295327}{1282969} a^{2} - \frac{221184}{1282969} a + \frac{540919}{1282969}$, $\frac{1}{1282969} a^{10} - \frac{2880}{1282969} a^{6} - \frac{458832}{1282969} a^{5} - \frac{76800}{1282969} a^{4} - \frac{537836}{1282969} a^{3} - \frac{552960}{1282969} a^{2} - \frac{225795}{1282969} a - \frac{589824}{1282969}$, $\frac{1}{1282969} a^{11} + \frac{42362}{1282969} a^{6} + \frac{84480}{1282969} a^{5} + \frac{426034}{1282969} a^{4} - \frac{538418}{1282969} a^{3} - \frac{206076}{1282969} a^{2} - \frac{531656}{1282969} a + \frac{35056}{1282969}$

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

Class group and class number

$C_{2}\times C_{22}\times C_{44}$, which has order $1936$ (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:  \( 120.784031363 \) (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

\(\Q(\sqrt{13}) \), 3.3.169.1, 4.0.2392533.1, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.12.6.2$x^{12} + 14641 x^{4} - 322102 x^{2} + 14172488$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$