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Properties

Label 12.0.25955658105...8125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 7^{10}\cdot 19^{6}$
Root discriminant $73.77$
Ramified primes $5, 7, 19$
Class number $13088$ (GRH)
Class group $[2, 4, 1636]$ (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![110528251, -27964467, 32079035, -4872746, 4213235, -311604, 283912, -10018, 9854, -164, 163, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 163*x^10 - 164*x^9 + 9854*x^8 - 10018*x^7 + 283912*x^6 - 311604*x^5 + 4213235*x^4 - 4872746*x^3 + 32079035*x^2 - 27964467*x + 110528251)
 
gp: K = bnfinit(x^12 - x^11 + 163*x^10 - 164*x^9 + 9854*x^8 - 10018*x^7 + 283912*x^6 - 311604*x^5 + 4213235*x^4 - 4872746*x^3 + 32079035*x^2 - 27964467*x + 110528251, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 163 x^{10} - 164 x^{9} + 9854 x^{8} - 10018 x^{7} + 283912 x^{6} - 311604 x^{5} + 4213235 x^{4} - 4872746 x^{3} + 32079035 x^{2} - 27964467 x + 110528251 \)

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:  \(25955658105272205078125=5^{9}\cdot 7^{10}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.77$
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:  $5, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(665=5\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{665}(1,·)$, $\chi_{665}(227,·)$, $\chi_{665}(132,·)$, $\chi_{665}(134,·)$, $\chi_{665}(39,·)$, $\chi_{665}(493,·)$, $\chi_{665}(398,·)$, $\chi_{665}(208,·)$, $\chi_{665}(324,·)$, $\chi_{665}(607,·)$, $\chi_{665}(571,·)$, $\chi_{665}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{41} a^{9} + \frac{18}{41} a^{8} + \frac{5}{41} a^{7} - \frac{11}{41} a^{6} - \frac{2}{41} a^{5} - \frac{20}{41} a^{4} - \frac{16}{41} a^{3} - \frac{19}{41} a^{2} - \frac{16}{41} a$, $\frac{1}{160501921} a^{10} + \frac{695891}{160501921} a^{9} - \frac{26588773}{160501921} a^{8} - \frac{13115109}{160501921} a^{7} - \frac{31217797}{160501921} a^{6} - \frac{7456896}{160501921} a^{5} + \frac{46832470}{160501921} a^{4} - \frac{36959600}{160501921} a^{3} + \frac{10016623}{160501921} a^{2} - \frac{945550}{160501921} a + \frac{9609}{134989}$, $\frac{1}{253533582561078256894379929} a^{11} - \frac{523141913489129190}{253533582561078256894379929} a^{10} - \frac{1956599704770243998133624}{253533582561078256894379929} a^{9} - \frac{95152608067475545311971936}{253533582561078256894379929} a^{8} - \frac{27081479286394572057517168}{253533582561078256894379929} a^{7} - \frac{15168573505016863717403970}{253533582561078256894379929} a^{6} - \frac{24280488331902128096293039}{253533582561078256894379929} a^{5} - \frac{72289687028536081553008865}{253533582561078256894379929} a^{4} - \frac{113027432105648235028950932}{253533582561078256894379929} a^{3} - \frac{45415554505361439231143259}{253533582561078256894379929} a^{2} + \frac{88662586645603355521582416}{253533582561078256894379929} a + \frac{938590412398876169}{2293835107922573179}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{1636}$, which has order $13088$ (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:  \( 104.882003477 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.2211125.1, 6.6.300125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ R ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$19$19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$