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Properties

Label 15.15.2746410307...8161.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{20}\cdot 31^{12}$
Root discriminant $67.49$
Ramified primes $3, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{15}$ (as 15T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1637, 10398, -7836, -31481, 35805, 26625, -40813, -3687, 16959, -2236, -2841, 663, 168, -48, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 48*x^13 + 168*x^12 + 663*x^11 - 2841*x^10 - 2236*x^9 + 16959*x^8 - 3687*x^7 - 40813*x^6 + 26625*x^5 + 35805*x^4 - 31481*x^3 - 7836*x^2 + 10398*x - 1637)
 
gp: K = bnfinit(x^15 - 3*x^14 - 48*x^13 + 168*x^12 + 663*x^11 - 2841*x^10 - 2236*x^9 + 16959*x^8 - 3687*x^7 - 40813*x^6 + 26625*x^5 + 35805*x^4 - 31481*x^3 - 7836*x^2 + 10398*x - 1637, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 48 x^{13} + 168 x^{12} + 663 x^{11} - 2841 x^{10} - 2236 x^{9} + 16959 x^{8} - 3687 x^{7} - 40813 x^{6} + 26625 x^{5} + 35805 x^{4} - 31481 x^{3} - 7836 x^{2} + 10398 x - 1637 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2746410307762150989067078161=3^{20}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.49$
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:  $3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(279=3^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{279}(256,·)$, $\chi_{279}(1,·)$, $\chi_{279}(163,·)$, $\chi_{279}(4,·)$, $\chi_{279}(70,·)$, $\chi_{279}(97,·)$, $\chi_{279}(64,·)$, $\chi_{279}(202,·)$, $\chi_{279}(109,·)$, $\chi_{279}(16,·)$, $\chi_{279}(94,·)$, $\chi_{279}(250,·)$, $\chi_{279}(187,·)$, $\chi_{279}(157,·)$, $\chi_{279}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{482014848411505891714945} a^{14} + \frac{17557725826304093750486}{482014848411505891714945} a^{13} - \frac{2843982781205037944466}{96402969682301178342989} a^{12} - \frac{889571167651853138456}{482014848411505891714945} a^{11} + \frac{12002463554808680551434}{482014848411505891714945} a^{10} + \frac{184970375126972145664378}{482014848411505891714945} a^{9} + \frac{22901978135349587214133}{96402969682301178342989} a^{8} - \frac{22302228434216729309974}{96402969682301178342989} a^{7} - \frac{69128921954478077616473}{482014848411505891714945} a^{6} - \frac{118440890239581215763618}{482014848411505891714945} a^{5} + \frac{163396838012447747006359}{482014848411505891714945} a^{4} - \frac{154404421460349485177218}{482014848411505891714945} a^{3} - \frac{172679322483682833955256}{482014848411505891714945} a^{2} + \frac{195900644609208717051148}{482014848411505891714945} a - \frac{96932339707114428352801}{482014848411505891714945}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 552470977.222 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{15}$ (as 15T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 15
The 15 conjugacy class representatives for $C_{15}$
Character table for $C_{15}$

Intermediate fields

\(\Q(\zeta_{9})^+\), 5.5.923521.1

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 $15$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{5}$ $15$ $15$ $15$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ $15$ $15$ R ${\href{/LocalNumberField/37.1.0.1}{1} }^{15}$ $15$ $15$ $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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
3Data not computed
31Data not computed