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Properties

Label 15.15.1383382540...7881.1
Degree $15$
Signature $[15, 0]$
Discriminant $19^{10}\cdot 41^{12}$
Root discriminant $138.91$
Ramified primes $19, 41$
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![-7213, 294489, -658269, -86756, 826932, -168943, -350546, 100680, 65923, -21178, -5720, 1972, 208, -79, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 79*x^13 + 208*x^12 + 1972*x^11 - 5720*x^10 - 21178*x^9 + 65923*x^8 + 100680*x^7 - 350546*x^6 - 168943*x^5 + 826932*x^4 - 86756*x^3 - 658269*x^2 + 294489*x - 7213)
 
gp: K = bnfinit(x^15 - 2*x^14 - 79*x^13 + 208*x^12 + 1972*x^11 - 5720*x^10 - 21178*x^9 + 65923*x^8 + 100680*x^7 - 350546*x^6 - 168943*x^5 + 826932*x^4 - 86756*x^3 - 658269*x^2 + 294489*x - 7213, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 79 x^{13} + 208 x^{12} + 1972 x^{11} - 5720 x^{10} - 21178 x^{9} + 65923 x^{8} + 100680 x^{7} - 350546 x^{6} - 168943 x^{5} + 826932 x^{4} - 86756 x^{3} - 658269 x^{2} + 294489 x - 7213 \)

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:  \(138338254038795273955595483867881=19^{10}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $138.91$
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:  $19, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(779=19\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{779}(672,·)$, $\chi_{779}(1,·)$, $\chi_{779}(387,·)$, $\chi_{779}(324,·)$, $\chi_{779}(406,·)$, $\chi_{779}(657,·)$, $\chi_{779}(201,·)$, $\chi_{779}(748,·)$, $\chi_{779}(590,·)$, $\chi_{779}(305,·)$, $\chi_{779}(83,·)$, $\chi_{779}(467,·)$, $\chi_{779}(182,·)$, $\chi_{779}(666,·)$, $\chi_{779}(543,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} - \frac{4}{9} a^{4} - \frac{1}{9} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a - \frac{4}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{4}{9} a^{7} + \frac{1}{3} a^{6} - \frac{2}{9} a^{5} + \frac{1}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{9} a + \frac{2}{9}$, $\frac{1}{2790282687838477760979683004447} a^{14} + \frac{98769484062846587524465137404}{2790282687838477760979683004447} a^{13} + \frac{39242888580197283987530423845}{2790282687838477760979683004447} a^{12} - \frac{115809214757696063177772851777}{2790282687838477760979683004447} a^{11} + \frac{28088277521940640311820928419}{310031409759830862331075889383} a^{10} + \frac{51605568281426373858784471300}{930094229279492586993227668149} a^{9} - \frac{902439218336554917148989062537}{2790282687838477760979683004447} a^{8} + \frac{972040279991282972531729531501}{2790282687838477760979683004447} a^{7} - \frac{879763822742125523973106572538}{2790282687838477760979683004447} a^{6} + \frac{992692901650986704559366813329}{2790282687838477760979683004447} a^{5} + \frac{1121674353323720151440149220390}{2790282687838477760979683004447} a^{4} - \frac{181555567923623005491876204212}{930094229279492586993227668149} a^{3} + \frac{426889129467425687722983638720}{2790282687838477760979683004447} a^{2} + \frac{185162839533555494045388266612}{930094229279492586993227668149} a + \frac{62828833514657022034341968361}{310031409759830862331075889383}$

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:  \( 131418258185.00424 \) (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

3.3.361.1, 5.5.2825761.1

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 $15$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ $15$ $15$ R $15$ $15$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ R $15$ $15$ $15$ $15$

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
$19$19.15.10.1$x^{15} + 102885 x^{6} - 130321 x^{3} + 309512375$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
$41$41.15.12.1$x^{15} + 2665 x^{10} + 1418764 x^{5} + 25589884853$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$