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Properties

Label 15.15.3203887529...9601.1
Degree $15$
Signature $[15, 0]$
Discriminant $151^{14}$
Root discriminant $108.07$
Ramified prime $151$
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![361, 7714, -152667, 239121, 140547, -259504, -46623, 103129, 7308, -19122, -646, 1714, 37, -70, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 70*x^13 + 37*x^12 + 1714*x^11 - 646*x^10 - 19122*x^9 + 7308*x^8 + 103129*x^7 - 46623*x^6 - 259504*x^5 + 140547*x^4 + 239121*x^3 - 152667*x^2 + 7714*x + 361)
 
gp: K = bnfinit(x^15 - x^14 - 70*x^13 + 37*x^12 + 1714*x^11 - 646*x^10 - 19122*x^9 + 7308*x^8 + 103129*x^7 - 46623*x^6 - 259504*x^5 + 140547*x^4 + 239121*x^3 - 152667*x^2 + 7714*x + 361, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 70 x^{13} + 37 x^{12} + 1714 x^{11} - 646 x^{10} - 19122 x^{9} + 7308 x^{8} + 103129 x^{7} - 46623 x^{6} - 259504 x^{5} + 140547 x^{4} + 239121 x^{3} - 152667 x^{2} + 7714 x + 361 \)

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:  \(3203887529980253057019236799601=151^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $108.07$
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:  $151$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(151\)
Dirichlet character group:    $\lbrace$$\chi_{151}(32,·)$, $\chi_{151}(1,·)$, $\chi_{151}(2,·)$, $\chi_{151}(4,·)$, $\chi_{151}(38,·)$, $\chi_{151}(8,·)$, $\chi_{151}(64,·)$, $\chi_{151}(128,·)$, $\chi_{151}(76,·)$, $\chi_{151}(16,·)$, $\chi_{151}(19,·)$, $\chi_{151}(85,·)$, $\chi_{151}(118,·)$, $\chi_{151}(105,·)$, $\chi_{151}(59,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{38} a^{8} + \frac{5}{38} a^{7} + \frac{9}{38} a^{6} - \frac{9}{38} a^{5} + \frac{17}{38} a^{4} - \frac{13}{38} a^{3} - \frac{17}{38} a^{2} - \frac{6}{19} a$, $\frac{1}{76} a^{9} - \frac{1}{76} a^{8} - \frac{1}{38} a^{7} + \frac{13}{76} a^{6} - \frac{5}{76} a^{5} - \frac{1}{76} a^{4} + \frac{23}{76} a^{3} - \frac{6}{19} a^{2} - \frac{1}{19} a + \frac{1}{4}$, $\frac{1}{76} a^{10} - \frac{1}{76} a^{8} - \frac{17}{76} a^{7} - \frac{3}{19} a^{6} + \frac{7}{38} a^{5} + \frac{9}{38} a^{4} + \frac{11}{76} a^{3} - \frac{6}{19} a^{2} + \frac{29}{76} a + \frac{1}{4}$, $\frac{1}{76} a^{11} - \frac{1}{76} a^{6} - \frac{35}{76} a^{5} - \frac{13}{38} a^{4} + \frac{31}{76} a^{3} - \frac{35}{76} a^{2} - \frac{11}{76} a - \frac{1}{4}$, $\frac{1}{152} a^{12} - \frac{1}{152} a^{11} - \frac{1}{152} a^{10} - \frac{1}{152} a^{9} - \frac{1}{76} a^{8} - \frac{1}{76} a^{7} + \frac{5}{152} a^{6} + \frac{9}{38} a^{5} - \frac{7}{38} a^{4} - \frac{6}{19} a^{3} + \frac{8}{19} a^{2} + \frac{15}{152} a + \frac{3}{8}$, $\frac{1}{2888} a^{13} - \frac{5}{2888} a^{12} + \frac{7}{2888} a^{11} + \frac{9}{2888} a^{10} + \frac{3}{1444} a^{9} - \frac{3}{361} a^{8} - \frac{425}{2888} a^{7} - \frac{52}{361} a^{6} + \frac{96}{361} a^{5} + \frac{70}{361} a^{4} - \frac{9}{76} a^{3} + \frac{707}{2888} a^{2} - \frac{637}{2888} a - \frac{15}{76}$, $\frac{1}{69316218679988872} a^{14} + \frac{783687238210}{8664527334998609} a^{13} + \frac{33922654681195}{34658109339994436} a^{12} - \frac{179946747190725}{34658109339994436} a^{11} - \frac{251017291612215}{69316218679988872} a^{10} + \frac{105589893027543}{17329054669997218} a^{9} + \frac{284719059767535}{69316218679988872} a^{8} - \frac{3186922461638831}{69316218679988872} a^{7} + \frac{38896404752123}{293712791016902} a^{6} - \frac{1114562630524501}{8664527334998609} a^{5} + \frac{8124159101072861}{17329054669997218} a^{4} + \frac{16835600166797607}{69316218679988872} a^{3} + \frac{6559595555488705}{17329054669997218} a^{2} - \frac{20410435539837}{1174851164067608} a + \frac{181980558047323}{912055508947222}$

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:  \( 63003998024.8 \) (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

3.3.22801.1, 5.5.519885601.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ $15$ $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{15}$

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
151Data not computed