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Properties

Label 15.15.1509139764...3129.1
Degree $15$
Signature $[15, 0]$
Discriminant $11^{12}\cdot 37^{10}$
Root discriminant $75.61$
Ramified primes $11, 37$
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![42527, 207733, 104955, -709558, -1031110, -211189, 350706, 161814, -36813, -28112, 858, 2098, 52, -73, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 73*x^13 + 52*x^12 + 2098*x^11 + 858*x^10 - 28112*x^9 - 36813*x^8 + 161814*x^7 + 350706*x^6 - 211189*x^5 - 1031110*x^4 - 709558*x^3 + 104955*x^2 + 207733*x + 42527)
 
gp: K = bnfinit(x^15 - 2*x^14 - 73*x^13 + 52*x^12 + 2098*x^11 + 858*x^10 - 28112*x^9 - 36813*x^8 + 161814*x^7 + 350706*x^6 - 211189*x^5 - 1031110*x^4 - 709558*x^3 + 104955*x^2 + 207733*x + 42527, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 73 x^{13} + 52 x^{12} + 2098 x^{11} + 858 x^{10} - 28112 x^{9} - 36813 x^{8} + 161814 x^{7} + 350706 x^{6} - 211189 x^{5} - 1031110 x^{4} - 709558 x^{3} + 104955 x^{2} + 207733 x + 42527 \)

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:  \(15091397646253318362996493129=11^{12}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.61$
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:  $11, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(407=11\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{407}(1,·)$, $\chi_{407}(322,·)$, $\chi_{407}(100,·)$, $\chi_{407}(38,·)$, $\chi_{407}(232,·)$, $\chi_{407}(137,·)$, $\chi_{407}(75,·)$, $\chi_{407}(269,·)$, $\chi_{407}(334,·)$, $\chi_{407}(47,·)$, $\chi_{407}(306,·)$, $\chi_{407}(174,·)$, $\chi_{407}(26,·)$, $\chi_{407}(158,·)$, $\chi_{407}(223,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{23} a^{12} - \frac{8}{23} a^{11} + \frac{6}{23} a^{10} + \frac{3}{23} a^{9} - \frac{5}{23} a^{8} - \frac{1}{23} a^{7} - \frac{2}{23} a^{6} - \frac{11}{23} a^{5} + \frac{8}{23} a^{4} - \frac{6}{23} a^{3} - \frac{4}{23} a^{2} - \frac{4}{23} a$, $\frac{1}{23} a^{13} + \frac{11}{23} a^{11} + \frac{5}{23} a^{10} - \frac{4}{23} a^{9} + \frac{5}{23} a^{8} - \frac{10}{23} a^{7} - \frac{4}{23} a^{6} - \frac{11}{23} a^{5} - \frac{11}{23} a^{4} - \frac{6}{23} a^{3} + \frac{10}{23} a^{2} - \frac{9}{23} a$, $\frac{1}{5282472214221688255067449909} a^{14} + \frac{1009916484153763389139779}{229672704966160358915976083} a^{13} - \frac{79686408454973465217919403}{5282472214221688255067449909} a^{12} - \frac{675397789397226983450044111}{5282472214221688255067449909} a^{11} + \frac{2179078430399751077795290285}{5282472214221688255067449909} a^{10} - \frac{1895111839821147915384968376}{5282472214221688255067449909} a^{9} - \frac{505918052266083900910179011}{5282472214221688255067449909} a^{8} + \frac{1721571278031961328278377523}{5282472214221688255067449909} a^{7} - \frac{1084832005421155370582828563}{5282472214221688255067449909} a^{6} - \frac{61317934079429008099966026}{229672704966160358915976083} a^{5} + \frac{2110200883577049695138638655}{5282472214221688255067449909} a^{4} + \frac{2477929614416485292816374619}{5282472214221688255067449909} a^{3} + \frac{1988854039032715252104236826}{5282472214221688255067449909} a^{2} + \frac{285957385525283301484169649}{5282472214221688255067449909} a + \frac{1688816471027981976795133}{5341225696887450207348281}$

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:  \( 2900978466.527207 \) (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

3.3.1369.1, \(\Q(\zeta_{11})^+\)

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 $15$ $15$ $15$ $15$ R $15$ $15$ $15$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ R $15$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ $15$ $15$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$37$37.15.10.1$x^{15} + 1975467 x^{6} - 1874161 x^{3} + 152348673529$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$