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Properties

Label 15.15.6782608935...7529.1
Degree $15$
Signature $[15, 0]$
Discriminant $11^{12}\cdot 43^{10}$
Root discriminant $83.58$
Ramified primes $11, 43$
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![-184427, 508174, 1652146, 197478, -1732412, -861711, 409471, 301977, -30177, -41297, -114, 2678, 81, -83, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 83*x^13 + 81*x^12 + 2678*x^11 - 114*x^10 - 41297*x^9 - 30177*x^8 + 301977*x^7 + 409471*x^6 - 861711*x^5 - 1732412*x^4 + 197478*x^3 + 1652146*x^2 + 508174*x - 184427)
 
gp: K = bnfinit(x^15 - 2*x^14 - 83*x^13 + 81*x^12 + 2678*x^11 - 114*x^10 - 41297*x^9 - 30177*x^8 + 301977*x^7 + 409471*x^6 - 861711*x^5 - 1732412*x^4 + 197478*x^3 + 1652146*x^2 + 508174*x - 184427, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 83 x^{13} + 81 x^{12} + 2678 x^{11} - 114 x^{10} - 41297 x^{9} - 30177 x^{8} + 301977 x^{7} + 409471 x^{6} - 861711 x^{5} - 1732412 x^{4} + 197478 x^{3} + 1652146 x^{2} + 508174 x - 184427 \)

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:  \(67826089355015287563327567529=11^{12}\cdot 43^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.58$
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:  $11, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(473=11\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{473}(1,·)$, $\chi_{473}(130,·)$, $\chi_{473}(388,·)$, $\chi_{473}(135,·)$, $\chi_{473}(423,·)$, $\chi_{473}(302,·)$, $\chi_{473}(49,·)$, $\chi_{473}(466,·)$, $\chi_{473}(345,·)$, $\chi_{473}(122,·)$, $\chi_{473}(251,·)$, $\chi_{473}(92,·)$, $\chi_{473}(221,·)$, $\chi_{473}(350,·)$, $\chi_{473}(36,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{182039999163663817025241622054401014} a^{14} + \frac{16103067426895670584306395083981467}{182039999163663817025241622054401014} a^{13} + \frac{39867193590629629909561132447398231}{182039999163663817025241622054401014} a^{12} + \frac{36145035256042082985380648435209703}{182039999163663817025241622054401014} a^{11} - \frac{4575465149133827735530140193764019}{182039999163663817025241622054401014} a^{10} - \frac{77515625363228928923173526674569927}{182039999163663817025241622054401014} a^{9} + \frac{24394211956628699170148186989988096}{91019999581831908512620811027200507} a^{8} + \frac{79856362095725174570351292060745661}{182039999163663817025241622054401014} a^{7} - \frac{21253236947975429892757357417839631}{91019999581831908512620811027200507} a^{6} + \frac{24197943516542866557908156329064465}{91019999581831908512620811027200507} a^{5} + \frac{73847331728709608319148886745231873}{182039999163663817025241622054401014} a^{4} - \frac{20961756895948932983422252882809550}{91019999581831908512620811027200507} a^{3} - \frac{90580040984270457778029338341765335}{182039999163663817025241622054401014} a^{2} + \frac{31504997124801341491335582516535393}{182039999163663817025241622054401014} a - \frac{76069698703851065559864066348664585}{182039999163663817025241622054401014}$

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:  \( 3066900946.178709 \) (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

3.3.1849.1, \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ $15$ $15$ $15$ R $15$ $15$ $15$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ $15$ $15$ $15$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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$$[\ ]_{5}$
$43$43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$