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Properties

Label 15.15.3091133177...6129.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{20}\cdot 7^{10}\cdot 11^{12}$
Root discriminant $107.81$
Ramified primes $3, 7, 11$
Class number $3$ (GRH)
Class group $[3]$ (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![-1701701, -14510214, -38133582, -43817519, -20966655, 785697, 4464049, 1090119, -259107, -120562, 3249, 5361, 130, -114, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 114*x^13 + 130*x^12 + 5361*x^11 + 3249*x^10 - 120562*x^9 - 259107*x^8 + 1090119*x^7 + 4464049*x^6 + 785697*x^5 - 20966655*x^4 - 43817519*x^3 - 38133582*x^2 - 14510214*x - 1701701)
 
gp: K = bnfinit(x^15 - 3*x^14 - 114*x^13 + 130*x^12 + 5361*x^11 + 3249*x^10 - 120562*x^9 - 259107*x^8 + 1090119*x^7 + 4464049*x^6 + 785697*x^5 - 20966655*x^4 - 43817519*x^3 - 38133582*x^2 - 14510214*x - 1701701, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 114 x^{13} + 130 x^{12} + 5361 x^{11} + 3249 x^{10} - 120562 x^{9} - 259107 x^{8} + 1090119 x^{7} + 4464049 x^{6} + 785697 x^{5} - 20966655 x^{4} - 43817519 x^{3} - 38133582 x^{2} - 14510214 x - 1701701 \)

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:  \(3091133177133909578645502426129=3^{20}\cdot 7^{10}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $107.81$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(693=3^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{693}(256,·)$, $\chi_{693}(1,·)$, $\chi_{693}(130,·)$, $\chi_{693}(67,·)$, $\chi_{693}(4,·)$, $\chi_{693}(520,·)$, $\chi_{693}(64,·)$, $\chi_{693}(394,·)$, $\chi_{693}(331,·)$, $\chi_{693}(268,·)$, $\chi_{693}(16,·)$, $\chi_{693}(631,·)$, $\chi_{693}(379,·)$, $\chi_{693}(445,·)$, $\chi_{693}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{23} a^{13} - \frac{4}{23} a^{12} - \frac{6}{23} a^{11} - \frac{4}{23} a^{10} + \frac{3}{23} a^{9} + \frac{1}{23} a^{8} - \frac{7}{23} a^{7} + \frac{7}{23} a^{6} - \frac{11}{23} a^{5} + \frac{5}{23} a^{4} - \frac{5}{23} a^{3} + \frac{3}{23} a^{2} - \frac{6}{23} a$, $\frac{1}{48832310178344731711209476886942293} a^{14} - \frac{857214419196307774359970567103176}{48832310178344731711209476886942293} a^{13} - \frac{7692004180898148352130424576927493}{48832310178344731711209476886942293} a^{12} - \frac{9286600789288271915491094856827578}{48832310178344731711209476886942293} a^{11} - \frac{5560641986666316085277152692497336}{48832310178344731711209476886942293} a^{10} - \frac{7737993590016632843735441639901439}{48832310178344731711209476886942293} a^{9} - \frac{380942793957344745814920672978286}{48832310178344731711209476886942293} a^{8} - \frac{4423804318228992982799822019172730}{48832310178344731711209476886942293} a^{7} + \frac{17331774212752794386427487262750922}{48832310178344731711209476886942293} a^{6} - \frac{13112703197493905309745334169841776}{48832310178344731711209476886942293} a^{5} - \frac{2212856433132188205830716536389181}{48832310178344731711209476886942293} a^{4} - \frac{5001783006738380849211138819257270}{48832310178344731711209476886942293} a^{3} - \frac{19718348180800337841133814842322716}{48832310178344731711209476886942293} a^{2} - \frac{17453745302589758735946848255575249}{48832310178344731711209476886942293} a - \frac{1948715854695564795707485622794}{6915778243640381208215476120513}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 12250984499.438423 \) (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

3.3.3969.2, \(\Q(\zeta_{11})^+\)

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 $15$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ R R $15$ $15$ $15$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{15}$ $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ $15$ $15$ $15$

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
$3$3.15.20.99$x^{15} + 42 x^{14} + 27 x^{13} + 31 x^{12} + 45 x^{11} + 45 x^{10} + 41 x^{9} + 30 x^{8} + 42 x^{7} + 8 x^{6} + 75 x^{5} + 18 x^{4} + 42 x^{3} + 24 x^{2} + 51 x + 19$$3$$5$$20$$C_{15}$$[2]^{5}$
$7$7.15.10.2$x^{15} - 2401 x^{3} + 67228$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
$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}$