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Properties

Label 15.15.3467004251...0041.1
Degree $15$
Signature $[15, 0]$
Discriminant $211^{14}$
Root discriminant $147.68$
Ramified prime $211$
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![-67519, -1305471, 215635, 2557353, -146020, -1621482, 77800, 436039, -12878, -56185, 320, 3496, 33, -98, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 98*x^13 + 33*x^12 + 3496*x^11 + 320*x^10 - 56185*x^9 - 12878*x^8 + 436039*x^7 + 77800*x^6 - 1621482*x^5 - 146020*x^4 + 2557353*x^3 + 215635*x^2 - 1305471*x - 67519)
 
gp: K = bnfinit(x^15 - x^14 - 98*x^13 + 33*x^12 + 3496*x^11 + 320*x^10 - 56185*x^9 - 12878*x^8 + 436039*x^7 + 77800*x^6 - 1621482*x^5 - 146020*x^4 + 2557353*x^3 + 215635*x^2 - 1305471*x - 67519, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 98 x^{13} + 33 x^{12} + 3496 x^{11} + 320 x^{10} - 56185 x^{9} - 12878 x^{8} + 436039 x^{7} + 77800 x^{6} - 1621482 x^{5} - 146020 x^{4} + 2557353 x^{3} + 215635 x^{2} - 1305471 x - 67519 \)

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:  \(346700425107361035262655933030041=211^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $147.68$
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:  $211$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(211\)
Dirichlet character group:    $\lbrace$$\chi_{211}(19,·)$, $\chi_{211}(1,·)$, $\chi_{211}(100,·)$, $\chi_{211}(134,·)$, $\chi_{211}(71,·)$, $\chi_{211}(137,·)$, $\chi_{211}(107,·)$, $\chi_{211}(14,·)$, $\chi_{211}(83,·)$, $\chi_{211}(21,·)$, $\chi_{211}(150,·)$, $\chi_{211}(55,·)$, $\chi_{211}(196,·)$, $\chi_{211}(201,·)$, $\chi_{211}(188,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{67} a^{11} + \frac{4}{67} a^{10} + \frac{3}{67} a^{9} + \frac{6}{67} a^{8} - \frac{16}{67} a^{7} - \frac{8}{67} a^{6} - \frac{15}{67} a^{5} - \frac{30}{67} a^{4} - \frac{24}{67} a^{3} + \frac{14}{67} a^{2} + \frac{21}{67} a + \frac{30}{67}$, $\frac{1}{67} a^{12} - \frac{13}{67} a^{10} - \frac{6}{67} a^{9} + \frac{27}{67} a^{8} - \frac{11}{67} a^{7} + \frac{17}{67} a^{6} + \frac{30}{67} a^{5} + \frac{29}{67} a^{4} - \frac{24}{67} a^{3} + \frac{32}{67} a^{2} + \frac{13}{67} a + \frac{14}{67}$, $\frac{1}{67} a^{13} - \frac{21}{67} a^{10} - \frac{1}{67} a^{9} + \frac{10}{67} a^{7} - \frac{7}{67} a^{6} - \frac{32}{67} a^{5} - \frac{12}{67} a^{4} - \frac{12}{67} a^{3} - \frac{6}{67} a^{2} + \frac{19}{67} a - \frac{12}{67}$, $\frac{1}{57122859335712730799941208052701} a^{14} - \frac{276753200053625777877760660253}{57122859335712730799941208052701} a^{13} + \frac{368807489747736043635261875527}{57122859335712730799941208052701} a^{12} - \frac{248966158562678010995739761475}{57122859335712730799941208052701} a^{11} + \frac{15644424350074702227895378914189}{57122859335712730799941208052701} a^{10} + \frac{4855978392766559859619117645378}{57122859335712730799941208052701} a^{9} - \frac{18487364851098705904891791660091}{57122859335712730799941208052701} a^{8} + \frac{21032480882500717627807258561870}{57122859335712730799941208052701} a^{7} + \frac{5344657885838136674546545839940}{57122859335712730799941208052701} a^{6} + \frac{8050104484959398945668733360453}{57122859335712730799941208052701} a^{5} + \frac{5344590201048675439022564093488}{57122859335712730799941208052701} a^{4} + \frac{22284174352994938440971679943780}{57122859335712730799941208052701} a^{3} + \frac{841937309566237430408270401413}{57122859335712730799941208052701} a^{2} - \frac{1065439110849203378869372567848}{57122859335712730799941208052701} a + \frac{92814805820781741991006373034}{212352636935735058735840922129}$

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:  \( 175451316933 \) (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

3.3.44521.1, 5.5.1982119441.1

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 $15$ $15$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ $15$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ $15$ $15$ $15$

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
211Data not computed