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Properties

Label 15.15.2229659760...0961.1
Degree $15$
Signature $[15, 0]$
Discriminant $241^{14}$
Root discriminant $167.19$
Ramified prime $241$
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![-1342853, -548532, 5422493, 3569665, -3747299, -2710134, 913205, 746261, -82128, -88262, 1950, 4670, 27, -112, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 112*x^13 + 27*x^12 + 4670*x^11 + 1950*x^10 - 88262*x^9 - 82128*x^8 + 746261*x^7 + 913205*x^6 - 2710134*x^5 - 3747299*x^4 + 3569665*x^3 + 5422493*x^2 - 548532*x - 1342853)
 
gp: K = bnfinit(x^15 - x^14 - 112*x^13 + 27*x^12 + 4670*x^11 + 1950*x^10 - 88262*x^9 - 82128*x^8 + 746261*x^7 + 913205*x^6 - 2710134*x^5 - 3747299*x^4 + 3569665*x^3 + 5422493*x^2 - 548532*x - 1342853, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 112 x^{13} + 27 x^{12} + 4670 x^{11} + 1950 x^{10} - 88262 x^{9} - 82128 x^{8} + 746261 x^{7} + 913205 x^{6} - 2710134 x^{5} - 3747299 x^{4} + 3569665 x^{3} + 5422493 x^{2} - 548532 x - 1342853 \)

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:  \(2229659760502257132817812847740961=241^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $167.19$
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:  $241$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(241\)
Dirichlet character group:    $\lbrace$$\chi_{241}(160,·)$, $\chi_{241}(1,·)$, $\chi_{241}(98,·)$, $\chi_{241}(100,·)$, $\chi_{241}(225,·)$, $\chi_{241}(87,·)$, $\chi_{241}(205,·)$, $\chi_{241}(15,·)$, $\chi_{241}(54,·)$, $\chi_{241}(119,·)$, $\chi_{241}(24,·)$, $\chi_{241}(91,·)$, $\chi_{241}(183,·)$, $\chi_{241}(94,·)$, $\chi_{241}(231,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{4} a^{3} + \frac{3}{8} a^{2} + \frac{3}{8}$, $\frac{1}{216181189387461307942219911120652024} a^{14} - \frac{2353153585751178322412084231860809}{108090594693730653971109955560326012} a^{13} - \frac{503990659368410459589701076760713}{108090594693730653971109955560326012} a^{12} - \frac{12755415075169557448643506716802003}{108090594693730653971109955560326012} a^{11} + \frac{21773145793907298028460643712425809}{216181189387461307942219911120652024} a^{10} - \frac{13058818266192345600825477428314517}{108090594693730653971109955560326012} a^{9} + \frac{4244018145302431442126523871913421}{216181189387461307942219911120652024} a^{8} + \frac{650173868917721059387272942342443}{216181189387461307942219911120652024} a^{7} - \frac{2385706157553338005250065509847467}{27022648673432663492777488890081503} a^{6} + \frac{3138093184714161394126102494107887}{27022648673432663492777488890081503} a^{5} - \frac{32925195801414266343699989459105617}{108090594693730653971109955560326012} a^{4} + \frac{64512513238849155001186482448980571}{216181189387461307942219911120652024} a^{3} - \frac{4007143027514641801387117164034989}{108090594693730653971109955560326012} a^{2} - \frac{83132543001696584924774724447586319}{216181189387461307942219911120652024} a - \frac{7742062596832428118930494231061626}{27022648673432663492777488890081503}$

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:  \( 709889982969 \) (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

3.3.58081.1, 5.5.3373402561.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ $15$ $15$ $15$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ $15$ $15$

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
241Data not computed