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Properties

Label 15.15.4050419900...8921.1
Degree $15$
Signature $[15, 0]$
Discriminant $181^{14}$
Root discriminant $127.99$
Ramified prime $181$
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![-11813, 106911, 218070, -1016574, 851676, 188069, -472660, 79471, 85976, -25412, -6230, 2394, 157, -84, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 84*x^13 + 157*x^12 + 2394*x^11 - 6230*x^10 - 25412*x^9 + 85976*x^8 + 79471*x^7 - 472660*x^6 + 188069*x^5 + 851676*x^4 - 1016574*x^3 + 218070*x^2 + 106911*x - 11813)
 
gp: K = bnfinit(x^15 - x^14 - 84*x^13 + 157*x^12 + 2394*x^11 - 6230*x^10 - 25412*x^9 + 85976*x^8 + 79471*x^7 - 472660*x^6 + 188069*x^5 + 851676*x^4 - 1016574*x^3 + 218070*x^2 + 106911*x - 11813, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 84 x^{13} + 157 x^{12} + 2394 x^{11} - 6230 x^{10} - 25412 x^{9} + 85976 x^{8} + 79471 x^{7} - 472660 x^{6} + 188069 x^{5} + 851676 x^{4} - 1016574 x^{3} + 218070 x^{2} + 106911 x - 11813 \)

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:  \(40504199006061377874300161158921=181^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $127.99$
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:  $181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(181\)
Dirichlet character group:    $\lbrace$$\chi_{181}(1,·)$, $\chi_{181}(27,·)$, $\chi_{181}(132,·)$, $\chi_{181}(5,·)$, $\chi_{181}(135,·)$, $\chi_{181}(42,·)$, $\chi_{181}(82,·)$, $\chi_{181}(29,·)$, $\chi_{181}(48,·)$, $\chi_{181}(145,·)$, $\chi_{181}(114,·)$, $\chi_{181}(117,·)$, $\chi_{181}(25,·)$, $\chi_{181}(59,·)$, $\chi_{181}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{1}{7}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{49} a^{10} - \frac{2}{49} a^{9} + \frac{1}{49} a^{7} - \frac{2}{49} a^{6} + \frac{2}{7} a^{5} - \frac{15}{49} a^{4} + \frac{23}{49} a^{3} + \frac{2}{7} a^{2} - \frac{15}{49} a + \frac{23}{49}$, $\frac{1}{931} a^{11} + \frac{3}{931} a^{10} - \frac{17}{931} a^{9} + \frac{3}{49} a^{8} - \frac{60}{931} a^{7} + \frac{4}{931} a^{6} - \frac{10}{49} a^{5} + \frac{242}{931} a^{4} - \frac{158}{931} a^{3} - \frac{246}{931} a^{2} + \frac{305}{931} a - \frac{179}{931}$, $\frac{1}{931} a^{12} - \frac{1}{133} a^{10} - \frac{9}{133} a^{9} + \frac{5}{133} a^{8} - \frac{9}{133} a^{7} + \frac{26}{931} a^{6} + \frac{3}{19} a^{5} - \frac{34}{133} a^{4} - \frac{1}{7} a^{3} + \frac{16}{133} a^{2} - \frac{26}{133} a - \frac{223}{931}$, $\frac{1}{6517} a^{13} - \frac{2}{6517} a^{12} - \frac{4}{931} a^{10} + \frac{6}{931} a^{9} - \frac{135}{6517} a^{7} - \frac{10}{6517} a^{6} + \frac{1}{7} a^{5} + \frac{424}{931} a^{4} + \frac{428}{931} a^{3} + \frac{1}{7} a^{2} - \frac{650}{6517} a + \frac{3050}{6517}$, $\frac{1}{207099198265145851} a^{14} - \frac{13676314615444}{207099198265145851} a^{13} + \frac{23388108648673}{207099198265145851} a^{12} - \frac{10640390733417}{29585599752163693} a^{11} + \frac{57666918054593}{29585599752163693} a^{10} - \frac{1807509033648946}{29585599752163693} a^{9} - \frac{14540033245453641}{207099198265145851} a^{8} - \frac{13287640542566072}{207099198265145851} a^{7} - \frac{6232130400083418}{207099198265145851} a^{6} + \frac{6439692090136043}{29585599752163693} a^{5} - \frac{6254795910644604}{29585599752163693} a^{4} - \frac{308177204572411}{1557136829061247} a^{3} - \frac{71829353299804792}{207099198265145851} a^{2} + \frac{70558898913181996}{207099198265145851} a - \frac{21479519776194495}{207099198265145851}$

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:  \( 885563835346 \) (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

3.3.32761.1, 5.5.1073283121.1

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 $15$ $15$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{15}$ $15$ $15$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{15}$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ $15$ $15$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
181Data not computed