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Properties

Label 15.15.4829212716...5561.1
Degree $15$
Signature $[15, 0]$
Discriminant $19^{10}\cdot 31^{12}$
Root discriminant $111.07$
Ramified primes $19, 31$
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![1303, -2587, -32647, 39828, 76130, -78181, -55028, 52210, 15183, -13856, -1830, 1524, 102, -67, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 67*x^13 + 102*x^12 + 1524*x^11 - 1830*x^10 - 13856*x^9 + 15183*x^8 + 52210*x^7 - 55028*x^6 - 78181*x^5 + 76130*x^4 + 39828*x^3 - 32647*x^2 - 2587*x + 1303)
 
gp: K = bnfinit(x^15 - 2*x^14 - 67*x^13 + 102*x^12 + 1524*x^11 - 1830*x^10 - 13856*x^9 + 15183*x^8 + 52210*x^7 - 55028*x^6 - 78181*x^5 + 76130*x^4 + 39828*x^3 - 32647*x^2 - 2587*x + 1303, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 67 x^{13} + 102 x^{12} + 1524 x^{11} - 1830 x^{10} - 13856 x^{9} + 15183 x^{8} + 52210 x^{7} - 55028 x^{6} - 78181 x^{5} + 76130 x^{4} + 39828 x^{3} - 32647 x^{2} - 2587 x + 1303 \)

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:  \(4829212716211581952447142935561=19^{10}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $111.07$
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:  $19, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(589=19\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{589}(64,·)$, $\chi_{589}(1,·)$, $\chi_{589}(163,·)$, $\chi_{589}(39,·)$, $\chi_{589}(311,·)$, $\chi_{589}(140,·)$, $\chi_{589}(419,·)$, $\chi_{589}(349,·)$, $\chi_{589}(562,·)$, $\chi_{589}(467,·)$, $\chi_{589}(438,·)$, $\chi_{589}(343,·)$, $\chi_{589}(543,·)$, $\chi_{589}(125,·)$, $\chi_{589}(159,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{185} a^{13} + \frac{1}{185} a^{12} - \frac{82}{185} a^{11} - \frac{51}{185} a^{10} + \frac{72}{185} a^{9} + \frac{7}{185} a^{8} - \frac{7}{37} a^{7} + \frac{78}{185} a^{6} + \frac{18}{37} a^{5} - \frac{33}{185} a^{4} - \frac{91}{185} a^{3} - \frac{28}{185} a^{2} + \frac{53}{185} a - \frac{16}{185}$, $\frac{1}{842869673516753716426995685} a^{14} - \frac{586473537278266858114334}{842869673516753716426995685} a^{13} + \frac{56596159251462412498456944}{842869673516753716426995685} a^{12} - \frac{376870611639735014560206321}{842869673516753716426995685} a^{11} - \frac{20296296512652255290236912}{168573934703350743285399137} a^{10} + \frac{347720753821473954561254908}{842869673516753716426995685} a^{9} + \frac{235984271860119804723908236}{842869673516753716426995685} a^{8} - \frac{152451178240338285527597866}{842869673516753716426995685} a^{7} - \frac{254190129445516997960532906}{842869673516753716426995685} a^{6} - \frac{207746491167076891189471739}{842869673516753716426995685} a^{5} - \frac{75677840227208411222951001}{168573934703350743285399137} a^{4} + \frac{10659347340465062739979364}{22780261446398749092621505} a^{3} + \frac{376911797903879534916817156}{842869673516753716426995685} a^{2} + \frac{129526223376085666759814158}{842869673516753716426995685} a + \frac{64071886406626801006187194}{842869673516753716426995685}$

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:  \( 25852132818.113102 \) (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

3.3.361.1, 5.5.923521.1

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 $15$ $15$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ $15$ $15$ R $15$ $15$ R ${\href{/LocalNumberField/37.1.0.1}{1} }^{15}$ $15$ $15$ $15$ $15$ $15$

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
$19$19.15.10.1$x^{15} + 102885 x^{6} - 130321 x^{3} + 309512375$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
$31$31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$