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Properties

Label 15.15.1085860034...8089.1
Degree $15$
Signature $[15, 0]$
Discriminant $13^{10}\cdot 31^{12}$
Root discriminant $86.24$
Ramified primes $13, 31$
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![625, -7475, 17185, 40542, -56572, -90077, 23558, 48322, -2451, -10866, -216, 1160, 48, -57, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 57*x^13 + 48*x^12 + 1160*x^11 - 216*x^10 - 10866*x^9 - 2451*x^8 + 48322*x^7 + 23558*x^6 - 90077*x^5 - 56572*x^4 + 40542*x^3 + 17185*x^2 - 7475*x + 625)
 
gp: K = bnfinit(x^15 - 2*x^14 - 57*x^13 + 48*x^12 + 1160*x^11 - 216*x^10 - 10866*x^9 - 2451*x^8 + 48322*x^7 + 23558*x^6 - 90077*x^5 - 56572*x^4 + 40542*x^3 + 17185*x^2 - 7475*x + 625, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 57 x^{13} + 48 x^{12} + 1160 x^{11} - 216 x^{10} - 10866 x^{9} - 2451 x^{8} + 48322 x^{7} + 23558 x^{6} - 90077 x^{5} - 56572 x^{4} + 40542 x^{3} + 17185 x^{2} - 7475 x + 625 \)

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:  \(108586003458674436566349398089=13^{10}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.24$
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:  $13, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(403=13\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{403}(256,·)$, $\chi_{403}(1,·)$, $\chi_{403}(66,·)$, $\chi_{403}(35,·)$, $\chi_{403}(326,·)$, $\chi_{403}(295,·)$, $\chi_{403}(94,·)$, $\chi_{403}(16,·)$, $\chi_{403}(373,·)$, $\chi_{403}(250,·)$, $\chi_{403}(287,·)$, $\chi_{403}(380,·)$, $\chi_{403}(157,·)$, $\chi_{403}(126,·)$, $\chi_{403}(159,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{9} - \frac{1}{5} a$, $\frac{1}{25} a^{10} - \frac{2}{25} a^{6} + \frac{1}{25} a^{2}$, $\frac{1}{25} a^{11} - \frac{2}{25} a^{7} + \frac{1}{25} a^{3}$, $\frac{1}{125} a^{12} + \frac{2}{125} a^{11} - \frac{1}{25} a^{9} - \frac{12}{125} a^{8} - \frac{9}{125} a^{7} - \frac{2}{25} a^{6} + \frac{1}{25} a^{5} + \frac{36}{125} a^{4} - \frac{43}{125} a^{3} + \frac{12}{25} a^{2}$, $\frac{1}{125} a^{13} + \frac{1}{125} a^{11} - \frac{2}{125} a^{9} - \frac{2}{25} a^{8} - \frac{2}{125} a^{7} - \frac{2}{25} a^{6} + \frac{1}{125} a^{5} + \frac{7}{25} a^{4} + \frac{26}{125} a^{3} + \frac{7}{25} a^{2} + \frac{1}{5} a$, $\frac{1}{3508417023896875} a^{14} - \frac{7232014025819}{3508417023896875} a^{13} + \frac{3229157600891}{3508417023896875} a^{12} - \frac{26906787802524}{3508417023896875} a^{11} - \frac{43630961601032}{3508417023896875} a^{10} + \frac{260711252254928}{3508417023896875} a^{9} - \frac{283986370456317}{3508417023896875} a^{8} + \frac{88797428925163}{3508417023896875} a^{7} - \frac{252896459092499}{3508417023896875} a^{6} - \frac{201259114472634}{3508417023896875} a^{5} - \frac{1379586146417524}{3508417023896875} a^{4} + \frac{581032295576661}{3508417023896875} a^{3} - \frac{119033173521659}{701683404779375} a^{2} - \frac{47541941031047}{140336680955875} a - \frac{949586663417}{5613467238235}$

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:  \( 67543570601.28224 \) (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

3.3.169.1, 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 $15$ $15$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{15}$ $15$ $15$ R $15$ $15$ $15$ $15$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ $15$ $15$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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
$13$13.15.10.1$x^{15} + 79092 x^{6} - 228488 x^{3} + 80199288$$3$$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}$