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Properties

Label 15.1.42630068813...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 5^{7}\cdot 127^{7}$
Root discriminant $32.26$
Ramified primes $2, 5, 127$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{15}$ (as 15T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-500, 3200, -7720, 11320, -12842, 12046, -9379, 6467, -3764, 1982, -874, 352, -114, 32, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 + 32*x^13 - 114*x^12 + 352*x^11 - 874*x^10 + 1982*x^9 - 3764*x^8 + 6467*x^7 - 9379*x^6 + 12046*x^5 - 12842*x^4 + 11320*x^3 - 7720*x^2 + 3200*x - 500)
 
gp: K = bnfinit(x^15 - 7*x^14 + 32*x^13 - 114*x^12 + 352*x^11 - 874*x^10 + 1982*x^9 - 3764*x^8 + 6467*x^7 - 9379*x^6 + 12046*x^5 - 12842*x^4 + 11320*x^3 - 7720*x^2 + 3200*x - 500, 1)
 

Normalized defining polynomial

\( x^{15} - 7 x^{14} + 32 x^{13} - 114 x^{12} + 352 x^{11} - 874 x^{10} + 1982 x^{9} - 3764 x^{8} + 6467 x^{7} - 9379 x^{6} + 12046 x^{5} - 12842 x^{4} + 11320 x^{3} - 7720 x^{2} + 3200 x - 500 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-42630068813240240000000=-\,2^{10}\cdot 5^{7}\cdot 127^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.26$
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:  $2, 5, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{8} - \frac{1}{10} a^{6} - \frac{1}{2} a^{5} + \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{10} a^{9} - \frac{1}{10} a^{7} - \frac{1}{10} a^{5} + \frac{1}{10} a^{3}$, $\frac{1}{20} a^{10} - \frac{1}{20} a^{9} - \frac{1}{5} a^{7} + \frac{3}{20} a^{6} + \frac{1}{20} a^{5} - \frac{1}{2} a^{4} + \frac{1}{5} a^{3} + \frac{3}{10} a^{2}$, $\frac{1}{20} a^{11} - \frac{1}{20} a^{9} - \frac{1}{20} a^{7} - \frac{9}{20} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{40} a^{12} - \frac{1}{40} a^{10} - \frac{1}{20} a^{9} + \frac{1}{40} a^{8} - \frac{1}{5} a^{7} + \frac{9}{40} a^{6} + \frac{1}{20} a^{5} - \frac{1}{20} a^{4} + \frac{1}{5} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{363800} a^{13} - \frac{3113}{363800} a^{12} + \frac{201}{72760} a^{11} + \frac{5631}{363800} a^{10} - \frac{1759}{363800} a^{9} + \frac{2239}{72760} a^{8} + \frac{53867}{363800} a^{7} - \frac{72331}{363800} a^{6} + \frac{67079}{181900} a^{5} - \frac{49021}{181900} a^{4} - \frac{31213}{90950} a^{3} - \frac{8217}{18190} a^{2} - \frac{3878}{9095} a - \frac{455}{3638}$, $\frac{1}{9458800} a^{14} - \frac{1}{945880} a^{13} - \frac{9063}{1182350} a^{12} - \frac{93217}{4729400} a^{11} - \frac{2339}{2364700} a^{10} - \frac{104131}{4729400} a^{9} + \frac{51811}{4729400} a^{8} + \frac{147521}{945880} a^{7} + \frac{49323}{1891760} a^{6} - \frac{290161}{1182350} a^{5} + \frac{257611}{4729400} a^{4} + \frac{812561}{2364700} a^{3} + \frac{38745}{94588} a^{2} + \frac{214507}{472940} a + \frac{167}{884}$

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:  $7$
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:  \( 1804580.53513 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{15}$ (as 15T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.2540.1, 5.1.403225.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $15$ R $15$ $15$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
$127$$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$