↑

Properties

Label 15.15.6642481855...9209.1
Degree $15$
Signature $[15, 0]$
Discriminant $31^{6}\cdot 67^{2}\cdot 401^{7}$
Root discriminant $113.45$
Ramified primes $31, 67, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T79

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3597987, 1839338, -7830362, -2072401, 5042048, 759852, -1492431, -94753, 231590, -2454, -18828, 1347, 698, -82, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 - 82*x^13 + 698*x^12 + 1347*x^11 - 18828*x^10 - 2454*x^9 + 231590*x^8 - 94753*x^7 - 1492431*x^6 + 759852*x^5 + 5042048*x^4 - 2072401*x^3 - 7830362*x^2 + 1839338*x + 3597987)
 
gp: K = bnfinit(x^15 - 7*x^14 - 82*x^13 + 698*x^12 + 1347*x^11 - 18828*x^10 - 2454*x^9 + 231590*x^8 - 94753*x^7 - 1492431*x^6 + 759852*x^5 + 5042048*x^4 - 2072401*x^3 - 7830362*x^2 + 1839338*x + 3597987, 1)
 

Normalized defining polynomial

\( x^{15} - 7 x^{14} - 82 x^{13} + 698 x^{12} + 1347 x^{11} - 18828 x^{10} - 2454 x^{9} + 231590 x^{8} - 94753 x^{7} - 1492431 x^{6} + 759852 x^{5} + 5042048 x^{4} - 2072401 x^{3} - 7830362 x^{2} + 1839338 x + 3597987 \)

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:  \(6642481855143226481920101489209=31^{6}\cdot 67^{2}\cdot 401^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $113.45$
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:  $31, 67, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{9} a^{13} - \frac{4}{9} a^{12} - \frac{2}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{4}{9} a^{5} - \frac{1}{3} a^{4} + \frac{2}{9} a^{3} - \frac{1}{9} a^{2} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{32039868413720681323054897844538333} a^{14} - \frac{284673720564487867632229971675214}{10679956137906893774351632614846111} a^{13} - \frac{14913950741436443947542227201459170}{32039868413720681323054897844538333} a^{12} + \frac{13745025906439816640083545376438705}{32039868413720681323054897844538333} a^{11} - \frac{4171040101896635339607574076863844}{32039868413720681323054897844538333} a^{10} + \frac{6110794124184672599396450333087764}{32039868413720681323054897844538333} a^{9} + \frac{6763760726510948508181091238555004}{32039868413720681323054897844538333} a^{8} + \frac{488123051073080926393868875705864}{10679956137906893774351632614846111} a^{7} + \frac{963347823515196862747755083255705}{2464605262593898563311915218810641} a^{6} - \frac{6091887295777869694500072569226187}{32039868413720681323054897844538333} a^{5} + \frac{16014376384648746640198413514331966}{32039868413720681323054897844538333} a^{4} - \frac{9148209851796286506555800095728056}{32039868413720681323054897844538333} a^{3} - \frac{141415100873327934493778212572509}{464345919039430164102244896297657} a^{2} + \frac{10732281250437673193103997735080292}{32039868413720681323054897844538333} a + \frac{2466708773620948963085135602395017}{10679956137906893774351632614846111}$

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:  \( 105180404251 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T79:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 38880
The 45 conjugacy class representatives for 1/2[S(3)^5]D(5)
Character table for 1/2[S(3)^5]D(5) is not computed

Intermediate fields

5.5.160801.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 45 siblings: 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 $15$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
31Data not computed
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.6.0.1$x^{6} + x^{2} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
401Data not computed