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Properties

Label 15.1.32479853059...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{12}\cdot 5^{27}\cdot 29^{5}\cdot 139^{5}$
Root discriminant $502.08$
Ramified primes $2, 5, 29, 139$
Class number $14$ (GRH)
Class group $[14]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14746040, 20860000, -2431800, 7708000, -531700, 1485851, -515600, 277205, -79200, 17310, -4280, 1210, -100, 55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 55*x^13 - 100*x^12 + 1210*x^11 - 4280*x^10 + 17310*x^9 - 79200*x^8 + 277205*x^7 - 515600*x^6 + 1485851*x^5 - 531700*x^4 + 7708000*x^3 - 2431800*x^2 + 20860000*x + 14746040)
 
gp: K = bnfinit(x^15 + 55*x^13 - 100*x^12 + 1210*x^11 - 4280*x^10 + 17310*x^9 - 79200*x^8 + 277205*x^7 - 515600*x^6 + 1485851*x^5 - 531700*x^4 + 7708000*x^3 - 2431800*x^2 + 20860000*x + 14746040, 1)
 

Normalized defining polynomial

\( x^{15} + 55 x^{13} - 100 x^{12} + 1210 x^{11} - 4280 x^{10} + 17310 x^{9} - 79200 x^{8} + 277205 x^{7} - 515600 x^{6} + 1485851 x^{5} - 531700 x^{4} + 7708000 x^{3} - 2431800 x^{2} + 20860000 x + 14746040 \)

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:  \(-32479853059663365203857421875000000000000=-\,2^{12}\cdot 5^{27}\cdot 29^{5}\cdot 139^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $502.08$
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:  $2, 5, 29, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{9} + \frac{3}{16} a^{7} + \frac{3}{16} a^{5} - \frac{1}{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{16} a^{8} - \frac{1}{4} a^{7} + \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{13} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{7}{32} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4}$, $\frac{1}{3791388606536223489368015170024130501792} a^{14} + \frac{25032198227010464743225478371716281835}{3791388606536223489368015170024130501792} a^{13} - \frac{10431037006372595182257727824001590085}{947847151634055872342003792506032625448} a^{12} - \frac{1655173950261051118684174426596618188}{118480893954256984042750474063254078181} a^{11} - \frac{54197937142761987552079925023515136717}{1895694303268111744684007585012065250896} a^{10} + \frac{121920763506086543512426360909611028857}{1895694303268111744684007585012065250896} a^{9} - \frac{24936208574126306233344290467693045565}{236961787908513968085500948126508156362} a^{8} - \frac{6797886206792961482694627271909369087}{118480893954256984042750474063254078181} a^{7} - \frac{804827841490700941895529372376023747607}{3791388606536223489368015170024130501792} a^{6} - \frac{890395978024967150353469916107099393589}{3791388606536223489368015170024130501792} a^{5} - \frac{46093367140684981613519114852474693473}{947847151634055872342003792506032625448} a^{4} + \frac{27208552715745189051131120556996748277}{236961787908513968085500948126508156362} a^{3} + \frac{14938735594801037835183511355556059614}{118480893954256984042750474063254078181} a^{2} - \frac{130748146750695989815029413253994275123}{473923575817027936171001896253016312724} a - \frac{160164767194640405473206837583921322969}{473923575817027936171001896253016312724}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{14}$, which has order $14$ (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:  \( 92157527758838.23 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.4031.1, 5.1.31250000.3

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ R ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$5$5.5.9.4$x^{5} + 30$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.4$x^{5} + 30$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.4$x^{5} + 30$$5$$1$$9$$F_5$$[9/4]_{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.2.1.2$x^{2} + 556$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$