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Properties

Label 15.3.89891385986...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{12}\cdot 3^{13}\cdot 5^{17}\cdot 71^{5}$
Root discriminant $115.77$
Ramified primes $2, 3, 5, 71$
Class number $1$ (GRH)
Class group Trivial (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![-231831, -901395, -419175, -411255, 262305, 243723, -101775, 22635, -30705, 16875, -6493, 2195, -385, 85, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 85*x^13 - 385*x^12 + 2195*x^11 - 6493*x^10 + 16875*x^9 - 30705*x^8 + 22635*x^7 - 101775*x^6 + 243723*x^5 + 262305*x^4 - 411255*x^3 - 419175*x^2 - 901395*x - 231831)
 
gp: K = bnfinit(x^15 - 5*x^14 + 85*x^13 - 385*x^12 + 2195*x^11 - 6493*x^10 + 16875*x^9 - 30705*x^8 + 22635*x^7 - 101775*x^6 + 243723*x^5 + 262305*x^4 - 411255*x^3 - 419175*x^2 - 901395*x - 231831, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 85 x^{13} - 385 x^{12} + 2195 x^{11} - 6493 x^{10} + 16875 x^{9} - 30705 x^{8} + 22635 x^{7} - 101775 x^{6} + 243723 x^{5} + 262305 x^{4} - 411255 x^{3} - 419175 x^{2} - 901395 x - 231831 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8989138598669915625000000000000=2^{12}\cdot 3^{13}\cdot 5^{17}\cdot 71^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.77$
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:  $2, 3, 5, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{30} a^{8} + \frac{1}{15} a^{7} - \frac{1}{15} a^{6} - \frac{1}{30} a^{5} - \frac{1}{2} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{10} a + \frac{1}{5}$, $\frac{1}{30} a^{9} - \frac{1}{30} a^{7} - \frac{1}{15} a^{6} + \frac{1}{15} a^{5} + \frac{3}{10} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} - \frac{1}{10} a + \frac{1}{10}$, $\frac{1}{60} a^{10} - \frac{1}{60} a^{8} - \frac{1}{30} a^{7} + \frac{1}{30} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} + \frac{1}{10} a^{3} - \frac{1}{20} a^{2} + \frac{3}{10} a + \frac{1}{4}$, $\frac{1}{360} a^{11} + \frac{1}{360} a^{10} - \frac{1}{72} a^{9} + \frac{1}{120} a^{8} - \frac{1}{15} a^{7} - \frac{1}{20} a^{6} - \frac{1}{6} a^{5} - \frac{5}{12} a^{4} + \frac{3}{40} a^{3} + \frac{1}{40} a^{2} - \frac{7}{40} a - \frac{9}{40}$, $\frac{1}{720} a^{12} - \frac{1}{120} a^{10} + \frac{1}{90} a^{9} - \frac{1}{240} a^{8} - \frac{1}{120} a^{7} + \frac{1}{24} a^{6} - \frac{29}{120} a^{5} + \frac{59}{240} a^{4} + \frac{1}{40} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a + \frac{5}{16}$, $\frac{1}{4320} a^{13} - \frac{1}{1440} a^{12} - \frac{1}{720} a^{11} - \frac{11}{2160} a^{10} + \frac{23}{1440} a^{9} - \frac{7}{1440} a^{8} + \frac{11}{180} a^{7} + \frac{2}{45} a^{6} + \frac{97}{1440} a^{5} - \frac{1}{480} a^{4} - \frac{5}{16} a^{3} - \frac{9}{80} a^{2} - \frac{29}{480} a + \frac{149}{480}$, $\frac{1}{1054951515686228087297481929280} a^{14} + \frac{45099544741551680406394441}{527475757843114043648740964640} a^{13} - \frac{79245685634940818708509853}{117216835076247565255275769920} a^{12} + \frac{270360478981620132872814221}{263737878921557021824370482320} a^{11} + \frac{321230329651553295604901839}{1054951515686228087297481929280} a^{10} + \frac{1062948238950011951629610489}{87912626307185673941456827440} a^{9} + \frac{2034743495727965463234703469}{351650505228742695765827309760} a^{8} + \frac{173179227685832847043135213}{14652104384530945656909471240} a^{7} - \frac{101908668486141183170494235}{7814455671749837683685051328} a^{6} - \frac{42901441772199082460196606119}{175825252614371347882913654880} a^{5} + \frac{15927389615454896153471479463}{39072278358749188418425256640} a^{4} - \frac{4840038863639349932198624113}{9768069589687297104606314160} a^{3} - \frac{55352088512705603507033819651}{117216835076247565255275769920} a^{2} + \frac{11675787705918126756197147191}{29304208769061891313818942480} a - \frac{4607662372908837519263716435}{23443367015249513051055153984}$

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:  $8$
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:  \( 21642035469.69577 \) (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

3.3.5325.1, 5.1.4050000.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.15.17.3$x^{15} - 5 x^{13} + 10 x^{12} + 10 x^{11} + 10 x^{9} - 5 x^{8} - 10 x^{6} + 10 x^{5} + 5 x^{3} + 10$$15$$1$$17$$F_5 \times S_3$$[5/4]_{12}^{2}$
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$