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Properties

Label 15.15.1021061771...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 3^{20}\cdot 5^{24}\cdot 37^{4}$
Root discriminant $341.93$
Ramified primes $2, 3, 5, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T37

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-273800, 2053500, -5133750, 3556625, 3607500, -4397235, -810450, 1325250, 47100, -140875, -678, 6570, 0, -135, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 135*x^13 + 6570*x^11 - 678*x^10 - 140875*x^9 + 47100*x^8 + 1325250*x^7 - 810450*x^6 - 4397235*x^5 + 3607500*x^4 + 3556625*x^3 - 5133750*x^2 + 2053500*x - 273800)
 
gp: K = bnfinit(x^15 - 135*x^13 + 6570*x^11 - 678*x^10 - 140875*x^9 + 47100*x^8 + 1325250*x^7 - 810450*x^6 - 4397235*x^5 + 3607500*x^4 + 3556625*x^3 - 5133750*x^2 + 2053500*x - 273800, 1)
 

Normalized defining polynomial

\( x^{15} - 135 x^{13} + 6570 x^{11} - 678 x^{10} - 140875 x^{9} + 47100 x^{8} + 1325250 x^{7} - 810450 x^{6} - 4397235 x^{5} + 3607500 x^{4} + 3556625 x^{3} - 5133750 x^{2} + 2053500 x - 273800 \)

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:  \(102106177183790015625000000000000000000=2^{18}\cdot 3^{20}\cdot 5^{24}\cdot 37^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $341.93$
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:  $2, 3, 5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{40} a^{10} + \frac{1}{8} a^{6} + \frac{1}{20} a^{5} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{40} a^{11} - \frac{1}{8} a^{7} - \frac{1}{5} a^{6} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{40} a^{12} - \frac{1}{8} a^{8} + \frac{1}{20} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{14800} a^{13} - \frac{27}{2960} a^{11} - \frac{83}{1480} a^{9} + \frac{31}{7400} a^{8} - \frac{11}{592} a^{7} - \frac{5}{74} a^{6} + \frac{13}{296} a^{5} - \frac{77}{296} a^{4} - \frac{771}{2960} a^{3} - \frac{1}{2} a^{2} - \frac{7}{16} a + \frac{3}{8}$, $\frac{1}{24141675967813834145004800} a^{14} + \frac{175059849576635744697}{12070837983906917072502400} a^{13} - \frac{5845299761491336813851}{965667038712553365800192} a^{12} + \frac{2657806709405780399329}{2414167596781383414500480} a^{11} - \frac{29521369198331406504093}{2414167596781383414500480} a^{10} + \frac{454256358319630723807771}{12070837983906917072502400} a^{9} + \frac{1285768159864101316729953}{24141675967813834145004800} a^{8} - \frac{148987101482321908832133}{2414167596781383414500480} a^{7} - \frac{263469181124941511570509}{2414167596781383414500480} a^{6} - \frac{16462761475125510901109}{68976217050896668985728} a^{5} + \frac{2079467026544575102770669}{4828335193562766829000960} a^{4} - \frac{918876702043314308574717}{2414167596781383414500480} a^{3} + \frac{1441916342958260634535}{3728444164913333458688} a^{2} + \frac{1167037137724031606041}{6524777288598333552704} a - \frac{597723531298099684467}{6524777288598333552704}$

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:  \( 811055870167000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T37:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1500
The 19 conjugacy class representatives for 1/2[5^3:4]S(3)
Character table for 1/2[5^3:4]S(3)

Intermediate fields

3.3.1620.1

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

Sibling fields

Degree 15 siblings: data not computed
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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ R $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $15$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.16.18$x^{12} + x^{10} + 6 x^{8} - 3 x^{6} + 6 x^{4} + x^{2} - 3$$6$$2$$16$$C_3 : C_4$$[2]_{3}^{2}$
$3$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.12.16.30$x^{12} + 93 x^{11} + 351 x^{10} + 3 x^{9} + 126 x^{8} - 297 x^{7} + 171 x^{6} + 243 x^{5} - 324 x^{4} - 54 x^{3} + 162 x^{2} - 243 x + 324$$3$$4$$16$$C_3 : C_4$$[2]^{4}$
$5$5.5.5.2$x^{5} + 5 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.19.6$x^{10} - 10 x^{5} + 30$$10$$1$$19$$C_5^2 : C_4$$[7/4, 9/4]_{4}$
37Data not computed