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Properties

Label 15.5.72347708329...6672.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{18}\cdot 13^{3}\cdot 19^{5}\cdot 2252389^{2}$
Root discriminant $71.99$
Ramified primes $2, 13, 19, 2252389$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T82

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1677206, 1900, -1963172, 442805, 615206, -252571, -56076, 42708, 5278, -4359, -766, 384, 66, -27, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 27*x^13 + 66*x^12 + 384*x^11 - 766*x^10 - 4359*x^9 + 5278*x^8 + 42708*x^7 - 56076*x^6 - 252571*x^5 + 615206*x^4 + 442805*x^3 - 1963172*x^2 + 1900*x + 1677206)
 
gp: K = bnfinit(x^15 - 2*x^14 - 27*x^13 + 66*x^12 + 384*x^11 - 766*x^10 - 4359*x^9 + 5278*x^8 + 42708*x^7 - 56076*x^6 - 252571*x^5 + 615206*x^4 + 442805*x^3 - 1963172*x^2 + 1900*x + 1677206, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 27 x^{13} + 66 x^{12} + 384 x^{11} - 766 x^{10} - 4359 x^{9} + 5278 x^{8} + 42708 x^{7} - 56076 x^{6} - 252571 x^{5} + 615206 x^{4} + 442805 x^{3} - 1963172 x^{2} + 1900 x + 1677206 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7234770832944223158655516672=-\,2^{18}\cdot 13^{3}\cdot 19^{5}\cdot 2252389^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.99$
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:  $2, 13, 19, 2252389$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{609205075604551748553922146837459149314653} a^{14} + \frac{202306832404738493285422279624828143170120}{609205075604551748553922146837459149314653} a^{13} + \frac{116347125053813334831849389801517745515029}{609205075604551748553922146837459149314653} a^{12} - \frac{245682591882704385030800167228014215513269}{609205075604551748553922146837459149314653} a^{11} + \frac{23232181334100703568024850704298518685938}{609205075604551748553922146837459149314653} a^{10} - \frac{81841058750675501212290845856055643935274}{609205075604551748553922146837459149314653} a^{9} - \frac{52548863603428165189998658126558270501028}{609205075604551748553922146837459149314653} a^{8} - \frac{228259532656684508820460391906253181456471}{609205075604551748553922146837459149314653} a^{7} - \frac{84450138771032479079882237038169249734432}{609205075604551748553922146837459149314653} a^{6} + \frac{299347990967652182551981424026385863066676}{609205075604551748553922146837459149314653} a^{5} + \frac{232860010834348709300908495895851890110722}{609205075604551748553922146837459149314653} a^{4} + \frac{98009240749716519256886346972692357406579}{203068358534850582851307382279153049771551} a^{3} + \frac{215710443051658893784575046908545557615991}{609205075604551748553922146837459149314653} a^{2} - \frac{21990638423045118860050816356038462631658}{203068358534850582851307382279153049771551} a + \frac{15094564781168826280515142349366428821613}{32063425031818513081785376149339955227087}$

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:  $9$
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:  \( 582657530.885 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T82:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48000
The 65 conjugacy class representatives for [F(5)^3]S(3)=F(5)wrS(3) are not computed
Character table for [F(5)^3]S(3)=F(5)wrS(3) is not computed

Intermediate fields

3.1.76.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$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}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
2252389Data not computed