↑

Properties

Label 15.15.5668527027...3125.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{8}\cdot 5^{5}\cdot 19^{2}\cdot 401^{6}\cdot 769^{2}\cdot 5581^{2}$
Root discriminant $383.32$
Ramified primes $3, 5, 19, 401, 769, 5581$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T55

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-315628125, -937128125, -1103753125, -605233125, -96115625, 56075375, 27572625, 1235925, -1615725, -269960, 35689, 10346, -223, -164, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 164*x^13 - 223*x^12 + 10346*x^11 + 35689*x^10 - 269960*x^9 - 1615725*x^8 + 1235925*x^7 + 27572625*x^6 + 56075375*x^5 - 96115625*x^4 - 605233125*x^3 - 1103753125*x^2 - 937128125*x - 315628125)
 
gp: K = bnfinit(x^15 - x^14 - 164*x^13 - 223*x^12 + 10346*x^11 + 35689*x^10 - 269960*x^9 - 1615725*x^8 + 1235925*x^7 + 27572625*x^6 + 56075375*x^5 - 96115625*x^4 - 605233125*x^3 - 1103753125*x^2 - 937128125*x - 315628125, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 164 x^{13} - 223 x^{12} + 10346 x^{11} + 35689 x^{10} - 269960 x^{9} - 1615725 x^{8} + 1235925 x^{7} + 27572625 x^{6} + 56075375 x^{5} - 96115625 x^{4} - 605233125 x^{3} - 1103753125 x^{2} - 937128125 x - 315628125 \)

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:  \(566852702762309292388072502432118253125=3^{8}\cdot 5^{5}\cdot 19^{2}\cdot 401^{6}\cdot 769^{2}\cdot 5581^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $383.32$
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:  $3, 5, 19, 401, 769, 5581$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{8} - \frac{1}{25} a^{7} - \frac{9}{25} a^{6} - \frac{3}{25} a^{5} + \frac{1}{25} a^{4} - \frac{1}{25} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{125} a^{9} - \frac{1}{125} a^{8} - \frac{4}{125} a^{7} - \frac{8}{125} a^{6} + \frac{6}{125} a^{5} - \frac{16}{125} a^{4} + \frac{2}{5} a^{3} - \frac{12}{25} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{125} a^{10} + \frac{8}{125} a^{7} + \frac{53}{125} a^{6} - \frac{36}{125} a^{4} + \frac{2}{25} a^{3} - \frac{2}{25} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{625} a^{11} - \frac{1}{625} a^{10} + \frac{1}{625} a^{9} + \frac{12}{625} a^{8} - \frac{39}{625} a^{7} + \frac{94}{625} a^{6} + \frac{26}{125} a^{5} - \frac{23}{125} a^{4} + \frac{8}{25} a^{3} + \frac{12}{25} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{9375} a^{12} - \frac{1}{9375} a^{11} + \frac{1}{9375} a^{10} + \frac{37}{9375} a^{9} + \frac{62}{3125} a^{8} - \frac{252}{3125} a^{7} + \frac{87}{625} a^{6} - \frac{206}{625} a^{5} - \frac{21}{125} a^{4} - \frac{6}{125} a^{3} - \frac{4}{15} a^{2} - \frac{16}{75} a + \frac{1}{5}$, $\frac{1}{46875} a^{13} - \frac{1}{46875} a^{12} + \frac{31}{46875} a^{11} + \frac{82}{46875} a^{10} + \frac{47}{15625} a^{9} + \frac{143}{15625} a^{8} - \frac{231}{3125} a^{7} - \frac{538}{3125} a^{6} - \frac{36}{125} a^{5} - \frac{37}{625} a^{4} - \frac{32}{375} a^{3} - \frac{139}{375} a^{2} - \frac{7}{25} a - \frac{12}{25}$, $\frac{1}{46875} a^{14} - \frac{7}{46875} a^{11} - \frac{32}{46875} a^{10} + \frac{4}{3125} a^{9} + \frac{28}{15625} a^{8} - \frac{17}{3125} a^{7} - \frac{963}{3125} a^{6} - \frac{256}{625} a^{5} - \frac{931}{1875} a^{4} - \frac{41}{125} a^{3} - \frac{34}{375} a^{2} + \frac{8}{25} a - \frac{12}{25}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 184788226014000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T55:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4860
The 48 conjugacy class representatives for [3^5:2]D(5)
Character table for [3^5:2]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 15 siblings: data not computed
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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.6.8.4$x^{6} + 18 x^{2} + 63$$3$$2$$8$$C_6$$[2]^{2}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$5$5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
401Data not computed
769Data not computed
5581Data not computed