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Properties

Label 15.15.1934399424...5625.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{20}\cdot 5^{8}\cdot 61^{3}\cdot 397^{3}$
Root discriminant $76.87$
Ramified primes $3, 5, 61, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T63

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63125, 30000, -579375, 677750, 172875, -532125, 48050, 166500, -24300, -26915, 2970, 2295, -103, -87, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 87*x^13 - 103*x^12 + 2295*x^11 + 2970*x^10 - 26915*x^9 - 24300*x^8 + 166500*x^7 + 48050*x^6 - 532125*x^5 + 172875*x^4 + 677750*x^3 - 579375*x^2 + 30000*x + 63125)
 
gp: K = bnfinit(x^15 - 87*x^13 - 103*x^12 + 2295*x^11 + 2970*x^10 - 26915*x^9 - 24300*x^8 + 166500*x^7 + 48050*x^6 - 532125*x^5 + 172875*x^4 + 677750*x^3 - 579375*x^2 + 30000*x + 63125, 1)
 

Normalized defining polynomial

\( x^{15} - 87 x^{13} - 103 x^{12} + 2295 x^{11} + 2970 x^{10} - 26915 x^{9} - 24300 x^{8} + 166500 x^{7} + 48050 x^{6} - 532125 x^{5} + 172875 x^{4} + 677750 x^{3} - 579375 x^{2} + 30000 x + 63125 \)

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:  \(19343994249123115056059765625=3^{20}\cdot 5^{8}\cdot 61^{3}\cdot 397^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.87$
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:  $3, 5, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{9} - \frac{2}{25} a^{7} + \frac{2}{25} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{50} a^{10} - \frac{1}{50} a^{9} + \frac{3}{50} a^{8} + \frac{2}{25} a^{7} + \frac{3}{50} a^{6} + \frac{3}{10} a^{5} + \frac{1}{5} a^{4} - \frac{3}{10} a^{3} - \frac{1}{2}$, $\frac{1}{50} a^{11} - \frac{3}{50} a^{8} + \frac{1}{50} a^{7} + \frac{2}{25} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{3}{10} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{250} a^{12} - \frac{1}{125} a^{10} - \frac{3}{250} a^{9} - \frac{1}{10} a^{8} - \frac{1}{25} a^{7} - \frac{3}{50} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} + \frac{2}{5} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{250} a^{13} - \frac{1}{125} a^{11} + \frac{1}{125} a^{10} + \frac{1}{50} a^{8} - \frac{1}{50} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{83921322258652699750} a^{14} - \frac{103389594820989}{41960661129326349875} a^{13} - \frac{17431293473738087}{41960661129326349875} a^{12} - \frac{327339516595389837}{83921322258652699750} a^{11} + \frac{156216330720732593}{83921322258652699750} a^{10} + \frac{688779877714825523}{41960661129326349875} a^{9} - \frac{384903895435598719}{16784264451730539950} a^{8} + \frac{313834052740465447}{3356852890346107990} a^{7} + \frac{168352505100405129}{3356852890346107990} a^{6} - \frac{43677512100541796}{1678426445173053995} a^{5} + \frac{1038716032244122103}{3356852890346107990} a^{4} + \frac{392896433954197551}{3356852890346107990} a^{3} + \frac{164354595373937541}{335685289034610799} a^{2} - \frac{21142821584200051}{335685289034610799} a + \frac{86718543411755659}{335685289034610799}$

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:  \( 1159976798.54 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T63:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 9720
The 36 conjugacy class representatives for [3^4]S(5)
Character table for [3^4]S(5) is not computed

Intermediate fields

5.5.24217.1

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
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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ R R ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.12.8.2$x^{12} + 25 x^{6} - 250 x^{3} + 1250$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$
$61$61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.6.0.1$x^{6} - 4 x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
61.6.3.1$x^{6} - 122 x^{4} + 3721 x^{2} - 22698100$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
397Data not computed