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Properties

Label 15.5.56665935234...6080.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{16}\cdot 3^{4}\cdot 5\cdot 53^{9}\cdot 647$
Root discriminant $52.11$
Ramified primes $2, 3, 5, 53, 647$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T87

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![288, 720, 720, -1288, -3206, -2463, -648, 161, 132, 238, 216, 54, -30, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 15*x^13 - 30*x^12 + 54*x^11 + 216*x^10 + 238*x^9 + 132*x^8 + 161*x^7 - 648*x^6 - 2463*x^5 - 3206*x^4 - 1288*x^3 + 720*x^2 + 720*x + 288)
 
gp: K = bnfinit(x^15 - 15*x^13 - 30*x^12 + 54*x^11 + 216*x^10 + 238*x^9 + 132*x^8 + 161*x^7 - 648*x^6 - 2463*x^5 - 3206*x^4 - 1288*x^3 + 720*x^2 + 720*x + 288, 1)
 

Normalized defining polynomial

\( x^{15} - 15 x^{13} - 30 x^{12} + 54 x^{11} + 216 x^{10} + 238 x^{9} + 132 x^{8} + 161 x^{7} - 648 x^{6} - 2463 x^{5} - 3206 x^{4} - 1288 x^{3} + 720 x^{2} + 720 x + 288 \)

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:  \(-56665935234850614192046080=-\,2^{16}\cdot 3^{4}\cdot 5\cdot 53^{9}\cdot 647\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.11$
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:  $2, 3, 5, 53, 647$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{2} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{16} a^{7} + \frac{1}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{16} a - \frac{1}{8}$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{6} + \frac{3}{16} a^{4} - \frac{1}{4} a^{3} + \frac{1}{16} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{16} a^{9} + \frac{1}{8} a^{4} + \frac{7}{16} a + \frac{3}{8}$, $\frac{1}{32} a^{10} - \frac{1}{16} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{7}{32} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{128} a^{11} - \frac{1}{128} a^{10} - \frac{1}{32} a^{9} + \frac{1}{64} a^{7} - \frac{1}{64} a^{6} - \frac{3}{32} a^{5} + \frac{5}{32} a^{4} - \frac{3}{128} a^{3} - \frac{61}{128} a^{2} + \frac{1}{4} a + \frac{7}{32}$, $\frac{1}{256} a^{12} + \frac{3}{256} a^{10} - \frac{1}{64} a^{9} + \frac{1}{128} a^{8} - \frac{7}{128} a^{6} - \frac{1}{32} a^{5} - \frac{47}{256} a^{4} + \frac{27}{256} a^{2} + \frac{19}{64} a - \frac{9}{64}$, $\frac{1}{4096} a^{13} - \frac{1}{1024} a^{12} - \frac{3}{4096} a^{11} + \frac{31}{2048} a^{10} - \frac{59}{2048} a^{9} + \frac{7}{512} a^{8} - \frac{13}{2048} a^{7} + \frac{35}{1024} a^{6} + \frac{345}{4096} a^{5} + \frac{249}{1024} a^{4} + \frac{109}{4096} a^{3} + \frac{443}{2048} a^{2} + \frac{43}{1024} a + \frac{57}{512}$, $\frac{1}{196608} a^{14} + \frac{1}{32768} a^{13} + \frac{7}{65536} a^{12} + \frac{1}{2048} a^{11} + \frac{105}{32768} a^{10} + \frac{333}{16384} a^{9} - \frac{181}{98304} a^{8} - \frac{85}{8192} a^{7} - \frac{12079}{196608} a^{6} + \frac{101}{32768} a^{5} + \frac{8583}{65536} a^{4} - \frac{5665}{24576} a^{3} - \frac{9575}{24576} a^{2} + \frac{149}{512} a + \frac{1023}{4096}$

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:  \( 82166960.7152 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T87:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 155520
The 63 conjugacy class representatives for [S(3)^5]F(5)=S(3)wrF(5) are not computed
Character table for [S(3)^5]F(5)=S(3)wrF(5) is not computed

Intermediate fields

5.5.2382032.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ R ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.12.15$x^{8} + 2 x^{7} + 2 x^{4} + 12$$4$$2$$12$$C_2^2:C_4$$[2, 2]^{4}$
$3$3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$53$$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
647Data not computed