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Properties

Label 15.7.21081131822...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{12}\cdot 5^{6}\cdot 31^{10}\cdot 97^{2}\cdot 163^{4}\cdot 245981^{2}$
Root discriminant $1225.35$
Ramified primes $2, 5, 31, 97, 163, 245981$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T98

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5341184000, -28041216000, -63493324800, -80902246400, -63453265920, -31462307840, -9717254656, -1735002176, -126963072, 12209760, 3324992, 193772, -7488, -936, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 936*x^13 - 7488*x^12 + 193772*x^11 + 3324992*x^10 + 12209760*x^9 - 126963072*x^8 - 1735002176*x^7 - 9717254656*x^6 - 31462307840*x^5 - 63453265920*x^4 - 80902246400*x^3 - 63493324800*x^2 - 28041216000*x - 5341184000)
 
gp: K = bnfinit(x^15 - 936*x^13 - 7488*x^12 + 193772*x^11 + 3324992*x^10 + 12209760*x^9 - 126963072*x^8 - 1735002176*x^7 - 9717254656*x^6 - 31462307840*x^5 - 63453265920*x^4 - 80902246400*x^3 - 63493324800*x^2 - 28041216000*x - 5341184000, 1)
 

Normalized defining polynomial

\( x^{15} - 936 x^{13} - 7488 x^{12} + 193772 x^{11} + 3324992 x^{10} + 12209760 x^{9} - 126963072 x^{8} - 1735002176 x^{7} - 9717254656 x^{6} - 31462307840 x^{5} - 63453265920 x^{4} - 80902246400 x^{3} - 63493324800 x^{2} - 28041216000 x - 5341184000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21081131822384482680088939852611594757696000000=2^{12}\cdot 5^{6}\cdot 31^{10}\cdot 97^{2}\cdot 163^{4}\cdot 245981^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1225.35$
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:  $2, 5, 31, 97, 163, 245981$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{40} a^{5} - \frac{1}{20} a^{3} + \frac{1}{10} a^{2}$, $\frac{1}{80} a^{6} - \frac{1}{40} a^{4} + \frac{1}{20} a^{3} - \frac{1}{2} a$, $\frac{1}{320} a^{7} - \frac{1}{160} a^{6} + \frac{1}{40} a^{4} + \frac{7}{80} a^{3} - \frac{9}{40} a^{2}$, $\frac{1}{3200} a^{8} - \frac{1}{800} a^{6} - \frac{1}{100} a^{5} - \frac{29}{800} a^{4} + \frac{7}{100} a^{3} + \frac{31}{200} a^{2} - \frac{1}{2} a + \frac{1}{5}$, $\frac{1}{6400} a^{9} - \frac{1}{1600} a^{7} - \frac{1}{200} a^{6} - \frac{9}{1600} a^{5} + \frac{7}{200} a^{4} + \frac{21}{400} a^{3} + \frac{1}{20} a^{2} - \frac{2}{5} a$, $\frac{1}{25600} a^{10} - \frac{1}{6400} a^{8} - \frac{1}{800} a^{7} - \frac{9}{6400} a^{6} + \frac{1}{400} a^{5} + \frac{21}{1600} a^{4} - \frac{1}{10} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{512000} a^{11} - \frac{9}{128000} a^{9} + \frac{1}{16000} a^{8} + \frac{103}{128000} a^{7} - \frac{47}{8000} a^{6} + \frac{57}{6400} a^{5} + \frac{59}{1000} a^{4} + \frac{117}{1000} a^{3} + \frac{3}{250} a^{2} + \frac{41}{100} a + \frac{12}{25}$, $\frac{1}{10240000} a^{12} - \frac{1}{1024000} a^{11} - \frac{29}{2560000} a^{10} + \frac{9}{1280000} a^{9} - \frac{217}{2560000} a^{8} + \frac{389}{1280000} a^{7} - \frac{139}{128000} a^{6} - \frac{2201}{320000} a^{5} + \frac{1849}{40000} a^{4} + \frac{411}{10000} a^{3} + \frac{63}{400} a^{2} - \frac{251}{1000} a + \frac{9}{50}$, $\frac{1}{5120000000} a^{13} - \frac{3}{64000000} a^{12} + \frac{103}{640000000} a^{11} + \frac{3011}{160000000} a^{10} - \frac{85717}{1280000000} a^{9} - \frac{5331}{40000000} a^{8} - \frac{6841}{6400000} a^{7} + \frac{23377}{5000000} a^{6} + \frac{653411}{80000000} a^{5} + \frac{51711}{5000000} a^{4} + \frac{101387}{1000000} a^{3} + \frac{8807}{125000} a^{2} + \frac{11963}{50000} a - \frac{2331}{12500}$, $\frac{1}{1310720000000} a^{14} - \frac{11}{327680000000} a^{13} - \frac{1027}{163840000000} a^{12} - \frac{23817}{40960000000} a^{11} - \frac{3066469}{327680000000} a^{10} - \frac{3831781}{81920000000} a^{9} + \frac{578721}{40960000000} a^{8} - \frac{7189667}{5120000000} a^{7} - \frac{81708817}{20480000000} a^{6} - \frac{23730767}{5120000000} a^{5} - \frac{50747209}{1280000000} a^{4} - \frac{2009273}{64000000} a^{3} - \frac{14392341}{64000000} a^{2} + \frac{606203}{1600000} a - \frac{207469}{800000}$

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:  $10$
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:  \( 141303339610000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T98:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 71 conjugacy class representatives for [1/2.S(5)^3]3 are not computed
Character table for [1/2.S(5)^3]3 is not computed

Intermediate fields

3.3.961.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ $15$ $15$ ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $15$ $15$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.4.5$x^{4} + 2 x + 2$$4$$1$$4$$S_4$$[4/3, 4/3]_{3}^{2}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31Data not computed
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
163Data not computed
245981Data not computed