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Properties

Label 15.11.2606494027...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{18}\cdot 5^{6}\cdot 13^{10}\cdot 79^{2}\cdot 103^{4}\cdot 6547^{4}\cdot 1891234619^{2}$
Root discriminant $26{,}775.47$
Ramified primes $2, 5, 13, 79, 103, 6547, 1891234619$
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![-1414195576832000, -3712263389184000, -4202812479897600, -2677580453478400, -1050040215797760, -260417811445760, -40366806612992, -3704165887616, -171313992576, -819354600, 261499904, 7855432, -84384, -5274, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5274*x^13 - 84384*x^12 + 7855432*x^11 + 261499904*x^10 - 819354600*x^9 - 171313992576*x^8 - 3704165887616*x^7 - 40366806612992*x^6 - 260417811445760*x^5 - 1050040215797760*x^4 - 2677580453478400*x^3 - 4202812479897600*x^2 - 3712263389184000*x - 1414195576832000)
 
gp: K = bnfinit(x^15 - 5274*x^13 - 84384*x^12 + 7855432*x^11 + 261499904*x^10 - 819354600*x^9 - 171313992576*x^8 - 3704165887616*x^7 - 40366806612992*x^6 - 260417811445760*x^5 - 1050040215797760*x^4 - 2677580453478400*x^3 - 4202812479897600*x^2 - 3712263389184000*x - 1414195576832000, 1)
 

Normalized defining polynomial

\( x^{15} - 5274 x^{13} - 84384 x^{12} + 7855432 x^{11} + 261499904 x^{10} - 819354600 x^{9} - 171313992576 x^{8} - 3704165887616 x^{7} - 40366806612992 x^{6} - 260417811445760 x^{5} - 1050040215797760 x^{4} - 2677580453478400 x^{3} - 4202812479897600 x^{2} - 3712263389184000 x - 1414195576832000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2606494027779061551579411214493333175014953976893624414572544000000=2^{18}\cdot 5^{6}\cdot 13^{10}\cdot 79^{2}\cdot 103^{4}\cdot 6547^{4}\cdot 1891234619^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26{,}775.47$
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:  $2, 5, 13, 79, 103, 6547, 1891234619$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} + \frac{1}{10} a^{3} + \frac{1}{10} a^{2}$, $\frac{1}{40} a^{6} + \frac{1}{20} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{160} a^{7} - \frac{1}{80} a^{5} - \frac{1}{20} a^{4} - \frac{1}{20} a^{3} + \frac{1}{5} a^{2} - \frac{1}{4} a$, $\frac{1}{3200} a^{8} - \frac{13}{1600} a^{6} + \frac{1}{50} a^{5} - \frac{47}{400} a^{4} - \frac{4}{25} a^{3} - \frac{77}{400} a^{2} + \frac{1}{5}$, $\frac{1}{102400} a^{9} + \frac{67}{51200} a^{7} + \frac{27}{3200} a^{6} - \frac{287}{12800} a^{5} + \frac{9}{200} a^{4} + \frac{1763}{12800} a^{3} - \frac{27}{160} a^{2} + \frac{21}{160} a - \frac{1}{4}$, $\frac{1}{32358400} a^{10} - \frac{69}{647168} a^{8} - \frac{1093}{1011200} a^{7} - \frac{28959}{4044800} a^{6} + \frac{591}{63200} a^{5} - \frac{482589}{4044800} a^{4} + \frac{23721}{252800} a^{3} + \frac{36713}{252800} a^{2} + \frac{163}{1264} a - \frac{143}{395}$, $\frac{1}{5177344000} a^{11} - \frac{1}{129433600} a^{10} + \frac{3963}{2588672000} a^{9} - \frac{21369}{323584000} a^{8} + \frac{1954389}{647168000} a^{7} + \frac{357451}{80896000} a^{6} - \frac{482229}{25886720} a^{5} + \frac{6550691}{80896000} a^{4} - \frac{2567141}{20224000} a^{3} - \frac{62183}{316000} a^{2} - \frac{81513}{505600} a + \frac{16069}{63200}$, $\frac{1}{103546880000} a^{12} + \frac{603}{51773440000} a^{10} + \frac{11863}{3235840000} a^{9} + \frac{25989}{12943360000} a^{8} - \frac{95611}{404480000} a^{7} + \frac{19702839}{2588672000} a^{6} + \frac{8786273}{808960000} a^{5} + \frac{16180889}{404480000} a^{4} + \frac{12248211}{50560000} a^{3} + \frac{849559}{10112000} a^{2} - \frac{36621}{79000} a - \frac{4179}{31600}$, $\frac{1}{207093760000000} a^{13} - \frac{3}{1294336000000} a^{12} + \frac{883}{103546880000000} a^{11} + \frac{96273}{6471680000000} a^{10} + \frac{34431049}{25886720000000} a^{9} - \frac{47255161}{808960000000} a^{8} - \frac{13133890201}{5177344000000} a^{7} + \frac{2200411503}{1617920000000} a^{6} + \frac{23355112363}{1617920000000} a^{5} - \frac{5182533873}{202240000000} a^{4} - \frac{4367270991}{20224000000} a^{3} + \frac{21703537}{632000000} a^{2} - \frac{54258059}{252800000} a - \frac{8364717}{31600000}$, $\frac{1}{212064010240000000} a^{14} - \frac{43}{26508001280000000} a^{13} - \frac{467757}{106032005120000000} a^{12} + \frac{877557}{13254000640000000} a^{11} + \frac{394549561}{26508001280000000} a^{10} - \frac{2437155811}{3313500160000000} a^{9} + \frac{1916086320323}{26508001280000000} a^{8} + \frac{6859323783921}{3313500160000000} a^{7} - \frac{7867406390729}{1656750080000000} a^{6} - \frac{803775087363}{51773440000000} a^{5} - \frac{10733549874319}{103546880000000} a^{4} + \frac{514282104001}{2588672000000} a^{3} - \frac{42221829731}{1294336000000} a^{2} + \frac{1065192793}{3235840000} a - \frac{1268768939}{4044800000}$

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:  $12$
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:  \( 81794670831100000000000000000 \) (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

3.3.169.1

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 $15$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
13Data not computed
79Data not computed
103Data not computed
6547Data not computed
1891234619Data not computed