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Properties

Label 15.9.95997288591...1264.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{10}\cdot 3^{21}\cdot 137^{10}\cdot 1567^{3}$
Root discriminant $855.36$
Ramified primes $2, 3, 137, 1567$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T70

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![215817165520608256, -159305388838030848, 21988813850200512, 1930435159845344, -325916709388536, -22989137783499, 2260733557600, 170071858773, -7775771472, -716487806, 12612768, 1689210, -7672, -2055, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2055*x^13 - 7672*x^12 + 1689210*x^11 + 12612768*x^10 - 716487806*x^9 - 7775771472*x^8 + 170071858773*x^7 + 2260733557600*x^6 - 22989137783499*x^5 - 325916709388536*x^4 + 1930435159845344*x^3 + 21988813850200512*x^2 - 159305388838030848*x + 215817165520608256)
 
gp: K = bnfinit(x^15 - 2055*x^13 - 7672*x^12 + 1689210*x^11 + 12612768*x^10 - 716487806*x^9 - 7775771472*x^8 + 170071858773*x^7 + 2260733557600*x^6 - 22989137783499*x^5 - 325916709388536*x^4 + 1930435159845344*x^3 + 21988813850200512*x^2 - 159305388838030848*x + 215817165520608256, 1)
 

Normalized defining polynomial

\( x^{15} - 2055 x^{13} - 7672 x^{12} + 1689210 x^{11} + 12612768 x^{10} - 716487806 x^{9} - 7775771472 x^{8} + 170071858773 x^{7} + 2260733557600 x^{6} - 22989137783499 x^{5} - 325916709388536 x^{4} + 1930435159845344 x^{3} + 21988813850200512 x^{2} - 159305388838030848 x + 215817165520608256 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-95997288591377091612527656182883144142091264=-\,2^{10}\cdot 3^{21}\cdot 137^{10}\cdot 1567^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $855.36$
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:  $2, 3, 137, 1567$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{548} a^{3} + \frac{1}{4} a$, $\frac{1}{548} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{548} a^{5} - \frac{1}{4} a$, $\frac{1}{900912} a^{6} - \frac{1}{1096} a^{4} + \frac{1}{1644} a^{3} + \frac{3}{16} a^{2} - \frac{1}{4} a + \frac{1}{3}$, $\frac{1}{900912} a^{7} - \frac{1}{1096} a^{5} + \frac{1}{1644} a^{4} - \frac{1}{2192} a^{3} - \frac{1}{4} a^{2} - \frac{5}{12} a$, $\frac{1}{900912} a^{8} + \frac{1}{1644} a^{5} - \frac{1}{2192} a^{4} + \frac{11}{24} a^{2} + \frac{1}{4} a$, $\frac{1}{493699776} a^{9} - \frac{1}{3603648} a^{7} - \frac{7}{8768} a^{5} - \frac{1}{1644} a^{4} + \frac{7}{8768} a^{3} + \frac{1}{4} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{493699776} a^{10} - \frac{1}{3603648} a^{8} - \frac{1}{3603648} a^{6} - \frac{1}{1644} a^{5} - \frac{1}{8768} a^{4} - \frac{1}{1644} a^{3} + \frac{11}{48} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{493699776} a^{11} - \frac{1}{1801824} a^{7} + \frac{1}{1644} a^{4} - \frac{5}{8768} a^{3} + \frac{5}{12} a^{2} - \frac{5}{12} a$, $\frac{1}{6193643396638464} a^{12} - \frac{1}{3767422990656} a^{10} - \frac{1187}{5651134485984} a^{9} + \frac{3}{18332958592} a^{8} + \frac{1187}{4583239648} a^{7} - \frac{3175}{9166479296} a^{6} - \frac{3561}{33454304} a^{5} + \frac{67569}{267634432} a^{4} - \frac{162143}{301088736} a^{3} - \frac{6219}{122096} a^{2} + \frac{19675}{91572} a + \frac{10294}{68679}$, $\frac{1}{6193643396638464} a^{13} - \frac{1}{3767422990656} a^{11} - \frac{1187}{5651134485984} a^{10} + \frac{3}{18332958592} a^{9} + \frac{1187}{4583239648} a^{8} - \frac{3175}{9166479296} a^{7} + \frac{527}{4583239648} a^{6} + \frac{67569}{267634432} a^{5} - \frac{162143}{301088736} a^{4} + \frac{2669}{16727152} a^{3} + \frac{19675}{91572} a^{2} + \frac{10294}{68679} a$, $\frac{1}{1649608788617291863296} a^{14} - \frac{437}{68733699525720494304} a^{13} - \frac{7}{6020470031449970304} a^{12} + \frac{749248301}{3010235015724985152} a^{11} + \frac{1087916879}{2006823343816656768} a^{10} - \frac{272226605}{752558753931246288} a^{9} + \frac{11756271}{187799302247488} a^{8} - \frac{133273135}{1220695464608672} a^{7} - \frac{3357039931}{9765563716869376} a^{6} - \frac{35782020107}{80191672857504} a^{5} - \frac{6318206137}{106922230476672} a^{4} - \frac{39801704951}{160383345715008} a^{3} + \frac{494211589}{8607951144} a^{2} - \frac{2058783646}{6097298727} a - \frac{1627101799}{18291896181}$

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:  $11$
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:  \( 67205708918800000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T70:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 19440
The 36 conjugacy class representatives for [3^4:2]S(5)
Character table for [3^4:2]S(5) is not computed

Intermediate fields

5.3.14103.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.12.18.2$x^{12} + 6 x^{10} - 3 x^{9} - 9 x^{8} + 9 x^{7} + 6 x^{6} + 9 x^{5} - 9 x^{4} - 9 x^{3} - 9$$6$$2$$18$12T170$[3/2, 3/2, 2, 2]_{2}^{4}$
137Data not computed
1567Data not computed