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Properties

Label 15.11.4248332612...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 71^{4}\cdot 223^{2}\cdot 1709^{2}\cdot 31541^{2}$
Root discriminant $1101.24$
Ramified primes $2, 5, 7, 71, 223, 1709, 31541$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T92

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![568000, 7952000, 41322000, 95779000, 85779360, -9352320, -40210112, -1338292, 5324544, -112752, 2756, 24240, -288, -306, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 306*x^13 - 288*x^12 + 24240*x^11 + 2756*x^10 - 112752*x^9 + 5324544*x^8 - 1338292*x^7 - 40210112*x^6 - 9352320*x^5 + 85779360*x^4 + 95779000*x^3 + 41322000*x^2 + 7952000*x + 568000)
 
gp: K = bnfinit(x^15 - 306*x^13 - 288*x^12 + 24240*x^11 + 2756*x^10 - 112752*x^9 + 5324544*x^8 - 1338292*x^7 - 40210112*x^6 - 9352320*x^5 + 85779360*x^4 + 95779000*x^3 + 41322000*x^2 + 7952000*x + 568000, 1)
 

Normalized defining polynomial

\( x^{15} - 306 x^{13} - 288 x^{12} + 24240 x^{11} + 2756 x^{10} - 112752 x^{9} + 5324544 x^{8} - 1338292 x^{7} - 40210112 x^{6} - 9352320 x^{5} + 85779360 x^{4} + 95779000 x^{3} + 41322000 x^{2} + 7952000 x + 568000 \)

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:  \(4248332612098651806088183712995608825856000000=2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 71^{4}\cdot 223^{2}\cdot 1709^{2}\cdot 31541^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1101.24$
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:  $2, 5, 7, 71, 223, 1709, 31541$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{20} a^{11} - \frac{1}{20} a^{9} + \frac{1}{10} a^{8} - \frac{1}{5} a^{6} - \frac{1}{10} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{40} a^{12} + \frac{1}{10} a^{10} + \frac{1}{20} a^{9} - \frac{1}{10} a^{7} + \frac{1}{5} a^{6} + \frac{1}{10} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3}$, $\frac{1}{2800} a^{13} + \frac{1}{280} a^{12} + \frac{1}{200} a^{11} - \frac{3}{175} a^{10} + \frac{3}{35} a^{9} + \frac{7}{100} a^{8} - \frac{37}{175} a^{7} - \frac{16}{175} a^{6} + \frac{11}{100} a^{5} + \frac{81}{350} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{3}{14} a - \frac{2}{7}$, $\frac{1}{3208222619859055324840797854930265432471200} a^{14} - \frac{81342419514345872491587160657504632303}{1604111309929527662420398927465132716235600} a^{13} + \frac{8784051216047403300923499406775480637137}{1604111309929527662420398927465132716235600} a^{12} - \frac{13673014302832452067859317272029736449483}{802055654964763831210199463732566358117800} a^{11} + \frac{7696157213733541757090350776218827384584}{100256956870595478901274932966570794764725} a^{10} + \frac{35882836050408607198131128181428330601839}{802055654964763831210199463732566358117800} a^{9} + \frac{7290877962254665744354575546352255197836}{100256956870595478901274932966570794764725} a^{8} + \frac{19990861819888558151084636502234572268121}{200513913741190957802549865933141589529450} a^{7} - \frac{133886265299872567386278375431252387026149}{802055654964763831210199463732566358117800} a^{6} + \frac{929101175012429854826283287662396542453}{80205565496476383121019946373256635811780} a^{5} + \frac{33942204371563752977108608961524947816467}{200513913741190957802549865933141589529450} a^{4} - \frac{52925891118460434623683249057081815896}{2864484482017013682893569513330594136135} a^{3} + \frac{347091703152135200074394008937338568783}{16041113099295276624203989274651327162356} a^{2} - \frac{850911236029076459317353920467700707925}{4010278274823819156050997318662831790589} a + \frac{238470602725777484827081387358465151126}{4010278274823819156050997318662831790589}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 211856365529000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T92:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 648000
The 55 conjugacy class representatives for [A(5)^3]3=A(5)wr3 are not computed
Character table for [A(5)^3]3=A(5)wr3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 sibling: data not computed
Degree 36 sibling: 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 R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ $15$ $15$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$$[\ ]^{3}$
2.12.18.44$x^{12} - 6 x^{11} + 8 x^{10} + 6 x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 8$$4$$3$$18$12T166$[2, 2, 2, 2, 2, 2]^{9}$
$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$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
71.5.4.2$x^{5} + 142$$5$$1$$4$$C_5$$[\ ]_{5}$
71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
223Data not computed
1709Data not computed
31541Data not computed