Properties

Label 15.3.15315650494266781.1
Degree $15$
Signature $[3, 6]$
Discriminant $127^{2}\cdot 9829^{3}$
Root discriminant $12.00$
Ramified primes $127, 9829$
Class number $1$
Class group Trivial
Galois Group 15T78

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 6, -11, -2, 27, -19, -15, 16, 7, -10, -5, 7, -1, 1, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + x^13 - x^12 + 7*x^11 - 5*x^10 - 10*x^9 + 7*x^8 + 16*x^7 - 15*x^6 - 19*x^5 + 27*x^4 - 2*x^3 - 11*x^2 + 6*x - 1)
gp: K = bnfinit(x^15 - 2*x^14 + x^13 - x^12 + 7*x^11 - 5*x^10 - 10*x^9 + 7*x^8 + 16*x^7 - 15*x^6 - 19*x^5 + 27*x^4 - 2*x^3 - 11*x^2 + 6*x - 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut +\mathstrut x^{13} \) \(\mathstrut -\mathstrut x^{12} \) \(\mathstrut +\mathstrut 7 x^{11} \) \(\mathstrut -\mathstrut 5 x^{10} \) \(\mathstrut -\mathstrut 10 x^{9} \) \(\mathstrut +\mathstrut 7 x^{8} \) \(\mathstrut +\mathstrut 16 x^{7} \) \(\mathstrut -\mathstrut 15 x^{6} \) \(\mathstrut -\mathstrut 19 x^{5} \) \(\mathstrut +\mathstrut 27 x^{4} \) \(\mathstrut -\mathstrut 2 x^{3} \) \(\mathstrut -\mathstrut 11 x^{2} \) \(\mathstrut +\mathstrut 6 x \) \(\mathstrut -\mathstrut 1 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[3, 6]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(15315650494266781=127^{2}\cdot 9829^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $12.00$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $127, 9829$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2962} a^{14} + \frac{60}{1481} a^{13} - \frac{169}{2962} a^{12} - \frac{683}{1481} a^{11} + \frac{354}{1481} a^{10} + \frac{473}{2962} a^{9} + \frac{709}{1481} a^{8} + \frac{1207}{2962} a^{7} + \frac{651}{2962} a^{6} + \frac{457}{1481} a^{5} - \frac{1067}{2962} a^{4} - \frac{650}{1481} a^{3} - \frac{135}{2962} a^{2} + \frac{1291}{2962} a + \frac{261}{1481}$

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

Class group and class number

Trivial Abelian group, order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $8$
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:  \( \frac{8454}{1481} a^{14} - \frac{14815}{1481} a^{13} + \frac{3401}{1481} a^{12} - \frac{6731}{1481} a^{11} + \frac{58470}{1481} a^{10} - \frac{26616}{1481} a^{9} - \frac{98669}{1481} a^{8} + \frac{29508}{1481} a^{7} + \frac{157144}{1481} a^{6} - \frac{79395}{1481} a^{5} - \frac{199582}{1481} a^{4} + \frac{170616}{1481} a^{3} + \frac{56839}{1481} a^{2} - \frac{77868}{1481} a + \frac{17380}{1481} \),  \( a \),  \( \frac{11329}{2962} a^{14} - \frac{7483}{2962} a^{13} - \frac{7073}{2962} a^{12} - \frac{12293}{2962} a^{11} + \frac{32500}{1481} a^{10} + \frac{38865}{2962} a^{9} - \frac{112441}{2962} a^{8} - \frac{81425}{2962} a^{7} + \frac{79133}{1481} a^{6} + \frac{72123}{2962} a^{5} - \frac{245967}{2962} a^{4} - \frac{31737}{2962} a^{3} + \frac{120419}{2962} a^{2} - \frac{4011}{1481} a - \frac{14705}{2962} \),  \( \frac{10377}{1481} a^{14} - \frac{12129}{1481} a^{13} - \frac{1899}{2962} a^{12} - \frac{19915}{2962} a^{11} + \frac{64839}{1481} a^{10} + \frac{3249}{1481} a^{9} - \frac{108743}{1481} a^{8} - \frac{39543}{2962} a^{7} + \frac{325511}{2962} a^{6} - \frac{14556}{1481} a^{5} - \frac{458235}{2962} a^{4} + \frac{173935}{2962} a^{3} + \frac{78624}{1481} a^{2} - \frac{52254}{1481} a + \frac{14883}{2962} \),  \( \frac{3602}{1481} a^{14} - \frac{7829}{2962} a^{13} - \frac{47}{1481} a^{12} - \frac{8305}{2962} a^{11} + \frac{22149}{1481} a^{10} + \frac{2077}{1481} a^{9} - \frac{67311}{2962} a^{8} - \frac{9488}{1481} a^{7} + \frac{100185}{2962} a^{6} - \frac{7475}{2962} a^{5} - \frac{71227}{1481} a^{4} + \frac{49517}{2962} a^{3} + \frac{18751}{1481} a^{2} - \frac{28455}{2962} a + \frac{9115}{2962} \),  \( \frac{3160}{1481} a^{14} - \frac{16163}{2962} a^{13} + \frac{8607}{2962} a^{12} - \frac{2407}{1481} a^{11} + \frac{24666}{1481} a^{10} - \frac{26307}{1481} a^{9} - \frac{73821}{2962} a^{8} + \frac{73659}{2962} a^{7} + \frac{65215}{1481} a^{6} - \frac{140115}{2962} a^{5} - \frac{142623}{2962} a^{4} + \frac{118774}{1481} a^{3} + \frac{2890}{1481} a^{2} - \frac{97455}{2962} a + \frac{14496}{1481} \),  \( \frac{14803}{1481} a^{14} - \frac{44629}{2962} a^{13} + \frac{6809}{2962} a^{12} - \frac{12653}{1481} a^{11} + \frac{97233}{1481} a^{10} - \frac{25526}{1481} a^{9} - \frac{326419}{2962} a^{8} + \frac{46785}{2962} a^{7} + \frac{251656}{1481} a^{6} - \frac{194959}{2962} a^{5} - \frac{667803}{2962} a^{4} + \frac{235693}{1481} a^{3} + \frac{94248}{1481} a^{2} - \frac{238743}{2962} a + \frac{28928}{1481} \),  \( \frac{39037}{2962} a^{14} - \frac{53279}{2962} a^{13} + \frac{5045}{2962} a^{12} - \frac{36681}{2962} a^{11} + \frac{125772}{1481} a^{10} - \frac{36151}{2962} a^{9} - \frac{412589}{2962} a^{8} + \frac{4087}{2962} a^{7} + \frac{315757}{1481} a^{6} - \frac{176673}{2962} a^{5} - \frac{856853}{2962} a^{4} + \frac{495981}{2962} a^{3} + \frac{248211}{2962} a^{2} - \frac{128938}{1481} a + \frac{56513}{2962} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 87.5727798613 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T78:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 29160
The 108 conjugacy class representatives for [3^5]S(5)=3wrS(5) are not computed
Character table for [3^5]S(5)=3wrS(5) is not computed

Intermediate fields

5.1.9829.1

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

Sibling fields

Degree 15 sibling: data not computed
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 ${\href{/LocalNumberField/2.9.0.1}{9} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ $15$ ${\href{/LocalNumberField/5.9.0.1}{9} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ $15$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ $15$ $15$

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
127Data not computed
9829Data not computed