Properties

Label 15.3.10544075126076649.1
Degree $15$
Signature $[3, 6]$
Discriminant $37^{2}\cdot 43^{2}\cdot 1609^{3}$
Root discriminant $11.70$
Ramified primes $37, 43, 1609$
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, 9, -19, -27, 47, 42, -64, -30, 40, 18, -13, -10, -1, 11, -6, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 11*x^13 - x^12 - 10*x^11 - 13*x^10 + 18*x^9 + 40*x^8 - 30*x^7 - 64*x^6 + 42*x^5 + 47*x^4 - 27*x^3 - 19*x^2 + 9*x + 1)
gp: K = bnfinit(x^15 - 6*x^14 + 11*x^13 - x^12 - 10*x^11 - 13*x^10 + 18*x^9 + 40*x^8 - 30*x^7 - 64*x^6 + 42*x^5 + 47*x^4 - 27*x^3 - 19*x^2 + 9*x + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 6 x^{14} \) \(\mathstrut +\mathstrut 11 x^{13} \) \(\mathstrut -\mathstrut x^{12} \) \(\mathstrut -\mathstrut 10 x^{11} \) \(\mathstrut -\mathstrut 13 x^{10} \) \(\mathstrut +\mathstrut 18 x^{9} \) \(\mathstrut +\mathstrut 40 x^{8} \) \(\mathstrut -\mathstrut 30 x^{7} \) \(\mathstrut -\mathstrut 64 x^{6} \) \(\mathstrut +\mathstrut 42 x^{5} \) \(\mathstrut +\mathstrut 47 x^{4} \) \(\mathstrut -\mathstrut 27 x^{3} \) \(\mathstrut -\mathstrut 19 x^{2} \) \(\mathstrut +\mathstrut 9 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(10544075126076649=37^{2}\cdot 43^{2}\cdot 1609^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.70$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $37, 43, 1609$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{69781} a^{14} - \frac{26375}{69781} a^{13} - \frac{24841}{69781} a^{12} - \frac{1919}{69781} a^{11} + \frac{10876}{69781} a^{10} + \frac{10653}{69781} a^{9} + \frac{29367}{69781} a^{8} - \frac{18626}{69781} a^{7} + \frac{30286}{69781} a^{6} + \frac{31947}{69781} a^{5} - \frac{14169}{69781} a^{4} + \frac{14934}{69781} a^{3} - \frac{20490}{69781} a^{2} - \frac{13492}{69781} a + \frac{27019}{69781}$

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:  \( \frac{4485}{69781} a^{14} - \frac{13080}{69781} a^{13} - \frac{41409}{69781} a^{12} + \frac{185691}{69781} a^{11} - \frac{137621}{69781} a^{10} - \frac{91061}{69781} a^{9} - \frac{175095}{69781} a^{8} + \frac{408933}{69781} a^{7} + \frac{387789}{69781} a^{6} - \frac{606127}{69781} a^{5} - \frac{396160}{69781} a^{4} + \frac{477697}{69781} a^{3} + \frac{73708}{69781} a^{2} - \frac{81274}{69781} a + \frac{40399}{69781} \),  \( \frac{5885}{69781} a^{14} - \frac{23931}{69781} a^{13} + \frac{1910}{69781} a^{12} + \frac{80988}{69781} a^{11} + \frac{16083}{69781} a^{10} - \frac{249557}{69781} a^{9} - \frac{22742}{69781} a^{8} + \frac{291065}{69781} a^{7} + \frac{431122}{69781} a^{6} - \frac{540167}{69781} a^{5} - \frac{624299}{69781} a^{4} + \frac{520778}{69781} a^{3} + \frac{486385}{69781} a^{2} - \frac{268766}{69781} a - \frac{93865}{69781} \),  \( \frac{26584}{69781} a^{14} - \frac{133074}{69781} a^{13} + \frac{173802}{69781} a^{12} + \frac{64996}{69781} a^{11} - \frac{44880}{69781} a^{10} - \frac{460413}{69781} a^{9} - \frac{17500}{69781} a^{8} + \frac{919545}{69781} a^{7} + \frac{199189}{69781} a^{6} - \frac{1072218}{69781} a^{5} - \frac{339763}{69781} a^{4} + \frac{649376}{69781} a^{3} + \frac{283450}{69781} a^{2} - \frac{136550}{69781} a - \frac{122299}{69781} \),  \( \frac{1572}{69781} a^{14} - \frac{11586}{69781} a^{13} + \frac{27308}{69781} a^{12} - \frac{16085}{69781} a^{11} + \frac{727}{69781} a^{10} - \frac{70705}{69781} a^{9} + \frac{109464}{69781} a^{8} + \frac{27948}{69781} a^{7} - \frac{50831}{69781} a^{6} - \frac{161198}{69781} a^{5} + \frac{126033}{69781} a^{4} + \frac{169394}{69781} a^{3} - \frac{111020}{69781} a^{2} - \frac{135562}{69781} a + \frac{47020}{69781} \),  \( \frac{15796}{69781} a^{14} - \frac{96711}{69781} a^{13} + \frac{199470}{69781} a^{12} - \frac{97351}{69781} a^{11} - \frac{73307}{69781} a^{10} - \frac{246327}{69781} a^{9} + \frac{395730}{69781} a^{8} + \frac{538648}{69781} a^{7} - \frac{579128}{69781} a^{6} - \frac{928533}{69781} a^{5} + \frac{602172}{69781} a^{4} + \frac{735494}{69781} a^{3} - \frac{155324}{69781} a^{2} - \frac{357363}{69781} a + \frac{11528}{69781} \),  \( \frac{17731}{69781} a^{14} - \frac{122425}{69781} a^{13} + \frac{281025}{69781} a^{12} - \frac{182004}{69781} a^{11} - \frac{32328}{69781} a^{10} - \frac{357729}{69781} a^{9} + \frac{488922}{69781} a^{8} + \frac{643896}{69781} a^{7} - \frac{661539}{69781} a^{6} - \frac{867273}{69781} a^{5} + \frac{609090}{69781} a^{4} + \frac{464326}{69781} a^{3} - \frac{98085}{69781} a^{2} - \frac{226727}{69781} a + \frac{27324}{69781} \),  \( \frac{14946}{69781} a^{14} - \frac{77662}{69781} a^{13} + \frac{100896}{69781} a^{12} + \frac{68398}{69781} a^{11} - \frac{106815}{69781} a^{10} - \frac{229847}{69781} a^{9} + \frac{66473}{69781} a^{8} + \frac{530461}{69781} a^{7} - \frac{14791}{69781} a^{6} - \frac{799112}{69781} a^{5} + \frac{224804}{69781} a^{4} + \frac{602174}{69781} a^{3} - \frac{323636}{69781} a^{2} - \frac{263466}{69781} a + \frac{142889}{69781} \),  \( \frac{490}{2251} a^{14} - \frac{3010}{2251} a^{13} + \frac{5820}{2251} a^{12} - \frac{1643}{2251} a^{11} - \frac{3379}{2251} a^{10} - \frac{6852}{2251} a^{9} + \frac{8191}{2251} a^{8} + \frac{19073}{2251} a^{7} - \frac{14209}{2251} a^{6} - \frac{26436}{2251} a^{5} + \frac{15031}{2251} a^{4} + \frac{17667}{2251} a^{3} - \frac{5142}{2251} a^{2} - \frac{4395}{2251} a - \frac{1072}{2251} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 73.2403388939 \)
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

5.1.1609.1

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 $15$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ R $15$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$37$37.3.2.2$x^{3} + 74$$3$$1$$2$$C_3$$[\ ]_{3}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$43$43.3.2.2$x^{3} + 387$$3$$1$$2$$C_3$$[\ ]_{3}$
43.12.0.1$x^{12} - x + 33$$1$$12$$0$$C_{12}$$[\ ]^{12}$
1609Data not computed