Properties

Label 15.3.10544075126076649.1
Degree 15
Signature $[3, 6]$
Discriminant $37^{2}\cdot 43^{2}\cdot 1609^{3}$
Ramified primes $37, 43, 1609$
Class number 1
Class group Trivial
Galois Group 15T78

Related objects

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Show commands for: Magma / SageMath / Pari/GP

magma: K<a> := NumberField(PolynomialRing(Rationals())![1, 9, -19, -27, 47, 42, -64, -30, 40, 18, -13, -10, -1, 11, -6, 1]);
sage: K = 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,"a")
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
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 $\Q$.

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:  \( -1 \)
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
Conjugacy class representatives for 15T78
Character table for 15T78

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

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