Properties

Label 15.3.11354350039657729.1
Degree $15$
Signature $[3, 6]$
Discriminant $13^{2}\cdot 127^{2}\cdot 1609^{3}$
Root discriminant $11.76$
Ramified primes $13, 127, 1609$
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, -7, 11, -20, 35, -37, 22, -8, -2, 7, 1, -7, 1, 5, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 5*x^13 + x^12 - 7*x^11 + x^10 + 7*x^9 - 2*x^8 - 8*x^7 + 22*x^6 - 37*x^5 + 35*x^4 - 20*x^3 + 11*x^2 - 7*x + 1)
gp: K = bnfinit(x^15 - 4*x^14 + 5*x^13 + x^12 - 7*x^11 + x^10 + 7*x^9 - 2*x^8 - 8*x^7 + 22*x^6 - 37*x^5 + 35*x^4 - 20*x^3 + 11*x^2 - 7*x + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 4 x^{14} \) \(\mathstrut +\mathstrut 5 x^{13} \) \(\mathstrut +\mathstrut x^{12} \) \(\mathstrut -\mathstrut 7 x^{11} \) \(\mathstrut +\mathstrut x^{10} \) \(\mathstrut +\mathstrut 7 x^{9} \) \(\mathstrut -\mathstrut 2 x^{8} \) \(\mathstrut -\mathstrut 8 x^{7} \) \(\mathstrut +\mathstrut 22 x^{6} \) \(\mathstrut -\mathstrut 37 x^{5} \) \(\mathstrut +\mathstrut 35 x^{4} \) \(\mathstrut -\mathstrut 20 x^{3} \) \(\mathstrut +\mathstrut 11 x^{2} \) \(\mathstrut -\mathstrut 7 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(11354350039657729=13^{2}\cdot 127^{2}\cdot 1609^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.76$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $13, 127, 1609$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{679097} a^{14} - \frac{18368}{679097} a^{13} - \frac{201252}{679097} a^{12} + \frac{145855}{679097} a^{11} - \frac{122659}{679097} a^{10} - \frac{54872}{679097} a^{9} - \frac{110533}{679097} a^{8} + \frac{7077}{679097} a^{7} - \frac{254509}{679097} a^{6} + \frac{257744}{679097} a^{5} + \frac{95237}{679097} a^{4} - \frac{257458}{679097} a^{3} + \frac{85378}{679097} a^{2} + \frac{153392}{679097} a + \frac{3661}{679097}$

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:  \( \frac{174629}{679097} a^{14} - \frac{210341}{679097} a^{13} - \frac{486661}{679097} a^{12} + \frac{979810}{679097} a^{11} + \frac{259063}{679097} a^{10} - \frac{1542012}{679097} a^{9} - \frac{293226}{679097} a^{8} + \frac{1251087}{679097} a^{7} - \frac{469899}{679097} a^{6} + \frac{1065107}{679097} a^{5} + \frac{735640}{679097} a^{4} - \frac{3411682}{679097} a^{3} + \frac{2616515}{679097} a^{2} - \frac{289597}{679097} a + \frac{965589}{679097} \),  \( \frac{230384}{679097} a^{14} - \frac{239905}{679097} a^{13} - \frac{572190}{679097} a^{12} + \frac{938760}{679097} a^{11} + \frac{592405}{679097} a^{10} - \frac{1598387}{679097} a^{9} - \frac{934463}{679097} a^{8} + \frac{1273865}{679097} a^{7} - \frac{208282}{679097} a^{6} + \frac{1210210}{679097} a^{5} + \frac{815132}{679097} a^{4} - \frac{2550989}{679097} a^{3} + \frac{1038741}{679097} a^{2} + \frac{212842}{679097} a + \frac{676447}{679097} \),  \( \frac{39462}{679097} a^{14} + \frac{437580}{679097} a^{13} - \frac{1125203}{679097} a^{12} + \frac{382935}{679097} a^{11} + \frac{1592152}{679097} a^{10} - \frac{1076725}{679097} a^{9} - \frac{2050506}{679097} a^{8} + \frac{842804}{679097} a^{7} + \frac{1089569}{679097} a^{6} - \frac{2458429}{679097} a^{5} + \frac{5552472}{679097} a^{4} - \frac{5949252}{679097} a^{3} + \frac{2902807}{679097} a^{2} - \frac{994651}{679097} a + \frac{1860012}{679097} \),  \( \frac{91140}{679097} a^{14} - \frac{85415}{679097} a^{13} - \frac{376407}{679097} a^{12} + \frac{580022}{679097} a^{11} + \frac{153554}{679097} a^{10} - \frac{842869}{679097} a^{9} - \frac{252722}{679097} a^{8} + \frac{534727}{679097} a^{7} - \frac{34031}{679097} a^{6} + \frac{822930}{679097} a^{5} + \frac{1040520}{679097} a^{4} - \frac{2599867}{679097} a^{3} + \frac{2294785}{679097} a^{2} - \frac{1102156}{679097} a + \frac{906010}{679097} \),  \( \frac{140071}{679097} a^{14} - \frac{404692}{679097} a^{13} + \frac{426675}{679097} a^{12} + \frac{101557}{679097} a^{11} - \frac{493786}{679097} a^{10} + \frac{43934}{679097} a^{9} + \frac{264660}{679097} a^{8} - \frac{199153}{679097} a^{7} - \frac{812221}{679097} a^{6} + \frac{2342401}{679097} a^{5} - \frac{3635126}{679097} a^{4} + \frac{3763055}{679097} a^{3} - \frac{2632720}{679097} a^{2} + \frac{1858140}{679097} a - \frac{597401}{679097} \),  \( \frac{488175}{679097} a^{14} - \frac{1359806}{679097} a^{13} + \frac{805181}{679097} a^{12} + \frac{1481466}{679097} a^{11} - \frac{1716641}{679097} a^{10} - \frac{1515629}{679097} a^{9} + \frac{1600345}{679097} a^{8} + \frac{927133}{679097} a^{7} - \frac{2776731}{679097} a^{6} + \frac{7196913}{679097} a^{5} - \frac{9523697}{679097} a^{4} + \frac{6109095}{679097} a^{3} - \frac{2210516}{679097} a^{2} + \frac{2187992}{679097} a - \frac{174629}{679097} \),  \( \frac{152986}{679097} a^{14} - \frac{622559}{679097} a^{13} + \frac{840411}{679097} a^{12} + \frac{3804}{679097} a^{11} - \frac{980567}{679097} a^{10} + \frac{349322}{679097} a^{9} + \frac{871956}{679097} a^{8} - \frac{477793}{679097} a^{7} - \frac{966476}{679097} a^{6} + \frac{3530861}{679097} a^{5} - \frac{6210326}{679097} a^{4} + \frac{6268285}{679097} a^{3} - \frac{4187572}{679097} a^{2} + \frac{1989871}{679097} a - \frac{1531473}{679097} \),  \( \frac{238798}{679097} a^{14} - \frac{633238}{679097} a^{13} + \frac{440497}{679097} a^{12} + \frac{355354}{679097} a^{11} - \frac{591175}{679097} a^{10} - \frac{147241}{679097} a^{9} + \frac{82862}{679097} a^{8} - \frac{298987}{679097} a^{7} - \frac{454167}{679097} a^{6} + \frac{4227893}{679097} a^{5} - \frac{5307083}{679097} a^{4} + \frac{4307799}{679097} a^{3} - \frac{2470878}{679097} a^{2} + \frac{1247927}{679097} a - \frac{437458}{679097} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 79.4890458165 \)
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 $15$ $15$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$13$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.9.0.1$x^{9} - 2 x + 2$$1$$9$$0$$C_9$$[\ ]^{9}$
$127$127.3.2.2$x^{3} + 1143$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
127.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
127.6.0.1$x^{6} - x + 29$$1$$6$$0$$C_6$$[\ ]^{6}$
1609Data not computed