Properties

Label 15.3.351730374981194881.1
Degree $15$
Signature $[3, 6]$
Discriminant $7^{12}\cdot 71^{4}$
Root discriminant $14.78$
Ramified primes $7, 71$
Class number $1$
Class group Trivial
Galois Group 15T30

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 9, 0, -8, -15, 14, -34, 44, -37, 27, 0, -6, 1, 0, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + x^12 - 6*x^11 + 27*x^9 - 37*x^8 + 44*x^7 - 34*x^6 + 14*x^5 - 15*x^4 - 8*x^3 + 9*x - 1)
gp: K = bnfinit(x^15 - 2*x^14 + x^12 - 6*x^11 + 27*x^9 - 37*x^8 + 44*x^7 - 34*x^6 + 14*x^5 - 15*x^4 - 8*x^3 + 9*x - 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut +\mathstrut x^{12} \) \(\mathstrut -\mathstrut 6 x^{11} \) \(\mathstrut +\mathstrut 27 x^{9} \) \(\mathstrut -\mathstrut 37 x^{8} \) \(\mathstrut +\mathstrut 44 x^{7} \) \(\mathstrut -\mathstrut 34 x^{6} \) \(\mathstrut +\mathstrut 14 x^{5} \) \(\mathstrut -\mathstrut 15 x^{4} \) \(\mathstrut -\mathstrut 8 x^{3} \) \(\mathstrut +\mathstrut 9 x \) \(\mathstrut -\mathstrut 1 \)

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:  \(351730374981194881=7^{12}\cdot 71^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $14.78$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{11} + \frac{1}{7} a^{10} - \frac{1}{7} a^{9} + \frac{3}{7} a^{8} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7}$, $\frac{1}{7} a^{13} + \frac{2}{7} a^{9} + \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{2049272407} a^{14} + \frac{650660}{2049272407} a^{13} + \frac{38608274}{2049272407} a^{12} - \frac{215400421}{2049272407} a^{11} + \frac{13067319}{292753201} a^{10} + \frac{715149348}{2049272407} a^{9} - \frac{754312263}{2049272407} a^{8} - \frac{413178839}{2049272407} a^{7} - \frac{648259}{1899233} a^{6} + \frac{481581404}{2049272407} a^{5} - \frac{790941285}{2049272407} a^{4} + \frac{390461632}{2049272407} a^{3} - \frac{293010650}{2049272407} a^{2} - \frac{642956668}{2049272407} a + \frac{365753205}{2049272407}$

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:  \( \frac{255880921}{2049272407} a^{14} - \frac{485636051}{2049272407} a^{13} - \frac{197206}{2049272407} a^{12} + \frac{167332587}{2049272407} a^{11} - \frac{1544287239}{2049272407} a^{10} - \frac{116046345}{2049272407} a^{9} + \frac{6574703196}{2049272407} a^{8} - \frac{8781867005}{2049272407} a^{7} + \frac{11059264}{1899233} a^{6} - \frac{1313768850}{292753201} a^{5} + \frac{4266157921}{2049272407} a^{4} - \frac{4294261913}{2049272407} a^{3} - \frac{3596720318}{2049272407} a^{2} - \frac{346727385}{2049272407} a + \frac{540418521}{292753201} \),  \( \frac{255880921}{2049272407} a^{14} - \frac{485636051}{2049272407} a^{13} - \frac{197206}{2049272407} a^{12} + \frac{167332587}{2049272407} a^{11} - \frac{1544287239}{2049272407} a^{10} - \frac{116046345}{2049272407} a^{9} + \frac{6574703196}{2049272407} a^{8} - \frac{8781867005}{2049272407} a^{7} + \frac{11059264}{1899233} a^{6} - \frac{1313768850}{292753201} a^{5} + \frac{4266157921}{2049272407} a^{4} - \frac{4294261913}{2049272407} a^{3} - \frac{3596720318}{2049272407} a^{2} - \frac{346727385}{2049272407} a + \frac{247665320}{292753201} \),  \( \frac{12852236}{292753201} a^{14} - \frac{122422434}{2049272407} a^{13} - \frac{114538017}{2049272407} a^{12} + \frac{151620566}{2049272407} a^{11} - \frac{529271406}{2049272407} a^{10} - \frac{369366035}{2049272407} a^{9} + \frac{316189996}{292753201} a^{8} - \frac{2148532518}{2049272407} a^{7} + \frac{1578590}{1899233} a^{6} + \frac{439905178}{2049272407} a^{5} - \frac{1281000761}{2049272407} a^{4} + \frac{3334239357}{2049272407} a^{3} - \frac{684134664}{292753201} a^{2} + \frac{977574347}{2049272407} a - \frac{129883449}{292753201} \),  \( \frac{230234853}{2049272407} a^{14} - \frac{423783931}{2049272407} a^{13} + \frac{5909252}{292753201} a^{12} + \frac{21717355}{292753201} a^{11} - \frac{1643983526}{2049272407} a^{10} - \frac{154507558}{2049272407} a^{9} + \frac{823087748}{292753201} a^{8} - \frac{8487705271}{2049272407} a^{7} + \frac{10976977}{1899233} a^{6} - \frac{6001607020}{2049272407} a^{5} + \frac{1515621637}{2049272407} a^{4} - \frac{1306943656}{2049272407} a^{3} - \frac{2919595537}{2049272407} a^{2} - \frac{2254342921}{2049272407} a - \frac{513987218}{2049272407} \),  \( \frac{62648636}{2049272407} a^{14} + \frac{5792520}{2049272407} a^{13} - \frac{224814635}{2049272407} a^{12} - \frac{41965270}{2049272407} a^{11} - \frac{198590085}{2049272407} a^{10} - \frac{785041694}{2049272407} a^{9} + \frac{1556176147}{2049272407} a^{8} + \frac{1553912298}{2049272407} a^{7} - \frac{1058661}{1899233} a^{6} + \frac{1558469032}{2049272407} a^{5} - \frac{1083243712}{2049272407} a^{4} - \frac{3513994917}{2049272407} a^{3} - \frac{762093273}{2049272407} a^{2} - \frac{49549904}{292753201} a + \frac{572514282}{2049272407} \),  \( \frac{221055444}{2049272407} a^{14} - \frac{347237069}{2049272407} a^{13} - \frac{219151923}{2049272407} a^{12} + \frac{328922297}{2049272407} a^{11} - \frac{1314313100}{2049272407} a^{10} - \frac{504703060}{2049272407} a^{9} + \frac{6085241802}{2049272407} a^{8} - \frac{6047367721}{2049272407} a^{7} + \frac{4997054}{1899233} a^{6} - \frac{1520419438}{2049272407} a^{5} - \frac{3855266138}{2049272407} a^{4} + \frac{3073335172}{2049272407} a^{3} - \frac{1026437769}{292753201} a^{2} + \frac{986250905}{2049272407} a + \frac{1202736900}{2049272407} \),  \( \frac{33247931}{292753201} a^{14} - \frac{464036849}{2049272407} a^{13} + \frac{23194359}{292753201} a^{12} + \frac{998838}{292753201} a^{11} - \frac{205663050}{292753201} a^{10} + \frac{155140392}{2049272407} a^{9} + \frac{5132102050}{2049272407} a^{8} - \frac{9330341444}{2049272407} a^{7} + \frac{1825530}{271319} a^{6} - \frac{13121056801}{2049272407} a^{5} + \frac{8242550949}{2049272407} a^{4} - \frac{5076936499}{2049272407} a^{3} - \frac{1777739870}{2049272407} a^{2} - \frac{653214006}{2049272407} a + \frac{1124025486}{2049272407} \),  \( \frac{2518059}{157636339} a^{14} - \frac{4280195}{157636339} a^{13} + \frac{962023}{157636339} a^{12} + \frac{2944593}{157636339} a^{11} - \frac{14921471}{157636339} a^{10} - \frac{12044792}{157636339} a^{9} + \frac{50284490}{157636339} a^{8} - \frac{96676948}{157636339} a^{7} + \frac{1052425}{1899233} a^{6} - \frac{72658038}{157636339} a^{5} + \frac{92140511}{157636339} a^{4} - \frac{141206899}{157636339} a^{3} + \frac{86490816}{157636339} a^{2} + \frac{24219284}{157636339} a + \frac{90969295}{157636339} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 694.485336338 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T30:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 750
The 32 conjugacy class representatives for [5^3:2]3
Character table for [5^3:2]3 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 15 siblings: data not computed
Degree 30 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 $15$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R $15$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$71$71.5.4.2$x^{5} + 142$$5$$1$$4$$C_5$$[\ ]_{5}$
71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$