Properties

Label 15.5.35351257235385344.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{10}\cdot 11^{13}$
Root discriminant $12.68$
Ramified primes $2, 11$
Class number $1$
Class group Trivial
Galois Group $S_3 \times C_5$ (as 15T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, -8, -2, 22, 1, 4, -17, -13, 14, 1, 8, -5, -3, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^13 - 5*x^12 + 8*x^11 + x^10 + 14*x^9 - 13*x^8 - 17*x^7 + 4*x^6 + x^5 + 22*x^4 - 2*x^3 - 8*x^2 - 3*x - 1)
gp: K = bnfinit(x^15 - 3*x^13 - 5*x^12 + 8*x^11 + x^10 + 14*x^9 - 13*x^8 - 17*x^7 + 4*x^6 + x^5 + 22*x^4 - 2*x^3 - 8*x^2 - 3*x - 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 3 x^{13} \) \(\mathstrut -\mathstrut 5 x^{12} \) \(\mathstrut +\mathstrut 8 x^{11} \) \(\mathstrut +\mathstrut x^{10} \) \(\mathstrut +\mathstrut 14 x^{9} \) \(\mathstrut -\mathstrut 13 x^{8} \) \(\mathstrut -\mathstrut 17 x^{7} \) \(\mathstrut +\mathstrut 4 x^{6} \) \(\mathstrut +\mathstrut x^{5} \) \(\mathstrut +\mathstrut 22 x^{4} \) \(\mathstrut -\mathstrut 2 x^{3} \) \(\mathstrut -\mathstrut 8 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut -\mathstrut 1 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[5, 5]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-35351257235385344=-\,2^{10}\cdot 11^{13}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $12.68$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2769251} a^{14} + \frac{1237085}{2769251} a^{13} - \frac{190661}{2769251} a^{12} - \frac{1217018}{2769251} a^{11} + \frac{1209397}{2769251} a^{10} + \frac{265482}{2769251} a^{9} - \frac{1060863}{2769251} a^{8} + \frac{806293}{2769251} a^{7} - \frac{772551}{2769251} a^{6} - \frac{1194966}{2769251} a^{5} - \frac{1253042}{2769251} a^{4} - \frac{753537}{2769251} a^{3} - \frac{1278776}{2769251} a^{2} - \frac{589461}{2769251} a - \frac{341613}{2769251}$

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:  $9$
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:  \( \frac{516353}{2769251} a^{14} + \frac{499839}{2769251} a^{13} - \frac{1506283}{2769251} a^{12} - \frac{4150681}{2769251} a^{11} + \frac{1360888}{2769251} a^{10} + \frac{4502646}{2769251} a^{9} + \frac{8516922}{2769251} a^{8} - \frac{155162}{2769251} a^{7} - \frac{15035459}{2769251} a^{6} - \frac{4694437}{2769251} a^{5} + \frac{1346316}{2769251} a^{4} + \frac{10828947}{2769251} a^{3} + \frac{8692265}{2769251} a^{2} - \frac{7116825}{2769251} a - \frac{2685693}{2769251} \),  \( \frac{975382}{2769251} a^{14} + \frac{1318746}{2769251} a^{13} - \frac{1025848}{2769251} a^{12} - \frac{6932722}{2769251} a^{11} - \frac{2091569}{2769251} a^{10} - \frac{758384}{2769251} a^{9} + \frac{17192496}{2769251} a^{8} + \frac{6856687}{2769251} a^{7} - \frac{5296127}{2769251} a^{6} - \frac{14889128}{2769251} a^{5} - \frac{16144955}{2769251} a^{4} + \frac{6020278}{2769251} a^{3} + \frac{8788980}{2769251} a^{2} + \frac{10551271}{2769251} a + \frac{1416907}{2769251} \),  \( \frac{821903}{2769251} a^{14} + \frac{137093}{2769251} a^{13} - \frac{1241546}{2769251} a^{12} - \frac{3437799}{2769251} a^{11} + \frac{3760798}{2769251} a^{10} - \frac{5449550}{2769251} a^{9} + \frac{13733826}{2769251} a^{8} - \frac{9282729}{2769251} a^{7} + \frac{2346488}{2769251} a^{6} + \frac{1957864}{2769251} a^{5} - \frac{13916783}{2769251} a^{4} + \frac{11434490}{2769251} a^{3} - \frac{10460196}{2769251} a^{2} + \frac{9005920}{2769251} a + \frac{4378602}{2769251} \),  \( \frac{450856}{2769251} a^{14} + \frac{658603}{2769251} a^{13} - \frac{335525}{2769251} a^{12} - \frac{3243519}{2769251} a^{11} - \frac{1628068}{2769251} a^{10} - \frac{1183381}{2769251} a^{9} + \frac{8583992}{2769251} a^{8} + \frac{5227289}{2769251} a^{7} - \frac{1170629}{2769251} a^{6} - \frac{10885850}{2769251} a^{5} - \frac{8761450}{2769251} a^{4} + \frac{3342761}{2769251} a^{3} + \frac{8687442}{2769251} a^{2} + \frac{5759105}{2769251} a - \frac{837861}{2769251} \),  \( \frac{137093}{2769251} a^{14} + \frac{1224163}{2769251} a^{13} + \frac{671716}{2769251} a^{12} - \frac{2814426}{2769251} a^{11} - \frac{6271453}{2769251} a^{10} + \frac{2227184}{2769251} a^{9} + \frac{1402010}{2769251} a^{8} + \frac{16318839}{2769251} a^{7} - \frac{1329748}{2769251} a^{6} - \frac{14738686}{2769251} a^{5} - \frac{6647376}{2769251} a^{4} - \frac{8816390}{2769251} a^{3} + \frac{15581144}{2769251} a^{2} + \frac{6844311}{2769251} a + \frac{821903}{2769251} \),  \( \frac{30196}{2769251} a^{14} + \frac{591921}{2769251} a^{13} + \frac{73273}{2769251} a^{12} - \frac{1114758}{2769251} a^{11} - \frac{1930376}{2769251} a^{10} + \frac{2282078}{2769251} a^{9} - \frac{4662082}{2769251} a^{8} + \frac{7876389}{2769251} a^{7} + \frac{220428}{2769251} a^{6} - \frac{2622057}{2769251} a^{5} + \frac{2189432}{2769251} a^{4} - \frac{9944789}{2769251} a^{3} + \frac{515848}{2769251} a^{2} + \frac{4150323}{2769251} a + \frac{2883078}{2769251} \),  \( \frac{108007}{2769251} a^{14} + \frac{248096}{2769251} a^{13} - \frac{572191}{2769251} a^{12} - \frac{1195160}{2769251} a^{11} + \frac{541360}{2769251} a^{10} + \frac{3858771}{2769251} a^{9} - \frac{100665}{2769251} a^{8} + \frac{651854}{2769251} a^{7} - \frac{11690980}{2769251} a^{6} + \frac{1788595}{2769251} a^{5} + \frac{7066080}{2769251} a^{4} + \frac{3785382}{2769251} a^{3} + \frac{5172695}{2769251} a^{2} - \frac{11910741}{2769251} a - \frac{1864218}{2769251} \),  \( \frac{37727}{2769251} a^{14} - \frac{1450559}{2769251} a^{13} - \frac{1322700}{2769251} a^{12} + \frac{2512745}{2769251} a^{11} + \frac{9048896}{2769251} a^{10} - \frac{3310704}{2769251} a^{9} + \frac{806302}{2769251} a^{8} - \frac{20590981}{2769251} a^{7} + \frac{335198}{2769251} a^{6} + \frac{14770253}{2769251} a^{5} + \frac{5906789}{2769251} a^{4} + \frac{11517371}{2769251} a^{3} - \frac{15106736}{2769251} a^{2} - \frac{7048119}{2769251} a + \frac{60503}{2769251} \),  \( \frac{1659764}{2769251} a^{14} - \frac{313763}{2769251} a^{13} - \frac{4413732}{2769251} a^{12} - \frac{7291579}{2769251} a^{11} + \frac{13707205}{2769251} a^{10} - \frac{2983621}{2769251} a^{9} + \frac{26647261}{2769251} a^{8} - \frac{28758914}{2769251} a^{7} - \frac{17124438}{2769251} a^{6} + \frac{6479937}{2769251} a^{5} - \frac{2193072}{2769251} a^{4} + \frac{38628882}{2769251} a^{3} - \frac{15478679}{2769251} a^{2} - \frac{6384410}{2769251} a - \frac{3894086}{2769251} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 211.33204879 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_5\times S_3$ (as 15T4):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 30
The 15 conjugacy class representatives for $S_3 \times C_5$
Character table for $S_3 \times C_5$

Intermediate fields

3.1.44.1, \(\Q(\zeta_{11})^+\)

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

Sibling fields

Galois closure: 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 R $15$ $15$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ $15$ $15$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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
2Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$