Properties

Label 15.5.8565893077528823.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,23^{5}\cdot 191^{4}$
Root discriminant $11.54$
Ramified primes $23, 191$
Class number $1$
Class group Trivial
Galois Group 15T32

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 3, -15, 3, 24, -19, -17, 32, 1, -18, 9, 0, -6, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^13 + 9*x^11 - 18*x^10 + x^9 + 32*x^8 - 17*x^7 - 19*x^6 + 24*x^5 + 3*x^4 - 15*x^3 + 3*x^2 + 4*x - 1)
gp: K = bnfinit(x^15 - 6*x^13 + 9*x^11 - 18*x^10 + x^9 + 32*x^8 - 17*x^7 - 19*x^6 + 24*x^5 + 3*x^4 - 15*x^3 + 3*x^2 + 4*x - 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 6 x^{13} \) \(\mathstrut +\mathstrut 9 x^{11} \) \(\mathstrut -\mathstrut 18 x^{10} \) \(\mathstrut +\mathstrut x^{9} \) \(\mathstrut +\mathstrut 32 x^{8} \) \(\mathstrut -\mathstrut 17 x^{7} \) \(\mathstrut -\mathstrut 19 x^{6} \) \(\mathstrut +\mathstrut 24 x^{5} \) \(\mathstrut +\mathstrut 3 x^{4} \) \(\mathstrut -\mathstrut 15 x^{3} \) \(\mathstrut +\mathstrut 3 x^{2} \) \(\mathstrut +\mathstrut 4 x \) \(\mathstrut -\mathstrut 1 \)

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:  \(-8565893077528823=-\,23^{5}\cdot 191^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.54$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $23, 191$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{70253969} a^{14} - \frac{20341715}{70253969} a^{13} + \frac{29826569}{70253969} a^{12} - \frac{32662361}{70253969} a^{11} + \frac{3665471}{70253969} a^{10} - \frac{24043703}{70253969} a^{9} + \frac{6808710}{70253969} a^{8} + \frac{14022021}{70253969} a^{7} + \frac{2281596}{70253969} a^{6} + \frac{22947435}{70253969} a^{5} + \frac{19824924}{70253969} a^{4} + \frac{3012647}{70253969} a^{3} - \frac{10018858}{70253969} a^{2} + \frac{31833807}{70253969} a - \frac{2514324}{70253969}$

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:  \( \frac{7609013}{70253969} a^{14} + \frac{9194900}{70253969} a^{13} - \frac{58826149}{70253969} a^{12} - \frac{66598332}{70253969} a^{11} + \frac{142066968}{70253969} a^{10} + \frac{3755544}{70253969} a^{9} - \frac{242980254}{70253969} a^{8} + \frac{442434353}{70253969} a^{7} + \frac{305599127}{70253969} a^{6} - \frac{689003472}{70253969} a^{5} + \frac{46774654}{70253969} a^{4} + \frac{453912246}{70253969} a^{3} - \frac{266211595}{70253969} a^{2} - \frac{171650655}{70253969} a + \frac{107089837}{70253969} \),  \( \frac{12326426}{70253969} a^{14} - \frac{14061450}{70253969} a^{13} - \frac{42069538}{70253969} a^{12} + \frac{75375941}{70253969} a^{11} - \frac{67284417}{70253969} a^{10} - \frac{304965737}{70253969} a^{9} + \frac{474285587}{70253969} a^{8} - \frac{220465614}{70253969} a^{7} - \frac{355376091}{70253969} a^{6} + \frac{730032657}{70253969} a^{5} - \frac{33826379}{70253969} a^{4} - \frac{375164088}{70253969} a^{3} + \frac{281474677}{70253969} a^{2} + \frac{56053399}{70253969} a - \frac{160555643}{70253969} \),  \( \frac{31215888}{70253969} a^{14} - \frac{29440847}{70253969} a^{13} - \frac{157352401}{70253969} a^{12} + \frac{149178330}{70253969} a^{11} + \frac{119496142}{70253969} a^{10} - \frac{680649843}{70253969} a^{9} + \frac{736876718}{70253969} a^{8} + \frac{281405707}{70253969} a^{7} - \frac{866543200}{70253969} a^{6} + \frac{409843759}{70253969} a^{5} + \frac{358366909}{70253969} a^{4} - \frac{495123430}{70253969} a^{3} + \frac{77470264}{70253969} a^{2} + \frac{222166882}{70253969} a - \frac{105768478}{70253969} \),  \( \frac{44594302}{70253969} a^{14} + \frac{5121714}{70253969} a^{13} - \frac{255911361}{70253969} a^{12} - \frac{24841032}{70253969} a^{11} + \frac{335137446}{70253969} a^{10} - \frac{784918229}{70253969} a^{9} + \frac{24904886}{70253969} a^{8} + \frac{1237976291}{70253969} a^{7} - \frac{604871607}{70253969} a^{6} - \frac{578033933}{70253969} a^{5} + \frac{805229885}{70253969} a^{4} + \frac{225218694}{70253969} a^{3} - \frac{491301104}{70253969} a^{2} - \frac{22288137}{70253969} a + \frac{108202462}{70253969} \),  \( \frac{4796110}{70253969} a^{14} + \frac{21989898}{70253969} a^{13} - \frac{47355024}{70253969} a^{12} - \frac{116647448}{70253969} a^{11} + \frac{140693033}{70253969} a^{10} + \frac{25400650}{70253969} a^{9} - \frac{479022325}{70253969} a^{8} + \frac{613567029}{70253969} a^{7} + \frac{237942027}{70253969} a^{6} - \frac{742817860}{70253969} a^{5} + \frac{302823257}{70253969} a^{4} + \frac{274122754}{70253969} a^{3} - \frac{289135295}{70253969} a^{2} - \frac{84736883}{70253969} a + \frac{119299210}{70253969} \),  \( \frac{15450992}{70253969} a^{14} + \frac{1638036}{70253969} a^{13} - \frac{80236558}{70253969} a^{12} - \frac{4899062}{70253969} a^{11} + \frac{66495820}{70253969} a^{10} - \frac{289869516}{70253969} a^{9} + \frac{79893022}{70253969} a^{8} + \frac{307930678}{70253969} a^{7} - \frac{239085092}{70253969} a^{6} + \frac{23443591}{70253969} a^{5} + \frac{201125801}{70253969} a^{4} - \frac{68560382}{70253969} a^{3} - \frac{186814086}{70253969} a^{2} + \frac{53174519}{70253969} a + \frac{29006305}{70253969} \),  \( \frac{11954488}{70253969} a^{14} + \frac{52302672}{70253969} a^{13} - \frac{83007377}{70253969} a^{12} - \frac{289241735}{70253969} a^{11} + \frac{164047106}{70253969} a^{10} + \frac{115142543}{70253969} a^{9} - \frac{974207199}{70253969} a^{8} + \frac{814285938}{70253969} a^{7} + \frac{964887823}{70253969} a^{6} - \frac{1175717664}{70253969} a^{5} + \frac{109923180}{70253969} a^{4} + \frac{922313018}{70253969} a^{3} - \frac{367827938}{70253969} a^{2} - \frac{245058997}{70253969} a + \frac{142518786}{70253969} \),  \( \frac{27123610}{70253969} a^{14} - \frac{4532177}{70253969} a^{13} - \frac{156031704}{70253969} a^{12} + \frac{24948575}{70253969} a^{11} + \frac{199100270}{70253969} a^{10} - \frac{517059731}{70253969} a^{9} + \frac{186332800}{70253969} a^{8} + \frac{731675503}{70253969} a^{7} - \frac{547401005}{70253969} a^{6} - \frac{149550096}{70253969} a^{5} + \frac{541431237}{70253969} a^{4} - \frac{74635548}{70253969} a^{3} - \frac{256913581}{70253969} a^{2} + \frac{71380926}{70253969} a + \frac{21483761}{70253969} \),  \( \frac{39989981}{70253969} a^{14} + \frac{9360189}{70253969} a^{13} - \frac{230140221}{70253969} a^{12} - \frac{64855497}{70253969} a^{11} + \frac{300502187}{70253969} a^{10} - \frac{587294417}{70253969} a^{9} - \frac{32944154}{70253969} a^{8} + \frac{1063639294}{70253969} a^{7} - \frac{237485570}{70253969} a^{6} - \frac{672540491}{70253969} a^{5} + \frac{419445446}{70253969} a^{4} + \frac{256026243}{70253969} a^{3} - \frac{333124590}{70253969} a^{2} - \frac{54970445}{70253969} a + \frac{132968770}{70253969} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 87.8024975924 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T32:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 750
The 65 conjugacy class representatives for [5^3]S(3) are not computed
Character table for [5^3]S(3) is not computed

Intermediate fields

3.1.23.1

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 $15$ $15$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ R $15$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$23$23.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
23.10.5.2$x^{10} - 279841 x^{2} + 12872686$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$191$191.5.4.2$x^{5} + 382$$5$$1$$4$$C_5$$[\ ]_{5}$
191.10.0.1$x^{10} - x + 28$$1$$10$$0$$C_{10}$$[\ ]^{10}$