# Properties

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

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: K<a> := NumberField(PolynomialRing(Rationals())![-1, 4, 3, -15, 3, 24, -19, -17, 32, 1, -18, 9, 0, -6, 0, 1]);
sage: K = 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,"a")
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)

## Normalizeddefining 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 Ramified primes: $23, 191$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is not Galois over $\Q$.

## 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: $-1$ 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

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 750 Conjugacy class representatives for 15T32 Character table for 15T32

## Intermediate fields

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

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 $15$ $15$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ 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$$[\ ]^{5} 23.10.5.2x^{10} - 279841 x^{2} + 12872686$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$191$191.5.4.2$x^{5} + 382$$5$$1$$4$$C_5$$[\ ]_{5} 191.10.0.1x^{10} - x + 28$$1$$10$$0$$C_{10}$$[\ ]^{10}$