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Properties

Label 15.15.1586393893...4721.1
Degree $15$
Signature $[15, 0]$
Discriminant $631^{14}$
Root discriminant $410.54$
Ramified prime $631$
Class number $11$ (GRH)
Class group $[11]$ (GRH)
Galois group $C_{15}$ (as 15T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-110965429, -67361473, 487995784, 304223819, -237407410, -165257997, 22528734, 22372090, -638978, -1219824, -353, 29253, 127, -294, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 294*x^13 + 127*x^12 + 29253*x^11 - 353*x^10 - 1219824*x^9 - 638978*x^8 + 22372090*x^7 + 22528734*x^6 - 165257997*x^5 - 237407410*x^4 + 304223819*x^3 + 487995784*x^2 - 67361473*x - 110965429)
 
gp: K = bnfinit(x^15 - x^14 - 294*x^13 + 127*x^12 + 29253*x^11 - 353*x^10 - 1219824*x^9 - 638978*x^8 + 22372090*x^7 + 22528734*x^6 - 165257997*x^5 - 237407410*x^4 + 304223819*x^3 + 487995784*x^2 - 67361473*x - 110965429, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 294 x^{13} + 127 x^{12} + 29253 x^{11} - 353 x^{10} - 1219824 x^{9} - 638978 x^{8} + 22372090 x^{7} + 22528734 x^{6} - 165257997 x^{5} - 237407410 x^{4} + 304223819 x^{3} + 487995784 x^{2} - 67361473 x - 110965429 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1586393893300992738237335763411468444721=631^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $410.54$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $631$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(631\)
Dirichlet character group:    $\lbrace$$\chi_{631}(64,·)$, $\chi_{631}(1,·)$, $\chi_{631}(43,·)$, $\chi_{631}(228,·)$, $\chi_{631}(310,·)$, $\chi_{631}(8,·)$, $\chi_{631}(587,·)$, $\chi_{631}(242,·)$, $\chi_{631}(79,·)$, $\chi_{631}(562,·)$, $\chi_{631}(339,·)$, $\chi_{631}(512,·)$, $\chi_{631}(279,·)$, $\chi_{631}(344,·)$, $\chi_{631}(188,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{12599} a^{13} - \frac{2806}{12599} a^{12} - \frac{3292}{12599} a^{11} + \frac{1316}{12599} a^{10} + \frac{5115}{12599} a^{9} + \frac{94}{293} a^{8} + \frac{4496}{12599} a^{7} - \frac{2560}{12599} a^{6} - \frac{1857}{12599} a^{5} + \frac{5389}{12599} a^{4} - \frac{2118}{12599} a^{3} + \frac{2025}{12599} a^{2} - \frac{1648}{12599} a - \frac{3513}{12599}$, $\frac{1}{53512925189422502830963760774322648940574166656276593} a^{14} + \frac{118592812541805878405211305013210867429287520296}{53512925189422502830963760774322648940574166656276593} a^{13} + \frac{1591869520984156712396857367207228947820581220606535}{53512925189422502830963760774322648940574166656276593} a^{12} + \frac{18614673280162605575398937865824954375218765139179680}{53512925189422502830963760774322648940574166656276593} a^{11} + \frac{11477296889761020790999442316667576156688463530257523}{53512925189422502830963760774322648940574166656276593} a^{10} + \frac{6642148656507797175811235245530971219664241029373179}{53512925189422502830963760774322648940574166656276593} a^{9} - \frac{1287808107918947560166372516343332287101172407253318}{53512925189422502830963760774322648940574166656276593} a^{8} + \frac{1812254686439872189276857899658290725277687298488003}{53512925189422502830963760774322648940574166656276593} a^{7} + \frac{25993020611356930659661535697237170062356856682946762}{53512925189422502830963760774322648940574166656276593} a^{6} + \frac{1871254091810630290583099028992667167047801650912293}{53512925189422502830963760774322648940574166656276593} a^{5} + \frac{12964360386676446536279720515349186314414191634560810}{53512925189422502830963760774322648940574166656276593} a^{4} - \frac{4236399140608327441522213045225232747219746732455111}{53512925189422502830963760774322648940574166656276593} a^{3} + \frac{16281377050449914589279669049821422906026174067154519}{53512925189422502830963760774322648940574166656276593} a^{2} - \frac{24421946156330185622801142929883921830623905540688990}{53512925189422502830963760774322648940574166656276593} a - \frac{21686842058959925087239452852101286176949079677881750}{53512925189422502830963760774322648940574166656276593}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{11}$, which has order $11$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 40655123886363.37 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{15}$ (as 15T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 15
The 15 conjugacy class representatives for $C_{15}$
Character table for $C_{15}$

Intermediate fields

3.3.398161.1, 5.5.158532181921.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $15$ $15$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{15}$ $15$ $15$ $15$

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
631Data not computed