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Properties

Label 15.15.7867402601...2481.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{20}\cdot 41^{12}$
Root discriminant $84.41$
Ramified primes $3, 41$
Class number $1$ (GRH)
Class group Trivial (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![-8373, -22284, 39540, 91153, -49737, -121443, 12471, 62079, 3897, -13178, -1311, 1311, 114, -60, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 60*x^13 + 114*x^12 + 1311*x^11 - 1311*x^10 - 13178*x^9 + 3897*x^8 + 62079*x^7 + 12471*x^6 - 121443*x^5 - 49737*x^4 + 91153*x^3 + 39540*x^2 - 22284*x - 8373)
 
gp: K = bnfinit(x^15 - 3*x^14 - 60*x^13 + 114*x^12 + 1311*x^11 - 1311*x^10 - 13178*x^9 + 3897*x^8 + 62079*x^7 + 12471*x^6 - 121443*x^5 - 49737*x^4 + 91153*x^3 + 39540*x^2 - 22284*x - 8373, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 60 x^{13} + 114 x^{12} + 1311 x^{11} - 1311 x^{10} - 13178 x^{9} + 3897 x^{8} + 62079 x^{7} + 12471 x^{6} - 121443 x^{5} - 49737 x^{4} + 91153 x^{3} + 39540 x^{2} - 22284 x - 8373 \)

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:  \(78674026011431622215094122481=3^{20}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.41$
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:  $3, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(369=3^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{369}(256,·)$, $\chi_{369}(1,·)$, $\chi_{369}(100,·)$, $\chi_{369}(133,·)$, $\chi_{369}(262,·)$, $\chi_{369}(10,·)$, $\chi_{369}(139,·)$, $\chi_{369}(223,·)$, $\chi_{369}(16,·)$, $\chi_{369}(247,·)$, $\chi_{369}(160,·)$, $\chi_{369}(346,·)$, $\chi_{369}(283,·)$, $\chi_{369}(124,·)$, $\chi_{369}(37,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{9} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{27} a^{11} - \frac{1}{27} a^{10} + \frac{1}{27} a^{9} - \frac{1}{3} a^{8} + \frac{4}{9} a^{7} + \frac{1}{3} a^{6} - \frac{4}{27} a^{5} + \frac{13}{27} a^{4} - \frac{10}{27} a^{3} - \frac{1}{3} a^{2} - \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{27} a^{12} + \frac{1}{27} a^{9} + \frac{1}{9} a^{8} - \frac{2}{9} a^{7} + \frac{5}{27} a^{6} + \frac{1}{3} a^{5} + \frac{1}{9} a^{4} - \frac{1}{27} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{27} a^{13} + \frac{1}{27} a^{10} - \frac{2}{9} a^{8} - \frac{13}{27} a^{7} - \frac{1}{3} a^{6} - \frac{2}{9} a^{5} + \frac{8}{27} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{247777074773292112323183} a^{14} - \frac{705274630823863672748}{247777074773292112323183} a^{13} - \frac{2430625709956914947786}{247777074773292112323183} a^{12} - \frac{102561143434855437110}{247777074773292112323183} a^{11} - \frac{11386445993895967462721}{247777074773292112323183} a^{10} - \frac{10428995225825563346111}{247777074773292112323183} a^{9} - \frac{115962792829947720261808}{247777074773292112323183} a^{8} + \frac{48761865471794255094806}{247777074773292112323183} a^{7} + \frac{13750592488952216200529}{247777074773292112323183} a^{6} + \frac{59146732706882301215837}{247777074773292112323183} a^{5} - \frac{61222534302805884850207}{247777074773292112323183} a^{4} + \frac{440994498804996662426}{3394206503743727566071} a^{3} - \frac{3204243007970555119690}{27530786085921345813687} a^{2} - \frac{421259980422527741177}{1131402167914575855357} a - \frac{28277535747481739465198}{82592358257764037441061}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 18946465206.742867 \) (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

\(\Q(\zeta_{9})^+\), 5.5.2825761.1

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 $15$ R $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ $15$ $15$ $15$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ R $15$ $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
$41$41.15.12.1$x^{15} + 2665 x^{10} + 1418764 x^{5} + 25589884853$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$