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Properties

Label 15.15.8217006435...5625.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{24}\cdot 13^{10}$
Root discriminant $72.61$
Ramified primes $5, 13$
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![151, -490, -7780, 9345, 33245, -19249, -34955, 13045, 13695, -3980, -2349, 585, 180, -40, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 40*x^13 + 180*x^12 + 585*x^11 - 2349*x^10 - 3980*x^9 + 13695*x^8 + 13045*x^7 - 34955*x^6 - 19249*x^5 + 33245*x^4 + 9345*x^3 - 7780*x^2 - 490*x + 151)
 
gp: K = bnfinit(x^15 - 5*x^14 - 40*x^13 + 180*x^12 + 585*x^11 - 2349*x^10 - 3980*x^9 + 13695*x^8 + 13045*x^7 - 34955*x^6 - 19249*x^5 + 33245*x^4 + 9345*x^3 - 7780*x^2 - 490*x + 151, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 40 x^{13} + 180 x^{12} + 585 x^{11} - 2349 x^{10} - 3980 x^{9} + 13695 x^{8} + 13045 x^{7} - 34955 x^{6} - 19249 x^{5} + 33245 x^{4} + 9345 x^{3} - 7780 x^{2} - 490 x + 151 \)

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:  \(8217006435930728912353515625=5^{24}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.61$
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:  $5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(325=5^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{325}(256,·)$, $\chi_{325}(1,·)$, $\chi_{325}(66,·)$, $\chi_{325}(131,·)$, $\chi_{325}(196,·)$, $\chi_{325}(261,·)$, $\chi_{325}(321,·)$, $\chi_{325}(16,·)$, $\chi_{325}(81,·)$, $\chi_{325}(146,·)$, $\chi_{325}(211,·)$, $\chi_{325}(276,·)$, $\chi_{325}(61,·)$, $\chi_{325}(126,·)$, $\chi_{325}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{2}{7} a^{10} - \frac{1}{7} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{11} + \frac{1}{7} a^{10} - \frac{3}{7} a^{8} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{1065605562735719002843657} a^{14} + \frac{20139632931300850908753}{1065605562735719002843657} a^{13} - \frac{22545411783630783466865}{1065605562735719002843657} a^{12} + \frac{58378003859460592224025}{1065605562735719002843657} a^{11} - \frac{88583657096104786985665}{1065605562735719002843657} a^{10} + \frac{250725216407997293673969}{1065605562735719002843657} a^{9} + \frac{126442943407333605450017}{1065605562735719002843657} a^{8} - \frac{85257264310334073386566}{1065605562735719002843657} a^{7} + \frac{5398890418377332588816}{1065605562735719002843657} a^{6} - \frac{218704565538058300922745}{1065605562735719002843657} a^{5} - \frac{443758242612099692432935}{1065605562735719002843657} a^{4} - \frac{52387610593327281989726}{1065605562735719002843657} a^{3} + \frac{86196600580208868212130}{1065605562735719002843657} a^{2} - \frac{480307167585265399344488}{1065605562735719002843657} a + \frac{453502934308786204294724}{1065605562735719002843657}$

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:  \( 1285347259.8659444 \) (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

3.3.169.1, 5.5.390625.1

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 $15$ $15$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ $15$ R $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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
$5$5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
$13$13.15.10.1$x^{15} + 79092 x^{6} - 228488 x^{3} + 80199288$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$