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Properties

Label 15.15.4885365289...0625.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{24}\cdot 31^{10}$
Root discriminant $129.60$
Ramified primes $5, 31$
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![-28349, 206735, -202555, -414630, 603860, 102680, -361145, 32860, 83055, -13835, -8259, 1605, 345, -70, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 70*x^13 + 345*x^12 + 1605*x^11 - 8259*x^10 - 13835*x^9 + 83055*x^8 + 32860*x^7 - 361145*x^6 + 102680*x^5 + 603860*x^4 - 414630*x^3 - 202555*x^2 + 206735*x - 28349)
 
gp: K = bnfinit(x^15 - 5*x^14 - 70*x^13 + 345*x^12 + 1605*x^11 - 8259*x^10 - 13835*x^9 + 83055*x^8 + 32860*x^7 - 361145*x^6 + 102680*x^5 + 603860*x^4 - 414630*x^3 - 202555*x^2 + 206735*x - 28349, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 70 x^{13} + 345 x^{12} + 1605 x^{11} - 8259 x^{10} - 13835 x^{9} + 83055 x^{8} + 32860 x^{7} - 361145 x^{6} + 102680 x^{5} + 603860 x^{4} - 414630 x^{3} - 202555 x^{2} + 206735 x - 28349 \)

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:  \(48853652893352568149566650390625=5^{24}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.60$
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:  $5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(775=5^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{775}(1,·)$, $\chi_{775}(676,·)$, $\chi_{775}(521,·)$, $\chi_{775}(621,·)$, $\chi_{775}(366,·)$, $\chi_{775}(656,·)$, $\chi_{775}(466,·)$, $\chi_{775}(211,·)$, $\chi_{775}(501,·)$, $\chi_{775}(311,·)$, $\chi_{775}(56,·)$, $\chi_{775}(36,·)$, $\chi_{775}(346,·)$, $\chi_{775}(156,·)$, $\chi_{775}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{14} a^{12} + \frac{1}{14} a^{11} + \frac{1}{7} a^{10} - \frac{1}{7} a^{8} + \frac{2}{7} a^{7} - \frac{3}{7} a^{6} + \frac{3}{14} a^{5} + \frac{2}{7} a^{4} + \frac{1}{14} a^{3} - \frac{1}{14} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{14} a^{13} + \frac{1}{14} a^{11} - \frac{1}{7} a^{10} - \frac{1}{7} a^{9} + \frac{3}{7} a^{8} + \frac{2}{7} a^{7} - \frac{5}{14} a^{6} + \frac{1}{14} a^{5} - \frac{3}{14} a^{4} - \frac{1}{7} a^{3} + \frac{5}{14} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{1485863211819566863728091344614} a^{14} + \frac{49609664320574944028307439593}{1485863211819566863728091344614} a^{13} + \frac{51198644414192397406764381967}{1485863211819566863728091344614} a^{12} - \frac{246834979581279389276610902223}{1485863211819566863728091344614} a^{11} + \frac{79477534600456895367895087066}{742931605909783431864045672307} a^{10} + \frac{341348779454803028099599871770}{742931605909783431864045672307} a^{9} + \frac{322034571683834250718062805106}{742931605909783431864045672307} a^{8} - \frac{312528950697909874719869028229}{1485863211819566863728091344614} a^{7} + \frac{156076642890301745476540834464}{742931605909783431864045672307} a^{6} + \frac{93167147680646350454761932281}{742931605909783431864045672307} a^{5} - \frac{48948642113642547244148478637}{1485863211819566863728091344614} a^{4} - \frac{308347078617970000224165359147}{1485863211819566863728091344614} a^{3} + \frac{406867521951914415063602174495}{1485863211819566863728091344614} a^{2} + \frac{20985019381106921138672476717}{742931605909783431864045672307} a + \frac{44236216469176298685256737777}{742931605909783431864045672307}$

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:  \( 73815924361.4963 \) (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

3.3.961.1, 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ $15$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ R $15$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ $15$ $15$

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$5.15.24.88$x^{15} + 375 x^{14} + 415 x^{13} + 575 x^{12} + 520 x^{11} + 378 x^{10} + 145 x^{9} + 275 x^{8} + 85 x^{7} + 545 x^{6} + 127 x^{5} + 380 x^{4} + 470 x^{3} + 615 x + 368$$5$$3$$24$$C_{15}$$[2]^{3}$
$31$31.15.10.1$x^{15} + 893730 x^{6} - 923521 x^{3} + 28629151000$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$