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Properties

Label 15.15.3654400263...5625.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{24}\cdot 19^{10}$
Root discriminant $93.51$
Ramified primes $5, 19$
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![2401, 9800, -17710, -55625, 82545, 43089, -81705, -4955, 29685, -2980, -4439, 745, 260, -50, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 50*x^13 + 260*x^12 + 745*x^11 - 4439*x^10 - 2980*x^9 + 29685*x^8 - 4955*x^7 - 81705*x^6 + 43089*x^5 + 82545*x^4 - 55625*x^3 - 17710*x^2 + 9800*x + 2401)
 
gp: K = bnfinit(x^15 - 5*x^14 - 50*x^13 + 260*x^12 + 745*x^11 - 4439*x^10 - 2980*x^9 + 29685*x^8 - 4955*x^7 - 81705*x^6 + 43089*x^5 + 82545*x^4 - 55625*x^3 - 17710*x^2 + 9800*x + 2401, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 50 x^{13} + 260 x^{12} + 745 x^{11} - 4439 x^{10} - 2980 x^{9} + 29685 x^{8} - 4955 x^{7} - 81705 x^{6} + 43089 x^{5} + 82545 x^{4} - 55625 x^{3} - 17710 x^{2} + 9800 x + 2401 \)

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:  \(365440026390612125396728515625=5^{24}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.51$
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:  $5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(475=5^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{475}(96,·)$, $\chi_{475}(1,·)$, $\chi_{475}(391,·)$, $\chi_{475}(296,·)$, $\chi_{475}(201,·)$, $\chi_{475}(106,·)$, $\chi_{475}(11,·)$, $\chi_{475}(406,·)$, $\chi_{475}(311,·)$, $\chi_{475}(216,·)$, $\chi_{475}(121,·)$, $\chi_{475}(26,·)$, $\chi_{475}(381,·)$, $\chi_{475}(286,·)$, $\chi_{475}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{49} a^{11} + \frac{3}{49} a^{10} + \frac{3}{49} a^{9} + \frac{2}{49} a^{8} - \frac{3}{7} a^{6} + \frac{13}{49} a^{5} - \frac{10}{49} a^{4} + \frac{4}{49} a^{3} + \frac{19}{49} a^{2} - \frac{1}{7} a$, $\frac{1}{49} a^{12} + \frac{1}{49} a^{10} + \frac{1}{49} a^{8} - \frac{22}{49} a^{6} - \frac{22}{49} a^{4} - \frac{22}{49} a^{2}$, $\frac{1}{7399} a^{13} + \frac{2}{7399} a^{12} - \frac{68}{7399} a^{11} + \frac{173}{7399} a^{10} + \frac{508}{7399} a^{9} + \frac{74}{7399} a^{8} + \frac{6}{7399} a^{7} - \frac{1633}{7399} a^{6} - \frac{723}{7399} a^{5} + \frac{2032}{7399} a^{4} - \frac{3658}{7399} a^{3} + \frac{1277}{7399} a^{2} - \frac{474}{1057} a + \frac{44}{151}$, $\frac{1}{699192908127222607} a^{14} + \frac{46755261178056}{699192908127222607} a^{13} + \frac{93074637234192}{14269243023004543} a^{12} - \frac{2969946382433477}{699192908127222607} a^{11} + \frac{49276306216377287}{699192908127222607} a^{10} - \frac{4237620374872071}{699192908127222607} a^{9} - \frac{9522573190372747}{699192908127222607} a^{8} - \frac{4017121165362764}{699192908127222607} a^{7} + \frac{27260933122117795}{99884701161031801} a^{6} - \frac{112076033040745957}{699192908127222607} a^{5} - \frac{242221223136511589}{699192908127222607} a^{4} + \frac{337071987668508370}{699192908127222607} a^{3} + \frac{44780333870462580}{99884701161031801} a^{2} - \frac{2040148524762616}{14269243023004543} a + \frac{131540357786125}{291209041285807}$

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:  \( 47458423135.208374 \) (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

3.3.361.1, 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 ${\href{/LocalNumberField/7.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ $15$ $15$ R $15$ $15$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ $15$ $15$ $15$

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}$$[2]^{3}$
$19$19.15.10.1$x^{15} + 102885 x^{6} - 130321 x^{3} + 309512375$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$