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Properties

Label 15.15.1288144726...5625.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{24}\cdot 43^{10}$
Root discriminant $161.19$
Ramified primes $5, 43$
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![59599, -566105, -106595, 3237450, 750200, -1946324, -430345, 460840, 90315, -53115, -8439, 3125, 345, -90, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 90*x^13 + 345*x^12 + 3125*x^11 - 8439*x^10 - 53115*x^9 + 90315*x^8 + 460840*x^7 - 430345*x^6 - 1946324*x^5 + 750200*x^4 + 3237450*x^3 - 106595*x^2 - 566105*x + 59599)
 
gp: K = bnfinit(x^15 - 5*x^14 - 90*x^13 + 345*x^12 + 3125*x^11 - 8439*x^10 - 53115*x^9 + 90315*x^8 + 460840*x^7 - 430345*x^6 - 1946324*x^5 + 750200*x^4 + 3237450*x^3 - 106595*x^2 - 566105*x + 59599, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 90 x^{13} + 345 x^{12} + 3125 x^{11} - 8439 x^{10} - 53115 x^{9} + 90315 x^{8} + 460840 x^{7} - 430345 x^{6} - 1946324 x^{5} + 750200 x^{4} + 3237450 x^{3} - 106595 x^{2} - 566105 x + 59599 \)

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:  \(1288144726352944910526275634765625=5^{24}\cdot 43^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $161.19$
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:  $5, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1075=5^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{1075}(896,·)$, $\chi_{1075}(1,·)$, $\chi_{1075}(866,·)$, $\chi_{1075}(36,·)$, $\chi_{1075}(6,·)$, $\chi_{1075}(646,·)$, $\chi_{1075}(681,·)$, $\chi_{1075}(651,·)$, $\chi_{1075}(431,·)$, $\chi_{1075}(466,·)$, $\chi_{1075}(436,·)$, $\chi_{1075}(216,·)$, $\chi_{1075}(251,·)$, $\chi_{1075}(221,·)$, $\chi_{1075}(861,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{14} a^{12} + \frac{1}{14} a^{11} + \frac{3}{7} a^{9} - \frac{3}{7} a^{8} + \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{3}{14} a^{5} - \frac{5}{14} a^{3} + \frac{1}{14} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{14} a^{13} - \frac{1}{14} a^{11} - \frac{1}{14} a^{10} - \frac{5}{14} a^{9} - \frac{3}{7} a^{8} + \frac{1}{14} a^{6} - \frac{3}{14} a^{5} - \frac{5}{14} a^{4} - \frac{1}{14} a^{3} + \frac{1}{14} a^{2} + \frac{3}{14} a - \frac{5}{14}$, $\frac{1}{20685964642859652708060498605152214} a^{14} + \frac{43760940578912178814671208508287}{2955137806122807529722928372164602} a^{13} - \frac{11876688266676924464842616965932}{10342982321429826354030249302576107} a^{12} + \frac{108284145919625258242889361615773}{10342982321429826354030249302576107} a^{11} - \frac{3366138072410758738701525355247283}{20685964642859652708060498605152214} a^{10} + \frac{8244893605487038155745689749194655}{20685964642859652708060498605152214} a^{9} - \frac{702526017014858846400681961776015}{1477568903061403764861464186082301} a^{8} - \frac{5635139448570040994258798585493675}{20685964642859652708060498605152214} a^{7} + \frac{5160214238208488917788579026590724}{10342982321429826354030249302576107} a^{6} - \frac{7123017442487178750520068606610467}{20685964642859652708060498605152214} a^{5} - \frac{6284342708689480615362742194487679}{20685964642859652708060498605152214} a^{4} - \frac{4782255690970619023268894157653065}{20685964642859652708060498605152214} a^{3} - \frac{3024141475315742290458568365209815}{10342982321429826354030249302576107} a^{2} + \frac{9380215372305079646766092814456497}{20685964642859652708060498605152214} a + \frac{1158814829627688166547355398830035}{2955137806122807529722928372164602}$

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:  \( 402299381527.07513 \) (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

3.3.1849.1, 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 ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ $15$ $15$ $15$ $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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}$$[2]^{3}$
$43$43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$