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Properties

Label 15.15.1344408835...0625.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{24}\cdot 7^{10}\cdot 41^{8}$
Root discriminant $348.26$
Ramified primes $5, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5^2 : C_3$ (as 15T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4461221029, 13667219210, 13213071845, 2138248810, -2991213425, -980823484, 232522890, 88746755, -7203495, -3209275, 87400, 52195, -340, -380, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 380*x^13 - 340*x^12 + 52195*x^11 + 87400*x^10 - 3209275*x^9 - 7203495*x^8 + 88746755*x^7 + 232522890*x^6 - 980823484*x^5 - 2991213425*x^4 + 2138248810*x^3 + 13213071845*x^2 + 13667219210*x + 4461221029)
 
gp: K = bnfinit(x^15 - 380*x^13 - 340*x^12 + 52195*x^11 + 87400*x^10 - 3209275*x^9 - 7203495*x^8 + 88746755*x^7 + 232522890*x^6 - 980823484*x^5 - 2991213425*x^4 + 2138248810*x^3 + 13213071845*x^2 + 13667219210*x + 4461221029, 1)
 

Normalized defining polynomial

\( x^{15} - 380 x^{13} - 340 x^{12} + 52195 x^{11} + 87400 x^{10} - 3209275 x^{9} - 7203495 x^{8} + 88746755 x^{7} + 232522890 x^{6} - 980823484 x^{5} - 2991213425 x^{4} + 2138248810 x^{3} + 13213071845 x^{2} + 13667219210 x + 4461221029 \)

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:  \(134440883537670166857779026031494140625=5^{24}\cdot 7^{10}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $348.26$
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:  $5, 7, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{14} a^{7} + \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a + \frac{5}{14}$, $\frac{1}{14} a^{8} + \frac{3}{7} a^{5} - \frac{1}{7} a^{4} + \frac{2}{7} a^{3} - \frac{1}{2} a + \frac{2}{7}$, $\frac{1}{98} a^{9} + \frac{1}{98} a^{8} - \frac{1}{49} a^{7} - \frac{3}{49} a^{6} + \frac{17}{49} a^{5} + \frac{11}{49} a^{4} + \frac{3}{49} a^{3} + \frac{37}{98} a^{2} - \frac{1}{2} a + \frac{17}{49}$, $\frac{1}{4018} a^{10} - \frac{26}{2009} a^{8} - \frac{53}{4018} a^{7} + \frac{83}{2009} a^{6} - \frac{867}{2009} a^{5} - \frac{13}{49} a^{4} - \frac{45}{98} a^{3} - \frac{6}{49} a^{2} + \frac{11}{49} a - \frac{29}{98}$, $\frac{1}{56252} a^{11} - \frac{1}{14063} a^{10} + \frac{71}{56252} a^{9} - \frac{9}{56252} a^{8} - \frac{729}{56252} a^{7} - \frac{101}{4018} a^{6} - \frac{379}{2009} a^{5} - \frac{267}{1372} a^{4} + \frac{23}{686} a^{3} - \frac{29}{1372} a^{2} - \frac{439}{1372} a + \frac{533}{1372}$, $\frac{1}{337512} a^{12} - \frac{1}{112504} a^{11} - \frac{31}{337512} a^{10} + \frac{605}{168756} a^{9} + \frac{62}{14063} a^{8} + \frac{1591}{112504} a^{7} + \frac{215}{3444} a^{6} - \frac{15213}{112504} a^{5} - \frac{391}{2744} a^{4} - \frac{3833}{8232} a^{3} + \frac{71}{1029} a^{2} + \frac{106}{343} a - \frac{1063}{8232}$, $\frac{1}{337512} a^{13} + \frac{1}{168756} a^{11} + \frac{25}{337512} a^{10} + \frac{101}{28126} a^{9} + \frac{601}{112504} a^{8} - \frac{4805}{337512} a^{7} - \frac{75}{2744} a^{6} - \frac{3109}{8036} a^{5} + \frac{1469}{4116} a^{4} + \frac{997}{8232} a^{3} - \frac{251}{1372} a^{2} - \frac{1621}{8232} a - \frac{489}{2744}$, $\frac{1}{104398237477724388839302474764307364568} a^{14} - \frac{62863904468848458810792521277865}{52199118738862194419651237382153682284} a^{13} - \frac{12899188873976360658787464419657}{26099559369431097209825618691076841142} a^{12} + \frac{52153396656649759562144366104987}{34799412492574796279767491588102454856} a^{11} - \frac{3149876218076404119806594608534621}{26099559369431097209825618691076841142} a^{10} - \frac{279510907795136299408601002448834215}{104398237477724388839302474764307364568} a^{9} + \frac{956267239904855001655071151916060983}{104398237477724388839302474764307364568} a^{8} - \frac{447368847041347837200680585269531877}{14914033925389198405614639252043909224} a^{7} + \frac{750858505548977323130313961991305615}{13049779684715548604912809345538420571} a^{6} + \frac{3597903257293452456921673288300631566}{13049779684715548604912809345538420571} a^{5} + \frac{960556045880501548027097271688112753}{2546298475066448508275670116202618648} a^{4} - \frac{169262865499404415830041323997303667}{424383079177741418045945019367103108} a^{3} - \frac{1092444771019606762933244419326050591}{2546298475066448508275670116202618648} a^{2} + \frac{288524412110324785441775408470010309}{2546298475066448508275670116202618648} a + \frac{611606503171551363195259865863674623}{1273149237533224254137835058101309324}$

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:  \( 757850777329000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:C_3$ (as 15T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 75
The 11 conjugacy class representatives for $C_5^2 : C_3$
Character table for $C_5^2 : C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 15 sibling: data not computed
Degree 25 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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
5Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
41Data not computed