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Properties

Label 15.15.7497878874...6929.1
Degree $15$
Signature $[15, 0]$
Discriminant $7^{10}\cdot 61^{12}$
Root discriminant $98.10$
Ramified primes $7, 61$
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![-9659, -115927, -96207, 359552, 250272, -393515, -161206, 167356, 34801, -29682, -3320, 2380, 140, -83, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 83*x^13 + 140*x^12 + 2380*x^11 - 3320*x^10 - 29682*x^9 + 34801*x^8 + 167356*x^7 - 161206*x^6 - 393515*x^5 + 250272*x^4 + 359552*x^3 - 96207*x^2 - 115927*x - 9659)
 
gp: K = bnfinit(x^15 - 2*x^14 - 83*x^13 + 140*x^12 + 2380*x^11 - 3320*x^10 - 29682*x^9 + 34801*x^8 + 167356*x^7 - 161206*x^6 - 393515*x^5 + 250272*x^4 + 359552*x^3 - 96207*x^2 - 115927*x - 9659, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 83 x^{13} + 140 x^{12} + 2380 x^{11} - 3320 x^{10} - 29682 x^{9} + 34801 x^{8} + 167356 x^{7} - 161206 x^{6} - 393515 x^{5} + 250272 x^{4} + 359552 x^{3} - 96207 x^{2} - 115927 x - 9659 \)

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:  \(749787887447605250170677896929=7^{10}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.10$
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:  $7, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(427=7\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{427}(1,·)$, $\chi_{427}(386,·)$, $\chi_{427}(424,·)$, $\chi_{427}(9,·)$, $\chi_{427}(142,·)$, $\chi_{427}(400,·)$, $\chi_{427}(81,·)$, $\chi_{427}(302,·)$, $\chi_{427}(375,·)$, $\chi_{427}(184,·)$, $\chi_{427}(58,·)$, $\chi_{427}(123,·)$, $\chi_{427}(156,·)$, $\chi_{427}(253,·)$, $\chi_{427}(95,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{377} a^{11} + \frac{170}{377} a^{10} - \frac{75}{377} a^{9} - \frac{34}{377} a^{8} - \frac{28}{377} a^{7} + \frac{164}{377} a^{6} - \frac{139}{377} a^{5} - \frac{6}{377} a^{4} + \frac{28}{377} a^{3} + \frac{55}{377} a^{2} - \frac{19}{377} a - \frac{9}{29}$, $\frac{1}{377} a^{12} + \frac{54}{377} a^{10} - \frac{102}{377} a^{9} + \frac{97}{377} a^{8} + \frac{23}{377} a^{7} - \frac{121}{377} a^{6} - \frac{127}{377} a^{5} - \frac{83}{377} a^{4} - \frac{181}{377} a^{3} + \frac{56}{377} a^{2} + \frac{97}{377} a - \frac{7}{29}$, $\frac{1}{377} a^{13} + \frac{11}{29} a^{10} - \frac{2}{29} a^{8} - \frac{9}{29} a^{7} + \frac{5}{29} a^{6} - \frac{9}{29} a^{5} + \frac{11}{29} a^{4} + \frac{4}{29} a^{3} + \frac{11}{29} a^{2} + \frac{181}{377} a - \frac{7}{29}$, $\frac{1}{11596580113307505149456555491} a^{14} - \frac{241475482704049086735156}{892044624100577319188965807} a^{13} + \frac{10416199444912154592911307}{11596580113307505149456555491} a^{12} - \frac{8541617538589262841880801}{11596580113307505149456555491} a^{11} - \frac{82853649157921065159488639}{11596580113307505149456555491} a^{10} - \frac{4405138108537933846919201114}{11596580113307505149456555491} a^{9} + \frac{3032111070392277667345599640}{11596580113307505149456555491} a^{8} + \frac{5059748894850686130185710512}{11596580113307505149456555491} a^{7} + \frac{13912758340282615619646384}{30760159451744045489274683} a^{6} + \frac{5723878275970685696867206186}{11596580113307505149456555491} a^{5} + \frac{3682561226113516585242287228}{11596580113307505149456555491} a^{4} + \frac{1433625488645005312176185960}{11596580113307505149456555491} a^{3} + \frac{4367661785916006644898844813}{11596580113307505149456555491} a^{2} + \frac{5443749825697130557396316464}{11596580113307505149456555491} a + \frac{16802890968627675469513680}{892044624100577319188965807}$

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:  \( 32336716619.23167 \) (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

\(\Q(\zeta_{7})^+\), 5.5.13845841.1

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 $15$ $15$ $15$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{15}$ $15$ $15$ $15$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{15}$ $15$ $15$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ $15$ $15$

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
$7$7.15.10.1$x^{15} + 4116 x^{6} - 2401 x^{3} + 1075648$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
$61$61.15.12.1$x^{15} + 3050 x^{10} + 1856779 x^{5} + 22698100000$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$