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Properties

Label 15.15.1043511493...3529.1
Degree $15$
Signature $[15, 0]$
Discriminant $13^{10}\cdot 31^{14}$
Root discriminant $136.32$
Ramified primes $13, 31$
Class number $3$ (GRH)
Class group $[3]$ (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![-183581, 3647700, -18839782, 16617944, 12870179, -9064776, -2937677, 1754651, 284032, -157948, -12063, 6929, 199, -138, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 138*x^13 + 199*x^12 + 6929*x^11 - 12063*x^10 - 157948*x^9 + 284032*x^8 + 1754651*x^7 - 2937677*x^6 - 9064776*x^5 + 12870179*x^4 + 16617944*x^3 - 18839782*x^2 + 3647700*x - 183581)
 
gp: K = bnfinit(x^15 - x^14 - 138*x^13 + 199*x^12 + 6929*x^11 - 12063*x^10 - 157948*x^9 + 284032*x^8 + 1754651*x^7 - 2937677*x^6 - 9064776*x^5 + 12870179*x^4 + 16617944*x^3 - 18839782*x^2 + 3647700*x - 183581, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 138 x^{13} + 199 x^{12} + 6929 x^{11} - 12063 x^{10} - 157948 x^{9} + 284032 x^{8} + 1754651 x^{7} - 2937677 x^{6} - 9064776 x^{5} + 12870179 x^{4} + 16617944 x^{3} - 18839782 x^{2} + 3647700 x - 183581 \)

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:  \(104351149323786133540261771563529=13^{10}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $136.32$
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:  $13, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(403=13\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{403}(224,·)$, $\chi_{403}(1,·)$, $\chi_{403}(66,·)$, $\chi_{403}(165,·)$, $\chi_{403}(326,·)$, $\chi_{403}(81,·)$, $\chi_{403}(9,·)$, $\chi_{403}(107,·)$, $\chi_{403}(204,·)$, $\chi_{403}(113,·)$, $\chi_{403}(211,·)$, $\chi_{403}(276,·)$, $\chi_{403}(287,·)$, $\chi_{403}(157,·)$, $\chi_{403}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{67} a^{13} - \frac{12}{67} a^{12} + \frac{2}{67} a^{11} + \frac{14}{67} a^{10} + \frac{24}{67} a^{9} - \frac{21}{67} a^{8} - \frac{8}{67} a^{7} + \frac{6}{67} a^{6} - \frac{8}{67} a^{5} + \frac{7}{67} a^{4} - \frac{18}{67} a^{3} + \frac{1}{67} a^{2} - \frac{20}{67} a - \frac{25}{67}$, $\frac{1}{2191900006551318931632581428333912221197743} a^{14} + \frac{769358562879041103000874733855696902293}{2191900006551318931632581428333912221197743} a^{13} + \frac{948586074998002088563246417480986727876631}{2191900006551318931632581428333912221197743} a^{12} + \frac{1011454260984341441157152132868768549225839}{2191900006551318931632581428333912221197743} a^{11} - \frac{495237693500142474653798013029792658330915}{2191900006551318931632581428333912221197743} a^{10} - \frac{21674609127542680470227305366185631505302}{2191900006551318931632581428333912221197743} a^{9} + \frac{915907098518281986594526631384191212459658}{2191900006551318931632581428333912221197743} a^{8} + \frac{211703503729956157229173043168345329706608}{2191900006551318931632581428333912221197743} a^{7} + \frac{924510314449189869277289471762721794010938}{2191900006551318931632581428333912221197743} a^{6} - \frac{559295351453456040404749127822876725347010}{2191900006551318931632581428333912221197743} a^{5} + \frac{472661936875499064038195577311932363092851}{2191900006551318931632581428333912221197743} a^{4} + \frac{417017141529902678845562660345330728228594}{2191900006551318931632581428333912221197743} a^{3} + \frac{662019180693765819668066216337091097054202}{2191900006551318931632581428333912221197743} a^{2} + \frac{967810189790380167119261932716916354137259}{2191900006551318931632581428333912221197743} a - \frac{804910609293605501884204662388784217482544}{2191900006551318931632581428333912221197743}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 35183529541.906876 \) (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

3.3.162409.1, 5.5.923521.1

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 $15$ $15$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ $15$ $15$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$13$13.15.10.3$x^{15} - 676 x^{9} + 114244 x^{3} - 13366548$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
$31$31.15.14.1$x^{15} - 31$$15$$1$$14$$C_{15}$$[\ ]_{15}$