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Properties

Label 15.15.1348779085...8929.1
Degree $15$
Signature $[15, 0]$
Discriminant $11^{12}\cdot 73^{10}$
Root discriminant $118.94$
Ramified primes $11, 73$
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![3358343, 15844999, -22419019, -23518862, 32472178, -2779461, -6655484, 1451632, 529243, -150182, -19414, 6618, 326, -133, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 133*x^13 + 326*x^12 + 6618*x^11 - 19414*x^10 - 150182*x^9 + 529243*x^8 + 1451632*x^7 - 6655484*x^6 - 2779461*x^5 + 32472178*x^4 - 23518862*x^3 - 22419019*x^2 + 15844999*x + 3358343)
 
gp: K = bnfinit(x^15 - 2*x^14 - 133*x^13 + 326*x^12 + 6618*x^11 - 19414*x^10 - 150182*x^9 + 529243*x^8 + 1451632*x^7 - 6655484*x^6 - 2779461*x^5 + 32472178*x^4 - 23518862*x^3 - 22419019*x^2 + 15844999*x + 3358343, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 133 x^{13} + 326 x^{12} + 6618 x^{11} - 19414 x^{10} - 150182 x^{9} + 529243 x^{8} + 1451632 x^{7} - 6655484 x^{6} - 2779461 x^{5} + 32472178 x^{4} - 23518862 x^{3} - 22419019 x^{2} + 15844999 x + 3358343 \)

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:  \(13487790856470777216989713088929=11^{12}\cdot 73^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $118.94$
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:  $11, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(803=11\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{803}(64,·)$, $\chi_{803}(1,·)$, $\chi_{803}(738,·)$, $\chi_{803}(356,·)$, $\chi_{803}(137,·)$, $\chi_{803}(300,·)$, $\chi_{803}(658,·)$, $\chi_{803}(366,·)$, $\chi_{803}(592,·)$, $\chi_{803}(81,·)$, $\chi_{803}(210,·)$, $\chi_{803}(147,·)$, $\chi_{803}(665,·)$, $\chi_{803}(731,·)$, $\chi_{803}(575,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{62489531289280267687631580873035950153873089} a^{14} + \frac{2608114117805246938306875037042530990619400}{20829843763093422562543860291011983384624363} a^{13} - \frac{455847671199271903657939172866848046446462}{20829843763093422562543860291011983384624363} a^{12} + \frac{3005512008984236341231918743351909466175698}{20829843763093422562543860291011983384624363} a^{11} - \frac{4657465652685871239787170357992499977395420}{62489531289280267687631580873035950153873089} a^{10} - \frac{19574897864621605729195719190900527581957323}{62489531289280267687631580873035950153873089} a^{9} + \frac{27887472371032199938958357321582395143096251}{62489531289280267687631580873035950153873089} a^{8} + \frac{10096071436595513741194812375300421747002493}{20829843763093422562543860291011983384624363} a^{7} - \frac{9103984088054973875435639586373227591733699}{62489531289280267687631580873035950153873089} a^{6} - \frac{8142734416498310754463240380691561038040477}{20829843763093422562543860291011983384624363} a^{5} + \frac{19063956614667395806571180650317760292819537}{62489531289280267687631580873035950153873089} a^{4} + \frac{18627998245009777631574999269561173790096060}{62489531289280267687631580873035950153873089} a^{3} - \frac{3487276855978325745338484219582872119457938}{20829843763093422562543860291011983384624363} a^{2} - \frac{3583393489370929826540601204675898735604386}{20829843763093422562543860291011983384624363} a - \frac{272921787988745541801454940016391527068833}{1453244913704192271805385601698510468694723}$

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:  \( 35246330370.40924 \) (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

3.3.5329.1, \(\Q(\zeta_{11})^+\)

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 $15$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ R $15$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{15}$ $15$ $15$ $15$

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
$11$11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$
$73$73.15.10.1$x^{15} + 5835255 x^{6} - 28398241 x^{3} + 259133949125$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$