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Properties

Label 13.13.7175429735...6241.1
Degree $13$
Signature $[13, 0]$
Discriminant $547^{12}$
Root discriminant $336.80$
Ramified prime $547$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{13}$ (as 13T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1165671, 47088126, -116380476, 91453411, -22393129, -4853400, 3094114, -204415, -107844, 15626, 1123, -252, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 252*x^11 + 1123*x^10 + 15626*x^9 - 107844*x^8 - 204415*x^7 + 3094114*x^6 - 4853400*x^5 - 22393129*x^4 + 91453411*x^3 - 116380476*x^2 + 47088126*x - 1165671)
 
gp: K = bnfinit(x^13 - x^12 - 252*x^11 + 1123*x^10 + 15626*x^9 - 107844*x^8 - 204415*x^7 + 3094114*x^6 - 4853400*x^5 - 22393129*x^4 + 91453411*x^3 - 116380476*x^2 + 47088126*x - 1165671, 1)
 

Normalized defining polynomial

\( x^{13} - x^{12} - 252 x^{11} + 1123 x^{10} + 15626 x^{9} - 107844 x^{8} - 204415 x^{7} + 3094114 x^{6} - 4853400 x^{5} - 22393129 x^{4} + 91453411 x^{3} - 116380476 x^{2} + 47088126 x - 1165671 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $13$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[13, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(717542973516054083971838830896241=547^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $336.80$
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:  $547$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(547\)
Dirichlet character group:    $\lbrace$$\chi_{547}(1,·)$, $\chi_{547}(261,·)$, $\chi_{547}(519,·)$, $\chi_{547}(237,·)$, $\chi_{547}(46,·)$, $\chi_{547}(353,·)$, $\chi_{547}(375,·)$, $\chi_{547}(440,·)$, $\chi_{547}(293,·)$, $\chi_{547}(475,·)$, $\chi_{547}(509,·)$, $\chi_{547}(350,·)$, $\chi_{547}(517,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{117} a^{8} - \frac{2}{39} a^{6} + \frac{2}{13} a^{5} - \frac{2}{13} a^{4} - \frac{1}{13} a^{3} + \frac{41}{117} a^{2} - \frac{3}{13} a$, $\frac{1}{1053} a^{9} + \frac{11}{351} a^{7} + \frac{2}{117} a^{6} - \frac{32}{351} a^{5} - \frac{1}{117} a^{4} + \frac{80}{1053} a^{3} - \frac{55}{117} a^{2} + \frac{4}{9} a$, $\frac{1}{1053} a^{10} - \frac{1}{351} a^{8} + \frac{2}{117} a^{7} + \frac{1}{351} a^{6} + \frac{5}{117} a^{5} - \frac{7}{81} a^{4} - \frac{19}{117} a^{3} - \frac{8}{117} a^{2} + \frac{10}{39} a$, $\frac{1}{5051241} a^{11} - \frac{482}{1683747} a^{10} - \frac{1522}{5051241} a^{9} + \frac{55}{14391} a^{8} + \frac{16658}{1683747} a^{7} - \frac{53}{3321} a^{6} - \frac{294574}{5051241} a^{5} - \frac{159502}{1683747} a^{4} - \frac{821315}{5051241} a^{3} + \frac{68591}{561249} a^{2} - \frac{131335}{561249} a + \frac{5}{13}$, $\frac{1}{4549726744169906169} a^{12} + \frac{320927103838}{4549726744169906169} a^{11} - \frac{1717871911281493}{4549726744169906169} a^{10} - \frac{1087067231024560}{4549726744169906169} a^{9} - \frac{1513681934193424}{1516575581389968723} a^{8} - \frac{73128131352297823}{1516575581389968723} a^{7} - \frac{31841120184163141}{4549726744169906169} a^{6} - \frac{457276327436161849}{4549726744169906169} a^{5} - \frac{12208873978818125}{349978980320762013} a^{4} - \frac{679028580994186922}{4549726744169906169} a^{3} - \frac{12488085479727290}{505525193796656241} a^{2} + \frac{141606424664831462}{505525193796656241} a + \frac{391174973420}{3903096795039}$

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:  $12$
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:  \( 468882596177901.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{13}$ (as 13T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 13
The 13 conjugacy class representatives for $C_{13}$
Character table for $C_{13}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/43.13.0.1}{13} }$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
547Data not computed