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Properties

Label 13.13.3999932657...2641.1
Degree $13$
Signature $[13, 0]$
Discriminant $521^{12}$
Root discriminant $322.00$
Ramified prime $521$
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![4516075, 31673025, -68466088, -29107980, 23224049, 7791196, -1795569, -616830, 45777, 19153, -293, -240, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 240*x^11 - 293*x^10 + 19153*x^9 + 45777*x^8 - 616830*x^7 - 1795569*x^6 + 7791196*x^5 + 23224049*x^4 - 29107980*x^3 - 68466088*x^2 + 31673025*x + 4516075)
 
gp: K = bnfinit(x^13 - x^12 - 240*x^11 - 293*x^10 + 19153*x^9 + 45777*x^8 - 616830*x^7 - 1795569*x^6 + 7791196*x^5 + 23224049*x^4 - 29107980*x^3 - 68466088*x^2 + 31673025*x + 4516075, 1)
 

Normalized defining polynomial

\( x^{13} - x^{12} - 240 x^{11} - 293 x^{10} + 19153 x^{9} + 45777 x^{8} - 616830 x^{7} - 1795569 x^{6} + 7791196 x^{5} + 23224049 x^{4} - 29107980 x^{3} - 68466088 x^{2} + 31673025 x + 4516075 \)

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:  \(399993265701109317068886081212641=521^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $322.00$
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:  $521$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(521\)
Dirichlet character group:    $\lbrace$$\chi_{521}(320,·)$, $\chi_{521}(1,·)$, $\chi_{521}(226,·)$, $\chi_{521}(324,·)$, $\chi_{521}(101,·)$, $\chi_{521}(422,·)$, $\chi_{521}(423,·)$, $\chi_{521}(302,·)$, $\chi_{521}(18,·)$, $\chi_{521}(255,·)$, $\chi_{521}(284,·)$, $\chi_{521}(29,·)$, $\chi_{521}(421,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{9} - \frac{1}{5} a$, $\frac{1}{25} a^{10} - \frac{2}{25} a^{6} + \frac{1}{25} a^{2}$, $\frac{1}{5875} a^{11} + \frac{76}{5875} a^{10} + \frac{114}{1175} a^{9} - \frac{4}{1175} a^{8} + \frac{98}{5875} a^{7} + \frac{238}{5875} a^{6} + \frac{26}{1175} a^{5} - \frac{446}{1175} a^{4} + \frac{976}{5875} a^{3} - \frac{964}{5875} a^{2} + \frac{1}{5} a + \frac{74}{235}$, $\frac{1}{873548021880746171101344791108125} a^{12} - \frac{55380244948630784680174182687}{873548021880746171101344791108125} a^{11} - \frac{12774055582612104100222033432823}{873548021880746171101344791108125} a^{10} - \frac{10926666358122469029522798056976}{174709604376149234220268958221625} a^{9} - \frac{70287731689703960438633762035092}{873548021880746171101344791108125} a^{8} + \frac{14038894249173767394123795935939}{873548021880746171101344791108125} a^{7} + \frac{37892205047085753178633124904496}{873548021880746171101344791108125} a^{6} - \frac{11454955821605589186679459978494}{174709604376149234220268958221625} a^{5} + \frac{436383839300475998577714012970166}{873548021880746171101344791108125} a^{4} - \frac{64053142726787280973430812878002}{873548021880746171101344791108125} a^{3} - \frac{419545594701004932787718691384973}{873548021880746171101344791108125} a^{2} - \frac{10032303823140760961779677334292}{34941920875229846844053791644325} a + \frac{228608814748413414337784686696}{812602811051856903350088177775}$

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:  \( 15429359577074.87 \) (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 ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
521Data not computed