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Properties

Label 13.13.2133643557...2801.1
Degree $13$
Signature $[13, 0]$
Discriminant $599^{12}$
Root discriminant $366.25$
Ramified prime $599$
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![57031547, -55304818, -20502539, 34969245, -5465967, -4924742, 1684091, 114077, -109375, 8169, 1967, -276, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 276*x^11 + 1967*x^10 + 8169*x^9 - 109375*x^8 + 114077*x^7 + 1684091*x^6 - 4924742*x^5 - 5465967*x^4 + 34969245*x^3 - 20502539*x^2 - 55304818*x + 57031547)
 
gp: K = bnfinit(x^13 - x^12 - 276*x^11 + 1967*x^10 + 8169*x^9 - 109375*x^8 + 114077*x^7 + 1684091*x^6 - 4924742*x^5 - 5465967*x^4 + 34969245*x^3 - 20502539*x^2 - 55304818*x + 57031547, 1)
 

Normalized defining polynomial

\( x^{13} - x^{12} - 276 x^{11} + 1967 x^{10} + 8169 x^{9} - 109375 x^{8} + 114077 x^{7} + 1684091 x^{6} - 4924742 x^{5} - 5465967 x^{4} + 34969245 x^{3} - 20502539 x^{2} - 55304818 x + 57031547 \)

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:  \(2133643557240451317422184503752801=599^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $366.25$
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:  $599$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(599\)
Dirichlet character group:    $\lbrace$$\chi_{599}(1,·)$, $\chi_{599}(421,·)$, $\chi_{599}(361,·)$, $\chi_{599}(459,·)$, $\chi_{599}(338,·)$, $\chi_{599}(270,·)$, $\chi_{599}(335,·)$, $\chi_{599}(432,·)$, $\chi_{599}(434,·)$, $\chi_{599}(19,·)$, $\chi_{599}(212,·)$, $\chi_{599}(375,·)$, $\chi_{599}(536,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{128109284441928108968429961621641041} a^{12} + \frac{59758820702796741030183147213264288}{128109284441928108968429961621641041} a^{11} + \frac{20611808671482864972647267699180156}{128109284441928108968429961621641041} a^{10} + \frac{31169912239317186086708644895547706}{128109284441928108968429961621641041} a^{9} - \frac{6304058097457155210357265093578076}{128109284441928108968429961621641041} a^{8} + \frac{50503195829424519998606746746513066}{128109284441928108968429961621641041} a^{7} - \frac{50948838601702200056188297912023305}{128109284441928108968429961621641041} a^{6} - \frac{26326866063919171382431247220419188}{128109284441928108968429961621641041} a^{5} + \frac{9883217898291327366882993947128655}{128109284441928108968429961621641041} a^{4} + \frac{56538401497877282618378393873013980}{128109284441928108968429961621641041} a^{3} + \frac{34991072369985416435391707779506781}{128109284441928108968429961621641041} a^{2} + \frac{37085215665199627181185807068611159}{128109284441928108968429961621641041} a + \frac{29904214785857672478273274219438499}{128109284441928108968429961621641041}$

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:  \( 1689788670896.741 \) (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 ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.13.0.1}{13} }$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
599Data not computed