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Properties

Label 13.13.1614053648...3281.1
Degree $13$
Signature $[13, 0]$
Discriminant $859^{12}$
Root discriminant $510.86$
Ramified prime $859$
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![-2211739517, 391778734, 1359880245, -381260855, -162308625, 47857178, 7665285, -2232951, -158783, 45719, 1235, -396, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 396*x^11 + 1235*x^10 + 45719*x^9 - 158783*x^8 - 2232951*x^7 + 7665285*x^6 + 47857178*x^5 - 162308625*x^4 - 381260855*x^3 + 1359880245*x^2 + 391778734*x - 2211739517)
 
gp: K = bnfinit(x^13 - x^12 - 396*x^11 + 1235*x^10 + 45719*x^9 - 158783*x^8 - 2232951*x^7 + 7665285*x^6 + 47857178*x^5 - 162308625*x^4 - 381260855*x^3 + 1359880245*x^2 + 391778734*x - 2211739517, 1)
 

Normalized defining polynomial

\( x^{13} - x^{12} - 396 x^{11} + 1235 x^{10} + 45719 x^{9} - 158783 x^{8} - 2232951 x^{7} + 7665285 x^{6} + 47857178 x^{5} - 162308625 x^{4} - 381260855 x^{3} + 1359880245 x^{2} + 391778734 x - 2211739517 \)

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:  \(161405364891475526005003176560483281=859^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $510.86$
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:  $859$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(859\)
Dirichlet character group:    $\lbrace$$\chi_{859}(1,·)$, $\chi_{859}(100,·)$, $\chi_{859}(773,·)$, $\chi_{859}(551,·)$, $\chi_{859}(555,·)$, $\chi_{859}(524,·)$, $\chi_{859}(718,·)$, $\chi_{859}(463,·)$, $\chi_{859}(849,·)$, $\chi_{859}(374,·)$, $\chi_{859}(503,·)$, $\chi_{859}(124,·)$, $\chi_{859}(478,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{13} a^{7} - \frac{5}{13} a^{6} - \frac{1}{13} a^{5} - \frac{1}{13} a^{3} + \frac{5}{13} a^{2} + \frac{1}{13} a$, $\frac{1}{13} a^{8} - \frac{5}{13} a^{5} - \frac{1}{13} a^{4} + \frac{5}{13} a$, $\frac{1}{13} a^{9} - \frac{5}{13} a^{6} - \frac{1}{13} a^{5} + \frac{5}{13} a^{2}$, $\frac{1}{13} a^{10} - \frac{5}{13} a^{5} - \frac{1}{13} a^{2} + \frac{5}{13} a$, $\frac{1}{138073} a^{11} - \frac{2762}{138073} a^{10} - \frac{4923}{138073} a^{9} + \frac{1842}{138073} a^{8} - \frac{3676}{138073} a^{7} - \frac{54809}{138073} a^{6} + \frac{20570}{138073} a^{5} - \frac{17624}{138073} a^{4} + \frac{43104}{138073} a^{3} + \frac{26566}{138073} a^{2} - \frac{67920}{138073} a + \frac{212}{559}$, $\frac{1}{305691644957705720852691536460925473701} a^{12} + \frac{912580108308589324221101760190755}{305691644957705720852691536460925473701} a^{11} - \frac{8005849721853050070831513697695635895}{305691644957705720852691536460925473701} a^{10} - \frac{10153359039024277405295773617395404127}{305691644957705720852691536460925473701} a^{9} + \frac{4842513277263478108103890167698496181}{305691644957705720852691536460925473701} a^{8} + \frac{9191435795233787455000484918289928575}{305691644957705720852691536460925473701} a^{7} - \frac{70608549062610406423696322390033302169}{305691644957705720852691536460925473701} a^{6} + \frac{78758620202357390475049717532230050166}{305691644957705720852691536460925473701} a^{5} + \frac{85767248751362875280431298551033766200}{305691644957705720852691536460925473701} a^{4} - \frac{3724466169815273764219023060641349942}{305691644957705720852691536460925473701} a^{3} + \frac{5032625045026113012089591302260955517}{305691644957705720852691536460925473701} a^{2} - \frac{65161206351532277691333243197115648248}{305691644957705720852691536460925473701} a - \frac{445124450370505974869278466651657168}{1237617995780185104666767354092815683}$

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:  \( 107792443172161.95 \) (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 ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
859Data not computed