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Properties

Label 13.13.8841624172...5681.1
Degree $13$
Signature $[13, 0]$
Discriminant $313^{12}$
Root discriminant $201.18$
Ramified prime $313$
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![-409375, 253950, 1134730, -650724, -917970, 512628, 196143, -109398, -9620, 6530, 161, -144, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 144*x^11 + 161*x^10 + 6530*x^9 - 9620*x^8 - 109398*x^7 + 196143*x^6 + 512628*x^5 - 917970*x^4 - 650724*x^3 + 1134730*x^2 + 253950*x - 409375)
 
gp: K = bnfinit(x^13 - x^12 - 144*x^11 + 161*x^10 + 6530*x^9 - 9620*x^8 - 109398*x^7 + 196143*x^6 + 512628*x^5 - 917970*x^4 - 650724*x^3 + 1134730*x^2 + 253950*x - 409375, 1)
 

Normalized defining polynomial

\( x^{13} - x^{12} - 144 x^{11} + 161 x^{10} + 6530 x^{9} - 9620 x^{8} - 109398 x^{7} + 196143 x^{6} + 512628 x^{5} - 917970 x^{4} - 650724 x^{3} + 1134730 x^{2} + 253950 x - 409375 \)

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:  \(884162417215006648162206715681=313^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $201.18$
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:  $313$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(313\)
Dirichlet character group:    $\lbrace$$\chi_{313}(1,·)$, $\chi_{313}(294,·)$, $\chi_{313}(103,·)$, $\chi_{313}(234,·)$, $\chi_{313}(44,·)$, $\chi_{313}(48,·)$, $\chi_{313}(113,·)$, $\chi_{313}(277,·)$, $\chi_{313}(150,·)$, $\chi_{313}(280,·)$, $\chi_{313}(249,·)$, $\chi_{313}(58,·)$, $\chi_{313}(27,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{7} + \frac{1}{25} a^{6} + \frac{2}{25} a^{5} + \frac{2}{25} a^{4} - \frac{4}{25} a^{3} + \frac{1}{25} a^{2} - \frac{1}{5} a$, $\frac{1}{25} a^{8} + \frac{1}{25} a^{6} - \frac{1}{25} a^{4} + \frac{2}{5} a^{3} - \frac{1}{25} a^{2} + \frac{2}{5} a$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{6} + \frac{2}{25} a^{5} - \frac{2}{25} a^{4} - \frac{7}{25} a^{3} - \frac{1}{25} a^{2} - \frac{2}{5} a$, $\frac{1}{3625} a^{10} - \frac{7}{725} a^{9} - \frac{16}{3625} a^{8} - \frac{67}{3625} a^{7} - \frac{19}{725} a^{6} + \frac{341}{3625} a^{5} + \frac{172}{3625} a^{4} - \frac{38}{125} a^{3} + \frac{8}{29} a^{2} - \frac{59}{145} a + \frac{6}{29}$, $\frac{1}{2628125} a^{11} - \frac{217}{2628125} a^{10} - \frac{49036}{2628125} a^{9} + \frac{1819}{105125} a^{8} + \frac{15434}{2628125} a^{7} + \frac{166111}{2628125} a^{6} - \frac{4233}{105125} a^{5} + \frac{198289}{2628125} a^{4} + \frac{958754}{2628125} a^{3} + \frac{194174}{525625} a^{2} + \frac{48497}{105125} a + \frac{70}{841}$, $\frac{1}{94867598624609375} a^{12} - \frac{9218116429}{94867598624609375} a^{11} + \frac{6077567184843}{94867598624609375} a^{10} - \frac{217130475748143}{94867598624609375} a^{9} + \frac{1079543007441609}{94867598624609375} a^{8} - \frac{406086214662272}{94867598624609375} a^{7} - \frac{5611714837050607}{94867598624609375} a^{6} - \frac{4276434169717686}{94867598624609375} a^{5} + \frac{500807941726986}{94867598624609375} a^{4} - \frac{22397077178286478}{94867598624609375} a^{3} - \frac{95637243330903}{18973519724921875} a^{2} + \frac{1605571579111786}{3794703944984375} a - \frac{781608495677}{30357631559875}$

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:  \( 927977436616.0109 \) (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 ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
313Data not computed