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Properties

Label 13.13.3267537072...5521.1
Degree $13$
Signature $[13, 0]$
Discriminant $911^{12}$
Root discriminant $539.34$
Ramified prime $911$
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![-57869, -40719, 1733969, 4833237, 1482536, -3569396, 31458, 735262, -193733, 2721, 4253, -420, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 420*x^11 + 4253*x^10 + 2721*x^9 - 193733*x^8 + 735262*x^7 + 31458*x^6 - 3569396*x^5 + 1482536*x^4 + 4833237*x^3 + 1733969*x^2 - 40719*x - 57869)
 
gp: K = bnfinit(x^13 - x^12 - 420*x^11 + 4253*x^10 + 2721*x^9 - 193733*x^8 + 735262*x^7 + 31458*x^6 - 3569396*x^5 + 1482536*x^4 + 4833237*x^3 + 1733969*x^2 - 40719*x - 57869, 1)
 

Normalized defining polynomial

\( x^{13} - x^{12} - 420 x^{11} + 4253 x^{10} + 2721 x^{9} - 193733 x^{8} + 735262 x^{7} + 31458 x^{6} - 3569396 x^{5} + 1482536 x^{4} + 4833237 x^{3} + 1733969 x^{2} - 40719 x - 57869 \)

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:  \(326753707264991140811478515720435521=911^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $539.34$
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:  $911$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(911\)
Dirichlet character group:    $\lbrace$$\chi_{911}(128,·)$, $\chi_{911}(1,·)$, $\chi_{911}(196,·)$, $\chi_{911}(581,·)$, $\chi_{911}(897,·)$, $\chi_{911}(65,·)$, $\chi_{911}(491,·)$, $\chi_{911}(414,·)$, $\chi_{911}(577,·)$, $\chi_{911}(121,·)$, $\chi_{911}(900,·)$, $\chi_{911}(154,·)$, $\chi_{911}(30,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{133} a^{9} - \frac{1}{19} a^{8} + \frac{5}{133} a^{7} - \frac{5}{19} a^{5} - \frac{7}{19} a^{4} - \frac{15}{133} a^{3} + \frac{6}{19} a^{2} + \frac{44}{133} a + \frac{2}{19}$, $\frac{1}{133} a^{10} - \frac{6}{133} a^{8} - \frac{3}{133} a^{7} + \frac{3}{133} a^{6} + \frac{29}{133} a^{5} + \frac{60}{133} a^{4} + \frac{32}{133} a^{3} - \frac{23}{133} a^{2} - \frac{58}{133} a - \frac{5}{19}$, $\frac{1}{184471} a^{11} - \frac{62}{26353} a^{10} + \frac{253}{184471} a^{9} - \frac{10631}{184471} a^{8} - \frac{12391}{184471} a^{7} - \frac{127}{9709} a^{6} - \frac{44182}{184471} a^{5} - \frac{51904}{184471} a^{4} - \frac{13065}{184471} a^{3} + \frac{59999}{184471} a^{2} + \frac{5466}{26353} a + \frac{2441}{26353}$, $\frac{1}{3587736488104423580954189} a^{12} - \frac{9056712455923260132}{3587736488104423580954189} a^{11} - \frac{76717390438333704877}{188828236216022293734431} a^{10} - \frac{5911496384615498838466}{3587736488104423580954189} a^{9} + \frac{186708149157043488269977}{3587736488104423580954189} a^{8} - \frac{185437075879006022726693}{3587736488104423580954189} a^{7} - \frac{19313923114232422605400}{3587736488104423580954189} a^{6} + \frac{1572532224705314179691257}{3587736488104423580954189} a^{5} + \frac{1312688499392061731281648}{3587736488104423580954189} a^{4} + \frac{788882040843221895958057}{3587736488104423580954189} a^{3} - \frac{1079696915952071278487589}{3587736488104423580954189} a^{2} - \frac{3681738228157691025593}{10459873143161584784123} a + \frac{6069142723659558906730}{73219112002131093488861}$

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:  \( 345319680754792.2 \) (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 ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.13.0.1}{13} }$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
911Data not computed