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Properties

Label 11.11.2187417278...3401.1
Degree $11$
Signature $[11, 0]$
Discriminant $859^{10}$
Root discriminant $464.80$
Ramified prime $859$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{11}$ (as 11T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-145865807, 385923388, -250960565, 36326009, 11558015, -2723930, -146438, 52046, 653, -390, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 390*x^9 + 653*x^8 + 52046*x^7 - 146438*x^6 - 2723930*x^5 + 11558015*x^4 + 36326009*x^3 - 250960565*x^2 + 385923388*x - 145865807)
 
gp: K = bnfinit(x^11 - x^10 - 390*x^9 + 653*x^8 + 52046*x^7 - 146438*x^6 - 2723930*x^5 + 11558015*x^4 + 36326009*x^3 - 250960565*x^2 + 385923388*x - 145865807, 1)
 

Normalized defining polynomial

\( x^{11} - x^{10} - 390 x^{9} + 653 x^{8} + 52046 x^{7} - 146438 x^{6} - 2723930 x^{5} + 11558015 x^{4} + 36326009 x^{3} - 250960565 x^{2} + 385923388 x - 145865807 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $11$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(218741727855135890482344953401=859^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $464.80$
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$$\chi_{859}(1,·)$, $\chi_{859}(61,·)$, $\chi_{859}(169,·)$, $\chi_{859}(13,·)$, $\chi_{859}(205,·)$, $\chi_{859}(269,·)$, $\chi_{859}(214,·)$, $\chi_{859}(88,·)$, $\chi_{859}(793,·)$, $\chi_{859}(285,·)$, $\chi_{859}(479,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{1114782749592579720774895786190407} a^{10} + \frac{371592414753387183275868589899869}{1114782749592579720774895786190407} a^{9} - \frac{101540489658247583070083316283308}{1114782749592579720774895786190407} a^{8} - \frac{21998966196854822836042217062752}{1114782749592579720774895786190407} a^{7} + \frac{71282596110962285738091969365430}{1114782749592579720774895786190407} a^{6} - \frac{2937618334867801484437215196020}{48468815199677379164125903747409} a^{5} + \frac{396121849722711493239214107836}{6443830922500460813727721307459} a^{4} + \frac{277648673684933061910578102113434}{1114782749592579720774895786190407} a^{3} + \frac{129310737935916669156173659777766}{1114782749592579720774895786190407} a^{2} + \frac{538164579388386738740248449543812}{1114782749592579720774895786190407} a - \frac{36362784110151111824432102666807}{1114782749592579720774895786190407}$

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:  $10$
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:  \( 80664010957.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{11}$ (as 11T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 11
The 11 conjugacy class representatives for $C_{11}$
Character table for $C_{11}$

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 ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

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