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Properties

Label 11.11.6727499949...9201.1
Degree $11$
Signature $[11, 0]$
Discriminant $11^{20}$
Root discriminant $78.24$
Ramified prime $11$
Class number $1$
Class group Trivial
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![-243, -10989, 924, 12760, -1287, -4972, 396, 825, -33, -55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 55*x^9 - 33*x^8 + 825*x^7 + 396*x^6 - 4972*x^5 - 1287*x^4 + 12760*x^3 + 924*x^2 - 10989*x - 243)
 
gp: K = bnfinit(x^11 - 55*x^9 - 33*x^8 + 825*x^7 + 396*x^6 - 4972*x^5 - 1287*x^4 + 12760*x^3 + 924*x^2 - 10989*x - 243, 1)
 

Normalized defining polynomial

\( x^{11} - 55 x^{9} - 33 x^{8} + 825 x^{7} + 396 x^{6} - 4972 x^{5} - 1287 x^{4} + 12760 x^{3} + 924 x^{2} - 10989 x - 243 \)

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:  \(672749994932560009201=11^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.24$
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:  $11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(121=11^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{121}(1,·)$, $\chi_{121}(34,·)$, $\chi_{121}(67,·)$, $\chi_{121}(100,·)$, $\chi_{121}(12,·)$, $\chi_{121}(45,·)$, $\chi_{121}(78,·)$, $\chi_{121}(111,·)$, $\chi_{121}(23,·)$, $\chi_{121}(56,·)$, $\chi_{121}(89,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{81} a^{8} - \frac{1}{27} a^{7} - \frac{1}{27} a^{6} - \frac{1}{9} a^{5} - \frac{2}{27} a^{4} - \frac{19}{81} a^{2} + \frac{4}{27} a + \frac{1}{3}$, $\frac{1}{243} a^{9} + \frac{1}{243} a^{8} - \frac{2}{81} a^{7} - \frac{4}{81} a^{6} + \frac{7}{81} a^{5} + \frac{13}{81} a^{4} - \frac{10}{243} a^{3} + \frac{26}{243} a^{2} - \frac{29}{81} a + \frac{4}{9}$, $\frac{1}{333153} a^{10} + \frac{40}{37017} a^{9} + \frac{2042}{333153} a^{8} + \frac{991}{111051} a^{7} - \frac{4909}{111051} a^{6} + \frac{4607}{37017} a^{5} + \frac{28535}{333153} a^{4} + \frac{58}{457} a^{3} - \frac{148340}{333153} a^{2} - \frac{10396}{111051} a - \frac{2662}{12339}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 278432564.723 \)
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 ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ R ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

In the table, R denotes a ramified prime. 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
$11$11.11.20.9$x^{11} - 11 x^{10} + 11$$11$$1$$20$$C_{11}$$[2]$