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Properties

Label 11.1.17706569736...8543.1
Degree $11$
Signature $[1, 5]$
Discriminant $-\,4463^{5}$
Root discriminant $45.60$
Ramified prime $4463$
Class number $1$
Class group Trivial
Galois group $D_{11}$ (as 11T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8575, 23275, -27566, 18677, -8282, 2936, -981, 268, -49, 11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 4*x^10 + 11*x^9 - 49*x^8 + 268*x^7 - 981*x^6 + 2936*x^5 - 8282*x^4 + 18677*x^3 - 27566*x^2 + 23275*x - 8575)
 
gp: K = bnfinit(x^11 - 4*x^10 + 11*x^9 - 49*x^8 + 268*x^7 - 981*x^6 + 2936*x^5 - 8282*x^4 + 18677*x^3 - 27566*x^2 + 23275*x - 8575, 1)
 

Normalized defining polynomial

\( x^{11} - 4 x^{10} + 11 x^{9} - 49 x^{8} + 268 x^{7} - 981 x^{6} + 2936 x^{5} - 8282 x^{4} + 18677 x^{3} - 27566 x^{2} + 23275 x - 8575 \)

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

Invariants

Degree:  $11$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1770656973616778543=-\,4463^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.60$
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:  $4463$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{7} - \frac{2}{7} a^{5} - \frac{2}{7} a^{3} - \frac{3}{7} a$, $\frac{1}{595} a^{8} + \frac{29}{595} a^{7} + \frac{6}{85} a^{6} + \frac{78}{595} a^{5} - \frac{29}{119} a^{4} + \frac{116}{595} a^{3} + \frac{17}{35} a^{2} + \frac{169}{595} a - \frac{1}{17}$, $\frac{1}{595} a^{9} - \frac{2}{35} a^{7} - \frac{1}{17} a^{6} - \frac{197}{595} a^{5} - \frac{2}{85} a^{4} - \frac{20}{119} a^{3} + \frac{29}{85} a^{2} - \frac{261}{595} a - \frac{5}{17}$, $\frac{1}{257376175} a^{10} + \frac{2886}{51475235} a^{9} - \frac{213699}{257376175} a^{8} + \frac{422766}{7353605} a^{7} + \frac{3810403}{257376175} a^{6} - \frac{48865979}{257376175} a^{5} - \frac{23817638}{51475235} a^{4} - \frac{96826472}{257376175} a^{3} - \frac{34159446}{257376175} a^{2} - \frac{86117}{7353605} a + \frac{44683}{210103}$

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:  $5$
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:  \( 80486.6635622 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{11}$ (as 11T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 22
The 7 conjugacy class representatives for $D_{11}$
Character table for $D_{11}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

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
4463Data not computed