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Properties

Label 11.1.129891985607.1
Degree $11$
Signature $[1, 5]$
Discriminant $-129891985607$
Root discriminant $10.24$
Ramified prime $167$
Class number $1$
Class group trivial
Galois group $D_{11}$ (as 11T2)

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Normalized defining polynomial

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

\( x^{11} - x^{10} + 5 x^{9} - 4 x^{8} + 10 x^{7} - 6 x^{6} + 11 x^{5} - 7 x^{4} + 9 x^{3} - 4 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $11$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-129891985607\)\(\medspace = -\,167^{5}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $10.24$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $167$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} + \frac{1}{17} a^{9} + \frac{7}{17} a^{8} - \frac{7}{17} a^{7} - \frac{4}{17} a^{6} + \frac{3}{17} a^{5} - \frac{7}{17} a^{3} - \frac{5}{17} a^{2} + \frac{3}{17} a + \frac{8}{17}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 9.67372434998 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{5}\cdot 9.67372434998 \cdot 1}{2\sqrt{129891985607}}\approx 0.262846302466$

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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: 22.0.2817611963463158891460983.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$167$$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$