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Properties

Label 11.1.26024049340425599.1
Degree $11$
Signature $[1, 5]$
Discriminant $-\,19^{5}\cdot 101^{5}$
Root discriminant $31.07$
Ramified primes $19, 101$
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![53, -142, 143, -87, 81, -53, 9, 0, 5, -1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 2*x^10 - x^9 + 5*x^8 + 9*x^6 - 53*x^5 + 81*x^4 - 87*x^3 + 143*x^2 - 142*x + 53)
 
gp: K = bnfinit(x^11 - 2*x^10 - x^9 + 5*x^8 + 9*x^6 - 53*x^5 + 81*x^4 - 87*x^3 + 143*x^2 - 142*x + 53, 1)
 

Normalized defining polynomial

\( x^{11} - 2 x^{10} - x^{9} + 5 x^{8} + 9 x^{6} - 53 x^{5} + 81 x^{4} - 87 x^{3} + 143 x^{2} - 142 x + 53 \)

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:  \(-26024049340425599=-\,19^{5}\cdot 101^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.07$
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:  $19, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{19} a^{8} + \frac{6}{19} a^{7} - \frac{6}{19} a^{6} + \frac{7}{19} a^{5} - \frac{2}{19} a^{4} - \frac{2}{19} a^{3} - \frac{9}{19} a^{2} + \frac{6}{19} a + \frac{6}{19}$, $\frac{1}{133} a^{9} + \frac{2}{133} a^{8} + \frac{65}{133} a^{7} + \frac{31}{133} a^{6} + \frac{65}{133} a^{5} + \frac{44}{133} a^{4} - \frac{20}{133} a^{3} + \frac{61}{133} a^{2} - \frac{37}{133} a + \frac{33}{133}$, $\frac{1}{2527} a^{10} + \frac{1}{361} a^{9} + \frac{5}{2527} a^{8} - \frac{862}{2527} a^{7} - \frac{1089}{2527} a^{6} + \frac{810}{2527} a^{5} + \frac{473}{2527} a^{4} + \frac{899}{2527} a^{3} + \frac{632}{2527} a^{2} - \frac{572}{2527} a - \frac{388}{2527}$

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:  \( 6213.58467616 \)
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 ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ R ${\href{/LocalNumberField/23.11.0.1}{11} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

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
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$