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Properties

Label 11.3.25140215886...8928.1
Degree $11$
Signature $[3, 4]$
Discriminant $2^{22}\cdot 3^{15}\cdot 11^{15}$
Root discriminant $470.72$
Ramified primes $2, 3, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $S_{11}$ (as 11T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-819072, -1596672, -5436288, 5588352, 749628, -1085832, -193644, -21780, -1881, 66, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 + 66*x^9 - 1881*x^8 - 21780*x^7 - 193644*x^6 - 1085832*x^5 + 749628*x^4 + 5588352*x^3 - 5436288*x^2 - 1596672*x - 819072)
 
gp: K = bnfinit(x^11 + 66*x^9 - 1881*x^8 - 21780*x^7 - 193644*x^6 - 1085832*x^5 + 749628*x^4 + 5588352*x^3 - 5436288*x^2 - 1596672*x - 819072, 1)
 

Normalized defining polynomial

\( x^{11} + 66 x^{9} - 1881 x^{8} - 21780 x^{7} - 193644 x^{6} - 1085832 x^{5} + 749628 x^{4} + 5588352 x^{3} - 5436288 x^{2} - 1596672 x - 819072 \)

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

Invariants

Degree:  $11$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(251402158861673228847103868928=2^{22}\cdot 3^{15}\cdot 11^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $470.72$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{3} a^{4}$, $\frac{1}{18} a^{5} + \frac{1}{6} a^{2}$, $\frac{1}{396} a^{6} - \frac{1}{66} a^{5} + \frac{4}{33} a^{4} - \frac{1}{44} a^{3} - \frac{1}{22} a^{2} + \frac{7}{22} a + \frac{4}{11}$, $\frac{1}{792} a^{7} + \frac{1}{66} a^{5} + \frac{5}{264} a^{4} + \frac{5}{66} a^{3} + \frac{1}{44} a^{2} - \frac{4}{11} a + \frac{1}{11}$, $\frac{1}{4752} a^{8} + \frac{29}{1584} a^{5} - \frac{7}{132} a^{4} - \frac{37}{264} a^{3} + \frac{16}{33} a^{2} - \frac{3}{22} a - \frac{4}{11}$, $\frac{1}{14256} a^{9} - \frac{1}{2376} a^{7} + \frac{5}{4752} a^{6} + \frac{1}{132} a^{5} + \frac{59}{396} a^{4} + \frac{7}{99} a^{3} - \frac{13}{44} a^{2} + \frac{1}{33} a - \frac{1}{11}$, $\frac{1}{67389156181269397728} a^{10} - \frac{879062900505089}{33694578090634698864} a^{9} + \frac{11238703108843}{3743842010070522096} a^{8} + \frac{12975403980152689}{22463052060423132576} a^{7} + \frac{2832975269561669}{11231526030211566288} a^{6} - \frac{1596150942716353}{311986834172543508} a^{5} - \frac{9130469690091649}{467980251258815262} a^{4} - \frac{36303319851228151}{1871921005035261048} a^{3} + \frac{3513120662339069}{9454146490077076} a^{2} - \frac{34166052455116436}{77996708543135877} a - \frac{11303144003352252}{25998902847711959}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $6$
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:  \( 391564659981 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_{11}$ (as 11T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 39916800
The 56 conjugacy class representatives for $S_{11}$ are not computed
Character table for $S_{11}$ is not computed

Intermediate fields

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

Sibling fields

Degree 22 sibling: 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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.11.19$x^{4} + 22$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.4.11.16$x^{4} + 14$$4$$1$$11$$D_{4}$$[2, 3, 4]$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.5.1$x^{3} + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.6.9.12$x^{6} + 3 x^{4} + 3$$6$$1$$9$$D_{6}$$[2]_{2}^{2}$
$11$11.11.15.3$x^{11} + 110 x^{5} + 11$$11$$1$$15$$F_{11}$$[3/2]_{2}^{5}$