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Properties

Label 11.1.67274999493...0000.1
Degree $11$
Signature $[1, 5]$
Discriminant $-\,2^{11}\cdot 5^{11}\cdot 11^{20}$
Root discriminant $782.42$
Ramified primes $2, 5, 11$
Class number $11$ (GRH)
Class group $[11]$ (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![109999999990, -99, -440, -1155, -1980, -2310, -1848, -990, -330, -55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 55*x^9 - 330*x^8 - 990*x^7 - 1848*x^6 - 2310*x^5 - 1980*x^4 - 1155*x^3 - 440*x^2 - 99*x + 109999999990)
 
gp: K = bnfinit(x^11 - 55*x^9 - 330*x^8 - 990*x^7 - 1848*x^6 - 2310*x^5 - 1980*x^4 - 1155*x^3 - 440*x^2 - 99*x + 109999999990, 1)
 

Normalized defining polynomial

\( x^{11} - 55 x^{9} - 330 x^{8} - 990 x^{7} - 1848 x^{6} - 2310 x^{5} - 1980 x^{4} - 1155 x^{3} - 440 x^{2} - 99 x + 109999999990 \)

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:  \(-67274999493256000920100000000000=-\,2^{11}\cdot 5^{11}\cdot 11^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $782.42$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $\frac{1}{10} a^{2} + \frac{1}{10} a$, $\frac{1}{100} a^{3} + \frac{1}{50} a^{2} - \frac{9}{100} a - \frac{1}{10}$, $\frac{1}{1000} a^{4} + \frac{3}{1000} a^{3} - \frac{7}{1000} a^{2} - \frac{119}{1000} a - \frac{11}{100}$, $\frac{1}{10000} a^{5} + \frac{1}{2500} a^{4} - \frac{1}{2500} a^{3} - \frac{63}{5000} a^{2} - \frac{1229}{10000} a - \frac{111}{1000}$, $\frac{1}{100000} a^{6} - \frac{1}{20000} a^{5} - \frac{1}{2500} a^{4} - \frac{9}{10000} a^{3} - \frac{19}{20000} a^{2} - \frac{49}{100000} a - \frac{1}{10000}$, $\frac{1}{1000000} a^{7} - \frac{1}{250000} a^{6} - \frac{9}{200000} a^{5} - \frac{13}{100000} a^{4} - \frac{37}{200000} a^{3} - \frac{9}{62500} a^{2} - \frac{59}{1000000} a - \frac{1}{100000}$, $\frac{1}{10000000} a^{8} - \frac{3}{10000000} a^{7} - \frac{49}{10000000} a^{6} - \frac{7}{400000} a^{5} - \frac{63}{2000000} a^{4} - \frac{329}{10000000} a^{3} - \frac{203}{10000000} a^{2} - \frac{69}{10000000} a - \frac{1}{1000000}$, $\frac{1}{100000000} a^{9} - \frac{1}{50000000} a^{8} + \frac{3}{6250000} a^{7} + \frac{47}{12500000} a^{6} + \frac{1}{10000000} a^{5} - \frac{3411}{25000000} a^{4} - \frac{18629}{12500000} a^{3} - \frac{171209}{12500000} a^{2} - \frac{12344979}{100000000} a - \frac{1111101}{10000000}$, $\frac{1}{1000000000} a^{10} - \frac{1}{1000000000} a^{9} + \frac{23}{500000000} a^{8} + \frac{53}{125000000} a^{7} + \frac{193}{500000000} a^{6} - \frac{6817}{500000000} a^{5} - \frac{40669}{250000000} a^{4} - \frac{94919}{62500000} a^{3} - \frac{13714651}{1000000000} a^{2} - \frac{123455989}{1000000000} a - \frac{11111101}{100000000}$

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

Class group and class number

$C_{11}$, which has order $11$ (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:  $5$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 159707280519 \) (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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.11.0.1}{11} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$0$Trivial$[\ ]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.5.5.4$x^{5} + 10 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.5.6.2$x^{5} + 15 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
$11$11.11.20.10$x^{11} - 11 x^{10} + 132$$11$$1$$20$$C_{11}$$[2]$