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Properties

Label 11.1.12708314623...4768.1
Degree $11$
Signature $[1, 5]$
Discriminant $-\,2^{15}\cdot 7^{8}\cdot 11^{20}$
Root discriminant $829.00$
Ramified primes $2, 7, 11$
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![-142879744, -147702016, -65231936, -16783536, -2978272, -259952, 35420, 9009, -308, -154, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 154*x^9 - 308*x^8 + 9009*x^7 + 35420*x^6 - 259952*x^5 - 2978272*x^4 - 16783536*x^3 - 65231936*x^2 - 147702016*x - 142879744)
 
gp: K = bnfinit(x^11 - 154*x^9 - 308*x^8 + 9009*x^7 + 35420*x^6 - 259952*x^5 - 2978272*x^4 - 16783536*x^3 - 65231936*x^2 - 147702016*x - 142879744, 1)
 

Normalized defining polynomial

\( x^{11} - 154 x^{9} - 308 x^{8} + 9009 x^{7} + 35420 x^{6} - 259952 x^{5} - 2978272 x^{4} - 16783536 x^{3} - 65231936 x^{2} - 147702016 x - 142879744 \)

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:  \(-127083146233027522514188173344768=-\,2^{15}\cdot 7^{8}\cdot 11^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $829.00$
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:  $2, 7, 11$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} + \frac{1}{16} a^{4}$, $\frac{1}{896} a^{8} - \frac{19}{896} a^{7} - \frac{9}{896} a^{6} - \frac{17}{896} a^{5} + \frac{15}{224} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{896} a^{9} + \frac{11}{448} a^{7} - \frac{5}{224} a^{6} + \frac{17}{896} a^{5} - \frac{9}{224} a^{4} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{23779087200009214976} a^{10} - \frac{2234139114210011}{5944771800002303744} a^{9} - \frac{2025999905245433}{11889543600004607488} a^{8} - \frac{29837683920542401}{5944771800002303744} a^{7} - \frac{18138295278871665}{3397012457144173568} a^{6} - \frac{103545069273896125}{2972385900001151872} a^{5} - \frac{129984189948222805}{1486192950000575936} a^{4} + \frac{14422419914201717}{106156639285755424} a^{3} + \frac{17084125801769555}{212313278571510848} a^{2} - \frac{9730187086169661}{26539159821438856} a - \frac{6242274712325209}{13269579910719428}$

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:  \( 1078218924920 \) (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 ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R R ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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}$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.4.9.3$x^{4} + 6 x^{2} + 10$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$11$11.11.20.6$x^{11} - 11 x^{10} + 979$$11$$1$$20$$C_{11}$$[2]$