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Properties

Label 11.3.75642061235...4976.2
Degree $11$
Signature $[3, 4]$
Discriminant $2^{8}\cdot 3^{12}\cdot 11^{18}$
Root discriminant $277.67$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\PSL(2,11)$ (as 11T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27801869, 15347409, -393800, -2919719, 149754, 81158, 30976, 198, -847, -55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 55*x^9 - 847*x^8 + 198*x^7 + 30976*x^6 + 81158*x^5 + 149754*x^4 - 2919719*x^3 - 393800*x^2 + 15347409*x + 27801869)
 
gp: K = bnfinit(x^11 - 55*x^9 - 847*x^8 + 198*x^7 + 30976*x^6 + 81158*x^5 + 149754*x^4 - 2919719*x^3 - 393800*x^2 + 15347409*x + 27801869, 1)
 

Normalized defining polynomial

\( x^{11} - 55 x^{9} - 847 x^{8} + 198 x^{7} + 30976 x^{6} + 81158 x^{5} + 149754 x^{4} - 2919719 x^{3} - 393800 x^{2} + 15347409 x + 27801869 \)

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:  \(756420612351903997566494976=2^{8}\cdot 3^{12}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $277.67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12} a^{5} + \frac{1}{6} a^{3} + \frac{5}{12} a^{2} - \frac{1}{4} a + \frac{1}{12}$, $\frac{1}{72} a^{6} + \frac{1}{72} a^{5} - \frac{5}{36} a^{4} + \frac{7}{72} a^{3} + \frac{1}{36} a^{2} - \frac{13}{36} a - \frac{11}{72}$, $\frac{1}{864} a^{7} + \frac{7}{864} a^{5} + \frac{17}{864} a^{4} - \frac{329}{864} a^{3} + \frac{31}{432} a^{2} - \frac{133}{288} a - \frac{115}{864}$, $\frac{1}{1728} a^{8} - \frac{1}{1728} a^{7} + \frac{7}{1728} a^{6} + \frac{5}{864} a^{5} - \frac{29}{864} a^{4} + \frac{679}{1728} a^{3} + \frac{403}{1728} a^{2} + \frac{215}{432} a - \frac{173}{1728}$, $\frac{1}{62208} a^{9} - \frac{1}{2592} a^{7} + \frac{17}{62208} a^{6} + \frac{53}{10368} a^{5} - \frac{1307}{20736} a^{4} + \frac{1009}{7776} a^{3} - \frac{4423}{20736} a^{2} - \frac{5957}{20736} a + \frac{2701}{62208}$, $\frac{1}{86190054827855095296} a^{10} + \frac{13976575711777}{86190054827855095296} a^{9} - \frac{808451630858749}{3591252284493962304} a^{8} + \frac{43386722043507065}{86190054827855095296} a^{7} - \frac{424280755569672817}{86190054827855095296} a^{6} + \frac{910432880939016431}{28730018275951698432} a^{5} + \frac{8381222492553913655}{86190054827855095296} a^{4} + \frac{33969374225833928819}{86190054827855095296} a^{3} + \frac{23498104373510663}{266018687740293504} a^{2} - \frac{15733460435832627073}{43095027413927547648} a + \frac{37765050308080018669}{86190054827855095296}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 29099104590.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(2,11)$ (as 11T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 660
The 8 conjugacy class representatives for $\PSL(2,11)$
Character table for $\PSL(2,11)$

Intermediate fields

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

Sibling fields

Degree 12 sibling: data not computed
Arithmetically equvalently sibling: 11.3.756420612351903997566494976.1

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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
$11$11.11.18.5$x^{11} + 110 x^{8} + 11$$11$$1$$18$$C_{11}:C_5$$[9/5]_{5}$