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Properties

Label 11.5.28798830305...0000.1
Degree $11$
Signature $[5, 3]$
Discriminant $-\,2^{20}\cdot 3^{15}\cdot 5^{8}\cdot 7^{2}$
Root discriminant $72.43$
Ramified primes $2, 3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (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![-1048, -6016, -3040, 14040, -12735, 4878, -444, -255, 105, -20, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 2*x^10 - 20*x^9 + 105*x^8 - 255*x^7 - 444*x^6 + 4878*x^5 - 12735*x^4 + 14040*x^3 - 3040*x^2 - 6016*x - 1048)
 
gp: K = bnfinit(x^11 - 2*x^10 - 20*x^9 + 105*x^8 - 255*x^7 - 444*x^6 + 4878*x^5 - 12735*x^4 + 14040*x^3 - 3040*x^2 - 6016*x - 1048, 1)
 

Normalized defining polynomial

\( x^{11} - 2 x^{10} - 20 x^{9} + 105 x^{8} - 255 x^{7} - 444 x^{6} + 4878 x^{5} - 12735 x^{4} + 14040 x^{3} - 3040 x^{2} - 6016 x - 1048 \)

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

Invariants

Degree:  $11$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-287988303052800000000=-\,2^{20}\cdot 3^{15}\cdot 5^{8}\cdot 7^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.43$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{30} a^{6} + \frac{1}{10} a^{5} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{2}{5} a + \frac{2}{15}$, $\frac{1}{30} a^{7} - \frac{3}{10} a^{5} - \frac{1}{6} a^{4} + \frac{1}{10} a^{2} + \frac{1}{3} a - \frac{2}{5}$, $\frac{1}{900} a^{8} + \frac{1}{225} a^{7} + \frac{7}{900} a^{6} - \frac{263}{900} a^{5} + \frac{1}{9} a^{4} + \frac{13}{900} a^{3} + \frac{71}{450} a^{2} - \frac{11}{225} a - \frac{71}{225}$, $\frac{1}{900} a^{9} - \frac{1}{100} a^{7} + \frac{1}{100} a^{6} + \frac{7}{25} a^{5} - \frac{43}{100} a^{4} + \frac{13}{30} a^{3} + \frac{8}{25} a^{2} - \frac{3}{25} a - \frac{91}{225}$, $\frac{1}{194953500} a^{10} + \frac{12787}{38990700} a^{9} + \frac{2347}{6498450} a^{8} - \frac{3599}{722050} a^{7} + \frac{1531}{1299690} a^{6} - \frac{6626119}{32492250} a^{5} + \frac{2875043}{12996900} a^{4} - \frac{6140251}{12996900} a^{3} - \frac{1292278}{3249225} a^{2} - \frac{2887229}{9747675} a + \frac{5490916}{48738375}$

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:  $7$
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:  \( 60162090.9342 \) (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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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}$
2.4.10.1$x^{4} + 2 x^{2} - 9$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.10.1$x^{4} + 2 x^{2} - 9$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.11.5$x^{6} + 3 x^{3} + 6$$6$$1$$11$$S_3^2$$[2, 5/2]_{2}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$