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Properties

Label 11.1.36030006250...0000.1
Degree $11$
Signature $[1, 5]$
Discriminant $-\,2^{15}\cdot 5^{19}\cdot 7^{8}$
Root discriminant $170.77$
Ramified primes $2, 5, 7$
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![985507, 510231, -169795, -55695, 39450, 4226, -3642, 270, 195, -25, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 3*x^10 - 25*x^9 + 195*x^8 + 270*x^7 - 3642*x^6 + 4226*x^5 + 39450*x^4 - 55695*x^3 - 169795*x^2 + 510231*x + 985507)
 
gp: K = bnfinit(x^11 - 3*x^10 - 25*x^9 + 195*x^8 + 270*x^7 - 3642*x^6 + 4226*x^5 + 39450*x^4 - 55695*x^3 - 169795*x^2 + 510231*x + 985507, 1)
 

Normalized defining polynomial

\( x^{11} - 3 x^{10} - 25 x^{9} + 195 x^{8} + 270 x^{7} - 3642 x^{6} + 4226 x^{5} + 39450 x^{4} - 55695 x^{3} - 169795 x^{2} + 510231 x + 985507 \)

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:  \(-3603000625000000000000000=-\,2^{15}\cdot 5^{19}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $170.77$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{8} a^{4} - \frac{1}{8}$, $\frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{1}{16} a + \frac{1}{16}$, $\frac{1}{32} a^{6} + \frac{1}{32} a^{4} - \frac{1}{8} a^{3} + \frac{3}{32} a^{2} - \frac{1}{8} a + \frac{3}{32}$, $\frac{1}{128} a^{7} + \frac{1}{128} a^{6} - \frac{3}{128} a^{5} - \frac{3}{128} a^{4} + \frac{7}{128} a^{3} - \frac{9}{128} a^{2} + \frac{11}{128} a - \frac{5}{128}$, $\frac{1}{512} a^{8} + \frac{1}{128} a^{6} - \frac{1}{32} a^{5} + \frac{9}{256} a^{4} - \frac{1}{32} a^{3} - \frac{29}{128} a^{2} + \frac{125}{512}$, $\frac{1}{32768} a^{9} + \frac{15}{32768} a^{8} - \frac{23}{8192} a^{7} - \frac{93}{8192} a^{6} + \frac{1}{16384} a^{5} - \frac{561}{16384} a^{4} - \frac{65}{8192} a^{3} + \frac{1845}{8192} a^{2} - \frac{7267}{32768} a - \frac{10573}{32768}$, $\frac{1}{2097152} a^{10} - \frac{13}{1048576} a^{9} + \frac{573}{2097152} a^{8} + \frac{425}{262144} a^{7} - \frac{6197}{1048576} a^{6} + \frac{4819}{524288} a^{5} - \frac{55721}{1048576} a^{4} + \frac{26319}{262144} a^{3} - \frac{507479}{2097152} a^{2} + \frac{508231}{1048576} a + \frac{134741}{2097152}$

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:  $5$
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:  \( 1216049840.91 \) (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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.11.13$x^{4} + 4 x^{2} + 14$$4$$1$$11$$D_{4}$$[3, 4]^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.10.19.28$x^{10} - 20 x^{5} + 60$$10$$1$$19$$(C_5^2 : C_4) : C_2$$[7/4, 9/4]_{4}^{2}$
$7$7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$