↑

Properties

Label 11.1.12010159043...3712.1
Degree $11$
Signature $[1, 5]$
Discriminant $-\,2^{9}\cdot 3^{20}\cdot 11^{20}$
Root discriminant $1016.79$
Ramified primes $2, 3, 11$
Class number $11$ (GRH)
Class group $[11]$ (GRH)
Galois group $S_{11}$ (as 11T8)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15630408, 177408, 332288, 354816, 234432, 96096, 22176, 1584, -462, -99, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 99*x^9 - 462*x^8 + 1584*x^7 + 22176*x^6 + 96096*x^5 + 234432*x^4 + 354816*x^3 + 332288*x^2 + 177408*x + 15630408)
 
gp: K = bnfinit(x^11 - 99*x^9 - 462*x^8 + 1584*x^7 + 22176*x^6 + 96096*x^5 + 234432*x^4 + 354816*x^3 + 332288*x^2 + 177408*x + 15630408, 1)
 

Normalized defining polynomial

\( x^{11} - 99 x^{9} - 462 x^{8} + 1584 x^{7} + 22176 x^{6} + 96096 x^{5} + 234432 x^{4} + 354816 x^{3} + 332288 x^{2} + 177408 x + 15630408 \)

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:  \(-1201015904309083794984173196083712=-\,2^{9}\cdot 3^{20}\cdot 11^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1016.79$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{3} + \frac{1}{9} a^{2} - \frac{2}{9} a$, $\frac{1}{54} a^{4} - \frac{1}{18} a^{3} - \frac{1}{9} a^{2} + \frac{13}{27} a - \frac{1}{3}$, $\frac{1}{162} a^{5} - \frac{1}{162} a^{4} + \frac{1}{27} a^{3} - \frac{11}{81} a^{2} - \frac{1}{81} a + \frac{1}{9}$, $\frac{1}{486} a^{6} + \frac{1}{486} a^{5} + \frac{2}{243} a^{4} - \frac{5}{243} a^{3} - \frac{23}{243} a^{2} - \frac{74}{243} a + \frac{11}{27}$, $\frac{1}{1458} a^{7} + \frac{1}{1458} a^{6} + \frac{2}{729} a^{5} - \frac{5}{729} a^{4} + \frac{4}{729} a^{3} - \frac{47}{729} a^{2} + \frac{5}{81} a - \frac{1}{3}$, $\frac{1}{8748} a^{8} + \frac{1}{2916} a^{6} - \frac{7}{4374} a^{5} + \frac{1}{486} a^{4} + \frac{32}{729} a^{3} - \frac{197}{2187} a^{2} + \frac{38}{243} a - \frac{1}{9}$, $\frac{1}{26244} a^{9} - \frac{1}{26244} a^{8} + \frac{1}{8748} a^{7} + \frac{1}{26244} a^{6} + \frac{25}{13122} a^{5} - \frac{4}{2187} a^{4} + \frac{539}{13122} a^{3} - \frac{883}{6561} a^{2} - \frac{301}{729} a + \frac{14}{81}$, $\frac{1}{3700404} a^{10} + \frac{7}{3700404} a^{9} - \frac{25}{1850202} a^{8} + \frac{457}{3700404} a^{7} - \frac{2831}{3700404} a^{6} - \frac{362}{925101} a^{5} - \frac{7697}{925101} a^{4} + \frac{46606}{925101} a^{3} + \frac{110386}{925101} a^{2} - \frac{7985}{102789} a + \frac{5617}{11421}$

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:  \( 750281171506 \) (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 ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ R ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.9.19.5$x^{9} + 18 x^{2} + 6$$9$$1$$19$$(C_3^2:C_8):C_2$$[19/8, 19/8]_{8}^{2}$
$11$11.11.20.5$x^{11} - 11 x^{10} + 858$$11$$1$$20$$C_{11}$$[2]$