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Properties

Label 11.3.31423265337...0625.1
Degree $11$
Signature $[3, 4]$
Discriminant $3^{14}\cdot 5^{10}\cdot 11^{20}$
Root discriminant $1368.09$
Ramified primes $3, 5, 11$
Class number $11$ (GRH)
Class group $[11]$ (GRH)
Galois group $A_{11}$ (as 11T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9205180875, 8463163500, 404956750, -916892625, 105215550, -2498650, 390225, -18150, -3025, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 3025*x^8 - 18150*x^7 + 390225*x^6 - 2498650*x^5 + 105215550*x^4 - 916892625*x^3 + 404956750*x^2 + 8463163500*x + 9205180875)
 
gp: K = bnfinit(x^11 - 3025*x^8 - 18150*x^7 + 390225*x^6 - 2498650*x^5 + 105215550*x^4 - 916892625*x^3 + 404956750*x^2 + 8463163500*x + 9205180875, 1)
 

Normalized defining polynomial

\( x^{11} - 3025 x^{8} - 18150 x^{7} + 390225 x^{6} - 2498650 x^{5} + 105215550 x^{4} - 916892625 x^{3} + 404956750 x^{2} + 8463163500 x + 9205180875 \)

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:  \(31423265337037027486797829775390625=3^{14}\cdot 5^{10}\cdot 11^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1368.09$
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:  $3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{55} a^{6}$, $\frac{1}{55} a^{7}$, $\frac{1}{825} a^{8} - \frac{1}{165} a^{7} + \frac{1}{15} a^{5} + \frac{1}{15} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{825} a^{9} + \frac{1}{165} a^{7} - \frac{1}{165} a^{6} - \frac{1}{15} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{716457974324762125688067254818575} a^{10} - \frac{74941419222492724479332177644}{716457974324762125688067254818575} a^{9} - \frac{36689949589872489283717877147}{716457974324762125688067254818575} a^{8} + \frac{64938393488309831702326565593}{28658318972990485027522690192743} a^{7} + \frac{241098587126357410165412404061}{28658318972990485027522690192743} a^{6} - \frac{119731233083515867622915207287}{2605301724817316820683880926613} a^{5} - \frac{180605384852556468294394545922}{2605301724817316820683880926613} a^{4} - \frac{617133304435389396389157847250}{2605301724817316820683880926613} a^{3} + \frac{1188699857642752033510633134524}{2605301724817316820683880926613} a^{2} - \frac{414671316132441253587754201952}{868433908272438940227960308871} a + \frac{142674827729121839403025088040}{289477969424146313409320102957}$

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:  $6$
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:  \( 3043690739730 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{11}$ (as 11T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 19958400
The 31 conjugacy class representatives for $A_{11}$
Character table for $A_{11}$ is not computed

Intermediate fields

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.11.0.1}{11} }$ R R ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.6.9.15$x^{6} + 6 x^{4} + 6 x^{3} + 12$$6$$1$$9$$S_3\times C_3$$[3/2, 2]_{2}$
$5$5.11.10.1$x^{11} - 5$$11$$1$$10$$C_{11}:C_5$$[\ ]_{11}^{5}$
$11$11.11.20.5$x^{11} - 11 x^{10} + 858$$11$$1$$20$$C_{11}$$[2]$