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Properties

Label 11.3.27555839792...0000.1
Degree $11$
Signature $[3, 4]$
Discriminant $2^{22}\cdot 5^{10}\cdot 11^{20}$
Root discriminant $1351.85$
Ramified primes $2, 5, 11$
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![59609831680, -34495524800, 5702365240, -552727725, 114912820, -6042410, -133980, 91025, -2860, 110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 + 110*x^9 - 2860*x^8 + 91025*x^7 - 133980*x^6 - 6042410*x^5 + 114912820*x^4 - 552727725*x^3 + 5702365240*x^2 - 34495524800*x + 59609831680)
 
gp: K = bnfinit(x^11 + 110*x^9 - 2860*x^8 + 91025*x^7 - 133980*x^6 - 6042410*x^5 + 114912820*x^4 - 552727725*x^3 + 5702365240*x^2 - 34495524800*x + 59609831680, 1)
 

Normalized defining polynomial

\( x^{11} + 110 x^{9} - 2860 x^{8} + 91025 x^{7} - 133980 x^{6} - 6042410 x^{5} + 114912820 x^{4} - 552727725 x^{3} + 5702365240 x^{2} - 34495524800 x + 59609831680 \)

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:  \(27555839792437657976872960000000000=2^{22}\cdot 5^{10}\cdot 11^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1351.85$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{2297479348272499104032043192509938480072987744} a^{10} + \frac{11803170518029905038210891933924445856514431}{287184918534062388004005399063742310009123468} a^{9} + \frac{270085867946764735128307655302265771351064775}{1148739674136249552016021596254969240036493872} a^{8} - \frac{140455760397169298287007654114538079391783119}{574369837068124776008010798127484620018246936} a^{7} + \frac{1126207964341180651590236898673052620427383281}{2297479348272499104032043192509938480072987744} a^{6} + \frac{201012351647324905420687682393313928781627359}{574369837068124776008010798127484620018246936} a^{5} + \frac{438867334742318412568202312802890806823576907}{1148739674136249552016021596254969240036493872} a^{4} + \frac{23002360906061022942307367558521704325069097}{574369837068124776008010798127484620018246936} a^{3} - \frac{555529393743811789009746207031391372776766989}{2297479348272499104032043192509938480072987744} a^{2} + \frac{35795670813089370605580219547519757805958458}{71796229633515597001001349765935577502280867} a - \frac{8581210668307972884617270633132252468637394}{71796229633515597001001349765935577502280867}$

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:  \( 3670507552850 \) (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 R ${\href{/LocalNumberField/3.7.0.1}{7} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.8.20.19$x^{8} + 8 x^{7} + 12$$4$$2$$20$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 3, 3, 7/2]^{4}$
$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]$