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Properties

Label 11.11.4124241695...3649.1
Degree $11$
Signature $[11, 0]$
Discriminant $727^{10}$
Root discriminant $399.39$
Ramified prime $727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{11}$ (as 11T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2279191, 19842261, 17584739, 183306, -3355168, -616689, 122078, 30782, -637, -330, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 330*x^9 - 637*x^8 + 30782*x^7 + 122078*x^6 - 616689*x^5 - 3355168*x^4 + 183306*x^3 + 17584739*x^2 + 19842261*x + 2279191)
 
gp: K = bnfinit(x^11 - x^10 - 330*x^9 - 637*x^8 + 30782*x^7 + 122078*x^6 - 616689*x^5 - 3355168*x^4 + 183306*x^3 + 17584739*x^2 + 19842261*x + 2279191, 1)
 

Normalized defining polynomial

\( x^{11} - x^{10} - 330 x^{9} - 637 x^{8} + 30782 x^{7} + 122078 x^{6} - 616689 x^{5} - 3355168 x^{4} + 183306 x^{3} + 17584739 x^{2} + 19842261 x + 2279191 \)

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

Invariants

Degree:  $11$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(41242416955341131537413053649=727^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $399.39$
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:  $727$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(727\)
Dirichlet character group:    $\lbrace$$\chi_{727}(1,·)$, $\chi_{727}(645,·)$, $\chi_{727}(648,·)$, $\chi_{727}(425,·)$, $\chi_{727}(46,·)$, $\chi_{727}(241,·)$, $\chi_{727}(594,·)$, $\chi_{727}(181,·)$, $\chi_{727}(662,·)$, $\chi_{727}(329,·)$, $\chi_{727}(590,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{29} a^{8} + \frac{7}{29} a^{6} + \frac{1}{29} a^{5} - \frac{5}{29} a^{4} + \frac{10}{29} a^{3} - \frac{4}{29} a^{2} - \frac{5}{29} a - \frac{5}{29}$, $\frac{1}{17893} a^{9} - \frac{42}{17893} a^{8} + \frac{6764}{17893} a^{7} - \frac{5049}{17893} a^{6} + \frac{707}{17893} a^{5} + \frac{6803}{17893} a^{4} + \frac{1258}{17893} a^{3} - \frac{4419}{17893} a^{2} - \frac{5740}{17893} a - \frac{4227}{17893}$, $\frac{1}{259647786661076784636995911} a^{10} + \frac{2888977559385799782366}{259647786661076784636995911} a^{9} + \frac{859918845365004049336941}{259647786661076784636995911} a^{8} - \frac{21814259164910969017808517}{259647786661076784636995911} a^{7} - \frac{8009797803234967270599123}{259647786661076784636995911} a^{6} + \frac{70743325749847278078711194}{259647786661076784636995911} a^{5} + \frac{113768596303670902163080847}{259647786661076784636995911} a^{4} + \frac{3249434766286444540802878}{259647786661076784636995911} a^{3} - \frac{2143568679633685579788051}{259647786661076784636995911} a^{2} - \frac{56130604632267528207903882}{259647786661076784636995911} a - \frac{11482433010280082696761781}{259647786661076784636995911}$

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:  $10$
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:  \( 49130453388.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{11}$ (as 11T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 11
The 11 conjugacy class representatives for $C_{11}$
Character table for $C_{11}$

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 ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

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
727Data not computed