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Properties

Label 11.11.3004325983...2049.1
Degree $11$
Signature $[11, 0]$
Discriminant $353^{10}$
Root discriminant $207.09$
Ramified prime $353$
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![170569, 63189, -405001, -42966, 265374, -48369, -26034, 6066, 525, -160, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 160*x^9 + 525*x^8 + 6066*x^7 - 26034*x^6 - 48369*x^5 + 265374*x^4 - 42966*x^3 - 405001*x^2 + 63189*x + 170569)
 
gp: K = bnfinit(x^11 - x^10 - 160*x^9 + 525*x^8 + 6066*x^7 - 26034*x^6 - 48369*x^5 + 265374*x^4 - 42966*x^3 - 405001*x^2 + 63189*x + 170569, 1)
 

Normalized defining polynomial

\( x^{11} - x^{10} - 160 x^{9} + 525 x^{8} + 6066 x^{7} - 26034 x^{6} - 48369 x^{5} + 265374 x^{4} - 42966 x^{3} - 405001 x^{2} + 63189 x + 170569 \)

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:  \(30043259834681392663962049=353^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $207.09$
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:  $353$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(353\)
Dirichlet character group:    $\lbrace$$\chi_{353}(256,·)$, $\chi_{353}(1,·)$, $\chi_{353}(131,·)$, $\chi_{353}(231,·)$, $\chi_{353}(140,·)$, $\chi_{353}(337,·)$, $\chi_{353}(22,·)$, $\chi_{353}(185,·)$, $\chi_{353}(217,·)$, $\chi_{353}(58,·)$, $\chi_{353}(187,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{343} a^{9} - \frac{12}{343} a^{8} - \frac{16}{343} a^{7} + \frac{18}{343} a^{6} + \frac{139}{343} a^{5} + \frac{142}{343} a^{4} + \frac{2}{343} a^{3} + \frac{55}{343} a^{2} - \frac{5}{49} a + \frac{1}{7}$, $\frac{1}{22606707880007901643} a^{10} + \frac{425355903710770}{3229529697143985949} a^{9} + \frac{1385234345521597656}{22606707880007901643} a^{8} - \frac{205467290213406304}{22606707880007901643} a^{7} + \frac{271596129071586338}{22606707880007901643} a^{6} + \frac{10739218554099059413}{22606707880007901643} a^{5} + \frac{7545291121782715607}{22606707880007901643} a^{4} - \frac{10736768046791550524}{22606707880007901643} a^{3} + \frac{7798810059569900170}{22606707880007901643} a^{2} + \frac{365738944794150084}{3229529697143985949} a - \frac{2775060832662338}{7819684496716673}$

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:  \( 5373530988.23 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/11.11.0.1}{11} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{11}$

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
353Data not computed