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Properties

Label 11.11.3118171992...3601.1
Degree $11$
Signature $[11, 0]$
Discriminant $89^{10}$
Root discriminant $59.18$
Ramified prime $89$
Class number $1$
Class group Trivial
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![1, 57, 781, 3152, -102, -2185, -84, 482, 19, -40, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 40*x^9 + 19*x^8 + 482*x^7 - 84*x^6 - 2185*x^5 - 102*x^4 + 3152*x^3 + 781*x^2 + 57*x + 1)
 
gp: K = bnfinit(x^11 - x^10 - 40*x^9 + 19*x^8 + 482*x^7 - 84*x^6 - 2185*x^5 - 102*x^4 + 3152*x^3 + 781*x^2 + 57*x + 1, 1)
 

Normalized defining polynomial

\( x^{11} - x^{10} - 40 x^{9} + 19 x^{8} + 482 x^{7} - 84 x^{6} - 2185 x^{5} - 102 x^{4} + 3152 x^{3} + 781 x^{2} + 57 x + 1 \)

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:  \(31181719929966183601=89^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.18$
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:  $89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(89\)
Dirichlet character group:    $\lbrace$$\chi_{89}(32,·)$, $\chi_{89}(1,·)$, $\chi_{89}(2,·)$, $\chi_{89}(67,·)$, $\chi_{89}(4,·)$, $\chi_{89}(39,·)$, $\chi_{89}(8,·)$, $\chi_{89}(64,·)$, $\chi_{89}(45,·)$, $\chi_{89}(78,·)$, $\chi_{89}(16,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{37} a^{9} - \frac{9}{37} a^{8} + \frac{15}{37} a^{7} + \frac{15}{37} a^{6} - \frac{4}{37} a^{5} - \frac{11}{37} a^{4} + \frac{6}{37} a^{3} + \frac{16}{37} a - \frac{13}{37}$, $\frac{1}{677618999} a^{10} + \frac{8725065}{677618999} a^{9} - \frac{320149378}{677618999} a^{8} - \frac{212441436}{677618999} a^{7} - \frac{167088298}{677618999} a^{6} - \frac{211577786}{677618999} a^{5} - \frac{235075458}{677618999} a^{4} - \frac{53218474}{677618999} a^{3} - \frac{234831214}{677618999} a^{2} + \frac{81504955}{677618999} a + \frac{337178982}{677618999}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 846366.485722 \)
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 ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

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
$89$89.11.10.1$x^{11} - 89$$11$$1$$10$$C_{11}$$[\ ]_{11}$