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Properties

Label 11.11.2209057583...1449.1
Degree $11$
Signature $[11, 0]$
Discriminant $683^{10}$
Root discriminant $377.35$
Ramified prime $683$
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![996543, -1360638, -5088204, 2568726, 2220564, -744575, -57583, 26441, 395, -310, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 310*x^9 + 395*x^8 + 26441*x^7 - 57583*x^6 - 744575*x^5 + 2220564*x^4 + 2568726*x^3 - 5088204*x^2 - 1360638*x + 996543)
 
gp: K = bnfinit(x^11 - x^10 - 310*x^9 + 395*x^8 + 26441*x^7 - 57583*x^6 - 744575*x^5 + 2220564*x^4 + 2568726*x^3 - 5088204*x^2 - 1360638*x + 996543, 1)
 

Normalized defining polynomial

\( x^{11} - x^{10} - 310 x^{9} + 395 x^{8} + 26441 x^{7} - 57583 x^{6} - 744575 x^{5} + 2220564 x^{4} + 2568726 x^{3} - 5088204 x^{2} - 1360638 x + 996543 \)

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:  \(22090575837180674640752471449=683^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $377.35$
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:  $683$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(683\)
Dirichlet character group:    $\lbrace$$\chi_{683}(64,·)$, $\chi_{683}(1,·)$, $\chi_{683}(555,·)$, $\chi_{683}(4,·)$, $\chi_{683}(681,·)$, $\chi_{683}(651,·)$, $\chi_{683}(256,·)$, $\chi_{683}(16,·)$, $\chi_{683}(675,·)$, $\chi_{683}(171,·)$, $\chi_{683}(341,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{7} + \frac{1}{27} a^{5} + \frac{1}{9} a^{4} - \frac{2}{27} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{8} + \frac{1}{81} a^{7} + \frac{4}{81} a^{6} + \frac{4}{81} a^{5} + \frac{13}{81} a^{4} + \frac{4}{81} a^{3} + \frac{1}{9} a^{2} - \frac{4}{9} a$, $\frac{1}{2997} a^{9} - \frac{5}{999} a^{8} - \frac{14}{999} a^{7} + \frac{13}{999} a^{6} + \frac{11}{111} a^{5} - \frac{56}{999} a^{4} - \frac{130}{2997} a^{3} - \frac{86}{333} a^{2} - \frac{164}{333} a - \frac{15}{37}$, $\frac{1}{3096157918956615348903} a^{10} + \frac{65672983335599183}{3096157918956615348903} a^{9} - \frac{14643011641585023205}{3096157918956615348903} a^{8} + \frac{38742542531418914798}{3096157918956615348903} a^{7} - \frac{1616444581971221230}{3096157918956615348903} a^{6} - \frac{181932441362609436238}{3096157918956615348903} a^{5} - \frac{305598504763960815500}{3096157918956615348903} a^{4} - \frac{132103551701768734396}{1032052639652205116301} a^{3} - \frac{167616275696478305290}{344017546550735038767} a^{2} + \frac{9149111634645699652}{114672515516911679589} a - \frac{5519091770903465777}{12741390612990186621}$

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:  \( 2276941413140 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

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
683Data not computed