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Properties

Label 11.11.2786968520...4449.4
Degree $11$
Signature $[11, 0]$
Discriminant $11^{20}\cdot 23^{10}$
Root discriminant $1353.24$
Ramified primes $11, 23$
Class number $11$ (GRH)
Class group $[11]$ (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![-22424006285, 32566872448, 16580057472, 942027284, -446246713, -48772328, 3280398, 478170, -4807, -1265, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 1265*x^9 - 4807*x^8 + 478170*x^7 + 3280398*x^6 - 48772328*x^5 - 446246713*x^4 + 942027284*x^3 + 16580057472*x^2 + 32566872448*x - 22424006285)
 
gp: K = bnfinit(x^11 - 1265*x^9 - 4807*x^8 + 478170*x^7 + 3280398*x^6 - 48772328*x^5 - 446246713*x^4 + 942027284*x^3 + 16580057472*x^2 + 32566872448*x - 22424006285, 1)
 

Normalized defining polynomial

\( x^{11} - 1265 x^{9} - 4807 x^{8} + 478170 x^{7} + 3280398 x^{6} - 48772328 x^{5} - 446246713 x^{4} + 942027284 x^{3} + 16580057472 x^{2} + 32566872448 x - 22424006285 \)

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:  \(27869685209056005146671841116784449=11^{20}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1353.24$
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:  $11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2783=11^{2}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{2783}(1,·)$, $\chi_{2783}(265,·)$, $\chi_{2783}(2762,·)$, $\chi_{2783}(1871,·)$, $\chi_{2783}(2410,·)$, $\chi_{2783}(2454,·)$, $\chi_{2783}(2487,·)$, $\chi_{2783}(441,·)$, $\chi_{2783}(2267,·)$, $\chi_{2783}(650,·)$, $\chi_{2783}(1343,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{325} a^{8} - \frac{11}{325} a^{7} - \frac{9}{325} a^{6} + \frac{9}{325} a^{5} + \frac{109}{325} a^{4} - \frac{29}{325} a^{3} + \frac{99}{325} a^{2} - \frac{49}{325} a - \frac{4}{65}$, $\frac{1}{5364125} a^{9} - \frac{1282}{5364125} a^{8} + \frac{361072}{5364125} a^{7} + \frac{300373}{5364125} a^{6} + \frac{72744}{1072825} a^{5} - \frac{2538368}{5364125} a^{4} + \frac{808833}{5364125} a^{3} + \frac{1380822}{5364125} a^{2} + \frac{1867634}{5364125} a - \frac{134861}{1072825}$, $\frac{1}{558565118792835395318513825276875} a^{10} - \frac{20225275287340329916845189}{558565118792835395318513825276875} a^{9} - \frac{188188218336492035437530449964}{558565118792835395318513825276875} a^{8} + \frac{11868120677752363629023817566879}{558565118792835395318513825276875} a^{7} + \frac{10928205487747813004696725583374}{558565118792835395318513825276875} a^{6} - \frac{25021037242677400560911987839273}{558565118792835395318513825276875} a^{5} - \frac{262556127730347755558594617092431}{558565118792835395318513825276875} a^{4} + \frac{211603189417338940791955014518381}{558565118792835395318513825276875} a^{3} - \frac{40256500411817758524191849815082}{111713023758567079063702765055375} a^{2} - \frac{5706105135397275378276186665131}{42966547599448876562962601944375} a - \frac{47770160349290753699146329545433}{111713023758567079063702765055375}$

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:  $10$
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:  \( 50372046418790.55 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ R ${\href{/LocalNumberField/13.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ R ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

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
$11$11.11.20.6$x^{11} - 11 x^{10} + 979$$11$$1$$20$$C_{11}$$[2]$
$23$23.11.10.2$x^{11} - 5888$$11$$1$$10$$C_{11}$$[\ ]_{11}$