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Properties

Label 11.11.2786968520...4449.7
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![236115769, 106500350, -487081672, 123836416, 90501389, -27487944, -436172, 344586, -759, -1265, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 1265*x^9 - 759*x^8 + 344586*x^7 - 436172*x^6 - 27487944*x^5 + 90501389*x^4 + 123836416*x^3 - 487081672*x^2 + 106500350*x + 236115769)
 
gp: K = bnfinit(x^11 - 1265*x^9 - 759*x^8 + 344586*x^7 - 436172*x^6 - 27487944*x^5 + 90501389*x^4 + 123836416*x^3 - 487081672*x^2 + 106500350*x + 236115769, 1)
 

Normalized defining polynomial

\( x^{11} - 1265 x^{9} - 759 x^{8} + 344586 x^{7} - 436172 x^{6} - 27487944 x^{5} + 90501389 x^{4} + 123836416 x^{3} - 487081672 x^{2} + 106500350 x + 236115769 \)

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$$\chi_{2783}(1,·)$, $\chi_{2783}(771,·)$, $\chi_{2783}(1508,·)$, $\chi_{2783}(1222,·)$, $\chi_{2783}(353,·)$, $\chi_{2783}(2157,·)$, $\chi_{2783}(430,·)$, $\chi_{2783}(2256,·)$, $\chi_{2783}(2212,·)$, $\chi_{2783}(1596,·)$, $\chi_{2783}(1662,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{12202839308924009187740529958973720688874133} a^{10} - \frac{2665897102316953963813738868373710755883141}{12202839308924009187740529958973720688874133} a^{9} - \frac{5005797458628929021453444367801838109017533}{12202839308924009187740529958973720688874133} a^{8} + \frac{6043137660658327049768666851909333875433600}{12202839308924009187740529958973720688874133} a^{7} + \frac{244065197026031896630505109729506640038222}{12202839308924009187740529958973720688874133} a^{6} - \frac{1554545930220961273693950411049097957203463}{12202839308924009187740529958973720688874133} a^{5} + \frac{2822884815869015211059620034559406019084195}{12202839308924009187740529958973720688874133} a^{4} + \frac{5012773082680026855799514292452531442205762}{12202839308924009187740529958973720688874133} a^{3} - \frac{3530289544523484090075701324722520765538195}{12202839308924009187740529958973720688874133} a^{2} - \frac{5740583734571606593473528651332716573995666}{12202839308924009187740529958973720688874133} a - \frac{1850952639108844853228195617726553529454736}{12202839308924009187740529958973720688874133}$

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:  \( 2963701738171.1094 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ R ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ R ${\href{/LocalNumberField/29.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

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$11.11.20.2$x^{11} - 11 x^{10} + 495$$11$$1$$20$$C_{11}$$[2]$
$23$23.11.10.3$x^{11} + 736$$11$$1$$10$$C_{11}$$[\ ]_{11}$