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Properties

Label 11.11.2786968520...4449.6
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![-5403811843, -12188156343, 9233082947, 3526980655, -171278723, -74117109, 810612, 511566, -759, -1265, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 1265*x^9 - 759*x^8 + 511566*x^7 + 810612*x^6 - 74117109*x^5 - 171278723*x^4 + 3526980655*x^3 + 9233082947*x^2 - 12188156343*x - 5403811843)
 
gp: K = bnfinit(x^11 - 1265*x^9 - 759*x^8 + 511566*x^7 + 810612*x^6 - 74117109*x^5 - 171278723*x^4 + 3526980655*x^3 + 9233082947*x^2 - 12188156343*x - 5403811843, 1)
 

Normalized defining polynomial

\( x^{11} - 1265 x^{9} - 759 x^{8} + 511566 x^{7} + 810612 x^{6} - 74117109 x^{5} - 171278723 x^{4} + 3526980655 x^{3} + 9233082947 x^{2} - 12188156343 x - 5403811843 \)

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$$\chi_{2783}(1024,·)$, $\chi_{2783}(1,·)$, $\chi_{2783}(2377,·)$, $\chi_{2783}(1706,·)$, $\chi_{2783}(331,·)$, $\chi_{2783}(2520,·)$, $\chi_{2783}(2003,·)$, $\chi_{2783}(2168,·)$, $\chi_{2783}(2201,·)$, $\chi_{2783}(1981,·)$, $\chi_{2783}(639,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{41} a^{9} - \frac{11}{41} a^{8} + \frac{17}{41} a^{7} + \frac{18}{41} a^{6} - \frac{9}{41} a^{5} + \frac{6}{41} a^{4} + \frac{7}{41} a^{3} + \frac{8}{41} a^{2} + \frac{20}{41} a + \frac{9}{41}$, $\frac{1}{254377575449844576760977514428297904994324783} a^{10} + \frac{1294049412874065938057568255728992921635611}{254377575449844576760977514428297904994324783} a^{9} + \frac{96033573895359775037014587089973679756094528}{254377575449844576760977514428297904994324783} a^{8} + \frac{115200263926811644706366553149009432781364291}{254377575449844576760977514428297904994324783} a^{7} + \frac{29620978465718972181651912921499070014936236}{254377575449844576760977514428297904994324783} a^{6} - \frac{103119689929849313100473955192749048305407805}{254377575449844576760977514428297904994324783} a^{5} + \frac{113537796399120713911421698513892007021451766}{254377575449844576760977514428297904994324783} a^{4} - \frac{109370623460943807013373631102401971137543446}{254377575449844576760977514428297904994324783} a^{3} - \frac{4912677963729596074632527207020349870344868}{254377575449844576760977514428297904994324783} a^{2} + \frac{80922460757606328679170356101804020745139822}{254377575449844576760977514428297904994324783} a - \frac{78498134196418508758730430817370546805916}{2858175005054433446752556341890987696565447}$

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:  \( 3514735062507.8184 \) (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 ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

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$11.11.20.10$x^{11} - 11 x^{10} + 132$$11$$1$$20$$C_{11}$$[2]$
$23$23.11.10.7$x^{11} + 11776$$11$$1$$10$$C_{11}$$[\ ]_{11}$