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Properties

Label 11.11.9725346143...7049.1
Degree $11$
Signature $[11, 0]$
Discriminant $397^{10}$
Root discriminant $230.43$
Ramified prime $397$
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![3098903, 10752776, 10551746, 2670510, -720142, -316973, 12159, 11655, 13, -180, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 180*x^9 + 13*x^8 + 11655*x^7 + 12159*x^6 - 316973*x^5 - 720142*x^4 + 2670510*x^3 + 10551746*x^2 + 10752776*x + 3098903)
 
gp: K = bnfinit(x^11 - x^10 - 180*x^9 + 13*x^8 + 11655*x^7 + 12159*x^6 - 316973*x^5 - 720142*x^4 + 2670510*x^3 + 10551746*x^2 + 10752776*x + 3098903, 1)
 

Normalized defining polynomial

\( x^{11} - x^{10} - 180 x^{9} + 13 x^{8} + 11655 x^{7} + 12159 x^{6} - 316973 x^{5} - 720142 x^{4} + 2670510 x^{3} + 10551746 x^{2} + 10752776 x + 3098903 \)

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:  \(97253461433805715000527049=397^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $230.43$
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:  $397$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(397\)
Dirichlet character group:    $\lbrace$$\chi_{397}(256,·)$, $\chi_{397}(1,·)$, $\chi_{397}(290,·)$, $\chi_{397}(99,·)$, $\chi_{397}(167,·)$, $\chi_{397}(393,·)$, $\chi_{397}(333,·)$, $\chi_{397}(16,·)$, $\chi_{397}(273,·)$, $\chi_{397}(126,·)$, $\chi_{397}(31,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{593} a^{9} - \frac{62}{593} a^{8} - \frac{216}{593} a^{7} + \frac{252}{593} a^{6} + \frac{259}{593} a^{5} + \frac{221}{593} a^{4} + \frac{109}{593} a^{3} + \frac{288}{593} a^{2} - \frac{16}{593} a + \frac{125}{593}$, $\frac{1}{3433962596340982322581} a^{10} - \frac{1864034678147035764}{3433962596340982322581} a^{9} - \frac{1560395736328182674475}{3433962596340982322581} a^{8} - \frac{880648007073015337340}{3433962596340982322581} a^{7} + \frac{1073869237618574445387}{3433962596340982322581} a^{6} - \frac{203443722202491656865}{3433962596340982322581} a^{5} - \frac{916178225319154199745}{3433962596340982322581} a^{4} + \frac{155204954424450808630}{3433962596340982322581} a^{3} - \frac{281030044754286555435}{3433962596340982322581} a^{2} - \frac{1486650429592343581908}{3433962596340982322581} a + \frac{34835375090824357318}{83755185276609324941}$

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:  \( 1592800927.62 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

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
397Data not computed