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Properties

Label 11.11.2786968520...4449.1
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![-98108643071, 58422539890, 31567681332, 714773070, -916234187, -81455880, 4031808, 544962, -4807, -1265, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 1265*x^9 - 4807*x^8 + 544962*x^7 + 4031808*x^6 - 81455880*x^5 - 916234187*x^4 + 714773070*x^3 + 31567681332*x^2 + 58422539890*x - 98108643071)
 
gp: K = bnfinit(x^11 - 1265*x^9 - 4807*x^8 + 544962*x^7 + 4031808*x^6 - 81455880*x^5 - 916234187*x^4 + 714773070*x^3 + 31567681332*x^2 + 58422539890*x - 98108643071, 1)
 

Normalized defining polynomial

\( x^{11} - 1265 x^{9} - 4807 x^{8} + 544962 x^{7} + 4031808 x^{6} - 81455880 x^{5} - 916234187 x^{4} + 714773070 x^{3} + 31567681332 x^{2} + 58422539890 x - 98108643071 \)

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$$\chi_{2783}(1,·)$, $\chi_{2783}(1442,·)$, $\chi_{2783}(1156,·)$, $\chi_{2783}(518,·)$, $\chi_{2783}(2509,·)$, $\chi_{2783}(78,·)$, $\chi_{2783}(463,·)$, $\chi_{2783}(496,·)$, $\chi_{2783}(1112,·)$, $\chi_{2783}(892,·)$, $\chi_{2783}(2718,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2793353322801397212938793784547835219825825179} a^{10} - \frac{761749316129434838578361374362961504413255803}{2793353322801397212938793784547835219825825179} a^{9} + \frac{1065879531240153559450012424958316568679166398}{2793353322801397212938793784547835219825825179} a^{8} - \frac{626910588327343912869178353005331451952117833}{2793353322801397212938793784547835219825825179} a^{7} + \frac{603813411516275881060683050581855467428124022}{2793353322801397212938793784547835219825825179} a^{6} - \frac{512360850517058821814041430977291578894602118}{2793353322801397212938793784547835219825825179} a^{5} - \frac{641366647962805373595878366368722145453525645}{2793353322801397212938793784547835219825825179} a^{4} - \frac{686508953559489136690441045785130770098064415}{2793353322801397212938793784547835219825825179} a^{3} + \frac{1101825803599906291941949799182848747368331282}{2793353322801397212938793784547835219825825179} a^{2} + \frac{150692603483737135931518586441402068735442410}{2793353322801397212938793784547835219825825179} a - \frac{504448223796643350788853535497518766906527}{7293350712275188545532098654171893524349413}$

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:  \( 2524169127627.655 \) (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 ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

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$11.11.20.7$x^{11} - 11 x^{10} + 1100$$11$$1$$20$$C_{11}$$[2]$
$23$23.11.10.9$x^{11} + 46$$11$$1$$10$$C_{11}$$[\ ]_{11}$