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Properties

Label 22.22.1856564799...9008.1
Degree $22$
Signature $[22, 0]$
Discriminant $2^{20}\cdot 7^{11}\cdot 11^{23}$
Root discriminant $60.95$
Ramified primes $2, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_{11}$ (as 22T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-77, 4235, -27214, -737, 385000, -1076977, 829103, 1103938, -2643443, 1617539, 478390, -1066395, 335500, 177265, -133661, 3872, 18029, -3795, -924, 363, 0, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 363*x^19 - 924*x^18 - 3795*x^17 + 18029*x^16 + 3872*x^15 - 133661*x^14 + 177265*x^13 + 335500*x^12 - 1066395*x^11 + 478390*x^10 + 1617539*x^9 - 2643443*x^8 + 1103938*x^7 + 829103*x^6 - 1076977*x^5 + 385000*x^4 - 737*x^3 - 27214*x^2 + 4235*x - 77)
 
gp: K = bnfinit(x^22 - 11*x^21 + 363*x^19 - 924*x^18 - 3795*x^17 + 18029*x^16 + 3872*x^15 - 133661*x^14 + 177265*x^13 + 335500*x^12 - 1066395*x^11 + 478390*x^10 + 1617539*x^9 - 2643443*x^8 + 1103938*x^7 + 829103*x^6 - 1076977*x^5 + 385000*x^4 - 737*x^3 - 27214*x^2 + 4235*x - 77, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{21} + 363 x^{19} - 924 x^{18} - 3795 x^{17} + 18029 x^{16} + 3872 x^{15} - 133661 x^{14} + 177265 x^{13} + 335500 x^{12} - 1066395 x^{11} + 478390 x^{10} + 1617539 x^{9} - 2643443 x^{8} + 1103938 x^{7} + 829103 x^{6} - 1076977 x^{5} + 385000 x^{4} - 737 x^{3} - 27214 x^{2} + 4235 x - 77 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1856564799974488125951035818931423019008=2^{20}\cdot 7^{11}\cdot 11^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.95$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{32126130364399796864514964010959} a^{21} - \frac{9570595158263370783031472720156}{32126130364399796864514964010959} a^{20} + \frac{15570256890989220918160643776079}{32126130364399796864514964010959} a^{19} - \frac{1847831915804910948970587482040}{32126130364399796864514964010959} a^{18} + \frac{7307132583471068154399454341277}{32126130364399796864514964010959} a^{17} + \frac{8946517579520250916093162292368}{32126130364399796864514964010959} a^{16} + \frac{6632106961737088288040364133850}{32126130364399796864514964010959} a^{15} + \frac{8579092833147911411506691090246}{32126130364399796864514964010959} a^{14} + \frac{15438772200653583376220752914235}{32126130364399796864514964010959} a^{13} - \frac{14448450664487152886786671149898}{32126130364399796864514964010959} a^{12} + \frac{4164489764901059602615939171107}{32126130364399796864514964010959} a^{11} + \frac{2127225523969160502385599866401}{32126130364399796864514964010959} a^{10} - \frac{6381131628460118550373387018393}{32126130364399796864514964010959} a^{9} + \frac{2779809921578899303139106738388}{32126130364399796864514964010959} a^{8} + \frac{6291730801561064154782500013018}{32126130364399796864514964010959} a^{7} + \frac{1691825317224357093197568383561}{32126130364399796864514964010959} a^{6} - \frac{5162551117921510272694636335829}{32126130364399796864514964010959} a^{5} + \frac{7379938475916965711237551043426}{32126130364399796864514964010959} a^{4} - \frac{13801400245576044534302567151267}{32126130364399796864514964010959} a^{3} + \frac{4037611365782535477458189288459}{32126130364399796864514964010959} a^{2} - \frac{7980711490640184820343191986326}{32126130364399796864514964010959} a - \frac{8855328108349626312796838559506}{32126130364399796864514964010959}$

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:  $21$
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:  \( 8449061390020 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_{11}$ (as 22T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 110
The 11 conjugacy class representatives for $F_{11}$
Character table for $F_{11}$

Intermediate fields

\(\Q(\sqrt{77}) \), 11.11.4910318845910094848.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 11 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
2Data not computed
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed