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Properties

Label 22.22.6101090313...5625.1
Degree $22$
Signature $[22, 0]$
Discriminant $3^{21}\cdot 5^{20}\cdot 11^{19}$
Root discriminant $97.78$
Ramified primes $3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_{11}:C_5$ (as 22T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9498, 60561, -312471, -952095, 4770660, -1127478, -13357071, 10413186, 12236985, -12475935, -4837830, 6354345, 763635, -1647660, 15870, 225726, -18933, -15807, 2025, 510, -78, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 6*x^21 - 78*x^20 + 510*x^19 + 2025*x^18 - 15807*x^17 - 18933*x^16 + 225726*x^15 + 15870*x^14 - 1647660*x^13 + 763635*x^12 + 6354345*x^11 - 4837830*x^10 - 12475935*x^9 + 12236985*x^8 + 10413186*x^7 - 13357071*x^6 - 1127478*x^5 + 4770660*x^4 - 952095*x^3 - 312471*x^2 + 60561*x + 9498)
 
gp: K = bnfinit(x^22 - 6*x^21 - 78*x^20 + 510*x^19 + 2025*x^18 - 15807*x^17 - 18933*x^16 + 225726*x^15 + 15870*x^14 - 1647660*x^13 + 763635*x^12 + 6354345*x^11 - 4837830*x^10 - 12475935*x^9 + 12236985*x^8 + 10413186*x^7 - 13357071*x^6 - 1127478*x^5 + 4770660*x^4 - 952095*x^3 - 312471*x^2 + 60561*x + 9498, 1)
 

Normalized defining polynomial

\( x^{22} - 6 x^{21} - 78 x^{20} + 510 x^{19} + 2025 x^{18} - 15807 x^{17} - 18933 x^{16} + 225726 x^{15} + 15870 x^{14} - 1647660 x^{13} + 763635 x^{12} + 6354345 x^{11} - 4837830 x^{10} - 12475935 x^{9} + 12236985 x^{8} + 10413186 x^{7} - 13357071 x^{6} - 1127478 x^{5} + 4770660 x^{4} - 952095 x^{3} - 312471 x^{2} + 60561 x + 9498 \)

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:  \(61010903135742168957410203940677642822265625=3^{21}\cdot 5^{20}\cdot 11^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.78$
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:  $3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{5} a^{8} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5}$, $\frac{1}{10} a^{17} - \frac{1}{10} a^{15} + \frac{1}{10} a^{11} - \frac{1}{10} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{2}{5} a^{7} - \frac{1}{10} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{3}{10} a^{2} - \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{10} a^{18} - \frac{1}{10} a^{16} - \frac{1}{10} a^{12} + \frac{1}{10} a^{11} + \frac{1}{10} a^{10} - \frac{1}{2} a^{9} - \frac{2}{5} a^{8} + \frac{1}{10} a^{7} + \frac{3}{10} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} - \frac{3}{10} a^{3} + \frac{3}{10} a^{2} + \frac{1}{5}$, $\frac{1}{10} a^{19} - \frac{1}{10} a^{15} - \frac{1}{10} a^{13} - \frac{1}{10} a^{12} + \frac{2}{5} a^{11} + \frac{1}{10} a^{9} - \frac{2}{5} a^{8} + \frac{1}{10} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{3}{10} a^{2} + \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{10} a^{20} - \frac{1}{10} a^{16} - \frac{1}{10} a^{14} - \frac{1}{10} a^{13} + \frac{2}{5} a^{11} + \frac{3}{10} a^{10} - \frac{2}{5} a^{9} + \frac{1}{10} a^{8} - \frac{1}{5} a^{4} + \frac{3}{10} a^{3} - \frac{1}{2} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{9021857208133845727970799812058790283332927079114207686630} a^{21} + \frac{114552106452221955019279185671030150970969895628473440407}{9021857208133845727970799812058790283332927079114207686630} a^{20} - \frac{14983803712384045251066409186230131588871487993082382828}{902185720813384572797079981205879028333292707911420768663} a^{19} + \frac{410901589841775403733699418048822675447504905387673264837}{9021857208133845727970799812058790283332927079114207686630} a^{18} + \frac{441075116151736471079879515939748036883466744567324442}{644418372009560409140771415147056448809494791365300549045} a^{17} - \frac{384080080228867882458112947016071542142048073881446297203}{4510928604066922863985399906029395141666463539557103843315} a^{16} + \frac{165877137745675237470543807924656951798170567202469223509}{4510928604066922863985399906029395141666463539557103843315} a^{15} - \frac{70779624581743393573619386996348583044279299830280856284}{4510928604066922863985399906029395141666463539557103843315} a^{14} + \frac{327991431718035867557354349673253653782492887729149750893}{9021857208133845727970799812058790283332927079114207686630} a^{13} - \frac{704093763436668395976036291928209648659014407155314048823}{9021857208133845727970799812058790283332927079114207686630} a^{12} - \frac{651881062623183997526756700498907071169632973963838048013}{9021857208133845727970799812058790283332927079114207686630} a^{11} - \frac{2095808768027089175940390012775363907612169058626685290427}{9021857208133845727970799812058790283332927079114207686630} a^{10} + \frac{521680203728061798962269973870095650075694943311139345623}{9021857208133845727970799812058790283332927079114207686630} a^{9} - \frac{2112914014880184059010461396085783712748228311763951956273}{4510928604066922863985399906029395141666463539557103843315} a^{8} + \frac{3914978881208521792872742590273501180780074765443244540721}{9021857208133845727970799812058790283332927079114207686630} a^{7} + \frac{403916582783874771170798648849968801260659992372794906948}{902185720813384572797079981205879028333292707911420768663} a^{6} - \frac{1002683609593922053964843745929463493382732955253067273282}{4510928604066922863985399906029395141666463539557103843315} a^{5} + \frac{4360265260810548620325591433151221548511848428246247387119}{9021857208133845727970799812058790283332927079114207686630} a^{4} + \frac{38935436510360165551697968252045597206726645557411753019}{902185720813384572797079981205879028333292707911420768663} a^{3} - \frac{736740119687113886963589834061035862083575385824467601757}{1804371441626769145594159962411758056666585415822841537326} a^{2} + \frac{704159167610040466161605429945548919438036343037039437093}{1804371441626769145594159962411758056666585415822841537326} a - \frac{675700242292066707198389027877461680256322852049309942851}{4510928604066922863985399906029395141666463539557103843315}$

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:  \( 6302138028050000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{11}:C_5$ (as 22T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 110
The 14 conjugacy class representatives for $C_2\times C_{11}:C_5$
Character table for $C_2\times C_{11}:C_5$

Intermediate fields

\(\Q(\sqrt{33}) \), 11.11.123610132462587890625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ $22$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
3Data not computed
5Data not computed
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$