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Properties

Label 22.6.33458177485...8125.1
Degree $22$
Signature $[6, 8]$
Discriminant $5^{11}\cdot 14851^{2}\cdot 1762627^{2}$
Root discriminant $19.80$
Ramified primes $5, 14851, 1762627$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T47

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 6, 0, -25, 22, 18, -74, 38, -26, -32, 233, 9, -177, -10, 85, 41, 1, -13, -4, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^20 - 4*x^19 - 13*x^18 + x^17 + 41*x^16 + 85*x^15 - 10*x^14 - 177*x^13 + 9*x^12 + 233*x^11 - 32*x^10 - 26*x^9 + 38*x^8 - 74*x^7 + 18*x^6 + 22*x^5 - 25*x^4 + 6*x^2 - 1)
 
gp: K = bnfinit(x^22 - 3*x^20 - 4*x^19 - 13*x^18 + x^17 + 41*x^16 + 85*x^15 - 10*x^14 - 177*x^13 + 9*x^12 + 233*x^11 - 32*x^10 - 26*x^9 + 38*x^8 - 74*x^7 + 18*x^6 + 22*x^5 - 25*x^4 + 6*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{22} - 3 x^{20} - 4 x^{19} - 13 x^{18} + x^{17} + 41 x^{16} + 85 x^{15} - 10 x^{14} - 177 x^{13} + 9 x^{12} + 233 x^{11} - 32 x^{10} - 26 x^{9} + 38 x^{8} - 74 x^{7} + 18 x^{6} + 22 x^{5} - 25 x^{4} + 6 x^{2} - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33458177485426043697705078125=5^{11}\cdot 14851^{2}\cdot 1762627^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.80$
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:  $5, 14851, 1762627$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{1921929890880957955207} a^{21} + \frac{911346814028213435366}{1921929890880957955207} a^{20} - \frac{861280763309450626178}{1921929890880957955207} a^{19} - \frac{507911277580249485916}{1921929890880957955207} a^{18} + \frac{422873437454997461328}{1921929890880957955207} a^{17} - \frac{750302950227690264878}{1921929890880957955207} a^{16} - \frac{515044381902743857384}{1921929890880957955207} a^{15} - \frac{659222645119437296321}{1921929890880957955207} a^{14} + \frac{327654798202111499051}{1921929890880957955207} a^{13} + \frac{118908246030162823450}{1921929890880957955207} a^{12} - \frac{595448431761506718320}{1921929890880957955207} a^{11} - \frac{645642709894653633280}{1921929890880957955207} a^{10} + \frac{96923288625915272211}{274561412982993993601} a^{9} + \frac{580308321585659098980}{1921929890880957955207} a^{8} + \frac{371867167769904956685}{1921929890880957955207} a^{7} - \frac{936948316479080053010}{1921929890880957955207} a^{6} - \frac{658677499665778906142}{1921929890880957955207} a^{5} - \frac{950386889194810361757}{1921929890880957955207} a^{4} + \frac{518362382559476924818}{1921929890880957955207} a^{3} + \frac{268554255648592046353}{1921929890880957955207} a^{2} + \frac{661373265213569821378}{1921929890880957955207} a - \frac{11672212616619522882}{1921929890880957955207}$

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:  $13$
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:  \( 408757.310974 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T47:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 79833600
The 112 conjugacy class representatives for t22n47 are not computed
Character table for t22n47 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 11.3.26176773577.1

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

Sibling fields

Degree 22 sibling: data not computed
Degree 44 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 $22$ $22$ R $18{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $18{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $18{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ $22$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
14851Data not computed
1762627Data not computed