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Properties

Label 22.12.1781764468...3423.7
Degree $22$
Signature $[12, 5]$
Discriminant $-\,23^{20}\cdot 47^{3}$
Root discriminant $29.24$
Ramified primes $23, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T28

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![139, -482, -2259, 14864, -35838, 44371, -19737, -19265, 33608, -17342, -5659, 14118, -7580, -920, 3786, -2299, 333, 346, -243, 58, 8, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 7*x^21 + 8*x^20 + 58*x^19 - 243*x^18 + 346*x^17 + 333*x^16 - 2299*x^15 + 3786*x^14 - 920*x^13 - 7580*x^12 + 14118*x^11 - 5659*x^10 - 17342*x^9 + 33608*x^8 - 19265*x^7 - 19737*x^6 + 44371*x^5 - 35838*x^4 + 14864*x^3 - 2259*x^2 - 482*x + 139)
 
gp: K = bnfinit(x^22 - 7*x^21 + 8*x^20 + 58*x^19 - 243*x^18 + 346*x^17 + 333*x^16 - 2299*x^15 + 3786*x^14 - 920*x^13 - 7580*x^12 + 14118*x^11 - 5659*x^10 - 17342*x^9 + 33608*x^8 - 19265*x^7 - 19737*x^6 + 44371*x^5 - 35838*x^4 + 14864*x^3 - 2259*x^2 - 482*x + 139, 1)
 

Normalized defining polynomial

\( x^{22} - 7 x^{21} + 8 x^{20} + 58 x^{19} - 243 x^{18} + 346 x^{17} + 333 x^{16} - 2299 x^{15} + 3786 x^{14} - 920 x^{13} - 7580 x^{12} + 14118 x^{11} - 5659 x^{10} - 17342 x^{9} + 33608 x^{8} - 19265 x^{7} - 19737 x^{6} + 44371 x^{5} - 35838 x^{4} + 14864 x^{3} - 2259 x^{2} - 482 x + 139 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-178176446876650757881387571453423=-\,23^{20}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.24$
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:  $23, 47$
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}{22410356993839193044087548333582360836909753} a^{21} + \frac{4940740233644329625536477227771304209154839}{22410356993839193044087548333582360836909753} a^{20} - \frac{5031742894746528021984502884218731697753803}{22410356993839193044087548333582360836909753} a^{19} - \frac{8719762375223667057392832505969472825597122}{22410356993839193044087548333582360836909753} a^{18} + \frac{6070247365525417162872489694530291386941979}{22410356993839193044087548333582360836909753} a^{17} - \frac{2285043639256299580228808573996572109349902}{22410356993839193044087548333582360836909753} a^{16} - \frac{2225002098487938537439486169219192106385065}{22410356993839193044087548333582360836909753} a^{15} + \frac{6134142392237308565762992638784101771634227}{22410356993839193044087548333582360836909753} a^{14} - \frac{3036989824735622033636971447256628528726814}{22410356993839193044087548333582360836909753} a^{13} - \frac{369191318469791082055681115309717886840134}{22410356993839193044087548333582360836909753} a^{12} + \frac{1632063793464250784216912359992591090082931}{22410356993839193044087548333582360836909753} a^{11} - \frac{6950792530050971477394385555560966984710271}{22410356993839193044087548333582360836909753} a^{10} + \frac{5628914714739912024953053025379934987185529}{22410356993839193044087548333582360836909753} a^{9} - \frac{5990669897096108862501674399527480413649716}{22410356993839193044087548333582360836909753} a^{8} + \frac{2240029729293320293067395912434750055302844}{22410356993839193044087548333582360836909753} a^{7} - \frac{7505753821862960538216516694946497057545503}{22410356993839193044087548333582360836909753} a^{6} - \frac{2432863357293003233776381624989267208680021}{22410356993839193044087548333582360836909753} a^{5} + \frac{8706284545140564689931910115976121567392022}{22410356993839193044087548333582360836909753} a^{4} + \frac{109036535011339186246601056490432668575576}{22410356993839193044087548333582360836909753} a^{3} + \frac{7726459001079883708859159200021631575749782}{22410356993839193044087548333582360836909753} a^{2} + \frac{7649380704379166061889618841039562275114943}{22410356993839193044087548333582360836909753} a - \frac{56751760173105822115347798726764964627302}{161225589883735201756025527579729214654027}$

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:  $16$
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:  \( 109627633.006 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T28:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 22528
The 208 conjugacy class representatives for t22n28 are not computed
Character table for t22n28 is not computed

Intermediate fields

\(\Q(\zeta_{23})^+\)

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

Sibling fields

Degree 22 siblings: data not computed
Degree 44 siblings: 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ R $22$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ $22$ R ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
23Data not computed
47Data not computed