↑

Properties

Label 22.22.8379667145...8125.1
Degree $22$
Signature $[22, 0]$
Discriminant $5^{11}\cdot 23^{20}$
Root discriminant $38.67$
Ramified primes $5, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{22}$ (as 22T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -6, 81, 50, -1100, -301, 5656, 1331, -14356, -3174, 20269, 4100, -16780, -2967, 8247, 1195, -2365, -254, 379, 26, -31, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 31*x^20 + 26*x^19 + 379*x^18 - 254*x^17 - 2365*x^16 + 1195*x^15 + 8247*x^14 - 2967*x^13 - 16780*x^12 + 4100*x^11 + 20269*x^10 - 3174*x^9 - 14356*x^8 + 1331*x^7 + 5656*x^6 - 301*x^5 - 1100*x^4 + 50*x^3 + 81*x^2 - 6*x - 1)
 
gp: K = bnfinit(x^22 - x^21 - 31*x^20 + 26*x^19 + 379*x^18 - 254*x^17 - 2365*x^16 + 1195*x^15 + 8247*x^14 - 2967*x^13 - 16780*x^12 + 4100*x^11 + 20269*x^10 - 3174*x^9 - 14356*x^8 + 1331*x^7 + 5656*x^6 - 301*x^5 - 1100*x^4 + 50*x^3 + 81*x^2 - 6*x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} - 31 x^{20} + 26 x^{19} + 379 x^{18} - 254 x^{17} - 2365 x^{16} + 1195 x^{15} + 8247 x^{14} - 2967 x^{13} - 16780 x^{12} + 4100 x^{11} + 20269 x^{10} - 3174 x^{9} - 14356 x^{8} + 1331 x^{7} + 5656 x^{6} - 301 x^{5} - 1100 x^{4} + 50 x^{3} + 81 x^{2} - 6 x - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(83796671451884098775580820361328125=5^{11}\cdot 23^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.67$
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:  $5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(115=5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{115}(64,·)$, $\chi_{115}(1,·)$, $\chi_{115}(4,·)$, $\chi_{115}(6,·)$, $\chi_{115}(71,·)$, $\chi_{115}(9,·)$, $\chi_{115}(16,·)$, $\chi_{115}(81,·)$, $\chi_{115}(24,·)$, $\chi_{115}(26,·)$, $\chi_{115}(29,·)$, $\chi_{115}(94,·)$, $\chi_{115}(31,·)$, $\chi_{115}(96,·)$, $\chi_{115}(36,·)$, $\chi_{115}(101,·)$, $\chi_{115}(39,·)$, $\chi_{115}(104,·)$, $\chi_{115}(41,·)$, $\chi_{115}(49,·)$, $\chi_{115}(54,·)$, $\chi_{115}(59,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{56434294023897120739} a^{21} - \frac{16715073269265397265}{56434294023897120739} a^{20} + \frac{25973587480747409457}{56434294023897120739} a^{19} + \frac{3741467039879829759}{56434294023897120739} a^{18} - \frac{135456302100991293}{56434294023897120739} a^{17} + \frac{22662899710884134446}{56434294023897120739} a^{16} + \frac{14095648608578273486}{56434294023897120739} a^{15} + \frac{14780106911863947683}{56434294023897120739} a^{14} + \frac{13411739415746507241}{56434294023897120739} a^{13} + \frac{10938396768848346865}{56434294023897120739} a^{12} - \frac{21408531734584798482}{56434294023897120739} a^{11} + \frac{20450647543062054973}{56434294023897120739} a^{10} - \frac{7631575739397224646}{56434294023897120739} a^{9} - \frac{7849150185992359754}{56434294023897120739} a^{8} + \frac{14408483517612357320}{56434294023897120739} a^{7} + \frac{1928511410022929178}{56434294023897120739} a^{6} - \frac{22908421253297592882}{56434294023897120739} a^{5} - \frac{27238553615507393480}{56434294023897120739} a^{4} - \frac{13225948143484377746}{56434294023897120739} a^{3} - \frac{7018683369595104660}{56434294023897120739} a^{2} - \frac{21790241648713404537}{56434294023897120739} a + \frac{10784499631451005690}{56434294023897120739}$

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:  \( 17322043684.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{22}$ (as 22T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{23})^+\)

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 $22$ $22$ R $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
5Data not computed
23Data not computed