# Properties

 Label 22.10.412...904.6 Degree $22$ Signature $[10, 6]$ Discriminant $4.129\times 10^{45}$ Root discriminant $118.43$ Ramified primes $2, 74843$ Class number $1$ (GRH) Class group trivial (GRH) Galois group 22T39

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^20 - 439*x^18 - 1549*x^16 + 5292*x^14 + 11519*x^12 - 25067*x^10 - 2315*x^8 + 10967*x^6 + 2679*x^4 - 156*x^2 - 49)

gp: K = bnfinit(x^22 - 4*x^20 - 439*x^18 - 1549*x^16 + 5292*x^14 + 11519*x^12 - 25067*x^10 - 2315*x^8 + 10967*x^6 + 2679*x^4 - 156*x^2 - 49, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-49, 0, -156, 0, 2679, 0, 10967, 0, -2315, 0, -25067, 0, 11519, 0, 5292, 0, -1549, 0, -439, 0, -4, 0, 1]);

$$x^{22} - 4 x^{20} - 439 x^{18} - 1549 x^{16} + 5292 x^{14} + 11519 x^{12} - 25067 x^{10} - 2315 x^{8} + 10967 x^{6} + 2679 x^{4} - 156 x^{2} - 49$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $22$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[10, 6]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$4129233136056857981979443884256982828952059904$$$$\medspace = 2^{22}\cdot 74843^{8}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $118.43$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 74843$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{9} a^{16} + \frac{2}{9} a^{14} + \frac{2}{9} a^{12} + \frac{4}{9} a^{10} - \frac{2}{9} a^{8} - \frac{4}{9}$, $\frac{1}{9} a^{17} + \frac{2}{9} a^{15} + \frac{2}{9} a^{13} + \frac{4}{9} a^{11} - \frac{2}{9} a^{9} - \frac{4}{9} a$, $\frac{1}{81} a^{18} + \frac{1}{27} a^{16} - \frac{5}{81} a^{14} - \frac{7}{27} a^{12} + \frac{2}{81} a^{10} + \frac{25}{81} a^{8} - \frac{1}{9} a^{6} + \frac{1}{9} a^{4} + \frac{23}{81} a^{2} - \frac{4}{81}$, $\frac{1}{81} a^{19} + \frac{1}{27} a^{17} - \frac{5}{81} a^{15} - \frac{7}{27} a^{13} + \frac{2}{81} a^{11} + \frac{25}{81} a^{9} - \frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{23}{81} a^{3} - \frac{4}{81} a$, $\frac{1}{84699917326499853699} a^{20} + \frac{163614170592025960}{84699917326499853699} a^{18} - \frac{1588651118065492220}{84699917326499853699} a^{16} + \frac{23386856370014623384}{84699917326499853699} a^{14} + \frac{33789359523139706024}{84699917326499853699} a^{12} + \frac{2193194036712619676}{9411101925166650411} a^{10} + \frac{12003215131073068171}{84699917326499853699} a^{8} + \frac{151149764229837146}{348559330561727793} a^{6} + \frac{2798155759044462323}{84699917326499853699} a^{4} + \frac{14863786280858977471}{84699917326499853699} a^{2} - \frac{1885072494379702558}{84699917326499853699}$, $\frac{1}{592899421285498975893} a^{21} + \frac{3300648145647576097}{592899421285498975893} a^{19} - \frac{1588651118065492220}{592899421285498975893} a^{17} + \frac{73579399970903425576}{592899421285498975893} a^{15} + \frac{4827051360448529432}{84699917326499853699} a^{13} + \frac{17529804581428642568}{65877713476166552877} a^{11} + \frac{3507335861613609817}{84699917326499853699} a^{9} + \frac{244027327277572267}{813305104644031517} a^{7} + \frac{31031461534544413556}{592899421285498975893} a^{5} + \frac{2315650380636776923}{592899421285498975893} a^{3} + \frac{277310951285564259635}{592899421285498975893} a$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $15$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); 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) sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$944497361130000$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{10}\cdot(2\pi)^{6}\cdot 944497361130000 \cdot 1}{2\sqrt{4129233136056857981979443884256982828952059904}}\approx 0.463036490821387$ (assuming GRH)

## Galois group

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 675840 The 56 conjugacy class representatives for t22n39 are not computed Character table for t22n39 is not computed

## Intermediate fields

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

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
74843Data not computed