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Properties

Label 22.10.8549930793...8125.1
Degree $22$
Signature $[10, 6]$
Discriminant $5^{11}\cdot 211441^{2}\cdot 625831^{2}$
Root discriminant $22.94$
Ramified primes $5, 211441, 625831$
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, -1, 7, -2, -4, 33, -3, -93, 22, 44, -237, -44, 292, 154, 4, -66, -40, 27, 14, 0, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 - 4*x^20 + 14*x^18 + 27*x^17 - 40*x^16 - 66*x^15 + 4*x^14 + 154*x^13 + 292*x^12 - 44*x^11 - 237*x^10 + 44*x^9 + 22*x^8 - 93*x^7 - 3*x^6 + 33*x^5 - 4*x^4 - 2*x^3 + 7*x^2 - x - 1)
 
gp: K = bnfinit(x^22 - 2*x^21 - 4*x^20 + 14*x^18 + 27*x^17 - 40*x^16 - 66*x^15 + 4*x^14 + 154*x^13 + 292*x^12 - 44*x^11 - 237*x^10 + 44*x^9 + 22*x^8 - 93*x^7 - 3*x^6 + 33*x^5 - 4*x^4 - 2*x^3 + 7*x^2 - x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} - 4 x^{20} + 14 x^{18} + 27 x^{17} - 40 x^{16} - 66 x^{15} + 4 x^{14} + 154 x^{13} + 292 x^{12} - 44 x^{11} - 237 x^{10} + 44 x^{9} + 22 x^{8} - 93 x^{7} - 3 x^{6} + 33 x^{5} - 4 x^{4} - 2 x^{3} + 7 x^{2} - x - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(854993079356720164347705078125=5^{11}\cdot 211441^{2}\cdot 625831^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.94$
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:  $5, 211441, 625831$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{1446162571472358893239} a^{21} - \frac{453218684289368001677}{1446162571472358893239} a^{20} - \frac{138138409166331311176}{1446162571472358893239} a^{19} - \frac{461787609574644934774}{1446162571472358893239} a^{18} + \frac{662593631268643311079}{1446162571472358893239} a^{17} + \frac{617667104895864354318}{1446162571472358893239} a^{16} + \frac{664484371895700157780}{1446162571472358893239} a^{15} + \frac{182825960640851050283}{1446162571472358893239} a^{14} + \frac{409696275628140166179}{1446162571472358893239} a^{13} + \frac{701009901610921160695}{1446162571472358893239} a^{12} + \frac{537942675048982514862}{1446162571472358893239} a^{11} + \frac{430329344460744083763}{1446162571472358893239} a^{10} - \frac{138679219269329118668}{1446162571472358893239} a^{9} - \frac{506404787291784191350}{1446162571472358893239} a^{8} - \frac{690471923733912260118}{1446162571472358893239} a^{7} - \frac{391252094889661426662}{1446162571472358893239} a^{6} + \frac{166170956768983646363}{1446162571472358893239} a^{5} - \frac{87237407178949558635}{1446162571472358893239} a^{4} + \frac{158452981784716963211}{1446162571472358893239} a^{3} + \frac{398660422005366364848}{1446162571472358893239} a^{2} + \frac{475204776038010051442}{1446162571472358893239} a + \frac{539742921059276553593}{1446162571472358893239}$

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:  $15$
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:  \( 5205736.25321 \) (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

\(\Q(\sqrt{5}) \), 11.5.132326332471.1

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 $22$ ${\href{/LocalNumberField/3.14.0.1}{14} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ $22$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ $18{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
211441Data not computed
625831Data not computed