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Properties

Label 22.18.3674011011...8125.1
Degree $22$
Signature $[18, 2]$
Discriminant $5^{11}\cdot 8674315276967^{2}$
Root discriminant $33.55$
Ramified primes $5, 8674315276967$
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, 2, 25, -38, -211, 259, 811, -1001, -1770, 2301, 2570, -3041, -2680, 2178, 1905, -753, -819, 88, 191, 5, -22, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 22*x^20 + 5*x^19 + 191*x^18 + 88*x^17 - 819*x^16 - 753*x^15 + 1905*x^14 + 2178*x^13 - 2680*x^12 - 3041*x^11 + 2570*x^10 + 2301*x^9 - 1770*x^8 - 1001*x^7 + 811*x^6 + 259*x^5 - 211*x^4 - 38*x^3 + 25*x^2 + 2*x - 1)
 
gp: K = bnfinit(x^22 - x^21 - 22*x^20 + 5*x^19 + 191*x^18 + 88*x^17 - 819*x^16 - 753*x^15 + 1905*x^14 + 2178*x^13 - 2680*x^12 - 3041*x^11 + 2570*x^10 + 2301*x^9 - 1770*x^8 - 1001*x^7 + 811*x^6 + 259*x^5 - 211*x^4 - 38*x^3 + 25*x^2 + 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} - 22 x^{20} + 5 x^{19} + 191 x^{18} + 88 x^{17} - 819 x^{16} - 753 x^{15} + 1905 x^{14} + 2178 x^{13} - 2680 x^{12} - 3041 x^{11} + 2570 x^{10} + 2301 x^{9} - 1770 x^{8} - 1001 x^{7} + 811 x^{6} + 259 x^{5} - 211 x^{4} - 38 x^{3} + 25 x^{2} + 2 x - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3674011011924955171910111767578125=5^{11}\cdot 8674315276967^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.55$
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:  $5, 8674315276967$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{369540355594477003} a^{21} + \frac{115165080443008912}{369540355594477003} a^{20} - \frac{71418943859759571}{369540355594477003} a^{19} + \frac{30209115726009397}{369540355594477003} a^{18} + \frac{73812730791956866}{369540355594477003} a^{17} - \frac{133262781610271877}{369540355594477003} a^{16} + \frac{13101688255035339}{369540355594477003} a^{15} - \frac{144890273846423545}{369540355594477003} a^{14} - \frac{71295104131578503}{369540355594477003} a^{13} + \frac{26250847816354153}{369540355594477003} a^{12} - \frac{102637677859723982}{369540355594477003} a^{11} - \frac{30055080657704771}{369540355594477003} a^{10} - \frac{7009460147210212}{369540355594477003} a^{9} + \frac{92979096705562336}{369540355594477003} a^{8} - \frac{65695742251215794}{369540355594477003} a^{7} - \frac{169682173131454761}{369540355594477003} a^{6} - \frac{1415545655051506}{369540355594477003} a^{5} + \frac{47826047398339413}{369540355594477003} a^{4} - \frac{121760178466082191}{369540355594477003} a^{3} + \frac{87918391184021752}{369540355594477003} a^{2} - \frac{97876517656687406}{369540355594477003} a + \frac{120852792329076817}{369540355594477003}$

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:  $19$
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:  \( 2040269368.26 \) (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

\(\Q(\sqrt{5}) \), 11.9.8674315276967.1

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 $18{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ $18{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
8674315276967Data not computed