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Properties

Label 22.18.2769192260...3125.1
Degree $22$
Signature $[18, 2]$
Discriminant $5^{11}\cdot 29^{2}\cdot 131^{2}\cdot 5399^{2}\cdot 367163^{2}$
Root discriminant $33.12$
Ramified primes $5, 29, 131, 5399, 367163$
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, 4, 22, -32, -159, -60, 274, 692, 348, -1576, -1405, 1601, 1239, -852, -160, 287, -261, -78, 121, 14, -19, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 19*x^20 + 14*x^19 + 121*x^18 - 78*x^17 - 261*x^16 + 287*x^15 - 160*x^14 - 852*x^13 + 1239*x^12 + 1601*x^11 - 1405*x^10 - 1576*x^9 + 348*x^8 + 692*x^7 + 274*x^6 - 60*x^5 - 159*x^4 - 32*x^3 + 22*x^2 + 4*x - 1)
 
gp: K = bnfinit(x^22 - x^21 - 19*x^20 + 14*x^19 + 121*x^18 - 78*x^17 - 261*x^16 + 287*x^15 - 160*x^14 - 852*x^13 + 1239*x^12 + 1601*x^11 - 1405*x^10 - 1576*x^9 + 348*x^8 + 692*x^7 + 274*x^6 - 60*x^5 - 159*x^4 - 32*x^3 + 22*x^2 + 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} - 19 x^{20} + 14 x^{19} + 121 x^{18} - 78 x^{17} - 261 x^{16} + 287 x^{15} - 160 x^{14} - 852 x^{13} + 1239 x^{12} + 1601 x^{11} - 1405 x^{10} - 1576 x^{9} + 348 x^{8} + 692 x^{7} + 274 x^{6} - 60 x^{5} - 159 x^{4} - 32 x^{3} + 22 x^{2} + 4 x - 1 \)

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:  \(2769192260679449143892527783203125=5^{11}\cdot 29^{2}\cdot 131^{2}\cdot 5399^{2}\cdot 367163^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.12$
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:  $5, 29, 131, 5399, 367163$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{51825388642914980827} a^{21} + \frac{867887972110861404}{51825388642914980827} a^{20} + \frac{21463418767132064407}{51825388642914980827} a^{19} - \frac{13559855472580599573}{51825388642914980827} a^{18} - \frac{5852881316764319311}{51825388642914980827} a^{17} - \frac{13851327166233812905}{51825388642914980827} a^{16} - \frac{12216415214082962729}{51825388642914980827} a^{15} - \frac{2879371924013047285}{51825388642914980827} a^{14} + \frac{21096885824379205329}{51825388642914980827} a^{13} + \frac{19803705509707263878}{51825388642914980827} a^{12} - \frac{2085922241926368029}{51825388642914980827} a^{11} + \frac{18863222018202606277}{51825388642914980827} a^{10} + \frac{9331157190771542608}{51825388642914980827} a^{9} - \frac{3236224967010466153}{51825388642914980827} a^{8} + \frac{18512485678580629045}{51825388642914980827} a^{7} - \frac{1675356890253345094}{51825388642914980827} a^{6} + \frac{460555608540700784}{3048552273112645931} a^{5} + \frac{5139111887549847164}{51825388642914980827} a^{4} - \frac{7846236122960252210}{51825388642914980827} a^{3} + \frac{10204401265099665961}{51825388642914980827} a^{2} + \frac{11178506946467967830}{51825388642914980827} a - \frac{10693169541091960823}{51825388642914980827}$

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:  \( 1570808982.29 \) (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

\(\Q(\sqrt{5}) \), 11.9.7530807227563.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ $18{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ $22$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.5.0.1$x^{5} - x + 11$$1$$5$$0$$C_5$$[\ ]^{5}$
29.5.0.1$x^{5} - x + 11$$1$$5$$0$$C_5$$[\ ]^{5}$
$131$131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
131.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
131.5.0.1$x^{5} - x + 3$$1$$5$$0$$C_5$$[\ ]^{5}$
131.5.0.1$x^{5} - x + 3$$1$$5$$0$$C_5$$[\ ]^{5}$
5399Data not computed
367163Data not computed