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Properties

Label 22.2.173...125.1
Degree $22$
Signature $[2, 10]$
Discriminant $1.735\times 10^{27}$
Root discriminant $17.30$
Ramified primes $5, 64661, 92179$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 22T47

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 2*x^20 + x^19 - 13*x^18 + 3*x^17 - 35*x^16 - 18*x^15 + 37*x^14 + 27*x^13 + 152*x^12 + 277*x^11 + 160*x^10 + 72*x^9 - 42*x^8 - 69*x^7 - 31*x^6 - 22*x^5 + 3*x^4 + 4*x^3 + x^2 + x - 1)
 
gp: K = bnfinit(x^22 - x^21 + 2*x^20 + x^19 - 13*x^18 + 3*x^17 - 35*x^16 - 18*x^15 + 37*x^14 + 27*x^13 + 152*x^12 + 277*x^11 + 160*x^10 + 72*x^9 - 42*x^8 - 69*x^7 - 31*x^6 - 22*x^5 + 3*x^4 + 4*x^3 + x^2 + x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 1, 4, 3, -22, -31, -69, -42, 72, 160, 277, 152, 27, 37, -18, -35, 3, -13, 1, 2, -1, 1]);
 

\( x^{22} - x^{21} + 2 x^{20} + x^{19} - 13 x^{18} + 3 x^{17} - 35 x^{16} - 18 x^{15} + 37 x^{14} + 27 x^{13} + 152 x^{12} + 277 x^{11} + 160 x^{10} + 72 x^{9} - 42 x^{8} - 69 x^{7} - 31 x^{6} - 22 x^{5} + 3 x^{4} + 4 x^{3} + x^{2} + x - 1 \)

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

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(1734677982017693835986328125\)\(\medspace = 5^{11}\cdot 64661^{2}\cdot 92179^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $17.30$
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:  $5, 64661, 92179$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2124470127291230454011} a^{21} + \frac{432162695892207561224}{2124470127291230454011} a^{20} - \frac{653641761206245262533}{2124470127291230454011} a^{19} + \frac{167982702717637181560}{2124470127291230454011} a^{18} + \frac{532665987662589956145}{2124470127291230454011} a^{17} + \frac{625983005593256221797}{2124470127291230454011} a^{16} - \frac{463273244278987969270}{2124470127291230454011} a^{15} - \frac{518926591257159751500}{2124470127291230454011} a^{14} + \frac{508818139788462937089}{2124470127291230454011} a^{13} + \frac{749396289026961292787}{2124470127291230454011} a^{12} + \frac{856956959840666211577}{2124470127291230454011} a^{11} + \frac{556372178049218564352}{2124470127291230454011} a^{10} + \frac{828731590687777909608}{2124470127291230454011} a^{9} - \frac{463905878846124332921}{2124470127291230454011} a^{8} + \frac{215007940286959093640}{2124470127291230454011} a^{7} - \frac{674937675082602811008}{2124470127291230454011} a^{6} - \frac{243201499321128170322}{2124470127291230454011} a^{5} - \frac{616326127851568785886}{2124470127291230454011} a^{4} - \frac{832659424940701796791}{2124470127291230454011} a^{3} - \frac{819262980435466447033}{2124470127291230454011} a^{2} + \frac{1046535151217522863157}{2124470127291230454011} a + \frac{930991225635280714244}{2124470127291230454011}$

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:  $11$
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:  \( 38336.3977164 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{10}\cdot 38336.3977164 \cdot 1}{2\sqrt{1734677982017693835986328125}}\approx 0.176534888715$ (assuming GRH)

Galois group

22T47:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{5}) \), 11.1.5960386319.1

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 $22$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R $22$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $18{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
64661Data not computed
92179Data not computed