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Properties

Label 22.10.4098058278...8729.6
Degree $22$
Signature $[10, 6]$
Discriminant $23^{21}\cdot 47^{3}$
Root discriminant $33.72$
Ramified primes $23, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T28

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![277, -4212, 14913, -12470, -38191, 124670, -184444, 178846, -124953, 66337, -31525, 19365, -14993, 9536, -3607, 218, 335, 30, -113, 17, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 8*x^21 + 18*x^20 + 17*x^19 - 113*x^18 + 30*x^17 + 335*x^16 + 218*x^15 - 3607*x^14 + 9536*x^13 - 14993*x^12 + 19365*x^11 - 31525*x^10 + 66337*x^9 - 124953*x^8 + 178846*x^7 - 184444*x^6 + 124670*x^5 - 38191*x^4 - 12470*x^3 + 14913*x^2 - 4212*x + 277)
 
gp: K = bnfinit(x^22 - 8*x^21 + 18*x^20 + 17*x^19 - 113*x^18 + 30*x^17 + 335*x^16 + 218*x^15 - 3607*x^14 + 9536*x^13 - 14993*x^12 + 19365*x^11 - 31525*x^10 + 66337*x^9 - 124953*x^8 + 178846*x^7 - 184444*x^6 + 124670*x^5 - 38191*x^4 - 12470*x^3 + 14913*x^2 - 4212*x + 277, 1)
 

Normalized defining polynomial

\( x^{22} - 8 x^{21} + 18 x^{20} + 17 x^{19} - 113 x^{18} + 30 x^{17} + 335 x^{16} + 218 x^{15} - 3607 x^{14} + 9536 x^{13} - 14993 x^{12} + 19365 x^{11} - 31525 x^{10} + 66337 x^{9} - 124953 x^{8} + 178846 x^{7} - 184444 x^{6} + 124670 x^{5} - 38191 x^{4} - 12470 x^{3} + 14913 x^{2} - 4212 x + 277 \)

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:  \(4098058278162967431271914143428729=23^{21}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.72$
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:  $23, 47$
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}{799466368301954011781828145962836239752903} a^{21} - \frac{77854696372531296531780038959392878704849}{799466368301954011781828145962836239752903} a^{20} - \frac{285607015043815130481582153220012526599385}{799466368301954011781828145962836239752903} a^{19} + \frac{331599787624165743912655465788412128419578}{799466368301954011781828145962836239752903} a^{18} + \frac{162633937697908653435031435036851477520506}{799466368301954011781828145962836239752903} a^{17} - \frac{190436177429480918259370333400578430723434}{799466368301954011781828145962836239752903} a^{16} + \frac{241672719833464103548336534566223411973467}{799466368301954011781828145962836239752903} a^{15} - \frac{89629960528697959147269939218923917862846}{799466368301954011781828145962836239752903} a^{14} - \frac{182526637358794602005538684946383154000320}{799466368301954011781828145962836239752903} a^{13} + \frac{323168967732348308463364560889529991848940}{799466368301954011781828145962836239752903} a^{12} - \frac{390947281476651074564531492476096812304592}{799466368301954011781828145962836239752903} a^{11} + \frac{329356105513931806608702517790967263849214}{799466368301954011781828145962836239752903} a^{10} - \frac{152033191668755613301978003040725174081248}{799466368301954011781828145962836239752903} a^{9} + \frac{258237283135962645320541038185073413757504}{799466368301954011781828145962836239752903} a^{8} + \frac{48965143686327907882269389872433085128672}{799466368301954011781828145962836239752903} a^{7} + \frac{192279016315228959024248902918204065467016}{799466368301954011781828145962836239752903} a^{6} + \frac{5017911377482124002376160363879378321754}{799466368301954011781828145962836239752903} a^{5} - \frac{191793400614598250745086731383252050885997}{799466368301954011781828145962836239752903} a^{4} + \frac{219615171730082430448545661650869552079422}{799466368301954011781828145962836239752903} a^{3} - \frac{368452189628459530763418172897288108043210}{799466368301954011781828145962836239752903} a^{2} + \frac{17053004257499815180896212233678521987996}{799466368301954011781828145962836239752903} a + \frac{290901299842802213754452956434596501656160}{799466368301954011781828145962836239752903}$

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:  \( 343593724.102 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T28:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 22528
The 208 conjugacy class representatives for t22n28 are not computed
Character table for t22n28 is not computed

Intermediate fields

\(\Q(\zeta_{23})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 22 siblings: data not computed
Degree 44 siblings: 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ R $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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

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
23Data not computed
47Data not computed