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Properties

Label 22.10.4098058278...8729.4
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![47, 2961, 22323, -177615, 415894, -501195, 342689, -96327, -63140, 97872, -59796, 14121, 4698, -4636, 2125, -859, 232, -12, -66, 51, -7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 - 7*x^20 + 51*x^19 - 66*x^18 - 12*x^17 + 232*x^16 - 859*x^15 + 2125*x^14 - 4636*x^13 + 4698*x^12 + 14121*x^11 - 59796*x^10 + 97872*x^9 - 63140*x^8 - 96327*x^7 + 342689*x^6 - 501195*x^5 + 415894*x^4 - 177615*x^3 + 22323*x^2 + 2961*x + 47)
 
gp: K = bnfinit(x^22 - 4*x^21 - 7*x^20 + 51*x^19 - 66*x^18 - 12*x^17 + 232*x^16 - 859*x^15 + 2125*x^14 - 4636*x^13 + 4698*x^12 + 14121*x^11 - 59796*x^10 + 97872*x^9 - 63140*x^8 - 96327*x^7 + 342689*x^6 - 501195*x^5 + 415894*x^4 - 177615*x^3 + 22323*x^2 + 2961*x + 47, 1)
 

Normalized defining polynomial

\( x^{22} - 4 x^{21} - 7 x^{20} + 51 x^{19} - 66 x^{18} - 12 x^{17} + 232 x^{16} - 859 x^{15} + 2125 x^{14} - 4636 x^{13} + 4698 x^{12} + 14121 x^{11} - 59796 x^{10} + 97872 x^{9} - 63140 x^{8} - 96327 x^{7} + 342689 x^{6} - 501195 x^{5} + 415894 x^{4} - 177615 x^{3} + 22323 x^{2} + 2961 x + 47 \)

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}{225263899210989544348667749099893892489406421735523} a^{21} + \frac{42537820487941650525945118036314312627968731907112}{225263899210989544348667749099893892489406421735523} a^{20} + \frac{66026861404698800743940624639360307650631578997336}{225263899210989544348667749099893892489406421735523} a^{19} - \frac{30285314114942456351973338440992727859751461289793}{225263899210989544348667749099893892489406421735523} a^{18} + \frac{22446010104506475634576139244797340944467051271683}{225263899210989544348667749099893892489406421735523} a^{17} + \frac{20357153653583256725931577302656615162449059435334}{225263899210989544348667749099893892489406421735523} a^{16} + \frac{43753427009417790210238282939899859356146705119727}{225263899210989544348667749099893892489406421735523} a^{15} + \frac{107101462056984874231527469861781098014910204794100}{225263899210989544348667749099893892489406421735523} a^{14} + \frac{31986943484145386019041454049616379122960769320614}{225263899210989544348667749099893892489406421735523} a^{13} + \frac{112193210306766025843083648698264185365915260029393}{225263899210989544348667749099893892489406421735523} a^{12} - \frac{73556432647806138728124815952249869139456709106766}{225263899210989544348667749099893892489406421735523} a^{11} + \frac{8118674994874837020446445375683953333253473977362}{225263899210989544348667749099893892489406421735523} a^{10} - \frac{85085042322978035047483160754041427425808257211670}{225263899210989544348667749099893892489406421735523} a^{9} + \frac{68491264202622816789457773177688133283204165074047}{225263899210989544348667749099893892489406421735523} a^{8} + \frac{34276660360971182063087932722124272350759302771863}{225263899210989544348667749099893892489406421735523} a^{7} - \frac{26600510394215000362747524541183447557084706115143}{225263899210989544348667749099893892489406421735523} a^{6} + \frac{96301380159598814930198218073624059884760328035494}{225263899210989544348667749099893892489406421735523} a^{5} + \frac{21098960877486256102519901254582678948617709117828}{225263899210989544348667749099893892489406421735523} a^{4} + \frac{83805871517501947030846304152953858575140533417621}{225263899210989544348667749099893892489406421735523} a^{3} - \frac{8017451223299609915231123251176758824871513169099}{225263899210989544348667749099893892489406421735523} a^{2} - \frac{1647279035801630674500494420528381330528130209614}{4792848919382756262737611682976465797646945143309} a - \frac{1978602352494095934245727431211498152782943739126}{4792848919382756262737611682976465797646945143309}$

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:  \( 320060666.813 \) (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
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$