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Properties

Label 22.12.8719272932...7207.1
Degree $22$
Signature $[12, 5]$
Discriminant $-\,23^{21}\cdot 47^{2}$
Root discriminant $28.30$
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![-137, 891, -1666, -446, 4907, -5039, 196, 1469, 2042, -5559, 5455, -461, -5192, 5185, -1072, -1457, 1152, -196, -135, 74, -7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 - 7*x^20 + 74*x^19 - 135*x^18 - 196*x^17 + 1152*x^16 - 1457*x^15 - 1072*x^14 + 5185*x^13 - 5192*x^12 - 461*x^11 + 5455*x^10 - 5559*x^9 + 2042*x^8 + 1469*x^7 + 196*x^6 - 5039*x^5 + 4907*x^4 - 446*x^3 - 1666*x^2 + 891*x - 137)
 
gp: K = bnfinit(x^22 - 4*x^21 - 7*x^20 + 74*x^19 - 135*x^18 - 196*x^17 + 1152*x^16 - 1457*x^15 - 1072*x^14 + 5185*x^13 - 5192*x^12 - 461*x^11 + 5455*x^10 - 5559*x^9 + 2042*x^8 + 1469*x^7 + 196*x^6 - 5039*x^5 + 4907*x^4 - 446*x^3 - 1666*x^2 + 891*x - 137, 1)
 

Normalized defining polynomial

\( x^{22} - 4 x^{21} - 7 x^{20} + 74 x^{19} - 135 x^{18} - 196 x^{17} + 1152 x^{16} - 1457 x^{15} - 1072 x^{14} + 5185 x^{13} - 5192 x^{12} - 461 x^{11} + 5455 x^{10} - 5559 x^{9} + 2042 x^{8} + 1469 x^{7} + 196 x^{6} - 5039 x^{5} + 4907 x^{4} - 446 x^{3} - 1666 x^{2} + 891 x - 137 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-87192729322616328324934343477207=-\,23^{21}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.30$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{139} a^{20} + \frac{68}{139} a^{19} + \frac{46}{139} a^{18} + \frac{17}{139} a^{17} + \frac{16}{139} a^{16} - \frac{60}{139} a^{15} - \frac{36}{139} a^{14} + \frac{52}{139} a^{13} - \frac{66}{139} a^{12} + \frac{48}{139} a^{11} + \frac{9}{139} a^{10} - \frac{8}{139} a^{9} - \frac{66}{139} a^{8} - \frac{62}{139} a^{7} + \frac{18}{139} a^{6} + \frac{11}{139} a^{5} - \frac{6}{139} a^{4} + \frac{53}{139} a^{3} - \frac{27}{139} a^{2} + \frac{27}{139} a - \frac{38}{139}$, $\frac{1}{2524314182633022376938738804867149} a^{21} - \frac{3220302427458116989870400367649}{2524314182633022376938738804867149} a^{20} - \frac{646420696716268498986415394589004}{2524314182633022376938738804867149} a^{19} + \frac{674432794603526415624968611807275}{2524314182633022376938738804867149} a^{18} + \frac{526875624641578005170974443156489}{2524314182633022376938738804867149} a^{17} + \frac{400033301538694692567738176245547}{2524314182633022376938738804867149} a^{16} + \frac{87821046984664839759930308865846}{2524314182633022376938738804867149} a^{15} + \frac{208827456193601977288334282664091}{2524314182633022376938738804867149} a^{14} + \frac{612327095790311465870561214567532}{2524314182633022376938738804867149} a^{13} - \frac{165190027159969906758986421905849}{2524314182633022376938738804867149} a^{12} + \frac{1185360400411340715896596393394054}{2524314182633022376938738804867149} a^{11} + \frac{13913794594544234761851480303162}{53708812396447284615717846912067} a^{10} + \frac{1188184762162518720591586515359256}{2524314182633022376938738804867149} a^{9} + \frac{32832136284905633720367486602125}{2524314182633022376938738804867149} a^{8} - \frac{287991535529712728007535908974021}{2524314182633022376938738804867149} a^{7} + \frac{604565528407804718307091632800791}{2524314182633022376938738804867149} a^{6} - \frac{1205728001302975504971473663446233}{2524314182633022376938738804867149} a^{5} + \frac{121350090830481672312211330291426}{2524314182633022376938738804867149} a^{4} + \frac{157064664296643098078335219144982}{2524314182633022376938738804867149} a^{3} - \frac{1167604736253429332926271736038046}{2524314182633022376938738804867149} a^{2} - \frac{464379558075889855777239673436162}{2524314182633022376938738804867149} a - \frac{831480943939912750816534634350709}{2524314182633022376938738804867149}$

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:  $16$
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:  \( 78962808.6369 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ $22$ $22$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ $22$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ $22$ 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$$0$Trivial$[\ ]$
$\Q_{47}$$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$$0$Trivial$[\ ]$
47.2.0.1$x^{2} - x + 13$$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}$