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Properties

Label 22.8.87192729322...7207.1
Degree $22$
Signature $[8, 7]$
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![47, -162, -459, 298, 622, 574, -735, 1264, -114, -461, 852, -1105, 484, -162, -229, 68, -91, 22, -22, -1, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + x^20 - x^19 - 22*x^18 + 22*x^17 - 91*x^16 + 68*x^15 - 229*x^14 - 162*x^13 + 484*x^12 - 1105*x^11 + 852*x^10 - 461*x^9 - 114*x^8 + 1264*x^7 - 735*x^6 + 574*x^5 + 622*x^4 + 298*x^3 - 459*x^2 - 162*x + 47)
 
gp: K = bnfinit(x^22 - x^21 + x^20 - x^19 - 22*x^18 + 22*x^17 - 91*x^16 + 68*x^15 - 229*x^14 - 162*x^13 + 484*x^12 - 1105*x^11 + 852*x^10 - 461*x^9 - 114*x^8 + 1264*x^7 - 735*x^6 + 574*x^5 + 622*x^4 + 298*x^3 - 459*x^2 - 162*x + 47, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + x^{20} - x^{19} - 22 x^{18} + 22 x^{17} - 91 x^{16} + 68 x^{15} - 229 x^{14} - 162 x^{13} + 484 x^{12} - 1105 x^{11} + 852 x^{10} - 461 x^{9} - 114 x^{8} + 1264 x^{7} - 735 x^{6} + 574 x^{5} + 622 x^{4} + 298 x^{3} - 459 x^{2} - 162 x + 47 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 7]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} + \frac{18}{47} a^{19} + \frac{7}{47} a^{18} + \frac{6}{47} a^{17} - \frac{4}{47} a^{16} - \frac{2}{47} a^{15} - \frac{7}{47} a^{14} - \frac{4}{47} a^{13} - \frac{21}{47} a^{12} - \frac{16}{47} a^{11} - \frac{2}{47} a^{10} + \frac{3}{47} a^{9} - \frac{17}{47} a^{8} - \frac{6}{47} a^{7} - \frac{15}{47} a^{6} - \frac{13}{47} a^{5} + \frac{16}{47} a^{4} - \frac{18}{47} a^{3} - \frac{20}{47} a^{2} - \frac{3}{47} a$, $\frac{1}{12132506454156656784356113315074242197} a^{21} + \frac{32769940805618568363253395671867840}{12132506454156656784356113315074242197} a^{20} - \frac{5681791058205206174779540935529938906}{12132506454156656784356113315074242197} a^{19} + \frac{187148943072113096564843600450548275}{12132506454156656784356113315074242197} a^{18} + \frac{832500375242768051043162203050034418}{12132506454156656784356113315074242197} a^{17} - \frac{208367858976908438306832848378772521}{12132506454156656784356113315074242197} a^{16} + \frac{1376152239318889066019403573551626149}{12132506454156656784356113315074242197} a^{15} - \frac{129640996962688199066469302857772948}{12132506454156656784356113315074242197} a^{14} - \frac{3005110957896931754155009410914077218}{12132506454156656784356113315074242197} a^{13} + \frac{1809602737864766259751931499618313688}{12132506454156656784356113315074242197} a^{12} + \frac{3473172837782126519082997771995767800}{12132506454156656784356113315074242197} a^{11} + \frac{2711311500912408751692187063274373888}{12132506454156656784356113315074242197} a^{10} + \frac{1482154432882620009610487588496205271}{12132506454156656784356113315074242197} a^{9} - \frac{3629226468923651309840522730254863898}{12132506454156656784356113315074242197} a^{8} + \frac{758414639519510287874362450394300114}{12132506454156656784356113315074242197} a^{7} + \frac{5901809273156786828734097684205710452}{12132506454156656784356113315074242197} a^{6} - \frac{3694503999847689261632061203874593106}{12132506454156656784356113315074242197} a^{5} - \frac{2578231753603067630177996209373771436}{12132506454156656784356113315074242197} a^{4} - \frac{2854835307322961172919241892431894449}{12132506454156656784356113315074242197} a^{3} - \frac{1806582931603760570180596986848174422}{12132506454156656784356113315074242197} a^{2} + \frac{5747086747439894624586704066644826609}{12132506454156656784356113315074242197} a - \frac{124499499546300753816350597359122297}{258138435194822484773534325852643451}$

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:  $14$
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:  \( 28009585.914 \) (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
47Data not computed