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Properties

Label 22.8.17817644687...3423.1
Degree $22$
Signature $[8, 7]$
Discriminant $-\,23^{20}\cdot 47^{3}$
Root discriminant $29.24$
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, 61, 40, -86, -739, -990, 52, 4390, 1747, -2957, -1802, 301, -405, -691, 777, 727, -279, -232, 70, 31, -14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 - 14*x^20 + 31*x^19 + 70*x^18 - 232*x^17 - 279*x^16 + 727*x^15 + 777*x^14 - 691*x^13 - 405*x^12 + 301*x^11 - 1802*x^10 - 2957*x^9 + 1747*x^8 + 4390*x^7 + 52*x^6 - 990*x^5 - 739*x^4 - 86*x^3 + 40*x^2 + 61*x - 47)
 
gp: K = bnfinit(x^22 - 2*x^21 - 14*x^20 + 31*x^19 + 70*x^18 - 232*x^17 - 279*x^16 + 727*x^15 + 777*x^14 - 691*x^13 - 405*x^12 + 301*x^11 - 1802*x^10 - 2957*x^9 + 1747*x^8 + 4390*x^7 + 52*x^6 - 990*x^5 - 739*x^4 - 86*x^3 + 40*x^2 + 61*x - 47, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} - 14 x^{20} + 31 x^{19} + 70 x^{18} - 232 x^{17} - 279 x^{16} + 727 x^{15} + 777 x^{14} - 691 x^{13} - 405 x^{12} + 301 x^{11} - 1802 x^{10} - 2957 x^{9} + 1747 x^{8} + 4390 x^{7} + 52 x^{6} - 990 x^{5} - 739 x^{4} - 86 x^{3} + 40 x^{2} + 61 x - 47 \)

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:  \(-178176446876650757881387571453423=-\,23^{20}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.24$
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}{2360091364371641515923809793275382128957} a^{21} - \frac{407226193778271876173578298839318273300}{2360091364371641515923809793275382128957} a^{20} + \frac{145013260375449883394330420355883584668}{2360091364371641515923809793275382128957} a^{19} + \frac{825630055414468892725428483016051748775}{2360091364371641515923809793275382128957} a^{18} + \frac{685420574193419488965821431530532634009}{2360091364371641515923809793275382128957} a^{17} + \frac{62636280584193781025826192940554293507}{2360091364371641515923809793275382128957} a^{16} - \frac{374002486456244043965740195958625699243}{2360091364371641515923809793275382128957} a^{15} - \frac{689551138080475473027896012470500204567}{2360091364371641515923809793275382128957} a^{14} + \frac{972108898385428541736612826957662120965}{2360091364371641515923809793275382128957} a^{13} + \frac{329065570418991763645066168529173184937}{2360091364371641515923809793275382128957} a^{12} + \frac{941225801298728204687741121541785042519}{2360091364371641515923809793275382128957} a^{11} - \frac{563382319456614610439079098711716880233}{2360091364371641515923809793275382128957} a^{10} + \frac{457550216022184365966309721494292193257}{2360091364371641515923809793275382128957} a^{9} - \frac{293253256359407985547450533028915435422}{2360091364371641515923809793275382128957} a^{8} + \frac{340995930693629326155191580537348420789}{2360091364371641515923809793275382128957} a^{7} - \frac{1172994484171519263514559702844943497206}{2360091364371641515923809793275382128957} a^{6} - \frac{805276399828988094011302833379280673930}{2360091364371641515923809793275382128957} a^{5} + \frac{622986655162945055027107234097090257971}{2360091364371641515923809793275382128957} a^{4} - \frac{775506262662066248169067926062042407323}{2360091364371641515923809793275382128957} a^{3} + \frac{94159347250238796055606340917056426495}{2360091364371641515923809793275382128957} a^{2} - \frac{305364853720098224803666422383601642577}{2360091364371641515923809793275382128957} a + \frac{22939908443252727680498655440150050219}{50214709880247691828166165814369832531}$

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:  \( 44250455.5687 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ R $22$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ $22$ R ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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