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Properties

Label 22.8.87192729322...7207.2
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![-1609, -2306, 3417, 566, -5397, 2988, 5601, -6627, -2466, 8241, -1238, -5521, 2076, 1988, -1256, -192, 416, -150, -43, 51, -7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 - 7*x^20 + 51*x^19 - 43*x^18 - 150*x^17 + 416*x^16 - 192*x^15 - 1256*x^14 + 1988*x^13 + 2076*x^12 - 5521*x^11 - 1238*x^10 + 8241*x^9 - 2466*x^8 - 6627*x^7 + 5601*x^6 + 2988*x^5 - 5397*x^4 + 566*x^3 + 3417*x^2 - 2306*x - 1609)
 
gp: K = bnfinit(x^22 - 4*x^21 - 7*x^20 + 51*x^19 - 43*x^18 - 150*x^17 + 416*x^16 - 192*x^15 - 1256*x^14 + 1988*x^13 + 2076*x^12 - 5521*x^11 - 1238*x^10 + 8241*x^9 - 2466*x^8 - 6627*x^7 + 5601*x^6 + 2988*x^5 - 5397*x^4 + 566*x^3 + 3417*x^2 - 2306*x - 1609, 1)
 

Normalized defining polynomial

\( x^{22} - 4 x^{21} - 7 x^{20} + 51 x^{19} - 43 x^{18} - 150 x^{17} + 416 x^{16} - 192 x^{15} - 1256 x^{14} + 1988 x^{13} + 2076 x^{12} - 5521 x^{11} - 1238 x^{10} + 8241 x^{9} - 2466 x^{8} - 6627 x^{7} + 5601 x^{6} + 2988 x^{5} - 5397 x^{4} + 566 x^{3} + 3417 x^{2} - 2306 x - 1609 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{9160107466819862889071832243929911757} a^{21} + \frac{3664008150098448215631768739168918850}{9160107466819862889071832243929911757} a^{20} - \frac{3582301361320250355590339611355259589}{9160107466819862889071832243929911757} a^{19} + \frac{2023718677164715085494244778757891668}{9160107466819862889071832243929911757} a^{18} + \frac{1534884563347806417219499991512792688}{9160107466819862889071832243929911757} a^{17} + \frac{942665840229694079491578989612983804}{9160107466819862889071832243929911757} a^{16} - \frac{3378268854923134002746198218230658734}{9160107466819862889071832243929911757} a^{15} - \frac{2973604452988914944286776223461428678}{9160107466819862889071832243929911757} a^{14} - \frac{3876194226951138216103563655187462845}{9160107466819862889071832243929911757} a^{13} - \frac{3218478721244046773541320310639721839}{9160107466819862889071832243929911757} a^{12} + \frac{716901043532152835281272659572093592}{9160107466819862889071832243929911757} a^{11} - \frac{2296428946984123270849876002984672378}{9160107466819862889071832243929911757} a^{10} + \frac{94106263810151992834674465259230223}{9160107466819862889071832243929911757} a^{9} + \frac{2480906080244639714646810456417715587}{9160107466819862889071832243929911757} a^{8} + \frac{3250763562234657701696300000970154045}{9160107466819862889071832243929911757} a^{7} + \frac{3438852153302451219637416118365001776}{9160107466819862889071832243929911757} a^{6} - \frac{3063050164247612935749249435496043117}{9160107466819862889071832243929911757} a^{5} - \frac{370144812152687422931369801965708}{194895903549358784873868771147444931} a^{4} + \frac{439708455038813498128935947041691662}{9160107466819862889071832243929911757} a^{3} + \frac{1485937687741755247097847144223758527}{9160107466819862889071832243929911757} a^{2} + \frac{3818927114644332474119195872799104917}{9160107466819862889071832243929911757} a + \frac{380562140158201692247262424300599491}{9160107466819862889071832243929911757}$

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:  \( 27806400.2764 \) (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