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Properties

Label 22.4.17817644687...3423.1
Degree $22$
Signature $[4, 9]$
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![-2897, 3881, 1939, -6847, -2384, 7497, -1390, -4500, -360, 2535, 890, -2461, 469, 986, -626, -79, 209, -64, -34, 21, -4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 - 4*x^20 + 21*x^19 - 34*x^18 - 64*x^17 + 209*x^16 - 79*x^15 - 626*x^14 + 986*x^13 + 469*x^12 - 2461*x^11 + 890*x^10 + 2535*x^9 - 360*x^8 - 4500*x^7 - 1390*x^6 + 7497*x^5 - 2384*x^4 - 6847*x^3 + 1939*x^2 + 3881*x - 2897)
 
gp: K = bnfinit(x^22 - 3*x^21 - 4*x^20 + 21*x^19 - 34*x^18 - 64*x^17 + 209*x^16 - 79*x^15 - 626*x^14 + 986*x^13 + 469*x^12 - 2461*x^11 + 890*x^10 + 2535*x^9 - 360*x^8 - 4500*x^7 - 1390*x^6 + 7497*x^5 - 2384*x^4 - 6847*x^3 + 1939*x^2 + 3881*x - 2897, 1)
 

Normalized defining polynomial

\( x^{22} - 3 x^{21} - 4 x^{20} + 21 x^{19} - 34 x^{18} - 64 x^{17} + 209 x^{16} - 79 x^{15} - 626 x^{14} + 986 x^{13} + 469 x^{12} - 2461 x^{11} + 890 x^{10} + 2535 x^{9} - 360 x^{8} - 4500 x^{7} - 1390 x^{6} + 7497 x^{5} - 2384 x^{4} - 6847 x^{3} + 1939 x^{2} + 3881 x - 2897 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 9]$
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}{15141765365907803090604556432892046781164355933} a^{21} - \frac{352564206449829646559521974032580753689142599}{15141765365907803090604556432892046781164355933} a^{20} - \frac{221983485280962726998609723127622495161538457}{15141765365907803090604556432892046781164355933} a^{19} + \frac{6187691145206230956993152542458907723632679429}{15141765365907803090604556432892046781164355933} a^{18} - \frac{1700183661803413555508368894840178236030914957}{15141765365907803090604556432892046781164355933} a^{17} + \frac{4445281618680528016657829605171427843602328313}{15141765365907803090604556432892046781164355933} a^{16} - \frac{1988526930235837716172949700578608944995828492}{15141765365907803090604556432892046781164355933} a^{15} - \frac{4410905905603489411225984835331209974535948223}{15141765365907803090604556432892046781164355933} a^{14} - \frac{9142670649423523462140800141000574902640456}{322165220551229852991586307082809505982220339} a^{13} + \frac{696043843736455135819471507984489914362981691}{15141765365907803090604556432892046781164355933} a^{12} - \frac{4534811716938970373871436701814983224768969072}{15141765365907803090604556432892046781164355933} a^{11} - \frac{5407821327102780823272872170254264199661871540}{15141765365907803090604556432892046781164355933} a^{10} + \frac{7231697413126652547320758153571050138748890556}{15141765365907803090604556432892046781164355933} a^{9} + \frac{718295411600205942393190088530727901166713001}{15141765365907803090604556432892046781164355933} a^{8} + \frac{3101156820664651940465861715497566453186956458}{15141765365907803090604556432892046781164355933} a^{7} + \frac{30785525559921108192346331554105586392734019}{322165220551229852991586307082809505982220339} a^{6} - \frac{5823850638814697129607329587622999261238059836}{15141765365907803090604556432892046781164355933} a^{5} - \frac{899168788292962420656355254274760591254120439}{15141765365907803090604556432892046781164355933} a^{4} + \frac{87149648528613766369613369546321120419929309}{322165220551229852991586307082809505982220339} a^{3} - \frac{5377714192721055041329834318524438013080249006}{15141765365907803090604556432892046781164355933} a^{2} + \frac{3272707107350849589747191482005141924394541313}{15141765365907803090604556432892046781164355933} a - \frac{3821608333460023257590376740593150417047339978}{15141765365907803090604556432892046781164355933}$

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:  $12$
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:  \( 18036839.011 \) (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