↑

Properties

Label 22.10.4098058278...8729.2
Degree $22$
Signature $[10, 6]$
Discriminant $23^{21}\cdot 47^{3}$
Root discriminant $33.72$
Ramified primes $23, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T28

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-367, 5416, -28997, 74034, -82097, 1477, 72873, -55561, 26642, -21762, 853, 12005, -3622, -6, -1469, 378, 271, -9, 16, -31, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 4*x^20 - 31*x^19 + 16*x^18 - 9*x^17 + 271*x^16 + 378*x^15 - 1469*x^14 - 6*x^13 - 3622*x^12 + 12005*x^11 + 853*x^10 - 21762*x^9 + 26642*x^8 - 55561*x^7 + 72873*x^6 + 1477*x^5 - 82097*x^4 + 74034*x^3 - 28997*x^2 + 5416*x - 367)
 
gp: K = bnfinit(x^22 - 2*x^21 + 4*x^20 - 31*x^19 + 16*x^18 - 9*x^17 + 271*x^16 + 378*x^15 - 1469*x^14 - 6*x^13 - 3622*x^12 + 12005*x^11 + 853*x^10 - 21762*x^9 + 26642*x^8 - 55561*x^7 + 72873*x^6 + 1477*x^5 - 82097*x^4 + 74034*x^3 - 28997*x^2 + 5416*x - 367, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} + 4 x^{20} - 31 x^{19} + 16 x^{18} - 9 x^{17} + 271 x^{16} + 378 x^{15} - 1469 x^{14} - 6 x^{13} - 3622 x^{12} + 12005 x^{11} + 853 x^{10} - 21762 x^{9} + 26642 x^{8} - 55561 x^{7} + 72873 x^{6} + 1477 x^{5} - 82097 x^{4} + 74034 x^{3} - 28997 x^{2} + 5416 x - 367 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4098058278162967431271914143428729=23^{21}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.72$
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}{9293770151219698096245861734690797739374175460511} a^{21} - \frac{444388420701733581845953721685658659053633769435}{9293770151219698096245861734690797739374175460511} a^{20} - \frac{1609643444841586399771331733566719772966682666203}{9293770151219698096245861734690797739374175460511} a^{19} - \frac{38631904322953706248869734968866753259793008783}{9293770151219698096245861734690797739374175460511} a^{18} - \frac{3255696929746504101245359746302075822691946453753}{9293770151219698096245861734690797739374175460511} a^{17} + \frac{224044785794265210896324091942696398593175679519}{9293770151219698096245861734690797739374175460511} a^{16} - \frac{4588384186692746643441753086669575048704457401828}{9293770151219698096245861734690797739374175460511} a^{15} + \frac{2938640057110495942441916004130650402199883386318}{9293770151219698096245861734690797739374175460511} a^{14} - \frac{974270317013808009206263215194928940897854785138}{9293770151219698096245861734690797739374175460511} a^{13} + \frac{2375480790107671642955413452206529321921017110034}{9293770151219698096245861734690797739374175460511} a^{12} - \frac{3107038866404483878059725041211563623410191128134}{9293770151219698096245861734690797739374175460511} a^{11} - \frac{809246554069833782496057599660337079253674189740}{9293770151219698096245861734690797739374175460511} a^{10} + \frac{4406675882548304957068903487076290223165388912435}{9293770151219698096245861734690797739374175460511} a^{9} - \frac{762362972852055300203899675115466258487455179885}{9293770151219698096245861734690797739374175460511} a^{8} + \frac{3528954638145024913968927383061547593358604394583}{9293770151219698096245861734690797739374175460511} a^{7} + \frac{1479815183586489064736118037225878152386900446869}{9293770151219698096245861734690797739374175460511} a^{6} - \frac{3399419204613641239481161796028207618490661499151}{9293770151219698096245861734690797739374175460511} a^{5} + \frac{2606521865499539802679692847658026897842513907755}{9293770151219698096245861734690797739374175460511} a^{4} - \frac{836171380119626187002729300789288779224845061692}{9293770151219698096245861734690797739374175460511} a^{3} + \frac{2082093198161537349364928084931029281028383851445}{9293770151219698096245861734690797739374175460511} a^{2} - \frac{3348030203227827158314440430394576603934612546516}{9293770151219698096245861734690797739374175460511} a + \frac{3547368830350164192964532474278380938956380023137}{9293770151219698096245861734690797739374175460511}$

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:  $15$
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:  \( 338850726.154 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ 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$$0$Trivial$[\ ]$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$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$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$