↑

Properties

Label 22.22.1443504362...2832.1
Degree $22$
Signature $[22, 0]$
Discriminant $2^{43}\cdot 11^{22}\cdot 17^{10}$
Root discriminant $154.55$
Ramified primes $2, 11, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T15

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16203074, 14699696, 171802697, -110247720, -505302050, 84739424, 527297551, -23295712, -276604790, 2713744, 83059603, -113976, -15154920, 0, 1717628, 0, -120010, 0, 4961, 0, -110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 110*x^20 + 4961*x^18 - 120010*x^16 + 1717628*x^14 - 15154920*x^12 - 113976*x^11 + 83059603*x^10 + 2713744*x^9 - 276604790*x^8 - 23295712*x^7 + 527297551*x^6 + 84739424*x^5 - 505302050*x^4 - 110247720*x^3 + 171802697*x^2 + 14699696*x - 16203074)
 
gp: K = bnfinit(x^22 - 110*x^20 + 4961*x^18 - 120010*x^16 + 1717628*x^14 - 15154920*x^12 - 113976*x^11 + 83059603*x^10 + 2713744*x^9 - 276604790*x^8 - 23295712*x^7 + 527297551*x^6 + 84739424*x^5 - 505302050*x^4 - 110247720*x^3 + 171802697*x^2 + 14699696*x - 16203074, 1)
 

Normalized defining polynomial

\( x^{22} - 110 x^{20} + 4961 x^{18} - 120010 x^{16} + 1717628 x^{14} - 15154920 x^{12} - 113976 x^{11} + 83059603 x^{10} + 2713744 x^{9} - 276604790 x^{8} - 23295712 x^{7} + 527297551 x^{6} + 84739424 x^{5} - 505302050 x^{4} - 110247720 x^{3} + 171802697 x^{2} + 14699696 x - 16203074 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1443504362796122527044991464707560867604392312832=2^{43}\cdot 11^{22}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $154.55$
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:  $2, 11, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} - \frac{8}{17} a^{12} - \frac{3}{17} a^{10} - \frac{7}{17} a^{8} - \frac{1}{17} a^{6} + \frac{2}{17} a^{4} - \frac{8}{17} a^{3}$, $\frac{1}{17} a^{15} - \frac{8}{17} a^{13} - \frac{3}{17} a^{11} - \frac{7}{17} a^{9} - \frac{1}{17} a^{7} + \frac{2}{17} a^{5} - \frac{8}{17} a^{4}$, $\frac{1}{119} a^{16} + \frac{2}{119} a^{15} + \frac{2}{119} a^{14} - \frac{16}{119} a^{13} - \frac{7}{17} a^{12} + \frac{45}{119} a^{11} + \frac{48}{119} a^{10} + \frac{3}{119} a^{9} - \frac{3}{119} a^{8} - \frac{36}{119} a^{7} - \frac{8}{119} a^{6} - \frac{4}{119} a^{5} - \frac{30}{119} a^{4} + \frac{8}{17} a^{3} + \frac{2}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{2023} a^{17} + \frac{26}{2023} a^{15} - \frac{2}{119} a^{14} + \frac{116}{2023} a^{13} + \frac{50}{119} a^{12} - \frac{69}{289} a^{11} - \frac{45}{119} a^{10} - \frac{562}{2023} a^{9} - \frac{3}{119} a^{8} + \frac{393}{2023} a^{7} + \frac{502}{2023} a^{6} + \frac{58}{119} a^{5} - \frac{15}{119} a^{4} + \frac{16}{119} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{2023} a^{18} - \frac{8}{2023} a^{16} + \frac{1}{119} a^{15} + \frac{48}{2023} a^{14} + \frac{26}{119} a^{13} - \frac{120}{289} a^{12} - \frac{37}{119} a^{11} - \frac{171}{2023} a^{10} - \frac{58}{119} a^{9} + \frac{495}{2023} a^{8} - \frac{416}{2023} a^{7} - \frac{45}{119} a^{6} + \frac{1}{17} a^{5} + \frac{20}{119} a^{4} - \frac{10}{119} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{2023} a^{19} - \frac{16}{2023} a^{15} + \frac{1}{119} a^{14} + \frac{241}{2023} a^{13} - \frac{8}{119} a^{12} - \frac{40}{2023} a^{11} + \frac{31}{119} a^{10} - \frac{363}{2023} a^{9} + \frac{60}{2023} a^{8} - \frac{817}{2023} a^{7} + \frac{344}{2023} a^{6} - \frac{16}{119} a^{5} - \frac{2}{119} a^{4} - \frac{25}{119} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{34391} a^{20} - \frac{8}{34391} a^{18} + \frac{99}{34391} a^{16} + \frac{2}{119} a^{15} + \frac{197}{34391} a^{14} + \frac{52}{119} a^{13} - \frac{11884}{34391} a^{12} + \frac{4}{17} a^{11} - \frac{5353}{34391} a^{10} - \frac{110}{34391} a^{9} - \frac{864}{2023} a^{8} - \frac{879}{2023} a^{7} - \frac{1}{7} a^{6} - \frac{5}{17} a^{5} + \frac{46}{119} a^{4} + \frac{1}{17} a^{3} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{5533730993696888654107649476312755641482555887529} a^{21} + \frac{9242963909003598690387723588705525325719626}{790532999099555522015378496616107948783222269647} a^{20} - \frac{907753443679872265471892909148477618993529515}{5533730993696888654107649476312755641482555887529} a^{19} - \frac{91095819085548861645628982999392682953989153}{790532999099555522015378496616107948783222269647} a^{18} - \frac{1222075846601539130862238764699123879351010855}{5533730993696888654107649476312755641482555887529} a^{17} + \frac{15573603515909465402262572768043493756487375551}{5533730993696888654107649476312755641482555887529} a^{16} + \frac{68640047072629024279086653581187314097449293389}{5533730993696888654107649476312755641482555887529} a^{15} - \frac{228447067791188991892451295999396603670860721}{790532999099555522015378496616107948783222269647} a^{14} + \frac{1319627721082092732730691876905470946030975874551}{5533730993696888654107649476312755641482555887529} a^{13} + \frac{1743352635032145652325511089532038400808008227328}{5533730993696888654107649476312755641482555887529} a^{12} - \frac{1169778028110828446959057186837418843138201248300}{5533730993696888654107649476312755641482555887529} a^{11} - \frac{1646681375971551517313115197039512043321936918850}{5533730993696888654107649476312755641482555887529} a^{10} - \frac{309283402698446025413439889912285763717677688503}{5533730993696888654107649476312755641482555887529} a^{9} - \frac{17355939218936734795816605549301194635069424327}{325513587864522862006332322136044449498973875737} a^{8} - \frac{22850519515649073391937992053038299048032211081}{46501941123503266000904617448006349928424839391} a^{7} + \frac{143862601159610887644817552604785440613622919621}{325513587864522862006332322136044449498973875737} a^{6} + \frac{6547313630506766527858585029389041276383446855}{19147858109677815412137195419767320558763169161} a^{5} + \frac{1754923660823868383271209564933429270143109871}{19147858109677815412137195419767320558763169161} a^{4} - \frac{3836230263650759595927003091512977245843931652}{19147858109677815412137195419767320558763169161} a^{3} + \frac{326241880605755352311975174481772255652285082}{1126344594686930318361011495280430621103715833} a^{2} - \frac{75465088019317507740961641027430904000961729}{1126344594686930318361011495280430621103715833} a - \frac{279532511963151085032584625375701906022611950}{1126344594686930318361011495280430621103715833}$

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:  $21$
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:  \( 372931478202000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T15:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2420
The 29 conjugacy class representatives for t22n15
Character table for t22n15 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \)

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
2Data not computed
11Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.10.0.1$x^{10} - x + 7$$1$$10$$0$$C_{10}$$[\ ]^{10}$
17.11.10.1$x^{11} - 17$$11$$1$$10$$F_{11}$$[\ ]_{11}^{10}$