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Properties

Label 22.22.9610975147...5968.1
Degree $22$
Signature $[22, 0]$
Discriminant $2^{43}\cdot 11^{22}\cdot 41^{10}$
Root discriminant $230.60$
Ramified primes $2, 11, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T15

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2857488932, 2956714508, 14018600321, -7007773300, -23934195670, 2845252564, 15202800943, -418434192, -4917829378, 25806220, 920181163, -569132, -106106616, 0, 7743164, 0, -356510, 0, 9977, 0, -154, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 154*x^20 + 9977*x^18 - 356510*x^16 + 7743164*x^14 - 106106616*x^12 - 569132*x^11 + 920181163*x^10 + 25806220*x^9 - 4917829378*x^8 - 418434192*x^7 + 15202800943*x^6 + 2845252564*x^5 - 23934195670*x^4 - 7007773300*x^3 + 14018600321*x^2 + 2956714508*x - 2857488932)
 
gp: K = bnfinit(x^22 - 154*x^20 + 9977*x^18 - 356510*x^16 + 7743164*x^14 - 106106616*x^12 - 569132*x^11 + 920181163*x^10 + 25806220*x^9 - 4917829378*x^8 - 418434192*x^7 + 15202800943*x^6 + 2845252564*x^5 - 23934195670*x^4 - 7007773300*x^3 + 14018600321*x^2 + 2956714508*x - 2857488932, 1)
 

Normalized defining polynomial

\( x^{22} - 154 x^{20} + 9977 x^{18} - 356510 x^{16} + 7743164 x^{14} - 106106616 x^{12} - 569132 x^{11} + 920181163 x^{10} + 25806220 x^{9} - 4917829378 x^{8} - 418434192 x^{7} + 15202800943 x^{6} + 2845252564 x^{5} - 23934195670 x^{4} - 7007773300 x^{3} + 14018600321 x^{2} + 2956714508 x - 2857488932 \)

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:  \(9610975147402804808462869844433474471824562971475968=2^{43}\cdot 11^{22}\cdot 41^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $230.60$
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:  $2, 11, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{11} + \frac{3}{8} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{3}{8} a^{7} + \frac{3}{8} a^{5} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{11} + \frac{3}{8} a^{10} + \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{11} - \frac{3}{8} a^{9} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{541064} a^{17} - \frac{27291}{541064} a^{16} - \frac{14995}{541064} a^{15} + \frac{1327}{270532} a^{14} - \frac{24351}{541064} a^{13} + \frac{3907}{270532} a^{12} - \frac{18937}{135266} a^{11} + \frac{61201}{135266} a^{10} - \frac{124245}{541064} a^{9} + \frac{88649}{270532} a^{8} - \frac{65037}{135266} a^{7} - \frac{83265}{541064} a^{6} + \frac{187559}{541064} a^{5} - \frac{32177}{135266} a^{4} - \frac{101451}{541064} a^{3} - \frac{31591}{541064} a^{2} - \frac{31599}{135266} a + \frac{32933}{135266}$, $\frac{1}{541064} a^{18} + \frac{28553}{541064} a^{16} + \frac{2674}{67633} a^{15} - \frac{7245}{135266} a^{14} + \frac{6531}{541064} a^{13} - \frac{1545}{270532} a^{12} + \frac{73515}{541064} a^{11} - \frac{16827}{270532} a^{10} + \frac{93729}{270532} a^{9} - \frac{17617}{541064} a^{8} + \frac{194741}{541064} a^{7} - \frac{66921}{541064} a^{6} + \frac{213787}{541064} a^{5} + \frac{51275}{541064} a^{4} + \frac{96555}{541064} a^{3} + \frac{179639}{541064} a^{2} - \frac{7313}{67633} a + \frac{67199}{135266}$, $\frac{1}{541064} a^{19} - \frac{6111}{541064} a^{16} + \frac{6365}{541064} a^{15} - \frac{24171}{541064} a^{14} + \frac{23773}{541064} a^{13} + \frac{14007}{541064} a^{12} + \frac{7729}{135266} a^{11} + \frac{42349}{135266} a^{10} - \frac{35757}{270532} a^{9} - \frac{269}{541064} a^{8} + \frac{212331}{541064} a^{7} - \frac{3302}{67633} a^{6} - \frac{17489}{67633} a^{5} + \frac{54125}{541064} a^{4} + \frac{47163}{135266} a^{3} - \frac{264901}{541064} a^{2} + \frac{21593}{135266} a + \frac{33283}{135266}$, $\frac{1}{541064} a^{20} + \frac{7021}{270532} a^{16} - \frac{15901}{541064} a^{15} + \frac{10447}{541064} a^{14} - \frac{1177}{270532} a^{13} + \frac{8343}{135266} a^{12} - \frac{49279}{541064} a^{11} + \frac{4759}{135266} a^{10} + \frac{54227}{541064} a^{9} - \frac{70783}{541064} a^{8} - \frac{36203}{135266} a^{7} - \frac{101635}{541064} a^{6} + \frac{29589}{135266} a^{5} - \frac{22493}{67633} a^{4} - \frac{104985}{541064} a^{3} - \frac{212179}{541064} a^{2} - \frac{21812}{67633} a - \frac{22245}{135266}$, $\frac{1}{22671577065066080889069721300266763164905132309344152} a^{21} - \frac{19518858539883763799874310825526507107706396933}{22671577065066080889069721300266763164905132309344152} a^{20} - \frac{1517940335315800554177457599100388050618863809}{7557192355022026963023240433422254388301710769781384} a^{19} + \frac{3485428075014400236770263999874668628755948361}{3778596177511013481511620216711127194150855384890692} a^{18} + \frac{5059141484563596200329091662212107775383312609}{22671577065066080889069721300266763164905132309344152} a^{17} + \frac{125491773649292621080385517556257162591276297212615}{5667894266266520222267430325066690791226283077336038} a^{16} - \frac{23391881518265775313700540279978962679464177219219}{3778596177511013481511620216711127194150855384890692} a^{15} + \frac{346312678727906795864523423133038643075761754910351}{7557192355022026963023240433422254388301710769781384} a^{14} - \frac{507971460036539020200556960443730832884548335366411}{11335788532533040444534860650133381582452566154672076} a^{13} + \frac{544050542467850506167890002868511374561930098272973}{11335788532533040444534860650133381582452566154672076} a^{12} + \frac{332511547454789410056368530111450328127384332462897}{22671577065066080889069721300266763164905132309344152} a^{11} - \frac{485099898209704748053618392362598052437331186590961}{1889298088755506740755810108355563597075427692445346} a^{10} - \frac{8578533849396137166469142404176195044908466303588129}{22671577065066080889069721300266763164905132309344152} a^{9} - \frac{4094817110443261381122215016678133418087696579352463}{22671577065066080889069721300266763164905132309344152} a^{8} - \frac{251288930467980570008782006740258526856724458522315}{5667894266266520222267430325066690791226283077336038} a^{7} - \frac{1198747491959540256827604120907982529624826545696173}{3238796723580868698438531614323823309272161758477736} a^{6} + \frac{3851008315457163910699545036357790619594625008880683}{22671577065066080889069721300266763164905132309344152} a^{5} - \frac{1047166272221971296863150782289847040721559503768589}{3778596177511013481511620216711127194150855384890692} a^{4} - \frac{40898446210457497108336542224920037994160606499125}{22671577065066080889069721300266763164905132309344152} a^{3} - \frac{7727478833507793097310087330342090044400551012384649}{22671577065066080889069721300266763164905132309344152} a^{2} + \frac{312380902691942484046739808004787645540849509893797}{944649044377753370377905054177781798537713846222673} a + \frac{999782165253933911818810813905272542451534195574611}{5667894266266520222267430325066690791226283077336038}$

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:  \( 30149249254500000000 \) (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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ R $22$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
41Data not computed