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Properties

Label 22.14.8328705687...1312.1
Degree $22$
Signature $[14, 4]$
Discriminant $2^{45}\cdot 3^{28}\cdot 7^{4}\cdot 23^{4}\cdot 137^{16}$
Root discriminant $1507.33$
Ramified primes $2, 3, 7, 23, 137$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T46

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-530108, -2436432, 1997240, 14132688, -3630980, -25064112, 9743124, 22004508, -9151008, -11919636, 4901112, 4228056, -1608525, -990414, 323265, 144516, -38526, -11784, 2554, 456, -85, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 6*x^21 - 85*x^20 + 456*x^19 + 2554*x^18 - 11784*x^17 - 38526*x^16 + 144516*x^15 + 323265*x^14 - 990414*x^13 - 1608525*x^12 + 4228056*x^11 + 4901112*x^10 - 11919636*x^9 - 9151008*x^8 + 22004508*x^7 + 9743124*x^6 - 25064112*x^5 - 3630980*x^4 + 14132688*x^3 + 1997240*x^2 - 2436432*x - 530108)
 
gp: K = bnfinit(x^22 - 6*x^21 - 85*x^20 + 456*x^19 + 2554*x^18 - 11784*x^17 - 38526*x^16 + 144516*x^15 + 323265*x^14 - 990414*x^13 - 1608525*x^12 + 4228056*x^11 + 4901112*x^10 - 11919636*x^9 - 9151008*x^8 + 22004508*x^7 + 9743124*x^6 - 25064112*x^5 - 3630980*x^4 + 14132688*x^3 + 1997240*x^2 - 2436432*x - 530108, 1)
 

Normalized defining polynomial

\( x^{22} - 6 x^{21} - 85 x^{20} + 456 x^{19} + 2554 x^{18} - 11784 x^{17} - 38526 x^{16} + 144516 x^{15} + 323265 x^{14} - 990414 x^{13} - 1608525 x^{12} + 4228056 x^{11} + 4901112 x^{10} - 11919636 x^{9} - 9151008 x^{8} + 22004508 x^{7} + 9743124 x^{6} - 25064112 x^{5} - 3630980 x^{4} + 14132688 x^{3} + 1997240 x^{2} - 2436432 x - 530108 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8328705687908829703976940241094560024838491340873309254929718613901312=2^{45}\cdot 3^{28}\cdot 7^{4}\cdot 23^{4}\cdot 137^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1507.33$
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:  $2, 3, 7, 23, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{96} a^{18} - \frac{1}{16} a^{17} - \frac{1}{32} a^{16} - \frac{1}{8} a^{15} - \frac{3}{32} a^{14} + \frac{1}{16} a^{13} - \frac{3}{32} a^{12} + \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a - \frac{11}{24}$, $\frac{1}{96} a^{19} - \frac{1}{32} a^{17} - \frac{1}{16} a^{16} - \frac{3}{32} a^{15} - \frac{3}{32} a^{13} + \frac{1}{16} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{24} a + \frac{1}{4}$, $\frac{1}{432000} a^{20} - \frac{47}{72000} a^{19} + \frac{133}{43200} a^{18} - \frac{401}{14400} a^{17} - \frac{199}{18000} a^{16} + \frac{6317}{72000} a^{15} - \frac{827}{24000} a^{14} - \frac{983}{8000} a^{13} + \frac{7453}{144000} a^{12} + \frac{593}{6000} a^{11} + \frac{7223}{36000} a^{10} + \frac{57}{500} a^{9} - \frac{3269}{36000} a^{8} - \frac{809}{12000} a^{7} - \frac{951}{4000} a^{6} + \frac{3013}{9000} a^{5} - \frac{8161}{18000} a^{4} - \frac{109}{1125} a^{3} + \frac{49141}{108000} a^{2} - \frac{461}{1125} a + \frac{30169}{108000}$, $\frac{1}{14100367499027296139896309530000322720972545028176000} a^{21} + \frac{10327691698978512473393237492660791048883014417}{14100367499027296139896309530000322720972545028176000} a^{20} + \frac{32135107866832102491800658211570693324184354857}{881272968689206008743519345625020170060784064261000} a^{19} + \frac{55992926120973142949146705865359477639593971177}{44063648434460300437175967281251008503039203213050} a^{18} + \frac{125865354957482127011238500778524946789369078002959}{2350061249837882689982718255000053786828757504696000} a^{17} + \frac{23853938302306552100049799241975553744442648932913}{2350061249837882689982718255000053786828757504696000} a^{16} + \frac{66480858297684703409143596143461405348632741455213}{587515312459470672495679563750013446707189376174000} a^{15} - \frac{31052093323075368920326848150709177870006139026261}{391676874972980448330453042500008964471459584116000} a^{14} - \frac{469993300492685189080233646156887311574970779353153}{4700122499675765379965436510000107573657515009392000} a^{13} + \frac{26239191689554948180885320573920624016634015027879}{4700122499675765379965436510000107573657515009392000} a^{12} - \frac{3939236496170638060903941110976629946798964219847}{235006124983788268998271825500005378682875750469600} a^{11} + \frac{144546572040493810594188745716017128517939427142481}{1175030624918941344991359127500026893414378752348000} a^{10} - \frac{48375367534058256766179871609753004218849401341573}{1175030624918941344991359127500026893414378752348000} a^{9} - \frac{96482937993632326775767849735894687888530266943979}{587515312459470672495679563750013446707189376174000} a^{8} + \frac{43687121858732877306735952138365415732021250427289}{97919218743245112082613260625002241117864896029000} a^{7} - \frac{356950173681426724177833682161488374929004651440689}{1175030624918941344991359127500026893414378752348000} a^{6} - \frac{117180142430426526732671243317752498362449512514487}{587515312459470672495679563750013446707189376174000} a^{5} + \frac{263386242483101354370017467003156185982234457182717}{587515312459470672495679563750013446707189376174000} a^{4} - \frac{308176185611401927436112827849867578630067949031039}{705018374951364806994815476500016136048627251408800} a^{3} - \frac{1703387425062581364184350581141732757659732521366697}{3525091874756824034974077382500080680243136257044000} a^{2} + \frac{15353944680424701127718923452964245355273850251789}{141003674990272961398963095300003227209725450281760} a + \frac{693370457807847566190710507015708105461620820430131}{3525091874756824034974077382500080680243136257044000}$

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:  $17$
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:  \( 18736455246300000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T46:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 39916800
The 62 conjugacy class representatives for t22n46 are not computed
Character table for t22n46 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 11.7.63019333158425674204677255696384.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ R $22$ $18{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.8.18.53$x^{8} + 2 x^{6} + 4 x^{3} + 2$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
2.12.24.336$x^{12} + 4 x^{11} - 2 x^{10} + 4 x^{6} + 4 x^{5} + 4 x^{4} - 2 x^{2} + 4 x - 2$$12$$1$$24$$C_2 \times S_4$$[4/3, 4/3, 3]_{3}^{2}$
3Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
7.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.7.0.1$x^{7} - x + 8$$1$$7$$0$$C_7$$[\ ]^{7}$
23.7.0.1$x^{7} - x + 8$$1$$7$$0$$C_7$$[\ ]^{7}$
$137$$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
137.5.4.1$x^{5} - 137$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
137.5.4.1$x^{5} - 137$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
137.5.4.1$x^{5} - 137$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
137.5.4.1$x^{5} - 137$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$