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Properties

Label 22.8.31549754154...8416.1
Degree $22$
Signature $[8, 7]$
Discriminant $-\,2^{48}\cdot 3^{20}\cdot 79\cdot 337^{8}\cdot 25537\cdot 310501^{8}\cdot 1108640890417$
Root discriminant $69{,}320.78$
Ramified primes $2, 3, 79, 337, 25537, 310501, 1108640890417$
Class number Not computed
Class group Not computed
Galois group 22T44

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7476951174594, -11356029443979, 13843027641789, -4579083727977, 1216236277569, 1531016892369, -921763556307, 364667664168, -104670123606, 6576607873, 1221051993, -1522409545, 366194477, -48148705, 3961947, 1383522, -307032, 45987, -4261, -345, 73, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 73*x^20 - 345*x^19 - 4261*x^18 + 45987*x^17 - 307032*x^16 + 1383522*x^15 + 3961947*x^14 - 48148705*x^13 + 366194477*x^12 - 1522409545*x^11 + 1221051993*x^10 + 6576607873*x^9 - 104670123606*x^8 + 364667664168*x^7 - 921763556307*x^6 + 1531016892369*x^5 + 1216236277569*x^4 - 4579083727977*x^3 + 13843027641789*x^2 - 11356029443979*x + 7476951174594)
 
gp: K = bnfinit(x^22 - 11*x^21 + 73*x^20 - 345*x^19 - 4261*x^18 + 45987*x^17 - 307032*x^16 + 1383522*x^15 + 3961947*x^14 - 48148705*x^13 + 366194477*x^12 - 1522409545*x^11 + 1221051993*x^10 + 6576607873*x^9 - 104670123606*x^8 + 364667664168*x^7 - 921763556307*x^6 + 1531016892369*x^5 + 1216236277569*x^4 - 4579083727977*x^3 + 13843027641789*x^2 - 11356029443979*x + 7476951174594, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{21} + 73 x^{20} - 345 x^{19} - 4261 x^{18} + 45987 x^{17} - 307032 x^{16} + 1383522 x^{15} + 3961947 x^{14} - 48148705 x^{13} + 366194477 x^{12} - 1522409545 x^{11} + 1221051993 x^{10} + 6576607873 x^{9} - 104670123606 x^{8} + 364667664168 x^{7} - 921763556307 x^{6} + 1531016892369 x^{5} + 1216236277569 x^{4} - 4579083727977 x^{3} + 13843027641789 x^{2} - 11356029443979 x + 7476951174594 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-31549754154938736139605034462596274469412770924031278962771396260567384323209653421425695172275073152188416=-\,2^{48}\cdot 3^{20}\cdot 79\cdot 337^{8}\cdot 25537\cdot 310501^{8}\cdot 1108640890417\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69{,}320.78$
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:  $2, 3, 79, 337, 25537, 310501, 1108640890417$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{20} - \frac{5}{28943660303584240017862873266237250845263331199717806703258} a^{19} - \frac{737610584223940751356102048530880609070892994189179622805}{14471830151792120008931436633118625422631665599858903351629} a^{18} + \frac{26553981032061867048819673747111701926552147790810466421265}{57887320607168480035725746532474501690526662399435613406516} a^{17} - \frac{586860724951479526293649518167839847781819036475741033102}{1315620922890192728084676057556238674784696872714445759239} a^{16} + \frac{2473995676591125099716396130300640312759563594284891320221}{14471830151792120008931436633118625422631665599858903351629} a^{15} - \frac{201626948634834958040240051859366525231469762495376591742}{14471830151792120008931436633118625422631665599858903351629} a^{14} - \frac{71777298086336739403032951341673156430548613433947102415}{28943660303584240017862873266237250845263331199717806703258} a^{13} + \frac{24941956648467998252648493963976767457995250771258429199949}{57887320607168480035725746532474501690526662399435613406516} a^{12} - \frac{12841687910111895951037183097197048929683384075161616468737}{28943660303584240017862873266237250845263331199717806703258} a^{11} + \frac{4481593734962046273720394404725615723404558890057092463730}{14471830151792120008931436633118625422631665599858903351629} a^{10} - \frac{21077005455778262574412527945028274541246440252693679907575}{57887320607168480035725746532474501690526662399435613406516} a^{9} + \frac{2855866731678020413518640081046872437236270566910626353755}{28943660303584240017862873266237250845263331199717806703258} a^{8} - \frac{5489624315612776094710219373798349577215325272985413362438}{14471830151792120008931436633118625422631665599858903351629} a^{7} - \frac{1950857656348966698424457832454131078100859731309054393911}{14471830151792120008931436633118625422631665599858903351629} a^{6} - \frac{6358768140936813493221859227266859554484850275531464883665}{14471830151792120008931436633118625422631665599858903351629} a^{5} - \frac{24533229084395396778479410071165303866873244623983328928459}{57887320607168480035725746532474501690526662399435613406516} a^{4} - \frac{2397657681228293508054861432582376973667621265965112078817}{28943660303584240017862873266237250845263331199717806703258} a^{3} - \frac{5559281323579400765961827739301391134290668479882919079619}{14471830151792120008931436633118625422631665599858903351629} a^{2} - \frac{17559241910196809740042606207571598041070657975824712948371}{57887320607168480035725746532474501690526662399435613406516} a + \frac{583033358529455841748358103828492840585491229564348425843}{2631241845780385456169352115112477349569393745428891518478}$, $\frac{1}{173661961821505440107177239597423505071579987198306840219548} a^{21} + \frac{12996608983344238506219232536056864204489879611480544105969}{86830980910752720053588619798711752535789993599153420109774} a^{19} + \frac{25993217966688477012438465072113728408979759222961088212323}{173661961821505440107177239597423505071579987198306840219548} a^{18} - \frac{8286444102181471474558762331773407081585311949001001379403}{28943660303584240017862873266237250845263331199717806703258} a^{17} - \frac{1397787820301047585619768111895747084237956006203669638161}{14471830151792120008931436633118625422631665599858903351629} a^{16} - \frac{572035170851513454171836019642692111076691836946156771043}{1315620922890192728084676057556238674784696872714445759239} a^{15} - \frac{11015992191455758639356902418255418168774425021019761880171}{28943660303584240017862873266237250845263331199717806703258} a^{14} - \frac{1041845755164461108216300351373672647307344966571397765299}{5262483691560770912338704230224954699138787490857783036956} a^{13} - \frac{177570267368584761783918470102827955489111591715486240092}{3946862768670578184254028172668716024354090618143337277717} a^{12} - \frac{550026683898485663648817749414814542169789811287565342379}{2631241845780385456169352115112477349569393745428891518478} a^{11} + \frac{13468442424782388285088881912810100168619259351000985125335}{173661961821505440107177239597423505071579987198306840219548} a^{10} - \frac{7849089818230286202477689922691373866602968548702176537173}{43415490455376360026794309899355876267894996799576710054887} a^{9} - \frac{172185479061054364728741078838867054959564789039126816524}{1315620922890192728084676057556238674784696872714445759239} a^{8} + \frac{346739931563917463399698320678958280090849979424141796075}{14471830151792120008931436633118625422631665599858903351629} a^{7} - \frac{3798504850878120156178333639563181637620593996254368490382}{14471830151792120008931436633118625422631665599858903351629} a^{6} - \frac{1434384008884667162559114940361869069216988062045438504515}{5262483691560770912338704230224954699138787490857783036956} a^{5} + \frac{3275749735785987114810409090462892986380468602117879465863}{14471830151792120008931436633118625422631665599858903351629} a^{4} + \frac{12421337099839621144061637787055525035966926126625852603579}{28943660303584240017862873266237250845263331199717806703258} a^{3} + \frac{623104481178343023704549806560848914990078291706629824127}{5262483691560770912338704230224954699138787490857783036956} a^{2} + \frac{908023050598780935434587983827863929479517874206281342032}{14471830151792120008931436633118625422631665599858903351629} a + \frac{533181956585695493552371487195408509380919758369098789992}{1315620922890192728084676057556238674784696872714445759239}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T44:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 16220160
The 104 conjugacy class representatives for t22n44 are not computed
Character table for t22n44 is not computed

Intermediate fields

11.11.118769262421915560193703211428553337469927424.1

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

Sibling fields

Degree 22 sibling: 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 R $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ $16{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.9.10.1$x^{9} + 3 x^{2} + 3$$9$$1$$10$$C_3^2:Q_8$$[5/4, 5/4]_{4}^{2}$
3.9.10.1$x^{9} + 3 x^{2} + 3$$9$$1$$10$$C_3^2:Q_8$$[5/4, 5/4]_{4}^{2}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
79.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
79.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
79.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
337Data not computed
25537Data not computed
310501Data not computed
1108640890417Data not computed