↑

Properties

Label 22.6.13687637500...0000.1
Degree $22$
Signature $[6, 8]$
Discriminant $2^{36}\cdot 3^{16}\cdot 5^{27}\cdot 199^{6}$
Root discriminant $211.05$
Ramified primes $2, 3, 5, 199$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T48

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3254864, -9019808, 71987376, -74599880, -94548680, 172274184, -20546277, -83237556, 49064310, -5338040, -3684854, 116412, 105286, 88360, -25065, 26484, -10302, 1944, -80, -80, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 8*x^21 + 36*x^20 - 80*x^19 - 80*x^18 + 1944*x^17 - 10302*x^16 + 26484*x^15 - 25065*x^14 + 88360*x^13 + 105286*x^12 + 116412*x^11 - 3684854*x^10 - 5338040*x^9 + 49064310*x^8 - 83237556*x^7 - 20546277*x^6 + 172274184*x^5 - 94548680*x^4 - 74599880*x^3 + 71987376*x^2 - 9019808*x - 3254864)
 
gp: K = bnfinit(x^22 - 8*x^21 + 36*x^20 - 80*x^19 - 80*x^18 + 1944*x^17 - 10302*x^16 + 26484*x^15 - 25065*x^14 + 88360*x^13 + 105286*x^12 + 116412*x^11 - 3684854*x^10 - 5338040*x^9 + 49064310*x^8 - 83237556*x^7 - 20546277*x^6 + 172274184*x^5 - 94548680*x^4 - 74599880*x^3 + 71987376*x^2 - 9019808*x - 3254864, 1)
 

Normalized defining polynomial

\( x^{22} - 8 x^{21} + 36 x^{20} - 80 x^{19} - 80 x^{18} + 1944 x^{17} - 10302 x^{16} + 26484 x^{15} - 25065 x^{14} + 88360 x^{13} + 105286 x^{12} + 116412 x^{11} - 3684854 x^{10} - 5338040 x^{9} + 49064310 x^{8} - 83237556 x^{7} - 20546277 x^{6} + 172274184 x^{5} - 94548680 x^{4} - 74599880 x^{3} + 71987376 x^{2} - 9019808 x - 3254864 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1368763750033950505738752000000000000000000000000000=2^{36}\cdot 3^{16}\cdot 5^{27}\cdot 199^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $211.05$
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:  $2, 3, 5, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{30} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{4}{15} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{7}{15}$, $\frac{1}{30} a^{11} - \frac{1}{3} a^{8} + \frac{1}{6} a^{7} + \frac{1}{15} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{2}{15} a - \frac{1}{3}$, $\frac{1}{30} a^{12} - \frac{1}{6} a^{8} - \frac{4}{15} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{7}{15} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{30} a^{13} - \frac{1}{6} a^{9} - \frac{4}{15} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{7}{15} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{150} a^{14} - \frac{1}{150} a^{13} + \frac{1}{150} a^{12} - \frac{1}{150} a^{11} + \frac{1}{150} a^{10} - \frac{23}{150} a^{9} + \frac{13}{150} a^{8} - \frac{53}{150} a^{7} + \frac{43}{150} a^{6} - \frac{11}{50} a^{5} - \frac{19}{150} a^{4} + \frac{3}{50} a^{3} + \frac{13}{75} a^{2} - \frac{6}{25} a - \frac{9}{25}$, $\frac{1}{150} a^{15} - \frac{1}{75} a^{10} - \frac{1}{15} a^{9} + \frac{1}{15} a^{8} + \frac{4}{15} a^{7} + \frac{2}{5} a^{6} - \frac{31}{75} a^{5} + \frac{4}{15} a^{4} - \frac{13}{30} a^{3} + \frac{4}{15} a^{2} + \frac{2}{5} a - \frac{17}{75}$, $\frac{1}{150} a^{16} - \frac{1}{75} a^{11} + \frac{1}{15} a^{9} - \frac{1}{15} a^{8} + \frac{1}{15} a^{7} + \frac{19}{75} a^{6} - \frac{4}{15} a^{5} + \frac{7}{30} a^{4} - \frac{1}{15} a^{3} + \frac{1}{15} a^{2} - \frac{17}{75} a + \frac{1}{15}$, $\frac{1}{300} a^{17} - \frac{1}{300} a^{16} - \frac{1}{300} a^{14} + \frac{1}{300} a^{13} + \frac{1}{150} a^{12} + \frac{1}{100} a^{11} - \frac{1}{300} a^{10} - \frac{47}{300} a^{9} - \frac{17}{75} a^{8} + \frac{47}{100} a^{7} - \frac{121}{300} a^{6} + \frac{23}{50} a^{5} - \frac{101}{300} a^{4} - \frac{13}{100} a^{3} + \frac{11}{30} a^{2} - \frac{2}{5} a - \frac{29}{75}$, $\frac{1}{900} a^{18} + \frac{1}{900} a^{17} + \frac{1}{450} a^{16} - \frac{1}{300} a^{15} + \frac{1}{900} a^{14} + \frac{1}{450} a^{13} + \frac{1}{100} a^{12} - \frac{1}{180} a^{11} - \frac{13}{900} a^{10} + \frac{13}{225} a^{9} - \frac{229}{900} a^{8} - \frac{37}{180} a^{7} - \frac{1}{25} a^{6} - \frac{67}{900} a^{5} - \frac{419}{900} a^{4} + \frac{4}{15} a^{3} + \frac{181}{450} a^{2} + \frac{79}{225} a + \frac{112}{225}$, $\frac{1}{10800} a^{19} + \frac{1}{675} a^{17} - \frac{1}{1350} a^{16} + \frac{1}{2700} a^{15} + \frac{1}{675} a^{14} + \frac{11}{5400} a^{13} + \frac{1}{675} a^{12} - \frac{137}{10800} a^{11} + \frac{7}{540} a^{10} - \frac{253}{5400} a^{9} - \frac{49}{2700} a^{8} - \frac{383}{5400} a^{7} - \frac{137}{540} a^{6} + \frac{109}{5400} a^{5} + \frac{611}{2700} a^{4} + \frac{2567}{10800} a^{3} + \frac{131}{2700} a^{2} + \frac{277}{900} a + \frac{743}{2700}$, $\frac{1}{21600} a^{20} - \frac{1}{2700} a^{18} + \frac{1}{5400} a^{17} + \frac{2}{675} a^{16} + \frac{1}{1350} a^{15} - \frac{19}{10800} a^{14} + \frac{1}{5400} a^{13} - \frac{281}{21600} a^{12} + \frac{1}{270} a^{11} + \frac{137}{10800} a^{10} - \frac{61}{1350} a^{9} - \frac{3539}{10800} a^{8} - \frac{233}{2700} a^{7} + \frac{3439}{10800} a^{6} + \frac{1211}{5400} a^{5} + \frac{3947}{21600} a^{4} + \frac{1}{108} a^{3} + \frac{57}{200} a^{2} + \frac{863}{5400} a + \frac{23}{90}$, $\frac{1}{1055705653402723614159324116784029439534570990614083547992395201600} a^{21} + \frac{64400554418870059324071995806292240553893018735536823120021}{11730062815585817935103601297600327105939677673489817199915502240} a^{20} - \frac{1276274991677008695854649611508744056903642379292279954597489}{29325157038964544837759003244000817764849194183724542999788755600} a^{19} - \frac{35405825556361137186894849850364353529711990436947782721412017}{263926413350680903539831029196007359883642747653520886998098800400} a^{18} + \frac{3082711746871411824505134869543946811084241280733097140895621}{2932515703896454483775900324400081776484919418372454299978875560} a^{17} + \frac{1734286623394191909652614982204184929241192748196600300992443}{2932515703896454483775900324400081776484919418372454299978875560} a^{16} + \frac{277404379583849862218771454215256774102827834927017061886448547}{175950942233787269026554019464004906589095165102347257998732533600} a^{15} - \frac{68086292837267845366278179512270593384535321824361949511068767}{43987735558446817256638504866001226647273791275586814499683133400} a^{14} + \frac{5433100779761973123238161131696507281673491906499823208880425349}{351901884467574538053108038928009813178190330204694515997465067200} a^{13} - \frac{171144101456566013908393299297219335067275634620245540047214093}{527852826701361807079662058392014719767285495307041773996197600800} a^{12} + \frac{98761346884322350034772817106019553265874395649306263856630893}{7038037689351490761062160778560196263563806604093890319949301344} a^{11} + \frac{53139210974886882788667447861425684559262620312798167727418619}{17595094223378726902655401946400490658909516510234725799873253360} a^{10} - \frac{22396484922409923007888794494275071236147802702079565037442688587}{527852826701361807079662058392014719767285495307041773996197600800} a^{9} + \frac{12816024943412676342063630715171875023214369939360191026449717713}{29325157038964544837759003244000817764849194183724542999788755600} a^{8} + \frac{1586264331390363988789610181041829700456968106585267900238977171}{58650314077929089675518006488001635529698388367449085999577511200} a^{7} - \frac{2258299960952710268753129436605392163559375497643609960139058587}{21993867779223408628319252433000613323636895637793407249841566700} a^{6} - \frac{39505230217912403015024920909979191925955783507249904150556293087}{351901884467574538053108038928009813178190330204694515997465067200} a^{5} - \frac{1711542196121859219333923385679868005305715386851399005430775043}{7038037689351490761062160778560196263563806604093890319949301344} a^{4} - \frac{2279214714694471140460441577109030683526294884387160190793237978}{16495400834417556471239439324750459992727671728345055437381175025} a^{3} + \frac{9581068347894050828291975229290453919510198079354178120109327587}{87975471116893634513277009732002453294547582551173628999366266800} a^{2} + \frac{19617067607100876580029914518516648073708162809900052001213792517}{43987735558446817256638504866001226647273791275586814499683133400} a - \frac{24687607941993324723102508644417833748830603183485715161706997929}{65981603337670225884957757299001839970910686913380221749524700100}$

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:  $13$
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:  \( 396267155700000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T48:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 125452800
The 65 conjugacy class representatives for t22n48 are not computed
Character table for t22n48 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \)

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

Sibling fields

Degree 24 sibling: data not computed
Degree 44 sibling: 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 R $22$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $22$ $16{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $22$ ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ $22$ $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
3Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.10.13.1$x^{10} + 15 x^{4} + 5$$10$$1$$13$$D_5$$[3/2]_{2}$
5.10.13.1$x^{10} + 15 x^{4} + 5$$10$$1$$13$$D_5$$[3/2]_{2}$
$199$$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
199.5.0.1$x^{5} - x + 4$$1$$5$$0$$C_5$$[\ ]^{5}$
199.5.0.1$x^{5} - x + 4$$1$$5$$0$$C_5$$[\ ]^{5}$
199.8.6.2$x^{8} + 2189 x^{4} + 1425636$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$