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Properties

Label 20.12.7447830048...0736.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{40}\cdot 11^{16}\cdot 23^{8}\cdot 199^{4}\cdot 331^{4}$
Root discriminant $878.22$
Ramified primes $2, 11, 23, 199, 331$
Class number $16$ (GRH)
Class group $[2, 2, 2, 2]$ (GRH)
Galois group 20T254

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![78878740263118597254457489, 0, -9769599885004853387009708, 0, 135324984054173199006448, 0, -2999633096203383728, 0, -1039992379470643766, 0, 258620702416292, 0, 2688434077341, 0, -506591558, 0, -2835836, 0, 208, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 208*x^18 - 2835836*x^16 - 506591558*x^14 + 2688434077341*x^12 + 258620702416292*x^10 - 1039992379470643766*x^8 - 2999633096203383728*x^6 + 135324984054173199006448*x^4 - 9769599885004853387009708*x^2 + 78878740263118597254457489)
 
gp: K = bnfinit(x^20 + 208*x^18 - 2835836*x^16 - 506591558*x^14 + 2688434077341*x^12 + 258620702416292*x^10 - 1039992379470643766*x^8 - 2999633096203383728*x^6 + 135324984054173199006448*x^4 - 9769599885004853387009708*x^2 + 78878740263118597254457489, 1)
 

Normalized defining polynomial

\( x^{20} + 208 x^{18} - 2835836 x^{16} - 506591558 x^{14} + 2688434077341 x^{12} + 258620702416292 x^{10} - 1039992379470643766 x^{8} - 2999633096203383728 x^{6} + 135324984054173199006448 x^{4} - 9769599885004853387009708 x^{2} + 78878740263118597254457489 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(74478300488702629539932974793088386039042768856289940340736=2^{40}\cdot 11^{16}\cdot 23^{8}\cdot 199^{4}\cdot 331^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $878.22$
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:  $2, 11, 23, 199, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1514987} a^{14} - \frac{395006}{1514987} a^{12} + \frac{260007}{1514987} a^{10} - \frac{454162}{1514987} a^{8} + \frac{327477}{1514987} a^{6} + \frac{467224}{1514987} a^{4} - \frac{16488}{65869} a^{2}$, $\frac{1}{1514987} a^{15} - \frac{395006}{1514987} a^{13} + \frac{260007}{1514987} a^{11} - \frac{454162}{1514987} a^{9} + \frac{327477}{1514987} a^{7} + \frac{467224}{1514987} a^{5} - \frac{16488}{65869} a^{3}$, $\frac{1}{1097697465733} a^{16} - \frac{65661}{1097697465733} a^{14} + \frac{325387850487}{1097697465733} a^{12} - \frac{117423671344}{1097697465733} a^{10} + \frac{223582306544}{1097697465733} a^{8} + \frac{241064671712}{1097697465733} a^{6} - \frac{401333411717}{1097697465733} a^{4} - \frac{271813}{724559} a^{2} + \frac{4}{11}$, $\frac{1}{97695074450237} a^{17} + \frac{27467581}{97695074450237} a^{15} - \frac{32504357253625}{97695074450237} a^{13} - \frac{37964184113703}{97695074450237} a^{11} + \frac{46994693202922}{97695074450237} a^{9} - \frac{18184868481179}{97695074450237} a^{7} - \frac{20468065923499}{97695074450237} a^{5} + \frac{23267681}{64485751} a^{3} + \frac{15}{979} a$, $\frac{1}{43806750771412209867210740918708953277370628770832548754296513142854016982942224671482106621190584593} a^{18} - \frac{10995388913685096618173133769736377732351511209911411117487563553368215773107034902083526}{43806750771412209867210740918708953277370628770832548754296513142854016982942224671482106621190584593} a^{16} - \frac{1007307435341308405645184651312569203923201606621415065863982510461241704886039209130575828567}{3982431888310200897019158265337177570670057160984777159481501194804910634812929515589282420108234963} a^{14} + \frac{20093996818201951919368832704256520324349306491097804497963753486545156154832902390932726978384776647}{43806750771412209867210740918708953277370628770832548754296513142854016982942224671482106621190584593} a^{12} + \frac{242649800698561204969161069139246582521172856891274124880076244907873350829568826472356649731361910}{43806750771412209867210740918708953277370628770832548754296513142854016982942224671482106621190584593} a^{10} + \frac{1731475754727956434643338114756884855640983808144850500977034400394209106742457467414948587631523946}{43806750771412209867210740918708953277370628770832548754296513142854016982942224671482106621190584593} a^{8} + \frac{637132019275598941730259385603833455541417204916396163141337278193116968301752952784242301265671441}{1904641337887487385530901779074302316407418642210110815404196223602348564475748898760091592225677591} a^{6} + \frac{22338406347214175763850454052177033007575453794380694540501730911279659640529943873398026123253}{60459880798405940533773979646528375573791269959841156833738195430398376092136354211985644538527} a^{4} + \frac{72875299379642785951873261466977912561659665355407193959188909165790712135635097955921845}{438986399739710866081038171358442106309627719286206894957593794358883698020841034498541631} a^{2} - \frac{3436024391098182784486000331712762325775427116419229296986921213409059106307699530}{74882440264002190606284788168829159939626118522652792622877753845926686629256304691}$, $\frac{1}{43806750771412209867210740918708953277370628770832548754296513142854016982942224671482106621190584593} a^{19} + \frac{214682581200267684202713302746822122337654246835445239958651828625749957762245856306199}{43806750771412209867210740918708953277370628770832548754296513142854016982942224671482106621190584593} a^{17} + \frac{333791885576099428217435115239768299983433996353438614321635222914714496555737275153499129026}{1904641337887487385530901779074302316407418642210110815404196223602348564475748898760091592225677591} a^{15} - \frac{11030082197861760293783173728096747826117810751782928820759784546495955927505885260153822893020713173}{43806750771412209867210740918708953277370628770832548754296513142854016982942224671482106621190584593} a^{13} - \frac{209515521265319883940035534279689527506651876372535991119572971324436178389140094848179794478078348}{492210682824856290642817313693359025588434031132949986003331608346674348122946344623394456417871737} a^{11} + \frac{2767730179295887291312848994301741984012460408616994872566022148345741269408960726781434507058180681}{43806750771412209867210740918708953277370628770832548754296513142854016982942224671482106621190584593} a^{9} - \frac{83205764047408425061811983687582365441550734601810696497268808524612925547642014230621317639393368}{173149212535226125957354707188572937855219876564555528673108747600213505861431718069099235656879781} a^{7} + \frac{31735973222683142441482973483809119826448904173371771627099282672395750818165780266737400282244}{665058688782465345871513776111812131311703969558252725171120149734382137013499896331842089923797} a^{5} - \frac{136247305399401448798806333328870979020437292853203825625932600316397281894072200816543582}{438986399739710866081038171358442106309627719286206894957593794358883698020841034498541631} a^{3} + \frac{2247004292737790957395000112596740060038695572638215613802632893117730783717806947205}{6664537183496194963959346147025795234626724548516098543436120092287475110003811117499} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:  $15$
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:  \( 1715735176760000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T254:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 28 conjugacy class representatives for t20n254
Character table for t20n254 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.2670699013250048.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$199$$\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$Trivial$[\ ]$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.4.2.2$x^{4} - 199 x^{2} + 237606$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
199.4.2.2$x^{4} - 199 x^{2} + 237606$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
331Data not computed