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Properties

Label 20.0.96661053444...7344.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 3^{2}\cdot 97^{2}\cdot 401^{15}$
Root discriminant $223.49$
Ramified primes $2, 3, 97, 401$
Class number $2126629632$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 33228588]$ (GRH)
Galois group 20T350

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1431177952949883, 335190654909333, 1573396773373304, 87422158458786, 322532511927071, -10644189421621, 34524640552202, -2346917686507, 2068221953123, -144348494038, 72563180273, -4377184144, 1542496749, -74453400, 19969012, -724333, 151980, -3750, 613, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 613*x^18 - 3750*x^17 + 151980*x^16 - 724333*x^15 + 19969012*x^14 - 74453400*x^13 + 1542496749*x^12 - 4377184144*x^11 + 72563180273*x^10 - 144348494038*x^9 + 2068221953123*x^8 - 2346917686507*x^7 + 34524640552202*x^6 - 10644189421621*x^5 + 322532511927071*x^4 + 87422158458786*x^3 + 1573396773373304*x^2 + 335190654909333*x + 1431177952949883)
 
gp: K = bnfinit(x^20 - 8*x^19 + 613*x^18 - 3750*x^17 + 151980*x^16 - 724333*x^15 + 19969012*x^14 - 74453400*x^13 + 1542496749*x^12 - 4377184144*x^11 + 72563180273*x^10 - 144348494038*x^9 + 2068221953123*x^8 - 2346917686507*x^7 + 34524640552202*x^6 - 10644189421621*x^5 + 322532511927071*x^4 + 87422158458786*x^3 + 1573396773373304*x^2 + 335190654909333*x + 1431177952949883, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 613 x^{18} - 3750 x^{17} + 151980 x^{16} - 724333 x^{15} + 19969012 x^{14} - 74453400 x^{13} + 1542496749 x^{12} - 4377184144 x^{11} + 72563180273 x^{10} - 144348494038 x^{9} + 2068221953123 x^{8} - 2346917686507 x^{7} + 34524640552202 x^{6} - 10644189421621 x^{5} + 322532511927071 x^{4} + 87422158458786 x^{3} + 1573396773373304 x^{2} + 335190654909333 x + 1431177952949883 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(96661053444309405452136701474980370297825977344=2^{10}\cdot 3^{2}\cdot 97^{2}\cdot 401^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $223.49$
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:  $2, 3, 97, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{69} a^{17} - \frac{1}{23} a^{16} + \frac{28}{69} a^{15} + \frac{11}{23} a^{14} - \frac{1}{69} a^{13} - \frac{13}{69} a^{12} + \frac{10}{23} a^{11} + \frac{7}{69} a^{10} + \frac{4}{23} a^{9} + \frac{31}{69} a^{8} - \frac{7}{23} a^{7} + \frac{6}{23} a^{6} + \frac{17}{69} a^{5} - \frac{5}{23} a^{4} - \frac{4}{69} a^{3} - \frac{16}{69} a^{2} + \frac{31}{69} a + \frac{5}{23}$, $\frac{1}{69} a^{18} - \frac{4}{69} a^{16} + \frac{2}{69} a^{15} - \frac{17}{69} a^{14} + \frac{10}{23} a^{13} - \frac{32}{69} a^{12} + \frac{5}{69} a^{11} - \frac{13}{69} a^{10} + \frac{7}{23} a^{9} - \frac{20}{69} a^{8} + \frac{8}{23} a^{7} + \frac{2}{69} a^{6} - \frac{11}{23} a^{5} - \frac{26}{69} a^{4} + \frac{6}{23} a^{3} + \frac{29}{69} a^{2} + \frac{16}{69} a - \frac{8}{23}$, $\frac{1}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{19} + \frac{232827869753768084311835582689309148246146974916861847521300559128966841481784364608040326616173804015442873961034928097090674143184015}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{18} - \frac{32460067085083441604927904089989002094210946283813702693864165894601897310609786453704506927679421599738444146771920159449245320159566478}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{17} + \frac{392804771343630222830589649640319548805214971722722159315416044106379865010784972389560875404236957562692732651319791291222668267992330194}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{16} - \frac{357824787350958285408064247991689601563481434379948208031365184906561539014046009577308030448652561713039255458690819111723636609734885649}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{15} - \frac{731356334587462166141936801462446009334450515719258854820184679372712097034397830838431882750953607035266250683668394641720848426978861977}{1506997756127356257293914814775939004847927912317595457370133749104013690637622122582152391393090031837788508397815293128202934181740920089} a^{14} - \frac{2108873546511983527271144594159307007870771469820068533374868705158225185016034476977075908988938566571857739523824008759430211804807836022}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{13} + \frac{1671455944112152093045788565353960570327422073820919413811786181767885789243388826283626257422530943555531415706368129742853673276208822948}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{12} - \frac{1318763012136650824487515335552037236199189955499835871864496158509085673486959648079248612522509910268875426595724600006504251439345358614}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{11} + \frac{1002832875570087218839051725937758216073015071151319009332065536127679646979875601924296676783351635469146038611477035398486771074707082052}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{10} - \frac{276806359160770600338036442385922367507196400527819570850661578671022528849578350985246973093211296216713437233744688171957562307850713064}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{9} - \frac{607918211443447233715638378224215304899183765126976047527210389706212532079952463827435888011391382040501816705801856594006463308857629733}{1506997756127356257293914814775939004847927912317595457370133749104013690637622122582152391393090031837788508397815293128202934181740920089} a^{8} + \frac{1012626199679288879150026786050724437105274508249614694332694327380580400371442435093953908978607000620119312776953140197477400497704188173}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{7} + \frac{1051639215709933649359729922970308986666774056577288276722171379810423763711687549944370490797911089240515207794527993033618449115116870215}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{6} + \frac{1253244218356721743753093200257847087990493409006096454842403821411973692838900356504823953679114608635793012450689565190959135704632435839}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{5} - \frac{702789331265437639894460700255269842499836316475013894498828955393449938436385755452752202750779333036854114306182594063100275624597290885}{1506997756127356257293914814775939004847927912317595457370133749104013690637622122582152391393090031837788508397815293128202934181740920089} a^{4} - \frac{1903760774350180021550645655834096646401607466320921347746745382987550935885566651018054465918551713060985065364282392902283137013909420768}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{3} - \frac{1954341770651265956193906438850283285345965311701096018103288181597209543698779396872867009695705778619639056618175526144232923978953032796}{4520993268382068771881744444327817014543783736952786372110401247312041071912866367746457174179270095513365525193445879384608802545222760267} a^{2} - \frac{297972939444593312413861034588391098675283511995051054950275199684724811190584144459801440130886621984382415023329228959820250600529685286}{1506997756127356257293914814775939004847927912317595457370133749104013690637622122582152391393090031837788508397815293128202934181740920089} a + \frac{635267464591230437853753514055690209243034903113437802621386753698423837072873226881850149298586498328263132328260316366156098317203485772}{1506997756127356257293914814775939004847927912317595457370133749104013690637622122582152391393090031837788508397815293128202934181740920089}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{33228588}$, which has order $2126629632$ (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:  $9$
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:  \( 795087.603907 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T350:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n350 are not computed
Character table for t20n350 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.2$x^{10} - 5 x^{8} + 10 x^{6} - 2 x^{4} - 11 x^{2} + 39$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed