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Properties

Label 40.40.1547425049...0000.1
Degree $40$
Signature $[40, 0]$
Discriminant $2^{155}\cdot 5^{68}$
Root discriminant $226.33$
Ramified primes $2, 5$
Class number Not computed
Class group Not computed
Galois group $C_{40}$ (as 40T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20000, 0, -1600000, 0, 42000000, 0, -533600000, 0, 3851460000, 0, -17243312000, 0, 50709240000, 0, -101779400000, 0, 142941685000, 0, -142586040000, 0, 101730962000, 0, -51916276000, 0, 18826410000, 0, -4787672000, 0, 837321800, 0, -98237040, 0, 7514000, 0, -362000, 0, 10400, 0, -160, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 160*x^38 + 10400*x^36 - 362000*x^34 + 7514000*x^32 - 98237040*x^30 + 837321800*x^28 - 4787672000*x^26 + 18826410000*x^24 - 51916276000*x^22 + 101730962000*x^20 - 142586040000*x^18 + 142941685000*x^16 - 101779400000*x^14 + 50709240000*x^12 - 17243312000*x^10 + 3851460000*x^8 - 533600000*x^6 + 42000000*x^4 - 1600000*x^2 + 20000)
 
gp: K = bnfinit(x^40 - 160*x^38 + 10400*x^36 - 362000*x^34 + 7514000*x^32 - 98237040*x^30 + 837321800*x^28 - 4787672000*x^26 + 18826410000*x^24 - 51916276000*x^22 + 101730962000*x^20 - 142586040000*x^18 + 142941685000*x^16 - 101779400000*x^14 + 50709240000*x^12 - 17243312000*x^10 + 3851460000*x^8 - 533600000*x^6 + 42000000*x^4 - 1600000*x^2 + 20000, 1)
 

Normalized defining polynomial

\( x^{40} - 160 x^{38} + 10400 x^{36} - 362000 x^{34} + 7514000 x^{32} - 98237040 x^{30} + 837321800 x^{28} - 4787672000 x^{26} + 18826410000 x^{24} - 51916276000 x^{22} + 101730962000 x^{20} - 142586040000 x^{18} + 142941685000 x^{16} - 101779400000 x^{14} + 50709240000 x^{12} - 17243312000 x^{10} + 3851460000 x^{8} - 533600000 x^{6} + 42000000 x^{4} - 1600000 x^{2} + 20000 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[40, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15474250491067253436239052800000000000000000000000000000000000000000000000000000000000000000000=2^{155}\cdot 5^{68}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $226.33$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(800=2^{5}\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{800}(1,·)$, $\chi_{800}(389,·)$, $\chi_{800}(641,·)$, $\chi_{800}(521,·)$, $\chi_{800}(269,·)$, $\chi_{800}(109,·)$, $\chi_{800}(401,·)$, $\chi_{800}(789,·)$, $\chi_{800}(281,·)$, $\chi_{800}(29,·)$, $\chi_{800}(69,·)$, $\chi_{800}(161,·)$, $\chi_{800}(549,·)$, $\chi_{800}(41,·)$, $\chi_{800}(429,·)$, $\chi_{800}(669,·)$, $\chi_{800}(561,·)$, $\chi_{800}(309,·)$, $\chi_{800}(441,·)$, $\chi_{800}(189,·)$, $\chi_{800}(321,·)$, $\chi_{800}(709,·)$, $\chi_{800}(201,·)$, $\chi_{800}(589,·)$, $\chi_{800}(81,·)$, $\chi_{800}(469,·)$, $\chi_{800}(121,·)$, $\chi_{800}(601,·)$, $\chi_{800}(349,·)$, $\chi_{800}(481,·)$, $\chi_{800}(229,·)$, $\chi_{800}(721,·)$, $\chi_{800}(361,·)$, $\chi_{800}(749,·)$, $\chi_{800}(241,·)$, $\chi_{800}(629,·)$, $\chi_{800}(681,·)$, $\chi_{800}(761,·)$, $\chi_{800}(509,·)$, $\chi_{800}(149,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{10} a^{10}$, $\frac{1}{10} a^{11}$, $\frac{1}{10} a^{12}$, $\frac{1}{10} a^{13}$, $\frac{1}{10} a^{14}$, $\frac{1}{10} a^{15}$, $\frac{1}{20} a^{16}$, $\frac{1}{20} a^{17}$, $\frac{1}{20} a^{18}$, $\frac{1}{20} a^{19}$, $\frac{1}{100} a^{20}$, $\frac{1}{100} a^{21}$, $\frac{1}{100} a^{22}$, $\frac{1}{100} a^{23}$, $\frac{1}{1400} a^{24} + \frac{1}{140} a^{18} - \frac{1}{70} a^{12} + \frac{1}{7} a^{6} + \frac{2}{7}$, $\frac{1}{1400} a^{25} + \frac{1}{140} a^{19} - \frac{1}{70} a^{13} + \frac{1}{7} a^{7} + \frac{2}{7} a$, $\frac{1}{1400} a^{26} - \frac{1}{350} a^{20} - \frac{1}{70} a^{14} + \frac{1}{7} a^{8} + \frac{2}{7} a^{2}$, $\frac{1}{1400} a^{27} - \frac{1}{350} a^{21} - \frac{1}{70} a^{15} + \frac{1}{7} a^{9} + \frac{2}{7} a^{3}$, $\frac{1}{1400} a^{28} - \frac{1}{350} a^{22} - \frac{1}{70} a^{16} + \frac{3}{70} a^{10} + \frac{2}{7} a^{4}$, $\frac{1}{1400} a^{29} - \frac{1}{350} a^{23} - \frac{1}{70} a^{17} + \frac{3}{70} a^{11} + \frac{2}{7} a^{5}$, $\frac{1}{7000} a^{30} - \frac{1}{140} a^{18} - \frac{3}{70} a^{12} - \frac{3}{7} a^{6} + \frac{3}{7}$, $\frac{1}{7000} a^{31} - \frac{1}{140} a^{19} - \frac{3}{70} a^{13} - \frac{3}{7} a^{7} + \frac{3}{7} a$, $\frac{1}{98000} a^{32} + \frac{1}{24500} a^{30} - \frac{3}{9800} a^{28} - \frac{3}{9800} a^{26} + \frac{1}{4900} a^{24} + \frac{3}{2450} a^{22} + \frac{3}{700} a^{20} + \frac{3}{140} a^{18} + \frac{13}{980} a^{16} - \frac{1}{245} a^{14} - \frac{11}{245} a^{12} - \frac{8}{245} a^{10} - \frac{23}{98} a^{8} + \frac{17}{49} a^{6} - \frac{20}{49} a^{4} - \frac{1}{49} a^{2} + \frac{24}{49}$, $\frac{1}{98000} a^{33} + \frac{1}{24500} a^{31} - \frac{3}{9800} a^{29} - \frac{3}{9800} a^{27} + \frac{1}{4900} a^{25} + \frac{3}{2450} a^{23} + \frac{3}{700} a^{21} + \frac{3}{140} a^{19} + \frac{13}{980} a^{17} - \frac{1}{245} a^{15} - \frac{11}{245} a^{13} - \frac{8}{245} a^{11} - \frac{23}{98} a^{9} + \frac{17}{49} a^{7} - \frac{20}{49} a^{5} - \frac{1}{49} a^{3} + \frac{24}{49} a$, $\frac{1}{98000} a^{34} - \frac{1}{24500} a^{30} + \frac{1}{4900} a^{28} - \frac{3}{9800} a^{24} + \frac{11}{4900} a^{22} - \frac{1}{980} a^{18} + \frac{1}{140} a^{16} + \frac{8}{245} a^{12} - \frac{23}{490} a^{10} - \frac{11}{49} a^{6} + \frac{16}{49} a^{4} + \frac{2}{49}$, $\frac{1}{98000} a^{35} - \frac{1}{24500} a^{31} + \frac{1}{4900} a^{29} - \frac{3}{9800} a^{25} + \frac{11}{4900} a^{23} - \frac{1}{980} a^{19} + \frac{1}{140} a^{17} + \frac{8}{245} a^{13} - \frac{23}{490} a^{11} - \frac{11}{49} a^{7} + \frac{16}{49} a^{5} + \frac{2}{49} a$, $\frac{1}{98000} a^{36} - \frac{3}{49000} a^{30} + \frac{1}{4900} a^{28} - \frac{1}{9800} a^{26} + \frac{1}{4900} a^{24} - \frac{1}{1225} a^{22} + \frac{1}{2450} a^{20} - \frac{1}{70} a^{18} + \frac{6}{245} a^{16} - \frac{3}{245} a^{14} - \frac{2}{49} a^{12} - \frac{11}{245} a^{10} + \frac{6}{49} a^{8} + \frac{3}{7} a^{6} - \frac{3}{49} a^{4} - \frac{23}{49} a^{2} - \frac{23}{49}$, $\frac{1}{98000} a^{37} - \frac{3}{49000} a^{31} + \frac{1}{4900} a^{29} - \frac{1}{9800} a^{27} + \frac{1}{4900} a^{25} - \frac{1}{1225} a^{23} + \frac{1}{2450} a^{21} - \frac{1}{70} a^{19} + \frac{6}{245} a^{17} - \frac{3}{245} a^{15} - \frac{2}{49} a^{13} - \frac{11}{245} a^{11} + \frac{6}{49} a^{9} + \frac{3}{7} a^{7} - \frac{3}{49} a^{5} - \frac{23}{49} a^{3} - \frac{23}{49} a$, $\frac{1}{809976464334936566961519608364684347803876156997361698000} a^{38} + \frac{2208645232687684888215703216463413179322544955964281}{809976464334936566961519608364684347803876156997361698000} a^{36} + \frac{121431595908544217033146672330235110951286965420511}{40498823216746828348075980418234217390193807849868084900} a^{34} + \frac{1070366754218253112110710175006830474514648331422561}{404988232167468283480759804182342173901938078498680849000} a^{32} - \frac{3156606959982909177206641146052209080228637619698174}{50623529020933535435094975522792771737742259812335106125} a^{30} - \frac{246867755848376218871669390506207963637519588942157}{826506596260139354042366947310902395718240976527920100} a^{28} + \frac{23584218697604485141520135784060745283880740063071811}{80997646433493656696151960836468434780387615699736169800} a^{26} - \frac{1061384348955290374779398530312566647284231335234523}{4049882321674682834807598041823421739019380784986808490} a^{24} + \frac{12293759095684017333717499377049539664432886739296303}{4049882321674682834807598041823421739019380784986808490} a^{22} - \frac{23443802013675753569423312806030637690935587848273071}{20249411608373414174037990209117108695096903924934042450} a^{20} - \frac{9780666980702413689142377921059956162872874095028939}{4049882321674682834807598041823421739019380784986808490} a^{18} + \frac{46783898708781411815280371617488474307880476595001582}{2024941160837341417403799020911710869509690392493404245} a^{16} - \frac{6267311111458814079900948929413210174008791286610949}{404988232167468283480759804182342173901938078498680849} a^{14} - \frac{54724244111117118535114517664317803454211885235729598}{2024941160837341417403799020911710869509690392493404245} a^{12} - \frac{20168509466836138978069178283294816285681039372930079}{809976464334936566961519608364684347803876156997361698} a^{10} + \frac{60508266049909428660242968643274857932820615107075169}{404988232167468283480759804182342173901938078498680849} a^{8} + \frac{53204888003631460865836138148726192544806365717876482}{404988232167468283480759804182342173901938078498680849} a^{6} + \frac{199203916554849983157204177888365105772791157320603443}{404988232167468283480759804182342173901938078498680849} a^{4} + \frac{49557492313790683048872750557724554151966964452628652}{404988232167468283480759804182342173901938078498680849} a^{2} + \frac{35578248843913480510698911334647213618649381700648866}{404988232167468283480759804182342173901938078498680849}$, $\frac{1}{809976464334936566961519608364684347803876156997361698000} a^{39} + \frac{2208645232687684888215703216463413179322544955964281}{809976464334936566961519608364684347803876156997361698000} a^{37} + \frac{121431595908544217033146672330235110951286965420511}{40498823216746828348075980418234217390193807849868084900} a^{35} + \frac{1070366754218253112110710175006830474514648331422561}{404988232167468283480759804182342173901938078498680849000} a^{33} - \frac{3156606959982909177206641146052209080228637619698174}{50623529020933535435094975522792771737742259812335106125} a^{31} - \frac{246867755848376218871669390506207963637519588942157}{826506596260139354042366947310902395718240976527920100} a^{29} + \frac{23584218697604485141520135784060745283880740063071811}{80997646433493656696151960836468434780387615699736169800} a^{27} - \frac{1061384348955290374779398530312566647284231335234523}{4049882321674682834807598041823421739019380784986808490} a^{25} + \frac{12293759095684017333717499377049539664432886739296303}{4049882321674682834807598041823421739019380784986808490} a^{23} - \frac{23443802013675753569423312806030637690935587848273071}{20249411608373414174037990209117108695096903924934042450} a^{21} - \frac{9780666980702413689142377921059956162872874095028939}{4049882321674682834807598041823421739019380784986808490} a^{19} + \frac{46783898708781411815280371617488474307880476595001582}{2024941160837341417403799020911710869509690392493404245} a^{17} - \frac{6267311111458814079900948929413210174008791286610949}{404988232167468283480759804182342173901938078498680849} a^{15} - \frac{54724244111117118535114517664317803454211885235729598}{2024941160837341417403799020911710869509690392493404245} a^{13} - \frac{20168509466836138978069178283294816285681039372930079}{809976464334936566961519608364684347803876156997361698} a^{11} + \frac{60508266049909428660242968643274857932820615107075169}{404988232167468283480759804182342173901938078498680849} a^{9} + \frac{53204888003631460865836138148726192544806365717876482}{404988232167468283480759804182342173901938078498680849} a^{7} + \frac{199203916554849983157204177888365105772791157320603443}{404988232167468283480759804182342173901938078498680849} a^{5} + \frac{49557492313790683048872750557724554151966964452628652}{404988232167468283480759804182342173901938078498680849} a^{3} + \frac{35578248843913480510698911334647213618649381700648866}{404988232167468283480759804182342173901938078498680849} a$

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:  $39$
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

$C_{40}$ (as 40T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 40
The 40 conjugacy class representatives for $C_{40}$
Character table for $C_{40}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 5.5.390625.1, 8.8.1342177280000.1, 10.10.5000000000000000.1, 20.20.838860800000000000000000000000000000000.1

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 $40$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{10}$ $40$ $40$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{8}$ $40$ $20^{2}$ $40$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ $40$ $20^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{8}$ $40$ $40$

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
5Data not computed