↑

Properties

Label 40.40.1841563260...0625.1
Degree $40$
Signature $[40, 0]$
Discriminant $5^{20}\cdot 41^{38}$
Root discriminant $76.14$
Ramified primes $5, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_{20}$ (as 40T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, -265, 110, 11440, 2013, -195228, -46431, 1747746, 421058, -9440783, -2229292, 33400952, 7661653, -81534233, -18029877, 142090232, 30022768, -180748378, -36107412, 170133237, 31725068, -119327033, -20449687, 62453182, 9650330, -24306985, -3307776, 6977251, 812165, -1457140, -139848, 216813, 16342, -22237, -1226, 1486, 53, -58, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - 58*x^38 + 53*x^37 + 1486*x^36 - 1226*x^35 - 22237*x^34 + 16342*x^33 + 216813*x^32 - 139848*x^31 - 1457140*x^30 + 812165*x^29 + 6977251*x^28 - 3307776*x^27 - 24306985*x^26 + 9650330*x^25 + 62453182*x^24 - 20449687*x^23 - 119327033*x^22 + 31725068*x^21 + 170133237*x^20 - 36107412*x^19 - 180748378*x^18 + 30022768*x^17 + 142090232*x^16 - 18029877*x^15 - 81534233*x^14 + 7661653*x^13 + 33400952*x^12 - 2229292*x^11 - 9440783*x^10 + 421058*x^9 + 1747746*x^8 - 46431*x^7 - 195228*x^6 + 2013*x^5 + 11440*x^4 + 110*x^3 - 265*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^40 - x^39 - 58*x^38 + 53*x^37 + 1486*x^36 - 1226*x^35 - 22237*x^34 + 16342*x^33 + 216813*x^32 - 139848*x^31 - 1457140*x^30 + 812165*x^29 + 6977251*x^28 - 3307776*x^27 - 24306985*x^26 + 9650330*x^25 + 62453182*x^24 - 20449687*x^23 - 119327033*x^22 + 31725068*x^21 + 170133237*x^20 - 36107412*x^19 - 180748378*x^18 + 30022768*x^17 + 142090232*x^16 - 18029877*x^15 - 81534233*x^14 + 7661653*x^13 + 33400952*x^12 - 2229292*x^11 - 9440783*x^10 + 421058*x^9 + 1747746*x^8 - 46431*x^7 - 195228*x^6 + 2013*x^5 + 11440*x^4 + 110*x^3 - 265*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} - 58 x^{38} + 53 x^{37} + 1486 x^{36} - 1226 x^{35} - 22237 x^{34} + 16342 x^{33} + 216813 x^{32} - 139848 x^{31} - 1457140 x^{30} + 812165 x^{29} + 6977251 x^{28} - 3307776 x^{27} - 24306985 x^{26} + 9650330 x^{25} + 62453182 x^{24} - 20449687 x^{23} - 119327033 x^{22} + 31725068 x^{21} + 170133237 x^{20} - 36107412 x^{19} - 180748378 x^{18} + 30022768 x^{17} + 142090232 x^{16} - 18029877 x^{15} - 81534233 x^{14} + 7661653 x^{13} + 33400952 x^{12} - 2229292 x^{11} - 9440783 x^{10} + 421058 x^{9} + 1747746 x^{8} - 46431 x^{7} - 195228 x^{6} + 2013 x^{5} + 11440 x^{4} + 110 x^{3} - 265 x^{2} - 10 x + 1 \)

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:  \(1841563260274838293179920707241561880108912129957310865805656528472900390625=5^{20}\cdot 41^{38}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.14$
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:  $5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(205=5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{205}(1,·)$, $\chi_{205}(131,·)$, $\chi_{205}(4,·)$, $\chi_{205}(9,·)$, $\chi_{205}(139,·)$, $\chi_{205}(141,·)$, $\chi_{205}(16,·)$, $\chi_{205}(146,·)$, $\chi_{205}(21,·)$, $\chi_{205}(154,·)$, $\chi_{205}(156,·)$, $\chi_{205}(31,·)$, $\chi_{205}(36,·)$, $\chi_{205}(166,·)$, $\chi_{205}(39,·)$, $\chi_{205}(169,·)$, $\chi_{205}(46,·)$, $\chi_{205}(49,·)$, $\chi_{205}(51,·)$, $\chi_{205}(184,·)$, $\chi_{205}(159,·)$, $\chi_{205}(61,·)$, $\chi_{205}(64,·)$, $\chi_{205}(66,·)$, $\chi_{205}(196,·)$, $\chi_{205}(201,·)$, $\chi_{205}(74,·)$, $\chi_{205}(204,·)$, $\chi_{205}(81,·)$, $\chi_{205}(84,·)$, $\chi_{205}(86,·)$, $\chi_{205}(91,·)$, $\chi_{205}(144,·)$, $\chi_{205}(59,·)$, $\chi_{205}(174,·)$, $\chi_{205}(114,·)$, $\chi_{205}(189,·)$, $\chi_{205}(119,·)$, $\chi_{205}(121,·)$, $\chi_{205}(124,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{3} a^{36} + \frac{1}{3} a^{35} + \frac{1}{3} a^{34} - \frac{1}{3} a^{33} - \frac{1}{3} a^{32} - \frac{1}{3} a^{31} + \frac{1}{3} a^{30} - \frac{1}{3} a^{27} - \frac{1}{3} a^{26} - \frac{1}{3} a^{21} - \frac{1}{3} a^{20} - \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{37} + \frac{1}{3} a^{34} - \frac{1}{3} a^{31} - \frac{1}{3} a^{30} - \frac{1}{3} a^{28} + \frac{1}{3} a^{26} - \frac{1}{3} a^{22} + \frac{1}{3} a^{20} - \frac{1}{3} a^{18} - \frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{38} + \frac{1}{3} a^{35} - \frac{1}{3} a^{32} - \frac{1}{3} a^{31} - \frac{1}{3} a^{29} + \frac{1}{3} a^{27} - \frac{1}{3} a^{23} + \frac{1}{3} a^{21} - \frac{1}{3} a^{19} - \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{39} + \frac{699737603380849059257082763813307893634924164857872817429122128582508892}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{38} + \frac{415828203431548072655005384760644843589700657128826265981773258102126775}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{37} + \frac{2918923394682604629321948939197551240255516609695783277379280737384859829}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{36} - \frac{1142190940453762004562044418919723423405650124479190365300647447039168918}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{35} - \frac{3619007268490704376705135627411626126819615467051570008975598847574164944}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{34} - \frac{3463660624004406654672838019809263066380756060198473491924180679690441873}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{33} - \frac{10554908718567827276729063152304661688143588611691677868523238386319966530}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{32} + \frac{1889619049370121667252046887860706796080098980564287471532250709046736130}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{31} + \frac{8679994935829972311326731773000090966731170259710738954474689722048785868}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{30} - \frac{2381119804611472944555618067966644690715874475688368056370306303406214574}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{29} + \frac{3087729766381928419021549429355821539352901554101356805866330629077914617}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{28} + \frac{426735288052015872665611366059192203116060523297538907699375128270013358}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{27} + \frac{1024373795124671958434148553546062724151539055578049691893760476157258591}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{26} + \frac{931681467965556819586808263064509665032279075898435730886737745122576932}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{25} - \frac{11664899279519164719944723242348671508314174710136145549580506996094487625}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{24} + \frac{2120800060005888592689273483673811821687421593084491695310900757954566104}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{23} + \frac{1429955430259161865838444190327071178635466341789394285175570856913898150}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{22} + \frac{636833073340683523144832846423842875190279636140029661201437344896244365}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{21} + \frac{10047605057601433955543210933937267500968490042760562425525561871486766423}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{20} - \frac{1740198787615119831100107626337392531536647368347673830856576708649564045}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{19} + \frac{9474558142126520176670280930283748118246288947141303120652765008648658683}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{18} - \frac{4951446143969652299830980794556435255192474695491200030191569236445356787}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{17} + \frac{3385093048216223207178213982347179380142231513760978120463221575451969709}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{16} + \frac{11952712543991263928425832208166584364393607276457563541267977003470469231}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{15} - \frac{94864744702096178233585547598394305636744396774858140515300838558112661}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{14} - \frac{10802217372826453684820414969579906984919686333484037344895725713645236295}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{13} + \frac{2252029203136628338171289941274202744852547151438387339959099118388823720}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{12} - \frac{4274023019608422152570119028464199810169862442328689975765174509807595987}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{11} - \frac{11163830194030278570011653942938072920711907846984246681493687577330393495}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{10} + \frac{1070299271346457613450645849581256920236131597814953852804207613396677380}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{9} - \frac{7843715949789323952943922604799391609207046679079024913602752938353794282}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{8} + \frac{1068544202389032622854053854732415962978360173410150210255575795047169064}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{7} + \frac{3877588956058056975245315324995975542426647107970324516938679852016791781}{8047296183542554793877004857805755810293253973908820703536088342640760009} a^{6} - \frac{11396933653690351787659290770138849693496471548723409722317378857258895876}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{5} + \frac{3416569450862388256965304026889304515369169425178763814259141867725358083}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{4} + \frac{11370865765840305084537906266933510921086120427079879438712990086280385791}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{3} - \frac{10163306294544622358974664704102950410407961780205707606421492594688407712}{24141888550627664381631014573417267430879761921726462110608265027922280027} a^{2} - \frac{1243209794320167700983826409992272332263072349716280337743126650675149981}{8047296183542554793877004857805755810293253973908820703536088342640760009} a + \frac{2184764658194532951872003421543246986530884480569480933282779749450848473}{8047296183542554793877004857805755810293253973908820703536088342640760009}$

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:  $39$
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:  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:  \( 10069514429260745000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{20}$ (as 40T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{205}) \), \(\Q(\sqrt{5}, \sqrt{41})\), 4.4.68921.1, 4.4.1723025.1, 5.5.2825761.1, 8.8.2968815150625.1, 10.10.24952891341003125.1, 10.10.327381934393961.1, 10.10.1023068544981128125.1, 20.20.1046669247729802581605012397666015625.1, \(\Q(\zeta_{41})^+\), 20.20.42913439156921905845805508304306640625.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{10}$ R $20^{2}$ $20^{2}$ $20^{2}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

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
5Data not computed
41Data not computed