Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - 39*x^38 + 39*x^37 + 702*x^36 - 701*x^35 - 7736*x^34 + 7701*x^33 + 58378*x^32 - 57817*x^31 - 319703*x^30 + 314247*x^29 + 1313873*x^28 - 1277913*x^27 - 4133168*x^26 + 3963256*x^25 + 10063407*x^24 - 9469608*x^23 - 19054236*x^22 + 17493203*x^21 + 28043254*x^20 - 24933608*x^19 - 31915288*x^18 + 27222459*x^17 + 27803858*x^16 - 22476466*x^15 - 18249278*x^14 + 13764066*x^13 + 8815690*x^12 - 6077890*x^11 - 3029168*x^10 + 1857788*x^9 + 703905*x^8 - 369684*x^7 - 102396*x^6 + 43393*x^5 + 8232*x^4 - 2512*x^3 - 288*x^2 + 48*x + 1)
gp: K = bnfinit(y^40 - y^39 - 39*y^38 + 39*y^37 + 702*y^36 - 701*y^35 - 7736*y^34 + 7701*y^33 + 58378*y^32 - 57817*y^31 - 319703*y^30 + 314247*y^29 + 1313873*y^28 - 1277913*y^27 - 4133168*y^26 + 3963256*y^25 + 10063407*y^24 - 9469608*y^23 - 19054236*y^22 + 17493203*y^21 + 28043254*y^20 - 24933608*y^19 - 31915288*y^18 + 27222459*y^17 + 27803858*y^16 - 22476466*y^15 - 18249278*y^14 + 13764066*y^13 + 8815690*y^12 - 6077890*y^11 - 3029168*y^10 + 1857788*y^9 + 703905*y^8 - 369684*y^7 - 102396*y^6 + 43393*y^5 + 8232*y^4 - 2512*y^3 - 288*y^2 + 48*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - x^39 - 39*x^38 + 39*x^37 + 702*x^36 - 701*x^35 - 7736*x^34 + 7701*x^33 + 58378*x^32 - 57817*x^31 - 319703*x^30 + 314247*x^29 + 1313873*x^28 - 1277913*x^27 - 4133168*x^26 + 3963256*x^25 + 10063407*x^24 - 9469608*x^23 - 19054236*x^22 + 17493203*x^21 + 28043254*x^20 - 24933608*x^19 - 31915288*x^18 + 27222459*x^17 + 27803858*x^16 - 22476466*x^15 - 18249278*x^14 + 13764066*x^13 + 8815690*x^12 - 6077890*x^11 - 3029168*x^10 + 1857788*x^9 + 703905*x^8 - 369684*x^7 - 102396*x^6 + 43393*x^5 + 8232*x^4 - 2512*x^3 - 288*x^2 + 48*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - x^39 - 39*x^38 + 39*x^37 + 702*x^36 - 701*x^35 - 7736*x^34 + 7701*x^33 + 58378*x^32 - 57817*x^31 - 319703*x^30 + 314247*x^29 + 1313873*x^28 - 1277913*x^27 - 4133168*x^26 + 3963256*x^25 + 10063407*x^24 - 9469608*x^23 - 19054236*x^22 + 17493203*x^21 + 28043254*x^20 - 24933608*x^19 - 31915288*x^18 + 27222459*x^17 + 27803858*x^16 - 22476466*x^15 - 18249278*x^14 + 13764066*x^13 + 8815690*x^12 - 6077890*x^11 - 3029168*x^10 + 1857788*x^9 + 703905*x^8 - 369684*x^7 - 102396*x^6 + 43393*x^5 + 8232*x^4 - 2512*x^3 - 288*x^2 + 48*x + 1)
\( x^{40} - x^{39} - 39 x^{38} + 39 x^{37} + 702 x^{36} - 701 x^{35} - 7736 x^{34} + 7701 x^{33} + 58378 x^{32} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $40$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[40, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(100383397447978918530459891214693626269465146363712847232818603515625\)
\(\medspace = 3^{20}\cdot 5^{30}\cdot 11^{36}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(50.12\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $3^{1/2}5^{3/4}11^{9/10}\approx 50.12351825429183$ | ||
| Ramified primes: |
\(3\), \(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $40$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(165=3\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{165}(1,·)$, $\chi_{165}(131,·)$, $\chi_{165}(4,·)$, $\chi_{165}(134,·)$, $\chi_{165}(7,·)$, $\chi_{165}(136,·)$, $\chi_{165}(137,·)$, $\chi_{165}(13,·)$, $\chi_{165}(142,·)$, $\chi_{165}(16,·)$, $\chi_{165}(149,·)$, $\chi_{165}(23,·)$, $\chi_{165}(152,·)$, $\chi_{165}(28,·)$, $\chi_{165}(29,·)$, $\chi_{165}(158,·)$, $\chi_{165}(31,·)$, $\chi_{165}(161,·)$, $\chi_{165}(34,·)$, $\chi_{165}(164,·)$, $\chi_{165}(38,·)$, $\chi_{165}(41,·)$, $\chi_{165}(43,·)$, $\chi_{165}(47,·)$, $\chi_{165}(49,·)$, $\chi_{165}(52,·)$, $\chi_{165}(53,·)$, $\chi_{165}(64,·)$, $\chi_{165}(73,·)$, $\chi_{165}(74,·)$, $\chi_{165}(91,·)$, $\chi_{165}(92,·)$, $\chi_{165}(101,·)$, $\chi_{165}(112,·)$, $\chi_{165}(113,·)$, $\chi_{165}(116,·)$, $\chi_{165}(118,·)$, $\chi_{165}(122,·)$, $\chi_{165}(124,·)$, $\chi_{165}(127,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $39$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361790a^{21}-834900a^{19}+1427679a^{17}-1797818a^{15}+1641486a^{13}-1058148a^{11}+461890a^{9}-127908a^{7}+20196a^{5}-1496a^{3}+33a$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419900a^{12}+277134a^{10}-119340a^{8}+30940a^{6}-4200a^{4}+225a^{2}-2$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{39}+a^{38}-39a^{37}-38a^{36}+702a^{35}+665a^{34}-7735a^{33}-7106a^{32}+58344a^{31}+51832a^{30}-319176a^{29}-273296a^{28}+1308945a^{27}+1076103a^{26}-4102164a^{25}-3223350a^{24}+9924849a^{23}+7413705a^{22}-18601572a^{21}-13123110a^{20}+26947305a^{19}+17809935a^{18}-29942785a^{17}-18349630a^{16}+25179805a^{15}+14115100a^{14}-15705671a^{13}-7904456a^{12}+7061392a^{11}+3105322a^{10}-2198559a^{9}-810084a^{8}+447711a^{7}+128876a^{6}-54768a^{5}-10824a^{4}+3493a^{3}+352a^{2}-83a+1$, $a^{39}+a^{38}-39a^{37}-38a^{36}+702a^{35}+665a^{34}-7735a^{33}-7106a^{32}+58344a^{31}+51832a^{30}-319176a^{29}-273296a^{28}+1308945a^{27}+1076103a^{26}-4102164a^{25}-3223350a^{24}+9924849a^{23}+7413705a^{22}-18601572a^{21}-13123110a^{20}+26947305a^{19}+17809935a^{18}-29942785a^{17}-18349630a^{16}+25179805a^{15}+14115100a^{14}-15705671a^{13}-7904456a^{12}+7061392a^{11}+3105322a^{10}-2198559a^{9}-810084a^{8}+447711a^{7}+128876a^{6}-54768a^{5}-10824a^{4}+3493a^{3}+352a^{2}-83a$, $a^{5}-5a^{3}+5a+1$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}+25a+1$, $a^{39}+a^{38}-39a^{37}-38a^{36}+703a^{35}+665a^{34}-7770a^{33}-7106a^{32}+58904a^{31}+51833a^{30}-324601a^{29}-273326a^{28}+1344470a^{27}+1076508a^{26}-4268421a^{25}-3226600a^{24}+10498149a^{23}+7430955a^{22}-20081622a^{21}-13186867a^{20}+29825180a^{19}+17978200a^{18}-34148910a^{17}-18669570a^{16}+29756068a^{15}+14551950a^{14}-19345866a^{13}-8326631a^{12}+9118812a^{11}+3386459a^{10}-2989634a^{9}-933704a^{8}+641061a^{7}+162421a^{6}-81522a^{5}-15799a^{4}+5138a^{3}+652a^{2}-103a-2$, $a^{39}-a^{38}-39a^{37}+38a^{36}+701a^{35}-664a^{34}-7701a^{33}+7072a^{32}+57817a^{31}-51305a^{30}-314247a^{29}+268367a^{28}+1277913a^{27}-1045072a^{26}-3963256a^{25}+3084468a^{24}+9469608a^{23}-6958764a^{22}-17493203a^{21}+12016766a^{20}+24933608a^{19}-15805112a^{18}-27222459a^{17}+15655792a^{16}+22476466a^{15}-11466722a^{14}-13764066a^{13}+6042310a^{12}+6077890a^{11}-2200848a^{10}-1857788a^{9}+521880a^{8}+369684a^{7}-73164a^{6}-43393a^{5}+5068a^{4}+2512a^{3}-112a^{2}-48a$, $a^{39}-38a^{37}+665a^{35}+a^{34}-7106a^{33}-33a^{32}+51832a^{31}+496a^{30}-273296a^{29}-4495a^{28}+1076103a^{27}+27404a^{26}-3223349a^{25}-118731a^{24}+7413681a^{23}+376487a^{22}-13122858a^{21}-886489a^{20}+17808415a^{19}+1556271a^{18}-18343817a^{17}-2027318a^{16}+14100428a^{15}+1933445a^{14}-7879809a^{13}-1318448a^{12}+3078231a^{11}+619409a^{10}-791503a^{9}-189249a^{8}+121689a^{7}+34315a^{6}-9612a^{5}-3197a^{4}+355a^{3}+133a^{2}-11a-3$, $a^{38}-38a^{36}+665a^{34}-7106a^{32}+51832a^{30}-273297a^{28}+a^{27}+1076131a^{26}-27a^{25}-3223700a^{24}+324a^{23}+7416281a^{22}-2277a^{21}-13135507a^{20}+10395a^{19}+17850899a^{18}-32320a^{17}-18444591a^{16}+69785a^{15}+14270124a^{14}-104771a^{13}-8080710a^{12}+107848a^{11}+3241106a^{10}-73865a^{9}-877492a^{8}+32010a^{7}+148588a^{6}-8085a^{5}-13673a^{4}+1023a^{3}+484a^{2}-44a$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32320a^{17}+69785a^{15}-104771a^{13}+107848a^{11}-73865a^{9}+32010a^{7}-8085a^{5}+1023a^{3}-44a$, $a^{39}+a^{38}-39a^{37}-38a^{36}+702a^{35}+665a^{34}-7734a^{33}-7106a^{32}+58311a^{31}+51832a^{30}-318681a^{29}-273296a^{28}+1304479a^{27}+1076103a^{26}-4075137a^{25}-3223350a^{24}+9809019a^{23}+7413705a^{22}-18239782a^{21}-13123110a^{20}+26112405a^{19}+17809935a^{18}-28515106a^{17}-18349630a^{16}+23381987a^{15}+14115100a^{14}-14064185a^{13}-7904456a^{12}+6003244a^{11}+3105322a^{10}-1736669a^{9}-810084a^{8}+319803a^{7}+128876a^{6}-34572a^{5}-10824a^{4}+1997a^{3}+352a^{2}-50a$, $a^{39}+a^{38}-39a^{37}-38a^{36}+702a^{35}+665a^{34}-7736a^{33}-7106a^{32}+58377a^{31}+51832a^{30}-319671a^{29}-273296a^{28}+1313411a^{27}+1076103a^{26}-4129191a^{25}-3223350a^{24}+10040679a^{23}+7413705a^{22}-18963362a^{21}-13123110a^{20}+27782205a^{19}+17809935a^{18}-31370464a^{17}-18349630a^{16}+26977623a^{15}+14115100a^{14}-17347157a^{13}-7904456a^{12}+8119540a^{11}+3105322a^{10}-2660449a^{9}-810084a^{8}+575619a^{7}+128876a^{6}-74965a^{5}-10824a^{4}+4994a^{3}+352a^{2}-121a$, $a^{39}+a^{38}-39a^{37}-38a^{36}+703a^{35}+665a^{34}-7771a^{33}-7106a^{32}+58937a^{31}+51833a^{30}-325096a^{29}-273326a^{28}+1348936a^{27}+1076508a^{26}-4295448a^{25}-3226600a^{24}+10613979a^{23}+7430955a^{22}-20443412a^{21}-13186867a^{20}+30660080a^{19}+17978200a^{18}-35576589a^{17}-18669570a^{16}+31553886a^{15}+14551950a^{14}-20987352a^{13}-8326631a^{12}+10176960a^{11}+3386459a^{10}-3451524a^{9}-933704a^{8}+768969a^{7}+162421a^{6}-101718a^{5}-15799a^{4}+6634a^{3}+652a^{2}-136a$, $a^{39}+a^{38}-39a^{37}-38a^{36}+703a^{35}+665a^{34}-7769a^{33}-7106a^{32}+58871a^{31}+51833a^{30}-324106a^{29}-273326a^{28}+1340004a^{27}+1076508a^{26}-4241394a^{25}-3226600a^{24}+10382319a^{23}+7430955a^{22}-19719832a^{21}-13186867a^{20}+28990280a^{19}+17978200a^{18}-32721231a^{17}-18669570a^{16}+27958250a^{15}+14551950a^{14}-17704380a^{13}-8326631a^{12}+8060664a^{11}+3386459a^{10}-2527744a^{9}-933704a^{8}+513153a^{7}+162421a^{6}-61326a^{5}-15799a^{4}+3642a^{3}+652a^{2}-70a-2$, $a^{39}-a^{38}-39a^{37}+38a^{36}+701a^{35}-664a^{34}-7702a^{33}+7072a^{32}+57850a^{31}-51305a^{30}-314742a^{29}+268367a^{28}+1282379a^{27}-1045072a^{26}-3990283a^{25}+3084468a^{24}+9585438a^{23}-6958764a^{22}-17854993a^{21}+12016766a^{20}+25768508a^{19}-15805112a^{18}-28650138a^{17}+15655792a^{16}+24274283a^{15}-11466722a^{14}-15405537a^{13}+6042310a^{12}+7135948a^{11}-2200848a^{10}-2319403a^{9}+521880a^{8}+497142a^{7}-73164a^{6}-63211a^{5}+5068a^{4}+3868a^{3}-112a^{2}-66a+1$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27026a^{25}-115805a^{23}+361515a^{21}-833150a^{19}+1420554a^{17}-1778438a^{15}+1605786a^{13}-1013948a^{11}+426140a^{9}-110033a^{7}+15191a^{5}-846a^{3}+8a-1$, $a^{39}-38a^{37}+666a^{35}+a^{34}-7141a^{33}-33a^{32}+52392a^{31}+496a^{30}-278721a^{29}-4495a^{28}+1111628a^{27}+27404a^{26}-3389606a^{25}-118731a^{24}+7986981a^{23}+376487a^{22}-14602908a^{21}-886490a^{20}+20686289a^{19}+1556291a^{18}-22549923a^{17}-2027488a^{16}+18676539a^{15}+1934245a^{14}-11519339a^{13}-1320723a^{12}+5133922a^{11}+623413a^{10}-1579861a^{9}-193539a^{8}+312531a^{7}+36955a^{6}-35112a^{5}-4022a^{4}+1715a^{3}+233a^{2}-12a-5$, $a^{35}-36a^{33}+593a^{31}-5920a^{29}+39991a^{27}-193284a^{25}+689130a^{23}-1841840a^{21}-a^{20}+3712775a^{19}+20a^{18}-5633804a^{17}-170a^{16}+6374082a^{15}+800a^{14}-5281696a^{13}-2275a^{12}+3115658a^{11}+4004a^{10}-1253240a^{9}-4290a^{8}+321708a^{7}+2640a^{6}-47328a^{5}-825a^{4}+3281a^{3}+100a^{2}-68a-1$, $a^{33}-33a^{31}-a^{30}+495a^{29}+30a^{28}-4466a^{27}-405a^{26}+27027a^{25}+3250a^{24}-115830a^{23}-17250a^{22}+361790a^{21}+63756a^{20}-834900a^{19}-168245a^{18}+1427679a^{17}+319770a^{16}-1797818a^{15}-436050a^{14}+1641486a^{13}+419900a^{12}-1058148a^{11}-277134a^{10}+461890a^{9}+119340a^{8}-127908a^{7}-30940a^{6}+20196a^{5}+4200a^{4}-1496a^{3}-225a^{2}+33a+1$, $a^{4}-4a^{2}+2$, $a^{19}-19a^{17}+152a^{15}-665a^{13}+1729a^{11}-2717a^{9}+2508a^{7}-1254a^{5}+285a^{3}-19a$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}-a^{24}+20150a^{23}+24a^{22}-78430a^{21}-252a^{20}+219604a^{19}+1520a^{18}-447051a^{17}-5814a^{16}+660858a^{15}+14688a^{14}-700910a^{13}-24752a^{12}+520676a^{11}+27456a^{10}-260338a^{9}-19305a^{8}+82212a^{7}+8008a^{6}-14756a^{5}-1716a^{4}+1240a^{3}+144a^{2}-31a-2$, $3a^{39}+a^{38}-116a^{37}-37a^{36}+2069a^{35}+630a^{34}-22576a^{33}-6545a^{32}+168521a^{31}+46376a^{30}-911679a^{29}-237335a^{28}+3694425a^{27}+906163a^{26}-11431250a^{25}-2629201a^{24}+27282881a^{23}+5850096a^{22}-50399878a^{21}-10001067a^{20}+71901837a^{19}+13076142a^{18}-78611760a^{17}-12927278a^{16}+64981022a^{15}+9474880a^{14}-39781189a^{13}-4990506a^{12}+17502536a^{11}+1798510a^{10}-5295784a^{9}-409115a^{8}+1030243a^{7}+50834a^{6}-115150a^{5}-2597a^{4}+5915a^{3}+53a^{2}-75a-3$, $a^{39}+a^{38}-39a^{37}-38a^{36}+702a^{35}+665a^{34}-7734a^{33}-7106a^{32}+58311a^{31}+51832a^{30}-318681a^{29}-273295a^{28}+1304478a^{27}+1076075a^{26}-4075110a^{25}-3223000a^{24}+9808695a^{23}+7411129a^{22}-18237505a^{21}-13110713a^{20}+26102010a^{19}+17768971a^{18}-28482786a^{17}-18254669a^{16}+23312202a^{15}+13960076a^{14}-13959414a^{13}-7728202a^{12}+5895396a^{11}+2969538a^{10}-1662804a^{9}-742676a^{8}+287793a^{7}+109165a^{6}-26488a^{5}-7981a^{4}+979a^{3}+229a^{2}-11a-3$, $a^{37}-a^{36}-36a^{35}+36a^{34}+594a^{33}-592a^{32}-5954a^{31}+5889a^{30}+40517a^{29}-39557a^{28}-198183a^{27}+189656a^{26}+719757a^{25}-668953a^{24}-1977496a^{23}+1763086a^{22}+4150719a^{21}-3490897a^{20}-6677933a^{19}+5176419a^{18}+8216661a^{17}-5681432a^{16}-7671372a^{15}+4513539a^{14}+5355469a^{13}-2497627a^{12}-2727957a^{11}+898621a^{10}+973965a^{9}-181087a^{8}-227812a^{7}+11138a^{6}+30939a^{5}+2027a^{4}-1907a^{3}-269a^{2}+27a+4$, $3a^{39}+a^{38}-116a^{37}-37a^{36}+2069a^{35}+630a^{34}-22577a^{33}-6545a^{32}+168554a^{31}+46376a^{30}-912174a^{29}-237335a^{28}+3698891a^{27}+906163a^{26}-11458277a^{25}-2629202a^{24}+27398712a^{23}+5850120a^{22}-50761691a^{21}-10001319a^{20}+72736967a^{19}+13077662a^{18}-80040751a^{17}-12933091a^{16}+66783549a^{15}+9489553a^{14}-41433743a^{13}-5015168a^{12}+18577884a^{11}+1825691a^{10}-5775129a^{9}-427970a^{8}+1169332a^{7}+58463a^{6}-139588a^{5}-4167a^{4}+8251a^{3}+173a^{2}-171a-2$, $a^{39}+a^{38}-39a^{37}-38a^{36}+703a^{35}+666a^{34}-7770a^{33}-7139a^{32}+58904a^{31}+52328a^{30}-324601a^{29}-277792a^{28}+1344470a^{27}+1103535a^{26}-4268421a^{25}-3342430a^{24}+10498149a^{23}+7792745a^{22}-20081622a^{21}-14021767a^{20}+29825180a^{19}+19405879a^{18}-34148910a^{17}-20467388a^{16}+29756068a^{15}+16193436a^{14}-19345866a^{13}-9384780a^{12}+9118812a^{11}+3848361a^{10}-2989634a^{9}-1061666a^{8}+641061a^{7}+182729a^{6}-81522a^{5}-17400a^{4}+5138a^{3}+721a^{2}-104a-3$, $a^{34}-34a^{32}+a^{31}+527a^{30}-31a^{29}-4930a^{28}+434a^{27}+31059a^{26}-3627a^{25}-139230a^{24}+20150a^{23}+457470a^{22}-78430a^{21}-1118260a^{20}+219604a^{19}+2042975a^{18}-447051a^{17}-2778446a^{16}+660858a^{15}+2778446a^{14}-700910a^{13}-1998724a^{12}+520676a^{11}+999362a^{10}-260338a^{9}-329460a^{8}+82212a^{7}+65892a^{6}-14756a^{5}-6936a^{4}+1240a^{3}+289a^{2}-31a-2$, $a^{2}-2$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}-a^{18}+11560835a^{17}+18a^{16}-10994920a^{15}-135a^{14}+7696444a^{13}+546a^{12}-3848222a^{11}-1287a^{10}+1314610a^{9}+1781a^{8}-286824a^{7}-1378a^{6}+35853a^{5}+520a^{4}-2109a^{3}-65a^{2}+37a$, $a^{39}+2a^{38}-40a^{37}-76a^{36}+740a^{35}+1329a^{34}-8399a^{33}-14178a^{32}+65415a^{31}+103137a^{30}-370450a^{29}-541662a^{28}+1576878a^{27}+2121147a^{26}-5143608a^{25}-6307469a^{24}+12989142a^{23}+14369917a^{22}-25481630a^{21}-25127731a^{20}+38742695a^{19}+33575603a^{18}-45293513a^{17}-33916274a^{16}+40154255a^{15}+25441471a^{14}-26432178a^{13}-13795174a^{12}+12531364a^{11}+5197567a^{10}-4093594a^{9}-1283412a^{8}+861839a^{7}+189965a^{6}-104812a^{5}-14632a^{4}+5976a^{3}+465a^{2}-95a-2$, $2a^{39}-a^{38}-76a^{37}+39a^{36}+1329a^{35}-699a^{34}-14178a^{33}+7633a^{32}+103137a^{31}-56761a^{30}-541663a^{29}+304327a^{28}+2121175a^{27}-1214983a^{26}-6307818a^{25}+3678241a^{24}+14372469a^{23}-8519498a^{22}-25139876a^{21}+15124410a^{20}+33615047a^{19}-20489297a^{18}-34005422a^{17}+20958008a^{16}+25581822a^{15}-15901681a^{14}-13946766a^{13}+8711704a^{12}+5306170a^{11}-3310309a^{10}-1331964a^{9}+820703a^{8}+202040a^{7}-120540a^{6}-15891a^{5}+8962a^{4}+459a^{3}-264a^{2}+4a+2$, $a^{39}-38a^{37}+665a^{35}+a^{34}-7106a^{33}-33a^{32}+51832a^{31}+496a^{30}-273296a^{29}-4495a^{28}+1076103a^{27}+27404a^{26}-3223350a^{25}-118731a^{24}+7413706a^{23}+376487a^{22}-13123133a^{21}-886489a^{20}+17810165a^{19}+1556271a^{18}-18350942a^{17}-2027318a^{16}+14119808a^{15}+1933444a^{14}-7915509a^{13}-1318434a^{12}+3122431a^{11}+619332a^{10}-827253a^{9}-189038a^{8}+139564a^{7}+34013a^{6}-14617a^{5}-2981a^{4}+1005a^{3}+68a^{2}-36a+1$, $3a^{39}-116a^{37}+2068a^{35}+2a^{34}-22541a^{33}-67a^{32}+167961a^{31}+1022a^{30}-906255a^{29}-9394a^{28}+3658929a^{27}+58031a^{26}-11265369a^{25}-254389a^{24}+26712458a^{23}+814477a^{22}-48934274a^{21}-1931079a^{20}+69074052a^{19}+3401494a^{18}-74528602a^{17}-4426581a^{16}+60620522a^{15}+4196176a^{14}-36411126a^{13}-2829736a^{12}+15683603a^{11}+1310230a^{10}-4649653a^{9}-395743a^{8}+894905a^{7}+72478a^{6}-102438a^{5}-7259a^{4}+5992a^{3}+332a^{2}-126a-3$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58344a^{31}-319176a^{29}+a^{28}+1308944a^{27}-28a^{26}-4102137a^{25}+350a^{24}+9924526a^{23}-2576a^{22}-18599318a^{21}+12397a^{20}+26937140a^{19}-40964a^{18}-29911776a^{17}+94961a^{16}+25114712a^{15}-155024a^{14}-15611848a^{13}+176254a^{12}+6970288a^{11}-135784a^{10}-2141139a^{9}+67408a^{8}+425568a^{7}-19712a^{6}-49972a^{5}+2849a^{4}+2976a^{3}-132a^{2}-62a$, $2a^{39}-a^{38}-77a^{37}+38a^{36}+1367a^{35}-663a^{34}-14843a^{33}+7039a^{32}+110242a^{31}-50809a^{30}-593463a^{29}+263872a^{28}+2394007a^{27}-1017667a^{26}-7379889a^{25}+2965711a^{24}+17572419a^{23}-6581978a^{22}-32457901a^{21}+11128275a^{20}+46454797a^{19}-14240196a^{18}-51200061a^{17}+13603280a^{16}+42950824a^{15}-9482889a^{14}-26924906a^{13}+4654910a^{12}+12270050a^{11}-1518233a^{10}-3899814a^{9}+295451a^{8}+810572a^{7}-25836a^{6}-99072a^{5}-495a^{4}+5849a^{3}+189a^{2}-101a-1$, $2a^{39}+2a^{38}-79a^{37}-75a^{36}+1440a^{35}+1295a^{34}-16063a^{33}-13653a^{32}+122610a^{31}+98271a^{30}-678405a^{29}-511530a^{28}+2812040a^{27}+1989955a^{26}-8900658a^{25}-5896503a^{24}+21731190a^{23}+13441086a^{22}-41068226a^{21}-23643563a^{20}+59946852a^{19}+32006867a^{18}-67089284a^{17}-33064697a^{16}+56824253a^{15}+25686620a^{14}-35721569a^{13}-14676285a^{12}+16205786a^{11}+5971396a^{10}-5096391a^{9}-1650787a^{8}+1046355a^{7}+288756a^{6}-127462a^{5}-28367a^{4}+7760a^{3}+1214a^{2}-160a-6$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 1967499665744724500000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{40}\cdot(2\pi)^{0}\cdot 1967499665744724500000 \cdot 1}{2\cdot\sqrt{100383397447978918530459891214693626269465146363712847232818603515625}}\cr\approx \mathstrut & 0.107957682491059 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - 39*x^38 + 39*x^37 + 702*x^36 - 701*x^35 - 7736*x^34 + 7701*x^33 + 58378*x^32 - 57817*x^31 - 319703*x^30 + 314247*x^29 + 1313873*x^28 - 1277913*x^27 - 4133168*x^26 + 3963256*x^25 + 10063407*x^24 - 9469608*x^23 - 19054236*x^22 + 17493203*x^21 + 28043254*x^20 - 24933608*x^19 - 31915288*x^18 + 27222459*x^17 + 27803858*x^16 - 22476466*x^15 - 18249278*x^14 + 13764066*x^13 + 8815690*x^12 - 6077890*x^11 - 3029168*x^10 + 1857788*x^9 + 703905*x^8 - 369684*x^7 - 102396*x^6 + 43393*x^5 + 8232*x^4 - 2512*x^3 - 288*x^2 + 48*x + 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^40 - x^39 - 39*x^38 + 39*x^37 + 702*x^36 - 701*x^35 - 7736*x^34 + 7701*x^33 + 58378*x^32 - 57817*x^31 - 319703*x^30 + 314247*x^29 + 1313873*x^28 - 1277913*x^27 - 4133168*x^26 + 3963256*x^25 + 10063407*x^24 - 9469608*x^23 - 19054236*x^22 + 17493203*x^21 + 28043254*x^20 - 24933608*x^19 - 31915288*x^18 + 27222459*x^17 + 27803858*x^16 - 22476466*x^15 - 18249278*x^14 + 13764066*x^13 + 8815690*x^12 - 6077890*x^11 - 3029168*x^10 + 1857788*x^9 + 703905*x^8 - 369684*x^7 - 102396*x^6 + 43393*x^5 + 8232*x^4 - 2512*x^3 - 288*x^2 + 48*x + 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^40 - x^39 - 39*x^38 + 39*x^37 + 702*x^36 - 701*x^35 - 7736*x^34 + 7701*x^33 + 58378*x^32 - 57817*x^31 - 319703*x^30 + 314247*x^29 + 1313873*x^28 - 1277913*x^27 - 4133168*x^26 + 3963256*x^25 + 10063407*x^24 - 9469608*x^23 - 19054236*x^22 + 17493203*x^21 + 28043254*x^20 - 24933608*x^19 - 31915288*x^18 + 27222459*x^17 + 27803858*x^16 - 22476466*x^15 - 18249278*x^14 + 13764066*x^13 + 8815690*x^12 - 6077890*x^11 - 3029168*x^10 + 1857788*x^9 + 703905*x^8 - 369684*x^7 - 102396*x^6 + 43393*x^5 + 8232*x^4 - 2512*x^3 - 288*x^2 + 48*x + 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - x^39 - 39*x^38 + 39*x^37 + 702*x^36 - 701*x^35 - 7736*x^34 + 7701*x^33 + 58378*x^32 - 57817*x^31 - 319703*x^30 + 314247*x^29 + 1313873*x^28 - 1277913*x^27 - 4133168*x^26 + 3963256*x^25 + 10063407*x^24 - 9469608*x^23 - 19054236*x^22 + 17493203*x^21 + 28043254*x^20 - 24933608*x^19 - 31915288*x^18 + 27222459*x^17 + 27803858*x^16 - 22476466*x^15 - 18249278*x^14 + 13764066*x^13 + 8815690*x^12 - 6077890*x^11 - 3029168*x^10 + 1857788*x^9 + 703905*x^8 - 369684*x^7 - 102396*x^6 + 43393*x^5 + 8232*x^4 - 2512*x^3 - 288*x^2 + 48*x + 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2\times C_{20}$ (as 40T2):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| 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
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $20^{2}$ | R | R | $20^{2}$ | R | $20^{2}$ | $20^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{10}$ | ${\href{/padicField/29.5.0.1}{5} }^{8}$ | ${\href{/padicField/31.5.0.1}{5} }^{8}$ | $20^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{10}$ | $20^{2}$ | $20^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $40$ | $2$ | $20$ | $20$ | |||
|
\(5\)
| Deg $40$ | $4$ | $10$ | $30$ | |||
|
\(11\)
| 11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |
| 11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |