Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 39*x^38 + 702*x^36 - 7735*x^34 + 58344*x^32 - 319177*x^30 + 1308973*x^28 - 4102514*x^26 + 9927425*x^24 - 18613969*x^22 + 26988270*x^20 - 30037766*x^18 + 25334997*x^16 - 15882694*x^14 + 7199257*x^12 - 2269345*x^10 + 470613*x^8 - 59241*x^6 + 4018*x^4 - 120*x^2 + 1)
gp: K = bnfinit(y^40 - 39*y^38 + 702*y^36 - 7735*y^34 + 58344*y^32 - 319177*y^30 + 1308973*y^28 - 4102514*y^26 + 9927425*y^24 - 18613969*y^22 + 26988270*y^20 - 30037766*y^18 + 25334997*y^16 - 15882694*y^14 + 7199257*y^12 - 2269345*y^10 + 470613*y^8 - 59241*y^6 + 4018*y^4 - 120*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 39*x^38 + 702*x^36 - 7735*x^34 + 58344*x^32 - 319177*x^30 + 1308973*x^28 - 4102514*x^26 + 9927425*x^24 - 18613969*x^22 + 26988270*x^20 - 30037766*x^18 + 25334997*x^16 - 15882694*x^14 + 7199257*x^12 - 2269345*x^10 + 470613*x^8 - 59241*x^6 + 4018*x^4 - 120*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 39*x^38 + 702*x^36 - 7735*x^34 + 58344*x^32 - 319177*x^30 + 1308973*x^28 - 4102514*x^26 + 9927425*x^24 - 18613969*x^22 + 26988270*x^20 - 30037766*x^18 + 25334997*x^16 - 15882694*x^14 + 7199257*x^12 - 2269345*x^10 + 470613*x^8 - 59241*x^6 + 4018*x^4 - 120*x^2 + 1)
\( x^{40} - 39 x^{38} + 702 x^{36} - 7735 x^{34} + 58344 x^{32} - 319177 x^{30} + 1308973 x^{28} - 4102514 x^{26} + \cdots + 1 \)
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: |
\(31654584865659568778929513407372752241664000000000000000000000000000000\)
\(\medspace = 2^{40}\cdot 5^{30}\cdot 11^{36}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(57.88\) | 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: | $2\cdot 5^{3/4}11^{9/10}\approx 57.87765351369302$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
| 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: | \(220=2^{2}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(3,·)$, $\chi_{220}(9,·)$, $\chi_{220}(139,·)$, $\chi_{220}(13,·)$, $\chi_{220}(17,·)$, $\chi_{220}(131,·)$, $\chi_{220}(23,·)$, $\chi_{220}(153,·)$, $\chi_{220}(27,·)$, $\chi_{220}(69,·)$, $\chi_{220}(163,·)$, $\chi_{220}(39,·)$, $\chi_{220}(169,·)$, $\chi_{220}(171,·)$, $\chi_{220}(173,·)$, $\chi_{220}(47,·)$, $\chi_{220}(49,·)$, $\chi_{220}(51,·)$, $\chi_{220}(181,·)$, $\chi_{220}(73,·)$, $\chi_{220}(57,·)$, $\chi_{220}(151,·)$, $\chi_{220}(19,·)$, $\chi_{220}(193,·)$, $\chi_{220}(67,·)$, $\chi_{220}(197,·)$, $\chi_{220}(201,·)$, $\chi_{220}(79,·)$, $\chi_{220}(203,·)$, $\chi_{220}(207,·)$, $\chi_{220}(141,·)$, $\chi_{220}(81,·)$, $\chi_{220}(211,·)$, $\chi_{220}(89,·)$, $\chi_{220}(219,·)$, $\chi_{220}(103,·)$, $\chi_{220}(147,·)$, $\chi_{220}(117,·)$, $\chi_{220}(217,·)$$\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: |
$a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8008a^{12}+11011a^{10}-9438a^{8}+4719a^{6}-1210a^{4}+121a^{2}-2$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+2$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-2$, $a^{38}-38a^{36}+665a^{34}-7106a^{32}+51831a^{30}-273267a^{28}+1075726a^{26}-3220450a^{24}+7399031a^{22}-13071750a^{20}+17682634a^{18}-18124653a^{16}+13833306a^{14}-7658743a^{12}+2960671a^{10}-755172a^{8}+116319a^{6}-9282a^{4}+280a^{2}-2$, $a^{38}-38a^{36}+665a^{34}-7106a^{32}+51832a^{30}-273297a^{28}+1076131a^{26}-3223700a^{24}+7416281a^{22}-13135507a^{20}+17850899a^{18}-18444593a^{16}+14270156a^{14}-8080918a^{12}+3241809a^{10}-878802a^{8}+149899a^{6}-14307a^{4}+605a^{2}-5$, $a^{35}-35a^{33}+559a^{31}-5394a^{29}+35091a^{27}-162630a^{25}+553150a^{23}-1401620a^{21}+2658271a^{19}-3759074a^{17}+3915406a^{15}-2939301a^{13}+1536847a^{11}-531076a^{9}+111735a^{7}-12531a^{5}+606a^{3}-8a$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}+11560835a^{17}-10994920a^{15}+7696444a^{13}-3848222a^{11}+1314610a^{9}-286824a^{7}+35853a^{5}-2109a^{3}+37a$, $a$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58344a^{31}-319176a^{29}+1308944a^{27}-4102137a^{25}+9924525a^{23}-18599295a^{21}+26936910a^{19}-29910465a^{17}+25110020a^{15}-15600900a^{13}+6953544a^{11}-2124694a^{9}+415701a^{7}-46683a^{5}+2470a^{3}-39a$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58344a^{31}-319176a^{29}+1308943a^{27}-4102110a^{25}+9924201a^{23}-18597018a^{21}+26926515a^{19}-29878147a^{17}+25040269a^{15}-15496367a^{13}+6846580a^{11}-2052699a^{9}+385935a^{7}-40025a^{5}+1850a^{3}-24a$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a$, $2a^{39}-78a^{37}+1403a^{35}-15434a^{33}+116095a^{31}-632434a^{29}+2577955a^{27}-8011743a^{25}+19165694a^{23}-35385799a^{21}+50261764a^{19}-54433103a^{17}+44270887a^{15}-26434087a^{13}+11216062a^{11}-3225832a^{9}+585822a^{7}-59961a^{5}+2830a^{3}-39a$, $a^{19}-19a^{17}+152a^{15}-665a^{13}+1729a^{11}-2717a^{9}+2508a^{7}-1254a^{5}+285a^{3}-19a$, $a^{7}-7a^{5}+14a^{3}-7a$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45879a^{29}-232812a^{27}+878410a^{25}-2507920a^{23}+5461511a^{21}-9075616a^{19}+11433554a^{17}-10770113a^{15}+7415449a^{13}-3604770a^{11}+1173887a^{9}-236001a^{7}+25668a^{5}-1220a^{3}+16a$, $a^{3}-3a$, $a-1$, $a^{22}-a^{21}-22a^{20}+21a^{19}+209a^{18}-189a^{17}-1122a^{16}+952a^{15}+3740a^{14}-2940a^{13}-8008a^{12}+5733a^{11}+11011a^{10}-7007a^{9}-9438a^{8}+5148a^{7}+4719a^{6}-2079a^{5}-1210a^{4}+385a^{3}+121a^{2}-21a-2$, $2a^{39}-78a^{37}+1403a^{35}-15434a^{33}+116095a^{31}-632434a^{29}+2577955a^{27}-8011743a^{25}+19165694a^{23}-a^{22}-35385799a^{21}+22a^{20}+50261764a^{19}-209a^{18}-54433103a^{17}+1122a^{16}+44270887a^{15}-3740a^{14}-26434087a^{13}+8008a^{12}+11216062a^{11}-11011a^{10}-3225832a^{9}+9438a^{8}+585822a^{7}-4719a^{6}-59961a^{5}+1210a^{4}+2830a^{3}-121a^{2}-39a+3$, $a^{12}-12a^{10}+a^{9}+54a^{8}-9a^{7}-111a^{6}+27a^{5}+99a^{4}-29a^{3}-27a^{2}+6a+1$, $a^{39}-38a^{37}+665a^{35}-7106a^{33}+51832a^{31}-273296a^{29}+1076102a^{27}-3223323a^{25}+a^{24}+7413381a^{23}-24a^{22}-13120833a^{21}+252a^{20}+17799540a^{19}-1520a^{18}-18317312a^{17}+5814a^{16}+14045349a^{15}-14688a^{14}-7799923a^{13}+24751a^{12}+2998358a^{11}-27444a^{10}-738089a^{9}+19251a^{8}+99111a^{7}-7896a^{6}-4172a^{5}+1611a^{4}-259a^{3}-108a^{2}+13a+1$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}+11560835a^{17}-10994920a^{15}+7696444a^{13}-3848222a^{11}+1314610a^{9}-286824a^{7}+35853a^{5}-2109a^{3}+37a+1$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}-1015a^{3}+29a-1$, $a^{36}-36a^{34}+593a^{32}-5920a^{30}+39992a^{28}-a^{27}-193312a^{26}+27a^{25}+689479a^{24}-323a^{23}-1844392a^{22}+2254a^{21}+3724920a^{20}-10165a^{19}-5673248a^{18}+31009a^{17}+6463230a^{16}-65093a^{15}-5422049a^{14}+93822a^{13}+3267278a^{12}-91091a^{11}-1361996a^{10}+57355a^{9}+370672a^{8}-21988a^{7}-59971a^{6}+4621a^{5}+4915a^{4}-439a^{3}-156a^{2}+12a$, $a^{39}-a^{38}-38a^{37}+39a^{36}+665a^{35}-701a^{34}-7106a^{33}+7699a^{32}+51832a^{31}-57751a^{30}-273297a^{29}+313257a^{28}+1076131a^{27}-1268983a^{26}-3223700a^{25}+3909257a^{24}+7416281a^{23}-9238618a^{22}-13135506a^{21}+16774383a^{20}+17850878a^{19}-23285659a^{18}-18444403a^{17}+24434951a^{16}+14269188a^{15}-19025839a^{14}-8077875a^{13}+10694035a^{12}+3235738a^{11}-4172506a^{10}-871210a^{9}+1069743a^{8}+144269a^{7}-167106a^{6}-12117a^{5}+14377a^{4}+258a^{3}-582a^{2}+9a+7$, $a^{35}-35a^{33}+560a^{31}-5424a^{29}+35496a^{27}-165880a^{25}+570400a^{23}+a^{22}-1465376a^{21}-22a^{20}+2826515a^{19}+209a^{18}-4078825a^{17}-1122a^{16}+4351304a^{15}+3740a^{14}-3358536a^{13}-8008a^{12}+1812252a^{11}+11010a^{10}-647699a^{9}-9428a^{8}+140167a^{7}+4684a^{6}-15477a^{5}-1160a^{4}+546a^{3}+96a^{2}+9a$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58344a^{31}-319176a^{29}+1308944a^{27}-4102137a^{25}+9924525a^{23}-18599295a^{21}+26936910a^{19}-29910466a^{17}+a^{16}+25110037a^{15}-16a^{14}-15601019a^{13}+104a^{12}+6953986a^{11}-352a^{10}-2125629a^{9}+660a^{8}+416823a^{7}-673a^{6}-47396a^{5}+342a^{4}+2669a^{3}-73a^{2}-51a+4$, $3a^{39}-116a^{37}-a^{36}+2068a^{35}+36a^{34}-22540a^{33}-595a^{32}+167927a^{31}+5985a^{30}-905730a^{29}-40949a^{28}+3654058a^{27}+201753a^{26}-11235093a^{25}-739180a^{24}+26579399a^{23}+2050449a^{22}-48508909a^{21}-4343229a^{20}+68071699a^{19}+7032431a^{18}-72782733a^{17}-8653968a^{16}+58385987a^{15}+7987792a^{14}-34338543a^{13}-5408170a^{12}+14321384a^{11}+2593098a^{10}-4035917a^{9}-833153a^{8}+714708a^{7}+163892a^{6}-70818a^{5}-16877a^{4}+3222a^{3}+693a^{2}-52a-8$, $2a^{39}-77a^{37}+1367a^{35}-a^{34}-14841a^{33}+34a^{32}+110176a^{31}-527a^{30}-592474a^{29}+4930a^{28}+2385104a^{27}-31059a^{26}-7326214a^{25}+139231a^{24}+17343706a^{23}-457494a^{22}-31749476a^{21}+1118512a^{20}+44839169a^{19}-2044495a^{18}-48482358a^{17}+2784261a^{16}+39605136a^{15}-2793151a^{14}-23963493a^{13}+2023594a^{12}+10440625a^{11}-1027247a^{10}-3147222a^{9}+349635a^{8}+619424a^{7}-74867a^{6}-72877a^{5}+9191a^{4}+4430a^{3}-559a^{2}-104a+12$, $a^{22}-22a^{20}+209a^{18}-a^{17}-1122a^{16}+17a^{15}+3740a^{14}-119a^{13}-8008a^{12}+442a^{11}+11011a^{10}-935a^{9}-9438a^{8}+1122a^{7}+4719a^{6}-714a^{5}-1210a^{4}+204a^{3}+121a^{2}-17a-2$, $2a^{39}-2a^{38}-78a^{37}+75a^{36}+1402a^{35}-1293a^{34}-15398a^{33}+13584a^{32}+115503a^{31}-97183a^{30}-626546a^{29}+501148a^{28}+2538427a^{27}-1923050a^{26}-7822463a^{25}+5588817a^{24}+18499614a^{23}-12400819a^{22}-33637063a^{21}+21018963a^{20}+46819951a^{19}-27042415a^{18}-49373612a^{17}+26052383a^{16}+38782320a^{15}-18380463a^{14}-22133660a^{13}+9182563a^{12}+8862970a^{11}-3090581a^{10}-2371423a^{9}+651052a^{8}+395402a^{7}-77186a^{6}-36940a^{5}+4516a^{4}+1623a^{3}-112a^{2}-27a$, $a^{36}-36a^{34}+593a^{32}-5920a^{30}+39991a^{28}-193285a^{26}+689157a^{24}+a^{23}-1842162a^{22}-23a^{21}+3715007a^{20}+230a^{19}-5643760a^{18}-1311a^{17}+6403969a^{16}+4692a^{15}-5343049a^{14}-10948a^{13}+3201471a^{12}+16744a^{11}-1333307a^{10}-16445a^{9}+369561a^{8}+9867a^{7}-64450a^{6}-3289a^{5}+6537a^{4}+506a^{3}-325a^{2}-23a+6$, $a^{28}-a^{27}-27a^{26}+26a^{25}+324a^{24}-299a^{23}-2277a^{22}+2002a^{21}+10395a^{20}-8645a^{19}-32319a^{18}+25194a^{17}+69768a^{16}-50388a^{15}-104652a^{14}+68952a^{13}+107406a^{12}-63206a^{11}-72930a^{10}+37180a^{9}+30888a^{8}-13013a^{7}-7371a^{6}+2366a^{5}+819a^{4}-169a^{3}-27a^{2}+2a$, $a^{38}-37a^{36}+a^{35}+629a^{34}-36a^{33}-6512a^{32}+592a^{31}+45879a^{30}-5889a^{29}-232811a^{28}+39556a^{27}+878382a^{26}-189630a^{25}-2507570a^{24}+668656a^{23}+5458936a^{22}-1761133a^{21}-9063241a^{20}+3482776a^{19}+11392800a^{18}-5154434a^{17}-10676290a^{16}+5643456a^{15}+7264268a^{14}-4476136a^{13}-3436862a^{12}+2487602a^{11}+1049697a^{10}-920101a^{9}-178493a^{8}+208910a^{7}+10716a^{6}-25531a^{5}+557a^{4}+1360a^{3}-50a^{2}-20a$, $2a^{39}+a^{38}-77a^{37}-38a^{36}+1367a^{35}+665a^{34}-14841a^{33}-7106a^{32}+110175a^{31}+51831a^{30}-592443a^{29}-273267a^{28}+2384671a^{27}+1075727a^{26}-7322613a^{25}-3220476a^{24}+17323855a^{23}+7399330a^{22}-31673047a^{21}-13073752a^{20}+44628189a^{19}+17691279a^{18}-48060312a^{17}-18149847a^{16}+38993714a^{15}+13883694a^{14}-23328595a^{13}-7727695a^{12}+9977421a^{11}+3023877a^{10}-2917046a^{9}-792352a^{8}+545033a^{7}+129332a^{6}-58331a^{5}-11648a^{4}+2918a^{3}+449a^{2}-39a-3$, $a^{22}+a^{21}-22a^{20}-21a^{19}+209a^{18}+189a^{17}-1122a^{16}-952a^{15}+3740a^{14}+2940a^{13}-8008a^{12}-5733a^{11}+11011a^{10}+7007a^{9}-9438a^{8}-5148a^{7}+4719a^{6}+2079a^{5}-1210a^{4}-385a^{3}+121a^{2}+21a-3$
| 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: | \( 33894240488185236000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{40}\cdot(2\pi)^{0}\cdot 33894240488185236000000 \cdot 1}{2\cdot\sqrt{31654584865659568778929513407372752241664000000000000000000000000000000}}\cr\approx \mathstrut & 0.104731524741384 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^40 - 39*x^38 + 702*x^36 - 7735*x^34 + 58344*x^32 - 319177*x^30 + 1308973*x^28 - 4102514*x^26 + 9927425*x^24 - 18613969*x^22 + 26988270*x^20 - 30037766*x^18 + 25334997*x^16 - 15882694*x^14 + 7199257*x^12 - 2269345*x^10 + 470613*x^8 - 59241*x^6 + 4018*x^4 - 120*x^2 + 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 - 39*x^38 + 702*x^36 - 7735*x^34 + 58344*x^32 - 319177*x^30 + 1308973*x^28 - 4102514*x^26 + 9927425*x^24 - 18613969*x^22 + 26988270*x^20 - 30037766*x^18 + 25334997*x^16 - 15882694*x^14 + 7199257*x^12 - 2269345*x^10 + 470613*x^8 - 59241*x^6 + 4018*x^4 - 120*x^2 + 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 - 39*x^38 + 702*x^36 - 7735*x^34 + 58344*x^32 - 319177*x^30 + 1308973*x^28 - 4102514*x^26 + 9927425*x^24 - 18613969*x^22 + 26988270*x^20 - 30037766*x^18 + 25334997*x^16 - 15882694*x^14 + 7199257*x^12 - 2269345*x^10 + 470613*x^8 - 59241*x^6 + 4018*x^4 - 120*x^2 + 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 - 39*x^38 + 702*x^36 - 7735*x^34 + 58344*x^32 - 319177*x^30 + 1308973*x^28 - 4102514*x^26 + 9927425*x^24 - 18613969*x^22 + 26988270*x^20 - 30037766*x^18 + 25334997*x^16 - 15882694*x^14 + 7199257*x^12 - 2269345*x^10 + 470613*x^8 - 59241*x^6 + 4018*x^4 - 120*x^2 + 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 $C_2\times C_{20}$ |
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 | R | $20^{2}$ | R | $20^{2}$ | R | $20^{2}$ | $20^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{10}$ | $20^{2}$ | $20^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $40$ | $2$ | $20$ | $40$ | |||
|
\(5\)
| Deg $20$ | $4$ | $5$ | $15$ | |||
| Deg $20$ | $4$ | $5$ | $15$ | ||||
|
\(11\)
| 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 | $[\ ]_{10}^{2}$ |