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Properties

Label 40.40.234...625.1
Degree $40$
Signature $[40, 0]$
Discriminant $2.342\times 10^{90}$
Root discriminant \(181.65\)
Ramified primes $5,11,17$
Class number not computed
Class group not computed
Galois group $C_{40}$ (as 40T1)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^40 - 3*x^39 - 153*x^38 + 424*x^37 + 10435*x^36 - 26104*x^35 - 422087*x^34 + 928824*x^33 + 11344024*x^32 - 21340302*x^31 - 214673754*x^30 + 334559600*x^29 + 2952971467*x^28 - 3683043942*x^27 - 30025251733*x^26 + 28841721208*x^25 + 227220142820*x^24 - 161003561620*x^23 - 1279418767795*x^22 + 637078046638*x^21 + 5328173642850*x^20 - 1770142272603*x^19 - 16230752948952*x^18 + 3444167779449*x^17 + 35584533352769*x^16 - 4842006085510*x^15 - 54915618366102*x^14 + 5434842226712*x^13 + 57856494632629*x^12 - 5328728706700*x^11 - 39827070645512*x^10 + 4123513230427*x^9 + 16811606265180*x^8 - 2042316034583*x^7 - 3968102247745*x^6 + 551974490536*x^5 + 443309788137*x^4 - 62108079736*x^3 - 14574899705*x^2 + 582845329*x + 99452651)
 
Copy content gp:K = bnfinit(y^40 - 3*y^39 - 153*y^38 + 424*y^37 + 10435*y^36 - 26104*y^35 - 422087*y^34 + 928824*y^33 + 11344024*y^32 - 21340302*y^31 - 214673754*y^30 + 334559600*y^29 + 2952971467*y^28 - 3683043942*y^27 - 30025251733*y^26 + 28841721208*y^25 + 227220142820*y^24 - 161003561620*y^23 - 1279418767795*y^22 + 637078046638*y^21 + 5328173642850*y^20 - 1770142272603*y^19 - 16230752948952*y^18 + 3444167779449*y^17 + 35584533352769*y^16 - 4842006085510*y^15 - 54915618366102*y^14 + 5434842226712*y^13 + 57856494632629*y^12 - 5328728706700*y^11 - 39827070645512*y^10 + 4123513230427*y^9 + 16811606265180*y^8 - 2042316034583*y^7 - 3968102247745*y^6 + 551974490536*y^5 + 443309788137*y^4 - 62108079736*y^3 - 14574899705*y^2 + 582845329*y + 99452651, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 3*x^39 - 153*x^38 + 424*x^37 + 10435*x^36 - 26104*x^35 - 422087*x^34 + 928824*x^33 + 11344024*x^32 - 21340302*x^31 - 214673754*x^30 + 334559600*x^29 + 2952971467*x^28 - 3683043942*x^27 - 30025251733*x^26 + 28841721208*x^25 + 227220142820*x^24 - 161003561620*x^23 - 1279418767795*x^22 + 637078046638*x^21 + 5328173642850*x^20 - 1770142272603*x^19 - 16230752948952*x^18 + 3444167779449*x^17 + 35584533352769*x^16 - 4842006085510*x^15 - 54915618366102*x^14 + 5434842226712*x^13 + 57856494632629*x^12 - 5328728706700*x^11 - 39827070645512*x^10 + 4123513230427*x^9 + 16811606265180*x^8 - 2042316034583*x^7 - 3968102247745*x^6 + 551974490536*x^5 + 443309788137*x^4 - 62108079736*x^3 - 14574899705*x^2 + 582845329*x + 99452651);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^40 - 3*x^39 - 153*x^38 + 424*x^37 + 10435*x^36 - 26104*x^35 - 422087*x^34 + 928824*x^33 + 11344024*x^32 - 21340302*x^31 - 214673754*x^30 + 334559600*x^29 + 2952971467*x^28 - 3683043942*x^27 - 30025251733*x^26 + 28841721208*x^25 + 227220142820*x^24 - 161003561620*x^23 - 1279418767795*x^22 + 637078046638*x^21 + 5328173642850*x^20 - 1770142272603*x^19 - 16230752948952*x^18 + 3444167779449*x^17 + 35584533352769*x^16 - 4842006085510*x^15 - 54915618366102*x^14 + 5434842226712*x^13 + 57856494632629*x^12 - 5328728706700*x^11 - 39827070645512*x^10 + 4123513230427*x^9 + 16811606265180*x^8 - 2042316034583*x^7 - 3968102247745*x^6 + 551974490536*x^5 + 443309788137*x^4 - 62108079736*x^3 - 14574899705*x^2 + 582845329*x + 99452651)
 

\( x^{40} - 3 x^{39} - 153 x^{38} + 424 x^{37} + 10435 x^{36} - 26104 x^{35} - 422087 x^{34} + \cdots + 99452651 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $40$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[40, 0]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(234\!\cdots\!625\) \(\medspace = 5^{20}\cdot 11^{32}\cdot 17^{35}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(181.65\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $5^{1/2}11^{4/5}17^{7/8}\approx 181.65274145300944$
Ramified primes:   \(5\), \(11\), \(17\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{17}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{40}$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(935=5\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{935}(256,·)$, $\chi_{935}(1,·)$, $\chi_{935}(389,·)$, $\chi_{935}(774,·)$, $\chi_{935}(9,·)$, $\chi_{935}(526,·)$, $\chi_{935}(399,·)$, $\chi_{935}(16,·)$, $\chi_{935}(529,·)$, $\chi_{935}(786,·)$, $\chi_{935}(531,·)$, $\chi_{935}(276,·)$, $\chi_{935}(676,·)$, $\chi_{935}(421,·)$, $\chi_{935}(166,·)$, $\chi_{935}(559,·)$, $\chi_{935}(49,·)$, $\chi_{935}(434,·)$, $\chi_{935}(179,·)$, $\chi_{935}(696,·)$, $\chi_{935}(441,·)$, $\chi_{935}(59,·)$, $\chi_{935}(444,·)$, $\chi_{935}(191,·)$, $\chi_{935}(81,·)$, $\chi_{935}(851,·)$, $\chi_{935}(654,·)$, $\chi_{935}(86,·)$, $\chi_{935}(729,·)$, $\chi_{935}(474,·)$, $\chi_{935}(859,·)$, $\chi_{935}(784,·)$, $\chi_{935}(356,·)$, $\chi_{935}(229,·)$, $\chi_{935}(614,·)$, $\chi_{935}(104,·)$, $\chi_{935}(361,·)$, $\chi_{935}(144,·)$, $\chi_{935}(251,·)$, $\chi_{935}(511,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{13}a^{30}+\frac{1}{13}a^{29}+\frac{1}{13}a^{28}-\frac{3}{13}a^{27}+\frac{4}{13}a^{26}+\frac{5}{13}a^{25}-\frac{5}{13}a^{24}-\frac{3}{13}a^{23}+\frac{4}{13}a^{22}+\frac{1}{13}a^{21}+\frac{1}{13}a^{20}-\frac{2}{13}a^{18}+\frac{5}{13}a^{17}-\frac{2}{13}a^{16}+\frac{1}{13}a^{15}+\frac{6}{13}a^{14}-\frac{6}{13}a^{12}+\frac{3}{13}a^{11}-\frac{1}{13}a^{10}+\frac{6}{13}a^{9}+\frac{5}{13}a^{8}-\frac{6}{13}a^{7}-\frac{5}{13}a^{5}+\frac{6}{13}a^{4}+\frac{1}{13}a^{3}-\frac{2}{13}a^{2}+\frac{6}{13}a+\frac{3}{13}$, $\frac{1}{13}a^{31}-\frac{4}{13}a^{28}-\frac{6}{13}a^{27}+\frac{1}{13}a^{26}+\frac{3}{13}a^{25}+\frac{2}{13}a^{24}-\frac{6}{13}a^{23}-\frac{3}{13}a^{22}-\frac{1}{13}a^{20}-\frac{2}{13}a^{19}-\frac{6}{13}a^{18}+\frac{6}{13}a^{17}+\frac{3}{13}a^{16}+\frac{5}{13}a^{15}-\frac{6}{13}a^{14}-\frac{6}{13}a^{13}-\frac{4}{13}a^{12}-\frac{4}{13}a^{11}-\frac{6}{13}a^{10}-\frac{1}{13}a^{9}+\frac{2}{13}a^{8}+\frac{6}{13}a^{7}-\frac{5}{13}a^{6}-\frac{2}{13}a^{5}-\frac{5}{13}a^{4}-\frac{3}{13}a^{3}-\frac{5}{13}a^{2}-\frac{3}{13}a-\frac{3}{13}$, $\frac{1}{13}a^{32}-\frac{4}{13}a^{29}-\frac{6}{13}a^{28}+\frac{1}{13}a^{27}+\frac{3}{13}a^{26}+\frac{2}{13}a^{25}-\frac{6}{13}a^{24}-\frac{3}{13}a^{23}-\frac{1}{13}a^{21}-\frac{2}{13}a^{20}-\frac{6}{13}a^{19}+\frac{6}{13}a^{18}+\frac{3}{13}a^{17}+\frac{5}{13}a^{16}-\frac{6}{13}a^{15}-\frac{6}{13}a^{14}-\frac{4}{13}a^{13}-\frac{4}{13}a^{12}-\frac{6}{13}a^{11}-\frac{1}{13}a^{10}+\frac{2}{13}a^{9}+\frac{6}{13}a^{8}-\frac{5}{13}a^{7}-\frac{2}{13}a^{6}-\frac{5}{13}a^{5}-\frac{3}{13}a^{4}-\frac{5}{13}a^{3}-\frac{3}{13}a^{2}-\frac{3}{13}a$, $\frac{1}{13}a^{33}-\frac{2}{13}a^{29}+\frac{5}{13}a^{28}+\frac{4}{13}a^{27}+\frac{5}{13}a^{26}+\frac{1}{13}a^{25}+\frac{3}{13}a^{24}+\frac{1}{13}a^{23}+\frac{2}{13}a^{22}+\frac{2}{13}a^{21}-\frac{2}{13}a^{20}+\frac{6}{13}a^{19}-\frac{5}{13}a^{18}-\frac{1}{13}a^{17}-\frac{1}{13}a^{16}-\frac{2}{13}a^{15}-\frac{6}{13}a^{14}-\frac{4}{13}a^{13}-\frac{4}{13}a^{12}-\frac{2}{13}a^{11}-\frac{2}{13}a^{10}+\frac{4}{13}a^{9}+\frac{2}{13}a^{8}-\frac{5}{13}a^{6}+\frac{3}{13}a^{5}+\frac{6}{13}a^{4}+\frac{1}{13}a^{3}+\frac{2}{13}a^{2}-\frac{2}{13}a-\frac{1}{13}$, $\frac{1}{13}a^{34}-\frac{6}{13}a^{29}+\frac{6}{13}a^{28}-\frac{1}{13}a^{27}-\frac{4}{13}a^{26}+\frac{4}{13}a^{24}-\frac{4}{13}a^{23}-\frac{3}{13}a^{22}-\frac{5}{13}a^{20}-\frac{5}{13}a^{19}-\frac{5}{13}a^{18}-\frac{4}{13}a^{17}-\frac{6}{13}a^{16}-\frac{4}{13}a^{15}-\frac{5}{13}a^{14}-\frac{4}{13}a^{13}-\frac{1}{13}a^{12}+\frac{4}{13}a^{11}+\frac{2}{13}a^{10}+\frac{1}{13}a^{9}-\frac{3}{13}a^{8}-\frac{4}{13}a^{7}+\frac{3}{13}a^{6}-\frac{4}{13}a^{5}+\frac{4}{13}a^{3}-\frac{6}{13}a^{2}-\frac{2}{13}a+\frac{6}{13}$, $\frac{1}{611}a^{35}-\frac{6}{611}a^{34}-\frac{12}{611}a^{33}-\frac{17}{611}a^{32}+\frac{16}{611}a^{31}-\frac{2}{611}a^{30}+\frac{1}{13}a^{29}+\frac{23}{611}a^{28}+\frac{76}{611}a^{27}+\frac{166}{611}a^{26}+\frac{3}{47}a^{25}-\frac{171}{611}a^{24}-\frac{113}{611}a^{23}+\frac{261}{611}a^{22}+\frac{70}{611}a^{21}-\frac{46}{611}a^{20}+\frac{23}{611}a^{19}+\frac{88}{611}a^{18}-\frac{35}{611}a^{17}-\frac{131}{611}a^{16}+\frac{99}{611}a^{15}-\frac{288}{611}a^{14}-\frac{126}{611}a^{13}-\frac{144}{611}a^{12}-\frac{2}{47}a^{11}-\frac{96}{611}a^{10}-\frac{304}{611}a^{9}-\frac{112}{611}a^{8}-\frac{115}{611}a^{7}+\frac{187}{611}a^{6}+\frac{8}{611}a^{5}+\frac{96}{611}a^{4}+\frac{25}{611}a^{3}-\frac{92}{611}a^{2}-\frac{48}{611}a+\frac{122}{611}$, $\frac{1}{611}a^{36}-\frac{1}{611}a^{34}+\frac{5}{611}a^{33}+\frac{8}{611}a^{32}-\frac{12}{611}a^{30}+\frac{23}{611}a^{29}+\frac{120}{611}a^{28}-\frac{83}{611}a^{27}+\frac{95}{611}a^{26}-\frac{172}{611}a^{25}+\frac{36}{611}a^{24}-\frac{88}{611}a^{23}-\frac{56}{611}a^{22}-\frac{190}{611}a^{21}-\frac{206}{611}a^{20}+\frac{179}{611}a^{19}-\frac{212}{611}a^{18}+\frac{82}{611}a^{17}-\frac{170}{611}a^{16}+\frac{71}{611}a^{15}+\frac{120}{611}a^{14}-\frac{54}{611}a^{13}+\frac{191}{611}a^{12}+\frac{30}{611}a^{11}+\frac{154}{611}a^{10}-\frac{291}{611}a^{9}+\frac{12}{611}a^{8}-\frac{17}{47}a^{7}-\frac{139}{611}a^{6}+\frac{191}{611}a^{5}-\frac{151}{611}a^{4}+\frac{105}{611}a^{3}+\frac{199}{611}a^{2}-\frac{119}{611}a-\frac{161}{611}$, $\frac{1}{611}a^{37}-\frac{1}{611}a^{34}-\frac{4}{611}a^{33}-\frac{17}{611}a^{32}+\frac{4}{611}a^{31}+\frac{21}{611}a^{30}+\frac{167}{611}a^{29}-\frac{60}{611}a^{28}+\frac{171}{611}a^{27}-\frac{6}{611}a^{26}+\frac{75}{611}a^{25}-\frac{259}{611}a^{24}-\frac{13}{47}a^{23}+\frac{71}{611}a^{22}-\frac{136}{611}a^{21}+\frac{133}{611}a^{20}-\frac{189}{611}a^{19}+\frac{170}{611}a^{18}-\frac{205}{611}a^{17}-\frac{60}{611}a^{16}+\frac{219}{611}a^{15}+\frac{269}{611}a^{14}+\frac{5}{47}a^{13}-\frac{114}{611}a^{12}+\frac{128}{611}a^{11}+\frac{224}{611}a^{10}-\frac{292}{611}a^{9}+\frac{278}{611}a^{8}-\frac{254}{611}a^{7}-\frac{233}{611}a^{6}-\frac{11}{47}a^{5}+\frac{201}{611}a^{4}+\frac{224}{611}a^{3}-\frac{211}{611}a^{2}-\frac{209}{611}a+\frac{122}{611}$, $\frac{1}{611}a^{38}-\frac{10}{611}a^{34}+\frac{18}{611}a^{33}-\frac{1}{47}a^{32}-\frac{10}{611}a^{31}-\frac{23}{611}a^{30}-\frac{295}{611}a^{29}-\frac{14}{47}a^{28}-\frac{118}{611}a^{27}+\frac{288}{611}a^{26}-\frac{32}{611}a^{25}+\frac{36}{611}a^{24}+\frac{240}{611}a^{23}+\frac{219}{611}a^{22}+\frac{109}{611}a^{21}+\frac{3}{13}a^{20}-\frac{42}{611}a^{19}-\frac{305}{611}a^{18}-\frac{142}{611}a^{17}+\frac{276}{611}a^{16}-\frac{149}{611}a^{15}-\frac{129}{611}a^{14}-\frac{146}{611}a^{13}-\frac{110}{611}a^{12}-\frac{272}{611}a^{11}-\frac{12}{611}a^{10}+\frac{303}{611}a^{9}-\frac{84}{611}a^{8}-\frac{113}{611}a^{7}+\frac{44}{611}a^{6}+\frac{162}{611}a^{5}-\frac{291}{611}a^{4}-\frac{186}{611}a^{3}-\frac{207}{611}a^{2}+\frac{215}{611}a+\frac{263}{611}$, $\frac{1}{19\cdots 19}a^{39}-\frac{82\cdots 71}{19\cdots 19}a^{38}+\frac{19\cdots 19}{19\cdots 19}a^{37}-\frac{38\cdots 08}{19\cdots 19}a^{36}+\frac{98\cdots 39}{19\cdots 19}a^{35}+\frac{34\cdots 60}{19\cdots 19}a^{34}-\frac{64\cdots 80}{19\cdots 19}a^{33}+\frac{74\cdots 00}{19\cdots 19}a^{32}+\frac{62\cdots 31}{15\cdots 63}a^{31}+\frac{15\cdots 78}{19\cdots 19}a^{30}+\frac{51\cdots 34}{19\cdots 19}a^{29}-\frac{78\cdots 70}{19\cdots 19}a^{28}-\frac{82\cdots 73}{19\cdots 19}a^{27}-\frac{14\cdots 33}{19\cdots 19}a^{26}+\frac{86\cdots 36}{19\cdots 19}a^{25}+\frac{19\cdots 70}{19\cdots 19}a^{24}+\frac{90\cdots 79}{19\cdots 19}a^{23}+\frac{53\cdots 88}{19\cdots 19}a^{22}-\frac{30\cdots 43}{19\cdots 19}a^{21}+\frac{23\cdots 36}{19\cdots 19}a^{20}-\frac{70\cdots 41}{19\cdots 19}a^{19}+\frac{31\cdots 03}{19\cdots 19}a^{18}+\frac{66\cdots 43}{19\cdots 19}a^{17}-\frac{76\cdots 73}{19\cdots 19}a^{16}-\frac{23\cdots 14}{19\cdots 19}a^{15}+\frac{22\cdots 21}{19\cdots 19}a^{14}+\frac{67\cdots 39}{19\cdots 19}a^{13}+\frac{65\cdots 86}{19\cdots 19}a^{12}+\frac{83\cdots 43}{19\cdots 19}a^{11}+\frac{75\cdots 27}{19\cdots 19}a^{10}-\frac{69\cdots 90}{19\cdots 19}a^{9}-\frac{29\cdots 65}{19\cdots 19}a^{8}+\frac{46\cdots 94}{15\cdots 63}a^{7}-\frac{21\cdots 48}{19\cdots 19}a^{6}-\frac{47\cdots 78}{19\cdots 19}a^{5}+\frac{29\cdots 26}{19\cdots 19}a^{4}+\frac{45\cdots 62}{19\cdots 19}a^{3}-\frac{96\cdots 16}{19\cdots 19}a^{2}+\frac{25\cdots 38}{15\cdots 63}a-\frac{35\cdots 26}{19\cdots 19}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  not computed
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  not computed
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $39$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:  not computed
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{40}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{2342492774810879590625564693377242840437133595865567669360927718694362835612583160400390625}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^40 - 3*x^39 - 153*x^38 + 424*x^37 + 10435*x^36 - 26104*x^35 - 422087*x^34 + 928824*x^33 + 11344024*x^32 - 21340302*x^31 - 214673754*x^30 + 334559600*x^29 + 2952971467*x^28 - 3683043942*x^27 - 30025251733*x^26 + 28841721208*x^25 + 227220142820*x^24 - 161003561620*x^23 - 1279418767795*x^22 + 637078046638*x^21 + 5328173642850*x^20 - 1770142272603*x^19 - 16230752948952*x^18 + 3444167779449*x^17 + 35584533352769*x^16 - 4842006085510*x^15 - 54915618366102*x^14 + 5434842226712*x^13 + 57856494632629*x^12 - 5328728706700*x^11 - 39827070645512*x^10 + 4123513230427*x^9 + 16811606265180*x^8 - 2042316034583*x^7 - 3968102247745*x^6 + 551974490536*x^5 + 443309788137*x^4 - 62108079736*x^3 - 14574899705*x^2 + 582845329*x + 99452651) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^40 - 3*x^39 - 153*x^38 + 424*x^37 + 10435*x^36 - 26104*x^35 - 422087*x^34 + 928824*x^33 + 11344024*x^32 - 21340302*x^31 - 214673754*x^30 + 334559600*x^29 + 2952971467*x^28 - 3683043942*x^27 - 30025251733*x^26 + 28841721208*x^25 + 227220142820*x^24 - 161003561620*x^23 - 1279418767795*x^22 + 637078046638*x^21 + 5328173642850*x^20 - 1770142272603*x^19 - 16230752948952*x^18 + 3444167779449*x^17 + 35584533352769*x^16 - 4842006085510*x^15 - 54915618366102*x^14 + 5434842226712*x^13 + 57856494632629*x^12 - 5328728706700*x^11 - 39827070645512*x^10 + 4123513230427*x^9 + 16811606265180*x^8 - 2042316034583*x^7 - 3968102247745*x^6 + 551974490536*x^5 + 443309788137*x^4 - 62108079736*x^3 - 14574899705*x^2 + 582845329*x + 99452651, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 3*x^39 - 153*x^38 + 424*x^37 + 10435*x^36 - 26104*x^35 - 422087*x^34 + 928824*x^33 + 11344024*x^32 - 21340302*x^31 - 214673754*x^30 + 334559600*x^29 + 2952971467*x^28 - 3683043942*x^27 - 30025251733*x^26 + 28841721208*x^25 + 227220142820*x^24 - 161003561620*x^23 - 1279418767795*x^22 + 637078046638*x^21 + 5328173642850*x^20 - 1770142272603*x^19 - 16230752948952*x^18 + 3444167779449*x^17 + 35584533352769*x^16 - 4842006085510*x^15 - 54915618366102*x^14 + 5434842226712*x^13 + 57856494632629*x^12 - 5328728706700*x^11 - 39827070645512*x^10 + 4123513230427*x^9 + 16811606265180*x^8 - 2042316034583*x^7 - 3968102247745*x^6 + 551974490536*x^5 + 443309788137*x^4 - 62108079736*x^3 - 14574899705*x^2 + 582845329*x + 99452651); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^40 - 3*x^39 - 153*x^38 + 424*x^37 + 10435*x^36 - 26104*x^35 - 422087*x^34 + 928824*x^33 + 11344024*x^32 - 21340302*x^31 - 214673754*x^30 + 334559600*x^29 + 2952971467*x^28 - 3683043942*x^27 - 30025251733*x^26 + 28841721208*x^25 + 227220142820*x^24 - 161003561620*x^23 - 1279418767795*x^22 + 637078046638*x^21 + 5328173642850*x^20 - 1770142272603*x^19 - 16230752948952*x^18 + 3444167779449*x^17 + 35584533352769*x^16 - 4842006085510*x^15 - 54915618366102*x^14 + 5434842226712*x^13 + 57856494632629*x^12 - 5328728706700*x^11 - 39827070645512*x^10 + 4123513230427*x^9 + 16811606265180*x^8 - 2042316034583*x^7 - 3968102247745*x^6 + 551974490536*x^5 + 443309788137*x^4 - 62108079736*x^3 - 14574899705*x^2 + 582845329*x + 99452651); 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

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

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A cyclic group of order 40
The 40 conjugacy class representatives for $C_{40}$
Character table for $C_{40}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{11})^+\), 8.8.256461670625.1, 10.10.304358957700017.1, 20.20.131527565972137936816816034072938673.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content 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 $20^{2}$ $40$ R $40$ R ${\href{/padicField/13.5.0.1}{5} }^{8}$ R $20^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{5}$ $40$ $40$ $40$ $40$ ${\href{/padicField/43.4.0.1}{4} }^{10}$ ${\href{/padicField/47.5.0.1}{5} }^{8}$ $20^{2}$ $20^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display Deg $40$$2$$20$$20$
\(11\) Copy content Toggle raw display Deg $40$$5$$8$$32$
\(17\) Copy content Toggle raw display Deg $40$$8$$5$$35$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)