Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 60*x^18 + 1530*x^16 - 31*x^15 - 21600*x^14 + 1395*x^13 + 184275*x^12 - 25110*x^11 - 972254*x^10 + 230175*x^9 + 3105870*x^8 - 1129950*x^7 - 5547510*x^6 + 2832749*x^5 + 4443525*x^4 - 2942985*x^3 - 514350*x^2 + 354330*x + 51151)
gp: K = bnfinit(y^20 - 60*y^18 + 1530*y^16 - 31*y^15 - 21600*y^14 + 1395*y^13 + 184275*y^12 - 25110*y^11 - 972254*y^10 + 230175*y^9 + 3105870*y^8 - 1129950*y^7 - 5547510*y^6 + 2832749*y^5 + 4443525*y^4 - 2942985*y^3 - 514350*y^2 + 354330*y + 51151, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 60*x^18 + 1530*x^16 - 31*x^15 - 21600*x^14 + 1395*x^13 + 184275*x^12 - 25110*x^11 - 972254*x^10 + 230175*x^9 + 3105870*x^8 - 1129950*x^7 - 5547510*x^6 + 2832749*x^5 + 4443525*x^4 - 2942985*x^3 - 514350*x^2 + 354330*x + 51151);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 60*x^18 + 1530*x^16 - 31*x^15 - 21600*x^14 + 1395*x^13 + 184275*x^12 - 25110*x^11 - 972254*x^10 + 230175*x^9 + 3105870*x^8 - 1129950*x^7 - 5547510*x^6 + 2832749*x^5 + 4443525*x^4 - 2942985*x^3 - 514350*x^2 + 354330*x + 51151)
\( x^{20} - 60 x^{18} + 1530 x^{16} - 31 x^{15} - 21600 x^{14} + 1395 x^{13} + 184275 x^{12} - 25110 x^{11} + \cdots + 51151 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[20, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(75487840807181783020496368408203125\)
\(\medspace = 5^{35}\cdot 11^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(55.45\) | 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: | $5^{7/4}11^{1/2}\approx 55.449016844865795$ | ||
| Ramified primes: |
\(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(275=5^{2}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{275}(1,·)$, $\chi_{275}(197,·)$, $\chi_{275}(199,·)$, $\chi_{275}(34,·)$, $\chi_{275}(142,·)$, $\chi_{275}(144,·)$, $\chi_{275}(87,·)$, $\chi_{275}(153,·)$, $\chi_{275}(221,·)$, $\chi_{275}(32,·)$, $\chi_{275}(208,·)$, $\chi_{275}(98,·)$, $\chi_{275}(166,·)$, $\chi_{275}(43,·)$, $\chi_{275}(111,·)$, $\chi_{275}(89,·)$, $\chi_{275}(56,·)$, $\chi_{275}(252,·)$, $\chi_{275}(254,·)$, $\chi_{275}(263,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{282001}a^{13}-\frac{68734}{282001}a^{12}-\frac{39}{282001}a^{11}-\frac{63585}{282001}a^{10}+\frac{585}{282001}a^{9}-\frac{128606}{282001}a^{8}-\frac{4212}{282001}a^{7}+\frac{16879}{282001}a^{6}+\frac{14742}{282001}a^{5}+\frac{5403}{282001}a^{4}-\frac{22113}{282001}a^{3}+\frac{58900}{282001}a^{2}+\frac{9477}{282001}a-\frac{103817}{282001}$, $\frac{1}{282001}a^{14}-\frac{42}{282001}a^{12}+\frac{75799}{282001}a^{11}+\frac{693}{282001}a^{10}+\frac{36642}{282001}a^{9}-\frac{5670}{282001}a^{8}+\frac{124298}{282001}a^{7}+\frac{23814}{282001}a^{6}+\frac{52438}{282001}a^{5}-\frac{47628}{282001}a^{4}+\frac{129348}{282001}a^{3}+\frac{35721}{282001}a^{2}-\frac{134009}{282001}a-\frac{4374}{282001}$, $\frac{1}{282001}a^{15}+\frac{8981}{282001}a^{12}-\frac{945}{282001}a^{11}-\frac{95919}{282001}a^{10}+\frac{18900}{282001}a^{9}+\frac{80865}{282001}a^{8}+\frac{128911}{282001}a^{7}-\frac{84647}{282001}a^{6}+\frac{7534}{282001}a^{5}+\frac{74273}{282001}a^{4}-\frac{47022}{282001}a^{3}+\frac{83783}{282001}a^{2}+\frac{111659}{282001}a-\frac{130299}{282001}$, $\frac{1}{282001}a^{16}-\frac{1080}{282001}a^{12}-\frac{27661}{282001}a^{11}+\frac{23760}{282001}a^{10}-\frac{97002}{282001}a^{9}+\frac{63301}{282001}a^{8}-\frac{44809}{282001}a^{7}+\frac{133773}{282001}a^{6}-\frac{65160}{282001}a^{5}-\frac{67193}{282001}a^{4}-\frac{130069}{282001}a^{3}-\frac{117366}{282001}a^{2}-\frac{78934}{282001}a+\frac{85171}{282001}$, $\frac{1}{282001}a^{17}-\frac{94118}{282001}a^{12}-\frac{18360}{282001}a^{11}+\frac{39442}{282001}a^{10}+\frac{131099}{282001}a^{9}+\frac{87204}{282001}a^{8}+\frac{96829}{282001}a^{7}+\frac{116096}{282001}a^{6}+\frac{62111}{282001}a^{5}+\frac{65151}{282001}a^{4}-\frac{29321}{282001}a^{3}+\frac{82841}{282001}a^{2}-\frac{113706}{282001}a+\frac{114038}{282001}$, $\frac{1}{282001}a^{18}-\frac{22032}{282001}a^{12}+\frac{34853}{282001}a^{11}-\frac{18710}{282001}a^{10}-\frac{125962}{282001}a^{9}+\frac{4243}{282001}a^{8}-\frac{97515}{282001}a^{7}-\frac{113801}{282001}a^{6}+\frac{107787}{282001}a^{5}+\frac{42430}{282001}a^{4}+\frac{18887}{282001}a^{3}-\frac{139164}{282001}a^{2}+\frac{101161}{282001}a+\frac{4243}{282001}$, $\frac{1}{282001}a^{19}+\frac{32735}{282001}a^{12}-\frac{31955}{282001}a^{11}-\frac{49714}{282001}a^{10}-\frac{79083}{282001}a^{9}+\frac{1141}{282001}a^{8}-\frac{134256}{282001}a^{7}+\frac{26596}{282001}a^{6}-\frac{26978}{282001}a^{5}+\frac{53361}{282001}a^{4}-\frac{35052}{282001}a^{3}+\frac{17359}{282001}a^{2}+\frac{120767}{282001}a+\frac{13967}{282001}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| 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: | $19$ | 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: |
$\frac{31}{282001}a^{15}-\frac{1395}{282001}a^{13}+\frac{25110}{282001}a^{11}-\frac{718}{282001}a^{10}-\frac{230175}{282001}a^{9}+\frac{21540}{282001}a^{8}+\frac{1129950}{282001}a^{7}-\frac{226170}{282001}a^{6}-\frac{2847474}{282001}a^{5}+\frac{969300}{282001}a^{4}+\frac{3163860}{282001}a^{3}-\frac{1453950}{282001}a^{2}-\frac{1016955}{282001}a+\frac{66947}{282001}$, $\frac{16}{282001}a^{19}-\frac{912}{282001}a^{17}+\frac{21888}{282001}a^{15}-\frac{287280}{282001}a^{13}+\frac{2240784}{282001}a^{11}-\frac{10563696}{282001}a^{9}+\frac{29253312}{282001}a^{7}+\frac{15467}{282001}a^{6}-\frac{43879968}{282001}a^{5}-\frac{278406}{282001}a^{4}+\frac{29918160}{282001}a^{3}+\frac{1252827}{282001}a^{2}-\frac{5983632}{282001}a-\frac{835218}{282001}$, $\frac{16}{282001}a^{19}-\frac{877}{282001}a^{17}+\frac{20134}{282001}a^{15}-\frac{253}{282001}a^{14}-\frac{251350}{282001}a^{13}+\frac{10027}{282001}a^{12}+\frac{1853365}{282001}a^{11}-\frac{154483}{282001}a^{10}-\frac{8201139}{282001}a^{9}+\frac{1166737}{282001}a^{8}+\frac{21087288}{282001}a^{7}-\frac{4467493}{282001}a^{6}-\frac{28625211}{282001}a^{5}+\frac{7970463}{282001}a^{4}+\frac{16197975}{282001}a^{3}-\frac{4776516}{282001}a^{2}-\frac{1728945}{282001}a+\frac{38772}{282001}$, $\frac{15}{282001}a^{19}-\frac{855}{282001}a^{17}-\frac{155}{282001}a^{16}+\frac{20520}{282001}a^{15}+\frac{6975}{282001}a^{14}-\frac{269325}{282001}a^{13}-\frac{125550}{282001}a^{12}+\frac{2104325}{282001}a^{11}+\frac{1150875}{282001}a^{10}-\frac{10011165}{282001}a^{9}-\frac{5649750}{282001}a^{8}+\frac{28555830}{282001}a^{7}+\frac{14163745}{282001}a^{6}-\frac{45983970}{282001}a^{5}-\frac{14776051}{282001}a^{4}+\frac{35318025}{282001}a^{3}+\frac{2505162}{282001}a^{2}-\frac{5944390}{282001}a-\frac{1100268}{282001}$, $\frac{31}{282001}a^{18}-\frac{27}{282001}a^{17}-\frac{1674}{282001}a^{16}+\frac{1377}{282001}a^{15}+\frac{37665}{282001}a^{14}-\frac{29635}{282001}a^{13}-\frac{456165}{282001}a^{12}+\frac{350220}{282001}a^{11}+\frac{3201525}{282001}a^{10}-\frac{2464875}{282001}a^{9}-\frac{13002299}{282001}a^{8}+\frac{10366272}{282001}a^{7}+\frac{28437726}{282001}a^{6}-\frac{24231312}{282001}a^{5}-\frac{26841195}{282001}a^{4}+\frac{25198493}{282001}a^{3}+\frac{3355397}{282001}a^{2}-\frac{3614022}{282001}a-\frac{998562}{282001}$, $\frac{160}{282001}a^{13}+\frac{599}{282001}a^{12}-\frac{6240}{282001}a^{11}-\frac{21564}{282001}a^{10}+\frac{93600}{282001}a^{9}+\frac{291114}{282001}a^{8}-\frac{673920}{282001}a^{7}-\frac{1811376}{282001}a^{6}+\frac{2358720}{282001}a^{5}+\frac{5094495}{282001}a^{4}-\frac{3538080}{282001}a^{3}-\frac{5240052}{282001}a^{2}+\frac{1516320}{282001}a+\frac{873342}{282001}$, $\frac{15}{282001}a^{19}-\frac{18}{282001}a^{18}-\frac{793}{282001}a^{17}+\frac{891}{282001}a^{16}+\frac{17389}{282001}a^{15}-\frac{18447}{282001}a^{14}-\frac{203760}{282001}a^{13}+\frac{207634}{282001}a^{12}+\frac{1367765}{282001}a^{11}-\frac{1377889}{282001}a^{10}-\frac{5215385}{282001}a^{9}+\frac{5411478}{282001}a^{8}+\frac{10354097}{282001}a^{7}-\frac{11680781}{282001}a^{6}-\frac{7991421}{282001}a^{5}+\frac{11037245}{282001}a^{4}-\frac{729194}{282001}a^{3}-\frac{1291064}{282001}a^{2}+\frac{21251}{282001}a+\frac{124932}{282001}$, $\frac{35}{282001}a^{17}-\frac{1785}{282001}a^{15}+\frac{37485}{282001}a^{13}-\frac{417690}{282001}a^{11}+\frac{2650725}{282001}a^{9}+\frac{1801}{282001}a^{8}-\frac{9542610}{282001}a^{7}-\frac{43224}{282001}a^{6}+\frac{18217710}{282001}a^{5}+\frac{324180}{282001}a^{4}-\frac{15615180}{282001}a^{3}-\frac{778032}{282001}a^{2}+\frac{3903795}{282001}a+\frac{291762}{282001}$, $\frac{253}{282001}a^{14}-\frac{10626}{282001}a^{12}+\frac{1079}{282001}a^{11}+\frac{175329}{282001}a^{10}-\frac{35607}{282001}a^{9}-\frac{1434510}{282001}a^{8}+\frac{427284}{282001}a^{7}+\frac{6024942}{282001}a^{6}-\frac{2243241}{282001}a^{5}-\frac{12049884}{282001}a^{4}+\frac{4806945}{282001}a^{3}+\frac{9037413}{282001}a^{2}-\frac{2884167}{282001}a-\frac{1106622}{282001}$, $\frac{15}{282001}a^{19}+\frac{13}{282001}a^{18}-\frac{855}{282001}a^{17}-\frac{857}{282001}a^{16}+\frac{20551}{282001}a^{15}+\frac{22677}{282001}a^{14}-\frac{270720}{282001}a^{13}-\frac{313290}{282001}a^{12}+\frac{2130153}{282001}a^{11}+\frac{2440919}{282001}a^{10}-\frac{10265034}{282001}a^{9}-\frac{10730238}{282001}a^{8}+\frac{29980172}{282001}a^{7}+\frac{24857995}{282001}a^{6}-\frac{50535510}{282001}a^{5}-\frac{24730087}{282001}a^{4}+\frac{42948639}{282001}a^{3}+\frac{4637892}{282001}a^{2}-\frac{10782655}{282001}a-\frac{856296}{282001}$, $\frac{13}{282001}a^{19}-\frac{9}{282001}a^{18}-\frac{710}{282001}a^{17}+\frac{560}{282001}a^{16}+\frac{16203}{282001}a^{15}-\frac{14487}{282001}a^{14}-\frac{199935}{282001}a^{13}+\frac{201224}{282001}a^{12}+\frac{1439802}{282001}a^{11}-\frac{1615671}{282001}a^{10}-\frac{6068048}{282001}a^{9}+\frac{7497864}{282001}a^{8}+\frac{14049082}{282001}a^{7}-\frac{18820938}{282001}a^{6}-\frac{14751930}{282001}a^{5}+\frac{20941470}{282001}a^{4}+\frac{3242191}{282001}a^{3}-\frac{3904402}{282001}a^{2}-\frac{1197810}{282001}a-\frac{142726}{282001}$, $\frac{17}{282001}a^{18}+\frac{27}{282001}a^{17}-\frac{739}{282001}a^{16}-\frac{1377}{282001}a^{15}+\frac{12063}{282001}a^{14}+\frac{29635}{282001}a^{13}-\frac{83907}{282001}a^{12}-\frac{350220}{282001}a^{11}+\frac{101115}{282001}a^{10}+\frac{2470631}{282001}a^{9}+\frac{1786391}{282001}a^{8}-\frac{10506217}{282001}a^{7}-\frac{9168798}{282001}a^{6}+\frac{25305213}{282001}a^{5}+\frac{13999131}{282001}a^{4}-\frac{27912011}{282001}a^{3}-\frac{2900501}{282001}a^{2}+\frac{4886883}{282001}a+\frac{893830}{282001}$, $\frac{19}{282001}a^{19}+\frac{13}{282001}a^{18}-\frac{1083}{282001}a^{17}-\frac{702}{282001}a^{16}+\frac{25992}{282001}a^{15}+\frac{15795}{282001}a^{14}-\frac{340546}{282001}a^{13}-\frac{191166}{282001}a^{12}+\frac{2637570}{282001}a^{11}+\frac{1337931}{282001}a^{10}-\frac{12193974}{282001}a^{9}-\frac{5396058}{282001}a^{8}+\frac{32225384}{282001}a^{7}+\frac{11684344}{282001}a^{6}-\frac{43488348}{282001}a^{5}-\frac{11283696}{282001}a^{4}+\frac{23550192}{282001}a^{3}+\frac{2769795}{282001}a^{2}-\frac{3330936}{282001}a-\frac{415988}{282001}$, $\frac{17}{282001}a^{18}+\frac{27}{282001}a^{17}-\frac{844}{282001}a^{16}-\frac{1346}{282001}a^{15}+\frac{17103}{282001}a^{14}+\frac{28240}{282001}a^{13}-\frac{182187}{282001}a^{12}-\frac{325110}{282001}a^{11}+\frac{1098317}{282001}a^{10}+\frac{2238655}{282001}a^{9}-\frac{3805369}{282001}a^{8}-\frac{9327640}{282001}a^{7}+\frac{7751112}{282001}a^{6}+\frac{22020096}{282001}a^{5}-\frac{10750689}{282001}a^{4}-\frac{23289341}{282001}a^{3}+\frac{10342189}{282001}a^{2}+\frac{2556999}{282001}a+\frac{146969}{282001}$, $\frac{45}{282001}a^{18}-\frac{120}{282001}a^{17}-\frac{2430}{282001}a^{16}+\frac{6120}{282001}a^{15}+\frac{54262}{282001}a^{14}-\frac{129915}{282001}a^{13}-\frac{642454}{282001}a^{12}+\frac{1483609}{282001}a^{11}+\frac{4275666}{282001}a^{10}-\frac{9809367}{282001}a^{9}-\frac{15365609}{282001}a^{8}+\frac{37380739}{282001}a^{7}+\frac{23957724}{282001}a^{6}-\frac{75500481}{282001}a^{5}+\frac{3199194}{282001}a^{4}+\frac{61450062}{282001}a^{3}-\frac{35569195}{282001}a^{2}+\frac{2611845}{282001}a+\frac{1771162}{282001}$, $\frac{15}{282001}a^{19}+\frac{13}{282001}a^{18}-\frac{855}{282001}a^{17}-\frac{783}{282001}a^{16}+\frac{20489}{282001}a^{15}+\frac{18965}{282001}a^{14}-\frac{267930}{282001}a^{13}-\frac{237306}{282001}a^{12}+\frac{2078136}{282001}a^{11}+\frac{1628179}{282001}a^{10}-\frac{9741428}{282001}a^{9}-\frac{5910078}{282001}a^{8}+\frac{26901875}{282001}a^{7}+\frac{9416191}{282001}a^{6}-\frac{40143534}{282001}a^{5}-\frac{922351}{282001}a^{4}+\frac{25411734}{282001}a^{3}-\frac{8527200}{282001}a^{2}-\frac{780168}{282001}a+\frac{398677}{282001}$, $\frac{44}{282001}a^{18}+\frac{8}{282001}a^{17}-\frac{2302}{282001}a^{16}-\frac{155}{282001}a^{15}+\frac{50161}{282001}a^{14}-\frac{2776}{282001}a^{13}-\frac{588094}{282001}a^{12}+\frac{108938}{282001}a^{11}+\frac{3993162}{282001}a^{10}-\frac{1280312}{282001}a^{9}-\frac{15681022}{282001}a^{8}+\frac{7179167}{282001}a^{7}+\frac{33227421}{282001}a^{6}-\frac{19557006}{282001}a^{5}-\frac{30986838}{282001}a^{4}+\frac{21492185}{282001}a^{3}+\frac{4999711}{282001}a^{2}-\frac{3001918}{282001}a-\frac{508149}{282001}$, $\frac{31}{282001}a^{19}-\frac{1767}{282001}a^{17}-\frac{81}{282001}a^{16}+\frac{42408}{282001}a^{15}+\frac{3423}{282001}a^{14}-\frac{557044}{282001}a^{13}-\frac{56167}{282001}a^{12}+\frac{4362230}{282001}a^{11}+\frac{443295}{282001}a^{10}-\frac{20827721}{282001}a^{9}-\frac{1635876}{282001}a^{8}+\frac{59551425}{282001}a^{7}+\frac{1735452}{282001}a^{6}-\frac{95374611}{282001}a^{5}+\frac{4083494}{282001}a^{4}+\frac{71740242}{282001}a^{3}-\frac{7640655}{282001}a^{2}-\frac{12923229}{282001}a-\frac{1354958}{282001}$, $\frac{31}{282001}a^{15}-\frac{253}{282001}a^{14}-\frac{1555}{282001}a^{13}+\frac{10027}{282001}a^{12}+\frac{30271}{282001}a^{11}-\frac{154483}{282001}a^{10}-\frac{288168}{282001}a^{9}+\frac{1164936}{282001}a^{8}+\frac{1376586}{282001}a^{7}-\frac{4439736}{282001}a^{6}-\frac{2962953}{282001}a^{5}+\frac{7924689}{282001}a^{4}+\frac{1894995}{282001}a^{3}-\frac{5251311}{282001}a^{2}+\frac{632893}{282001}a+\frac{582228}{282001}$
| 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: | \( 73895548051.9 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 73895548051.9 \cdot 1}{2\cdot\sqrt{75487840807181783020496368408203125}}\cr\approx \mathstrut & 0.141009927859 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 60*x^18 + 1530*x^16 - 31*x^15 - 21600*x^14 + 1395*x^13 + 184275*x^12 - 25110*x^11 - 972254*x^10 + 230175*x^9 + 3105870*x^8 - 1129950*x^7 - 5547510*x^6 + 2832749*x^5 + 4443525*x^4 - 2942985*x^3 - 514350*x^2 + 354330*x + 51151)
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^20 - 60*x^18 + 1530*x^16 - 31*x^15 - 21600*x^14 + 1395*x^13 + 184275*x^12 - 25110*x^11 - 972254*x^10 + 230175*x^9 + 3105870*x^8 - 1129950*x^7 - 5547510*x^6 + 2832749*x^5 + 4443525*x^4 - 2942985*x^3 - 514350*x^2 + 354330*x + 51151, 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^20 - 60*x^18 + 1530*x^16 - 31*x^15 - 21600*x^14 + 1395*x^13 + 184275*x^12 - 25110*x^11 - 972254*x^10 + 230175*x^9 + 3105870*x^8 - 1129950*x^7 - 5547510*x^6 + 2832749*x^5 + 4443525*x^4 - 2942985*x^3 - 514350*x^2 + 354330*x + 51151);
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^20 - 60*x^18 + 1530*x^16 - 31*x^15 - 21600*x^14 + 1395*x^13 + 184275*x^12 - 25110*x^11 - 972254*x^10 + 230175*x^9 + 3105870*x^8 - 1129950*x^7 - 5547510*x^6 + 2832749*x^5 + 4443525*x^4 - 2942985*x^3 - 514350*x^2 + 354330*x + 51151);
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
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)
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.15125.1, 5.5.390625.1, \(\Q(\zeta_{25})^+\) |
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 | $20$ | $20$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | R | $20$ | $20$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $20$ | $20$ | $1$ | $35$ | |||
|
\(11\)
| 11.10.5.2 | $x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 11.10.5.2 | $x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.55.4t1.a.a | $1$ | $ 5 \cdot 11 $ | 4.4.15125.1 | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.55.4t1.a.b | $1$ | $ 5 \cdot 11 $ | 4.4.15125.1 | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.25.5t1.a.d | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.275.20t1.a.e | $1$ | $ 5^{2} \cdot 11 $ | 20.20.75487840807181783020496368408203125.1 | $C_{20}$ (as 20T1) | $0$ | $1$ |
| * | 1.25.10t1.a.a | $1$ | $ 5^{2}$ | \(\Q(\zeta_{25})^+\) | $C_{10}$ (as 10T1) | $0$ | $1$ |
| * | 1.275.20t1.a.d | $1$ | $ 5^{2} \cdot 11 $ | 20.20.75487840807181783020496368408203125.1 | $C_{20}$ (as 20T1) | $0$ | $1$ |
| * | 1.25.5t1.a.c | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.275.20t1.a.a | $1$ | $ 5^{2} \cdot 11 $ | 20.20.75487840807181783020496368408203125.1 | $C_{20}$ (as 20T1) | $0$ | $1$ |
| * | 1.25.10t1.a.b | $1$ | $ 5^{2}$ | \(\Q(\zeta_{25})^+\) | $C_{10}$ (as 10T1) | $0$ | $1$ |
| * | 1.275.20t1.a.h | $1$ | $ 5^{2} \cdot 11 $ | 20.20.75487840807181783020496368408203125.1 | $C_{20}$ (as 20T1) | $0$ | $1$ |
| * | 1.25.5t1.a.b | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.275.20t1.a.g | $1$ | $ 5^{2} \cdot 11 $ | 20.20.75487840807181783020496368408203125.1 | $C_{20}$ (as 20T1) | $0$ | $1$ |
| * | 1.25.10t1.a.c | $1$ | $ 5^{2}$ | \(\Q(\zeta_{25})^+\) | $C_{10}$ (as 10T1) | $0$ | $1$ |
| * | 1.275.20t1.a.b | $1$ | $ 5^{2} \cdot 11 $ | 20.20.75487840807181783020496368408203125.1 | $C_{20}$ (as 20T1) | $0$ | $1$ |
| * | 1.25.5t1.a.a | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.275.20t1.a.c | $1$ | $ 5^{2} \cdot 11 $ | 20.20.75487840807181783020496368408203125.1 | $C_{20}$ (as 20T1) | $0$ | $1$ |
| * | 1.25.10t1.a.d | $1$ | $ 5^{2}$ | \(\Q(\zeta_{25})^+\) | $C_{10}$ (as 10T1) | $0$ | $1$ |
| * | 1.275.20t1.a.f | $1$ | $ 5^{2} \cdot 11 $ | 20.20.75487840807181783020496368408203125.1 | $C_{20}$ (as 20T1) | $0$ | $1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.