Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 35*x^18 + 146*x^17 + 455*x^16 - 2044*x^15 - 2696*x^14 + 13960*x^13 + 6764*x^12 - 48836*x^11 - 2270*x^10 + 84688*x^9 - 15516*x^8 - 65992*x^7 + 17949*x^6 + 20748*x^5 - 6419*x^4 - 1814*x^3 + 787*x^2 - 72*x + 1)
gp: K = bnfinit(y^20 - 4*y^19 - 35*y^18 + 146*y^17 + 455*y^16 - 2044*y^15 - 2696*y^14 + 13960*y^13 + 6764*y^12 - 48836*y^11 - 2270*y^10 + 84688*y^9 - 15516*y^8 - 65992*y^7 + 17949*y^6 + 20748*y^5 - 6419*y^4 - 1814*y^3 + 787*y^2 - 72*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 - 35*x^18 + 146*x^17 + 455*x^16 - 2044*x^15 - 2696*x^14 + 13960*x^13 + 6764*x^12 - 48836*x^11 - 2270*x^10 + 84688*x^9 - 15516*x^8 - 65992*x^7 + 17949*x^6 + 20748*x^5 - 6419*x^4 - 1814*x^3 + 787*x^2 - 72*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 - 35*x^18 + 146*x^17 + 455*x^16 - 2044*x^15 - 2696*x^14 + 13960*x^13 + 6764*x^12 - 48836*x^11 - 2270*x^10 + 84688*x^9 - 15516*x^8 - 65992*x^7 + 17949*x^6 + 20748*x^5 - 6419*x^4 - 1814*x^3 + 787*x^2 - 72*x + 1)
\( x^{20} - 4 x^{19} - 35 x^{18} + 146 x^{17} + 455 x^{16} - 2044 x^{15} - 2696 x^{14} + 13960 x^{13} + \cdots + 1 \)
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: |
\(1470391355634309152000000000000000\)
\(\medspace = 2^{20}\cdot 5^{15}\cdot 11^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(45.54\) | 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: | $2\cdot 5^{3/4}11^{4/5}\approx 45.53775823431064$ | ||
| 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: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(220=2^{2}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(67,·)$, $\chi_{220}(69,·)$, $\chi_{220}(9,·)$, $\chi_{220}(203,·)$, $\chi_{220}(141,·)$, $\chi_{220}(207,·)$, $\chi_{220}(81,·)$, $\chi_{220}(3,·)$, $\chi_{220}(23,·)$, $\chi_{220}(89,·)$, $\chi_{220}(27,·)$, $\chi_{220}(163,·)$, $\chi_{220}(103,·)$, $\chi_{220}(169,·)$, $\chi_{220}(47,·)$, $\chi_{220}(49,·)$, $\chi_{220}(147,·)$, $\chi_{220}(181,·)$, $\chi_{220}(201,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{884372941}a^{18}-\frac{433934556}{884372941}a^{17}+\frac{282192243}{884372941}a^{16}-\frac{394641417}{884372941}a^{15}+\frac{139771924}{884372941}a^{14}-\frac{322974840}{884372941}a^{13}+\frac{30938985}{884372941}a^{12}-\frac{151298292}{884372941}a^{11}-\frac{112545307}{884372941}a^{10}-\frac{215392746}{884372941}a^{9}+\frac{388285222}{884372941}a^{8}+\frac{64050219}{884372941}a^{7}+\frac{71148378}{884372941}a^{6}+\frac{174835562}{884372941}a^{5}+\frac{438006565}{884372941}a^{4}-\frac{263417180}{884372941}a^{3}+\frac{55954187}{884372941}a^{2}-\frac{704594}{884372941}a-\frac{82193344}{884372941}$, $\frac{1}{23\!\cdots\!39}a^{19}+\frac{132546192707036}{23\!\cdots\!39}a^{18}-\frac{23\!\cdots\!25}{23\!\cdots\!39}a^{17}+\frac{79\!\cdots\!12}{23\!\cdots\!39}a^{16}-\frac{21\!\cdots\!62}{23\!\cdots\!39}a^{15}-\frac{27\!\cdots\!64}{23\!\cdots\!39}a^{14}-\frac{88\!\cdots\!02}{23\!\cdots\!39}a^{13}-\frac{27\!\cdots\!16}{23\!\cdots\!39}a^{12}-\frac{90\!\cdots\!53}{23\!\cdots\!39}a^{11}-\frac{10\!\cdots\!56}{23\!\cdots\!39}a^{10}-\frac{72\!\cdots\!42}{23\!\cdots\!39}a^{9}-\frac{45\!\cdots\!87}{23\!\cdots\!39}a^{8}+\frac{52\!\cdots\!81}{23\!\cdots\!39}a^{7}-\frac{83\!\cdots\!56}{23\!\cdots\!39}a^{6}+\frac{41\!\cdots\!37}{23\!\cdots\!39}a^{5}-\frac{11\!\cdots\!31}{23\!\cdots\!39}a^{4}-\frac{47\!\cdots\!53}{23\!\cdots\!39}a^{3}+\frac{90\!\cdots\!98}{23\!\cdots\!39}a^{2}+\frac{21\!\cdots\!43}{23\!\cdots\!39}a-\frac{41\!\cdots\!78}{23\!\cdots\!39}$
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: |
$\frac{174929876}{884372941}a^{19}-\frac{673899290}{884372941}a^{18}-\frac{6218535724}{884372941}a^{17}+\frac{24606686920}{884372941}a^{16}+\frac{83108702336}{884372941}a^{15}-\frac{344738896726}{884372941}a^{14}-\frac{521103623880}{884372941}a^{13}+\frac{2357487850459}{884372941}a^{12}+\frac{1524396168412}{884372941}a^{11}-\frac{8266906522891}{884372941}a^{10}-\frac{1610301276772}{884372941}a^{9}+\frac{14405100955089}{884372941}a^{8}-\frac{544506565172}{884372941}a^{7}-\frac{11348805468261}{884372941}a^{6}+\frac{1341886555748}{884372941}a^{5}+\frac{3648990445211}{884372941}a^{4}-\frac{498517107844}{884372941}a^{3}-\frac{354778587694}{884372941}a^{2}+\frac{70667211302}{884372941}a-\frac{1635833215}{884372941}$, $\frac{12\!\cdots\!64}{23\!\cdots\!39}a^{19}-\frac{48\!\cdots\!20}{23\!\cdots\!39}a^{18}-\frac{44\!\cdots\!36}{23\!\cdots\!39}a^{17}+\frac{17\!\cdots\!30}{23\!\cdots\!39}a^{16}+\frac{59\!\cdots\!24}{23\!\cdots\!39}a^{15}-\frac{24\!\cdots\!24}{23\!\cdots\!39}a^{14}-\frac{37\!\cdots\!00}{23\!\cdots\!39}a^{13}+\frac{17\!\cdots\!76}{23\!\cdots\!39}a^{12}+\frac{11\!\cdots\!16}{23\!\cdots\!39}a^{11}-\frac{60\!\cdots\!68}{23\!\cdots\!39}a^{10}-\frac{12\!\cdots\!28}{23\!\cdots\!39}a^{9}+\frac{10\!\cdots\!11}{23\!\cdots\!39}a^{8}-\frac{23\!\cdots\!68}{23\!\cdots\!39}a^{7}-\frac{87\!\cdots\!34}{23\!\cdots\!39}a^{6}+\frac{92\!\cdots\!76}{23\!\cdots\!39}a^{5}+\frac{30\!\cdots\!29}{23\!\cdots\!39}a^{4}-\frac{39\!\cdots\!36}{23\!\cdots\!39}a^{3}-\frac{35\!\cdots\!32}{23\!\cdots\!39}a^{2}+\frac{57\!\cdots\!32}{23\!\cdots\!39}a+\frac{36\!\cdots\!24}{23\!\cdots\!39}$, $\frac{47\!\cdots\!04}{23\!\cdots\!39}a^{19}-\frac{18\!\cdots\!40}{23\!\cdots\!39}a^{18}-\frac{16\!\cdots\!46}{23\!\cdots\!39}a^{17}+\frac{67\!\cdots\!40}{23\!\cdots\!39}a^{16}+\frac{22\!\cdots\!48}{23\!\cdots\!39}a^{15}-\frac{94\!\cdots\!54}{23\!\cdots\!39}a^{14}-\frac{14\!\cdots\!20}{23\!\cdots\!39}a^{13}+\frac{64\!\cdots\!56}{23\!\cdots\!39}a^{12}+\frac{41\!\cdots\!16}{23\!\cdots\!39}a^{11}-\frac{22\!\cdots\!04}{23\!\cdots\!39}a^{10}-\frac{42\!\cdots\!58}{23\!\cdots\!39}a^{9}+\frac{39\!\cdots\!16}{23\!\cdots\!39}a^{8}-\frac{19\!\cdots\!88}{23\!\cdots\!39}a^{7}-\frac{31\!\cdots\!64}{23\!\cdots\!39}a^{6}+\frac{44\!\cdots\!70}{23\!\cdots\!39}a^{5}+\frac{10\!\cdots\!39}{23\!\cdots\!39}a^{4}-\frac{18\!\cdots\!86}{23\!\cdots\!39}a^{3}-\frac{10\!\cdots\!97}{23\!\cdots\!39}a^{2}+\frac{25\!\cdots\!62}{23\!\cdots\!39}a-\frac{12\!\cdots\!68}{23\!\cdots\!39}$, $\frac{31\!\cdots\!24}{23\!\cdots\!39}a^{19}-\frac{12\!\cdots\!90}{23\!\cdots\!39}a^{18}-\frac{11\!\cdots\!26}{23\!\cdots\!39}a^{17}+\frac{43\!\cdots\!20}{23\!\cdots\!39}a^{16}+\frac{15\!\cdots\!64}{23\!\cdots\!39}a^{15}-\frac{61\!\cdots\!74}{23\!\cdots\!39}a^{14}-\frac{97\!\cdots\!00}{23\!\cdots\!39}a^{13}+\frac{42\!\cdots\!41}{23\!\cdots\!39}a^{12}+\frac{29\!\cdots\!16}{23\!\cdots\!39}a^{11}-\frac{14\!\cdots\!89}{23\!\cdots\!39}a^{10}-\frac{36\!\cdots\!78}{23\!\cdots\!39}a^{9}+\frac{25\!\cdots\!71}{23\!\cdots\!39}a^{8}+\frac{11\!\cdots\!32}{23\!\cdots\!39}a^{7}-\frac{20\!\cdots\!19}{23\!\cdots\!39}a^{6}+\frac{17\!\cdots\!62}{23\!\cdots\!39}a^{5}+\frac{67\!\cdots\!89}{23\!\cdots\!39}a^{4}-\frac{81\!\cdots\!26}{23\!\cdots\!39}a^{3}-\frac{66\!\cdots\!27}{23\!\cdots\!39}a^{2}+\frac{13\!\cdots\!92}{23\!\cdots\!39}a-\frac{61\!\cdots\!84}{23\!\cdots\!39}$, $\frac{18\!\cdots\!56}{23\!\cdots\!39}a^{19}-\frac{69\!\cdots\!20}{23\!\cdots\!39}a^{18}-\frac{67\!\cdots\!74}{23\!\cdots\!39}a^{17}+\frac{25\!\cdots\!55}{23\!\cdots\!39}a^{16}+\frac{93\!\cdots\!56}{23\!\cdots\!39}a^{15}-\frac{35\!\cdots\!66}{23\!\cdots\!39}a^{14}-\frac{62\!\cdots\!60}{23\!\cdots\!39}a^{13}+\frac{24\!\cdots\!64}{23\!\cdots\!39}a^{12}+\frac{20\!\cdots\!24}{23\!\cdots\!39}a^{11}-\frac{86\!\cdots\!86}{23\!\cdots\!39}a^{10}-\frac{31\!\cdots\!22}{23\!\cdots\!39}a^{9}+\frac{15\!\cdots\!94}{23\!\cdots\!39}a^{8}+\frac{17\!\cdots\!28}{23\!\cdots\!39}a^{7}-\frac{12\!\cdots\!16}{23\!\cdots\!39}a^{6}-\frac{24\!\cdots\!46}{23\!\cdots\!39}a^{5}+\frac{43\!\cdots\!96}{23\!\cdots\!39}a^{4}-\frac{83\!\cdots\!54}{23\!\cdots\!39}a^{3}-\frac{47\!\cdots\!73}{23\!\cdots\!39}a^{2}+\frac{45\!\cdots\!78}{23\!\cdots\!39}a+\frac{26\!\cdots\!19}{23\!\cdots\!39}$, $\frac{25\!\cdots\!60}{23\!\cdots\!39}a^{19}-\frac{94\!\cdots\!95}{23\!\cdots\!39}a^{18}-\frac{90\!\cdots\!70}{23\!\cdots\!39}a^{17}+\frac{34\!\cdots\!73}{23\!\cdots\!39}a^{16}+\frac{12\!\cdots\!80}{23\!\cdots\!39}a^{15}-\frac{48\!\cdots\!25}{23\!\cdots\!39}a^{14}-\frac{79\!\cdots\!60}{23\!\cdots\!39}a^{13}+\frac{33\!\cdots\!60}{23\!\cdots\!39}a^{12}+\frac{25\!\cdots\!00}{23\!\cdots\!39}a^{11}-\frac{11\!\cdots\!60}{23\!\cdots\!39}a^{10}-\frac{34\!\cdots\!00}{23\!\cdots\!39}a^{9}+\frac{21\!\cdots\!20}{23\!\cdots\!39}a^{8}+\frac{99\!\cdots\!24}{23\!\cdots\!39}a^{7}-\frac{17\!\cdots\!06}{23\!\cdots\!39}a^{6}+\frac{87\!\cdots\!86}{23\!\cdots\!39}a^{5}+\frac{60\!\cdots\!15}{23\!\cdots\!39}a^{4}-\frac{59\!\cdots\!70}{23\!\cdots\!39}a^{3}-\frac{67\!\cdots\!25}{23\!\cdots\!39}a^{2}+\frac{12\!\cdots\!70}{23\!\cdots\!39}a-\frac{26\!\cdots\!55}{23\!\cdots\!39}$, $\frac{63\!\cdots\!42}{23\!\cdots\!39}a^{19}-\frac{23\!\cdots\!17}{23\!\cdots\!39}a^{18}-\frac{22\!\cdots\!28}{23\!\cdots\!39}a^{17}+\frac{86\!\cdots\!53}{23\!\cdots\!39}a^{16}+\frac{30\!\cdots\!04}{23\!\cdots\!39}a^{15}-\frac{12\!\cdots\!52}{23\!\cdots\!39}a^{14}-\frac{20\!\cdots\!98}{23\!\cdots\!39}a^{13}+\frac{83\!\cdots\!71}{23\!\cdots\!39}a^{12}+\frac{63\!\cdots\!80}{23\!\cdots\!39}a^{11}-\frac{29\!\cdots\!26}{23\!\cdots\!39}a^{10}-\frac{85\!\cdots\!22}{23\!\cdots\!39}a^{9}+\frac{51\!\cdots\!27}{23\!\cdots\!39}a^{8}+\frac{24\!\cdots\!68}{23\!\cdots\!39}a^{7}-\frac{41\!\cdots\!58}{23\!\cdots\!39}a^{6}+\frac{18\!\cdots\!60}{23\!\cdots\!39}a^{5}+\frac{13\!\cdots\!80}{23\!\cdots\!39}a^{4}-\frac{10\!\cdots\!32}{23\!\cdots\!39}a^{3}-\frac{13\!\cdots\!89}{23\!\cdots\!39}a^{2}+\frac{21\!\cdots\!62}{23\!\cdots\!39}a-\frac{41\!\cdots\!53}{23\!\cdots\!39}$, $\frac{15\!\cdots\!22}{23\!\cdots\!39}a^{19}-\frac{61\!\cdots\!92}{23\!\cdots\!39}a^{18}-\frac{54\!\cdots\!38}{23\!\cdots\!39}a^{17}+\frac{22\!\cdots\!26}{23\!\cdots\!39}a^{16}+\frac{72\!\cdots\!44}{23\!\cdots\!39}a^{15}-\frac{31\!\cdots\!67}{23\!\cdots\!39}a^{14}-\frac{43\!\cdots\!22}{23\!\cdots\!39}a^{13}+\frac{21\!\cdots\!50}{23\!\cdots\!39}a^{12}+\frac{11\!\cdots\!32}{23\!\cdots\!39}a^{11}-\frac{74\!\cdots\!12}{23\!\cdots\!39}a^{10}-\frac{84\!\cdots\!42}{23\!\cdots\!39}a^{9}+\frac{12\!\cdots\!47}{23\!\cdots\!39}a^{8}-\frac{15\!\cdots\!72}{23\!\cdots\!39}a^{7}-\frac{98\!\cdots\!24}{23\!\cdots\!39}a^{6}+\frac{20\!\cdots\!70}{23\!\cdots\!39}a^{5}+\frac{29\!\cdots\!44}{23\!\cdots\!39}a^{4}-\frac{72\!\cdots\!14}{23\!\cdots\!39}a^{3}-\frac{23\!\cdots\!24}{23\!\cdots\!39}a^{2}+\frac{88\!\cdots\!96}{23\!\cdots\!39}a-\frac{91\!\cdots\!05}{23\!\cdots\!39}$, $\frac{23\!\cdots\!76}{23\!\cdots\!39}a^{19}-\frac{88\!\cdots\!92}{23\!\cdots\!39}a^{18}-\frac{85\!\cdots\!80}{23\!\cdots\!39}a^{17}+\frac{32\!\cdots\!26}{23\!\cdots\!39}a^{16}+\frac{11\!\cdots\!04}{23\!\cdots\!39}a^{15}-\frac{45\!\cdots\!01}{23\!\cdots\!39}a^{14}-\frac{75\!\cdots\!92}{23\!\cdots\!39}a^{13}+\frac{31\!\cdots\!27}{23\!\cdots\!39}a^{12}+\frac{24\!\cdots\!08}{23\!\cdots\!39}a^{11}-\frac{10\!\cdots\!82}{23\!\cdots\!39}a^{10}-\frac{33\!\cdots\!40}{23\!\cdots\!39}a^{9}+\frac{19\!\cdots\!33}{23\!\cdots\!39}a^{8}+\frac{10\!\cdots\!72}{23\!\cdots\!39}a^{7}-\frac{15\!\cdots\!45}{23\!\cdots\!39}a^{6}+\frac{75\!\cdots\!30}{23\!\cdots\!39}a^{5}+\frac{48\!\cdots\!68}{23\!\cdots\!39}a^{4}-\frac{52\!\cdots\!40}{23\!\cdots\!39}a^{3}-\frac{40\!\cdots\!32}{23\!\cdots\!39}a^{2}+\frac{11\!\cdots\!90}{23\!\cdots\!39}a-\frac{97\!\cdots\!75}{23\!\cdots\!39}$, $\frac{15\!\cdots\!84}{23\!\cdots\!39}a^{19}-\frac{63\!\cdots\!70}{23\!\cdots\!39}a^{18}-\frac{54\!\cdots\!86}{23\!\cdots\!39}a^{17}+\frac{23\!\cdots\!95}{23\!\cdots\!39}a^{16}+\frac{70\!\cdots\!64}{23\!\cdots\!39}a^{15}-\frac{32\!\cdots\!64}{23\!\cdots\!39}a^{14}-\frac{40\!\cdots\!60}{23\!\cdots\!39}a^{13}+\frac{21\!\cdots\!21}{23\!\cdots\!39}a^{12}+\frac{92\!\cdots\!08}{23\!\cdots\!39}a^{11}-\frac{75\!\cdots\!35}{23\!\cdots\!39}a^{10}+\frac{87\!\cdots\!62}{23\!\cdots\!39}a^{9}+\frac{12\!\cdots\!26}{23\!\cdots\!39}a^{8}-\frac{30\!\cdots\!48}{23\!\cdots\!39}a^{7}-\frac{90\!\cdots\!69}{23\!\cdots\!39}a^{6}+\frac{29\!\cdots\!62}{23\!\cdots\!39}a^{5}+\frac{24\!\cdots\!44}{23\!\cdots\!39}a^{4}-\frac{86\!\cdots\!86}{23\!\cdots\!39}a^{3}-\frac{12\!\cdots\!21}{23\!\cdots\!39}a^{2}+\frac{89\!\cdots\!87}{23\!\cdots\!39}a-\frac{98\!\cdots\!67}{23\!\cdots\!39}$, $\frac{38\!\cdots\!28}{23\!\cdots\!39}a^{19}-\frac{14\!\cdots\!03}{23\!\cdots\!39}a^{18}-\frac{13\!\cdots\!72}{23\!\cdots\!39}a^{17}+\frac{52\!\cdots\!32}{23\!\cdots\!39}a^{16}+\frac{18\!\cdots\!42}{23\!\cdots\!39}a^{15}-\frac{73\!\cdots\!98}{23\!\cdots\!39}a^{14}-\frac{12\!\cdots\!17}{23\!\cdots\!39}a^{13}+\frac{50\!\cdots\!28}{23\!\cdots\!39}a^{12}+\frac{38\!\cdots\!27}{23\!\cdots\!39}a^{11}-\frac{17\!\cdots\!65}{23\!\cdots\!39}a^{10}-\frac{52\!\cdots\!23}{23\!\cdots\!39}a^{9}+\frac{30\!\cdots\!13}{23\!\cdots\!39}a^{8}+\frac{19\!\cdots\!57}{23\!\cdots\!39}a^{7}-\frac{23\!\cdots\!67}{23\!\cdots\!39}a^{6}+\frac{41\!\cdots\!29}{23\!\cdots\!39}a^{5}+\frac{75\!\cdots\!85}{23\!\cdots\!39}a^{4}-\frac{25\!\cdots\!02}{23\!\cdots\!39}a^{3}-\frac{67\!\cdots\!03}{23\!\cdots\!39}a^{2}+\frac{70\!\cdots\!51}{23\!\cdots\!39}a+\frac{33\!\cdots\!83}{23\!\cdots\!39}$, $\frac{32\!\cdots\!04}{23\!\cdots\!39}a^{19}-\frac{27\!\cdots\!50}{23\!\cdots\!39}a^{18}-\frac{45\!\cdots\!16}{23\!\cdots\!39}a^{17}+\frac{92\!\cdots\!35}{23\!\cdots\!39}a^{16}-\frac{10\!\cdots\!16}{23\!\cdots\!39}a^{15}-\frac{11\!\cdots\!84}{23\!\cdots\!39}a^{14}+\frac{25\!\cdots\!60}{23\!\cdots\!39}a^{13}+\frac{61\!\cdots\!01}{23\!\cdots\!39}a^{12}-\frac{21\!\cdots\!24}{23\!\cdots\!39}a^{11}-\frac{10\!\cdots\!35}{23\!\cdots\!39}a^{10}+\frac{80\!\cdots\!72}{23\!\cdots\!39}a^{9}-\frac{18\!\cdots\!34}{23\!\cdots\!39}a^{8}-\frac{14\!\cdots\!28}{23\!\cdots\!39}a^{7}+\frac{77\!\cdots\!31}{23\!\cdots\!39}a^{6}+\frac{10\!\cdots\!32}{23\!\cdots\!39}a^{5}-\frac{66\!\cdots\!36}{23\!\cdots\!39}a^{4}-\frac{34\!\cdots\!36}{23\!\cdots\!39}a^{3}+\frac{16\!\cdots\!78}{23\!\cdots\!39}a^{2}+\frac{38\!\cdots\!21}{23\!\cdots\!39}a-\frac{92\!\cdots\!27}{23\!\cdots\!39}$, $a$, $\frac{15\!\cdots\!40}{23\!\cdots\!39}a^{19}-\frac{53\!\cdots\!90}{23\!\cdots\!39}a^{18}-\frac{55\!\cdots\!90}{23\!\cdots\!39}a^{17}+\frac{19\!\cdots\!65}{23\!\cdots\!39}a^{16}+\frac{79\!\cdots\!96}{23\!\cdots\!39}a^{15}-\frac{27\!\cdots\!10}{23\!\cdots\!39}a^{14}-\frac{55\!\cdots\!40}{23\!\cdots\!39}a^{13}+\frac{19\!\cdots\!25}{23\!\cdots\!39}a^{12}+\frac{20\!\cdots\!40}{23\!\cdots\!39}a^{11}-\frac{68\!\cdots\!39}{23\!\cdots\!39}a^{10}-\frac{37\!\cdots\!70}{23\!\cdots\!39}a^{9}+\frac{12\!\cdots\!60}{23\!\cdots\!39}a^{8}+\frac{36\!\cdots\!80}{23\!\cdots\!39}a^{7}-\frac{10\!\cdots\!05}{23\!\cdots\!39}a^{6}-\frac{19\!\cdots\!78}{23\!\cdots\!39}a^{5}+\frac{38\!\cdots\!75}{23\!\cdots\!39}a^{4}+\frac{50\!\cdots\!70}{23\!\cdots\!39}a^{3}-\frac{46\!\cdots\!35}{23\!\cdots\!39}a^{2}-\frac{19\!\cdots\!99}{23\!\cdots\!39}a+\frac{69\!\cdots\!88}{23\!\cdots\!39}$, $\frac{19\!\cdots\!60}{23\!\cdots\!39}a^{19}-\frac{71\!\cdots\!70}{23\!\cdots\!39}a^{18}-\frac{68\!\cdots\!90}{23\!\cdots\!39}a^{17}+\frac{26\!\cdots\!90}{23\!\cdots\!39}a^{16}+\frac{92\!\cdots\!40}{23\!\cdots\!39}a^{15}-\frac{36\!\cdots\!50}{23\!\cdots\!39}a^{14}-\frac{59\!\cdots\!00}{23\!\cdots\!39}a^{13}+\frac{25\!\cdots\!65}{23\!\cdots\!39}a^{12}+\frac{18\!\cdots\!00}{23\!\cdots\!39}a^{11}-\frac{87\!\cdots\!21}{23\!\cdots\!39}a^{10}-\frac{23\!\cdots\!50}{23\!\cdots\!39}a^{9}+\frac{15\!\cdots\!60}{23\!\cdots\!39}a^{8}+\frac{35\!\cdots\!00}{23\!\cdots\!39}a^{7}-\frac{11\!\cdots\!85}{23\!\cdots\!39}a^{6}+\frac{84\!\cdots\!86}{23\!\cdots\!39}a^{5}+\frac{37\!\cdots\!60}{23\!\cdots\!39}a^{4}-\frac{42\!\cdots\!90}{23\!\cdots\!39}a^{3}-\frac{31\!\cdots\!95}{23\!\cdots\!39}a^{2}+\frac{83\!\cdots\!99}{23\!\cdots\!39}a-\frac{65\!\cdots\!08}{23\!\cdots\!39}$, $\frac{96\!\cdots\!14}{23\!\cdots\!39}a^{19}-\frac{36\!\cdots\!00}{23\!\cdots\!39}a^{18}-\frac{34\!\cdots\!36}{23\!\cdots\!39}a^{17}+\frac{13\!\cdots\!71}{23\!\cdots\!39}a^{16}+\frac{46\!\cdots\!24}{23\!\cdots\!39}a^{15}-\frac{18\!\cdots\!54}{23\!\cdots\!39}a^{14}-\frac{29\!\cdots\!95}{23\!\cdots\!39}a^{13}+\frac{12\!\cdots\!86}{23\!\cdots\!39}a^{12}+\frac{91\!\cdots\!23}{23\!\cdots\!39}a^{11}-\frac{45\!\cdots\!84}{23\!\cdots\!39}a^{10}-\frac{11\!\cdots\!93}{23\!\cdots\!39}a^{9}+\frac{79\!\cdots\!36}{23\!\cdots\!39}a^{8}+\frac{75\!\cdots\!57}{23\!\cdots\!39}a^{7}-\frac{63\!\cdots\!42}{23\!\cdots\!39}a^{6}+\frac{48\!\cdots\!98}{23\!\cdots\!39}a^{5}+\frac{20\!\cdots\!94}{23\!\cdots\!39}a^{4}-\frac{21\!\cdots\!06}{23\!\cdots\!39}a^{3}-\frac{20\!\cdots\!36}{23\!\cdots\!39}a^{2}+\frac{36\!\cdots\!13}{23\!\cdots\!39}a-\frac{12\!\cdots\!48}{23\!\cdots\!39}$, $\frac{11\!\cdots\!20}{23\!\cdots\!39}a^{19}-\frac{47\!\cdots\!08}{23\!\cdots\!39}a^{18}-\frac{38\!\cdots\!89}{23\!\cdots\!39}a^{17}+\frac{17\!\cdots\!76}{23\!\cdots\!39}a^{16}+\frac{47\!\cdots\!96}{23\!\cdots\!39}a^{15}-\frac{24\!\cdots\!59}{23\!\cdots\!39}a^{14}-\frac{25\!\cdots\!16}{23\!\cdots\!39}a^{13}+\frac{16\!\cdots\!62}{23\!\cdots\!39}a^{12}+\frac{43\!\cdots\!48}{23\!\cdots\!39}a^{11}-\frac{57\!\cdots\!52}{23\!\cdots\!39}a^{10}+\frac{82\!\cdots\!99}{23\!\cdots\!39}a^{9}+\frac{98\!\cdots\!57}{23\!\cdots\!39}a^{8}-\frac{34\!\cdots\!75}{23\!\cdots\!39}a^{7}-\frac{76\!\cdots\!67}{23\!\cdots\!39}a^{6}+\frac{31\!\cdots\!81}{23\!\cdots\!39}a^{5}+\frac{24\!\cdots\!71}{23\!\cdots\!39}a^{4}-\frac{91\!\cdots\!43}{23\!\cdots\!39}a^{3}-\frac{28\!\cdots\!46}{23\!\cdots\!39}a^{2}+\frac{75\!\cdots\!43}{23\!\cdots\!39}a-\frac{28\!\cdots\!46}{23\!\cdots\!39}$, $\frac{16\!\cdots\!02}{23\!\cdots\!39}a^{19}-\frac{66\!\cdots\!74}{23\!\cdots\!39}a^{18}-\frac{59\!\cdots\!94}{23\!\cdots\!39}a^{17}+\frac{24\!\cdots\!86}{23\!\cdots\!39}a^{16}+\frac{77\!\cdots\!21}{23\!\cdots\!39}a^{15}-\frac{33\!\cdots\!36}{23\!\cdots\!39}a^{14}-\frac{46\!\cdots\!15}{23\!\cdots\!39}a^{13}+\frac{22\!\cdots\!28}{23\!\cdots\!39}a^{12}+\frac{12\!\cdots\!38}{23\!\cdots\!39}a^{11}-\frac{79\!\cdots\!38}{23\!\cdots\!39}a^{10}-\frac{77\!\cdots\!11}{23\!\cdots\!39}a^{9}+\frac{13\!\cdots\!76}{23\!\cdots\!39}a^{8}-\frac{16\!\cdots\!77}{23\!\cdots\!39}a^{7}-\frac{10\!\cdots\!87}{23\!\cdots\!39}a^{6}+\frac{19\!\cdots\!59}{23\!\cdots\!39}a^{5}+\frac{29\!\cdots\!71}{23\!\cdots\!39}a^{4}-\frac{53\!\cdots\!59}{23\!\cdots\!39}a^{3}-\frac{25\!\cdots\!07}{23\!\cdots\!39}a^{2}+\frac{51\!\cdots\!26}{23\!\cdots\!39}a-\frac{12\!\cdots\!92}{23\!\cdots\!39}$, $\frac{92\!\cdots\!61}{23\!\cdots\!39}a^{19}-\frac{29\!\cdots\!44}{23\!\cdots\!39}a^{18}-\frac{35\!\cdots\!19}{23\!\cdots\!39}a^{17}+\frac{10\!\cdots\!59}{23\!\cdots\!39}a^{16}+\frac{51\!\cdots\!28}{23\!\cdots\!39}a^{15}-\frac{15\!\cdots\!11}{23\!\cdots\!39}a^{14}-\frac{38\!\cdots\!49}{23\!\cdots\!39}a^{13}+\frac{10\!\cdots\!16}{23\!\cdots\!39}a^{12}+\frac{15\!\cdots\!21}{23\!\cdots\!39}a^{11}-\frac{36\!\cdots\!29}{23\!\cdots\!39}a^{10}-\frac{33\!\cdots\!19}{23\!\cdots\!39}a^{9}+\frac{63\!\cdots\!22}{23\!\cdots\!39}a^{8}+\frac{40\!\cdots\!58}{23\!\cdots\!39}a^{7}-\frac{49\!\cdots\!59}{23\!\cdots\!39}a^{6}-\frac{25\!\cdots\!34}{23\!\cdots\!39}a^{5}+\frac{13\!\cdots\!45}{23\!\cdots\!39}a^{4}+\frac{63\!\cdots\!39}{23\!\cdots\!39}a^{3}-\frac{57\!\cdots\!93}{23\!\cdots\!39}a^{2}-\frac{16\!\cdots\!68}{23\!\cdots\!39}a-\frac{44\!\cdots\!96}{23\!\cdots\!39}$
| 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: | \( 8953413920.01 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 8953413920.01 \cdot 1}{2\cdot\sqrt{1470391355634309152000000000000000}}\cr\approx \mathstrut & 0.122417106397 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 35*x^18 + 146*x^17 + 455*x^16 - 2044*x^15 - 2696*x^14 + 13960*x^13 + 6764*x^12 - 48836*x^11 - 2270*x^10 + 84688*x^9 - 15516*x^8 - 65992*x^7 + 17949*x^6 + 20748*x^5 - 6419*x^4 - 1814*x^3 + 787*x^2 - 72*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^20 - 4*x^19 - 35*x^18 + 146*x^17 + 455*x^16 - 2044*x^15 - 2696*x^14 + 13960*x^13 + 6764*x^12 - 48836*x^11 - 2270*x^10 + 84688*x^9 - 15516*x^8 - 65992*x^7 + 17949*x^6 + 20748*x^5 - 6419*x^4 - 1814*x^3 + 787*x^2 - 72*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^20 - 4*x^19 - 35*x^18 + 146*x^17 + 455*x^16 - 2044*x^15 - 2696*x^14 + 13960*x^13 + 6764*x^12 - 48836*x^11 - 2270*x^10 + 84688*x^9 - 15516*x^8 - 65992*x^7 + 17949*x^6 + 20748*x^5 - 6419*x^4 - 1814*x^3 + 787*x^2 - 72*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^20 - 4*x^19 - 35*x^18 + 146*x^17 + 455*x^16 - 2044*x^15 - 2696*x^14 + 13960*x^13 + 6764*x^12 - 48836*x^11 - 2270*x^10 + 84688*x^9 - 15516*x^8 - 65992*x^7 + 17949*x^6 + 20748*x^5 - 6419*x^4 - 1814*x^3 + 787*x^2 - 72*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
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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1 |
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 | $20$ | R | $20$ | R | $20$ | $20$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 $20$ | $2$ | $10$ | $20$ | |||
|
\(5\)
| Deg $20$ | $4$ | $5$ | $15$ | |||
|
\(11\)
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |