Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 64*x^18 + 78*x^17 + 1609*x^16 - 2224*x^15 - 20399*x^14 + 30774*x^13 + 139797*x^12 - 228288*x^11 - 510719*x^10 + 929822*x^9 + 878982*x^8 - 2003579*x^7 - 355510*x^6 + 2013497*x^5 - 574064*x^4 - 658628*x^3 + 441152*x^2 - 86946*x + 4621)
gp: K = bnfinit(y^20 - y^19 - 64*y^18 + 78*y^17 + 1609*y^16 - 2224*y^15 - 20399*y^14 + 30774*y^13 + 139797*y^12 - 228288*y^11 - 510719*y^10 + 929822*y^9 + 878982*y^8 - 2003579*y^7 - 355510*y^6 + 2013497*y^5 - 574064*y^4 - 658628*y^3 + 441152*y^2 - 86946*y + 4621, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 - 64*x^18 + 78*x^17 + 1609*x^16 - 2224*x^15 - 20399*x^14 + 30774*x^13 + 139797*x^12 - 228288*x^11 - 510719*x^10 + 929822*x^9 + 878982*x^8 - 2003579*x^7 - 355510*x^6 + 2013497*x^5 - 574064*x^4 - 658628*x^3 + 441152*x^2 - 86946*x + 4621);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 - 64*x^18 + 78*x^17 + 1609*x^16 - 2224*x^15 - 20399*x^14 + 30774*x^13 + 139797*x^12 - 228288*x^11 - 510719*x^10 + 929822*x^9 + 878982*x^8 - 2003579*x^7 - 355510*x^6 + 2013497*x^5 - 574064*x^4 - 658628*x^3 + 441152*x^2 - 86946*x + 4621)
\( x^{20} - x^{19} - 64 x^{18} + 78 x^{17} + 1609 x^{16} - 2224 x^{15} - 20399 x^{14} + 30774 x^{13} + \cdots + 4621 \)
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: |
\(396107830343483954099825714111328125\)
\(\medspace = 5^{15}\cdot 7^{10}\cdot 11^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(60.24\) | 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: | $5^{3/4}7^{1/2}11^{4/5}\approx 60.24079177568486$ | ||
| Ramified primes: |
\(5\), \(7\), \(11\)
| 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: | \(385=5\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{385}(64,·)$, $\chi_{385}(1,·)$, $\chi_{385}(258,·)$, $\chi_{385}(71,·)$, $\chi_{385}(328,·)$, $\chi_{385}(202,·)$, $\chi_{385}(141,·)$, $\chi_{385}(342,·)$, $\chi_{385}(344,·)$, $\chi_{385}(27,·)$, $\chi_{385}(223,·)$, $\chi_{385}(97,·)$, $\chi_{385}(36,·)$, $\chi_{385}(169,·)$, $\chi_{385}(48,·)$, $\chi_{385}(309,·)$, $\chi_{385}(246,·)$, $\chi_{385}(377,·)$, $\chi_{385}(379,·)$, $\chi_{385}(188,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{53\!\cdots\!49}a^{19}-\frac{52\!\cdots\!65}{53\!\cdots\!49}a^{18}+\frac{11\!\cdots\!02}{53\!\cdots\!49}a^{17}-\frac{22\!\cdots\!64}{53\!\cdots\!49}a^{16}+\frac{44\!\cdots\!21}{53\!\cdots\!49}a^{15}+\frac{31\!\cdots\!50}{53\!\cdots\!49}a^{14}-\frac{14\!\cdots\!96}{53\!\cdots\!49}a^{13}-\frac{57\!\cdots\!42}{53\!\cdots\!49}a^{12}+\frac{92\!\cdots\!09}{53\!\cdots\!49}a^{11}+\frac{41\!\cdots\!58}{53\!\cdots\!49}a^{10}+\frac{11\!\cdots\!12}{53\!\cdots\!49}a^{9}-\frac{16\!\cdots\!34}{53\!\cdots\!49}a^{8}+\frac{52\!\cdots\!31}{53\!\cdots\!49}a^{7}-\frac{33\!\cdots\!90}{53\!\cdots\!49}a^{6}-\frac{94\!\cdots\!70}{53\!\cdots\!49}a^{5}+\frac{15\!\cdots\!41}{53\!\cdots\!49}a^{4}-\frac{19\!\cdots\!36}{53\!\cdots\!49}a^{3}+\frac{58\!\cdots\!73}{53\!\cdots\!49}a^{2}+\frac{96\!\cdots\!31}{53\!\cdots\!49}a-\frac{15\!\cdots\!64}{53\!\cdots\!49}$
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: |
$\frac{16\!\cdots\!78}{66\!\cdots\!21}a^{19}-\frac{29\!\cdots\!01}{66\!\cdots\!21}a^{18}-\frac{10\!\cdots\!93}{66\!\cdots\!21}a^{17}+\frac{41\!\cdots\!61}{66\!\cdots\!21}a^{16}+\frac{26\!\cdots\!84}{66\!\cdots\!21}a^{15}-\frac{14\!\cdots\!63}{66\!\cdots\!21}a^{14}-\frac{34\!\cdots\!74}{66\!\cdots\!21}a^{13}+\frac{21\!\cdots\!34}{66\!\cdots\!21}a^{12}+\frac{24\!\cdots\!64}{66\!\cdots\!21}a^{11}-\frac{17\!\cdots\!90}{66\!\cdots\!21}a^{10}-\frac{95\!\cdots\!29}{66\!\cdots\!21}a^{9}+\frac{72\!\cdots\!73}{66\!\cdots\!21}a^{8}+\frac{19\!\cdots\!06}{66\!\cdots\!21}a^{7}-\frac{16\!\cdots\!83}{66\!\cdots\!21}a^{6}-\frac{18\!\cdots\!68}{66\!\cdots\!21}a^{5}+\frac{17\!\cdots\!71}{66\!\cdots\!21}a^{4}+\frac{41\!\cdots\!60}{66\!\cdots\!21}a^{3}-\frac{69\!\cdots\!67}{66\!\cdots\!21}a^{2}+\frac{18\!\cdots\!99}{66\!\cdots\!21}a-\frac{11\!\cdots\!12}{66\!\cdots\!21}$, $\frac{18\!\cdots\!30}{53\!\cdots\!49}a^{19}-\frac{75\!\cdots\!50}{53\!\cdots\!49}a^{18}-\frac{11\!\cdots\!30}{53\!\cdots\!49}a^{17}+\frac{74\!\cdots\!75}{53\!\cdots\!49}a^{16}+\frac{29\!\cdots\!00}{53\!\cdots\!49}a^{15}-\frac{23\!\cdots\!40}{53\!\cdots\!49}a^{14}-\frac{38\!\cdots\!10}{53\!\cdots\!49}a^{13}+\frac{34\!\cdots\!40}{53\!\cdots\!49}a^{12}+\frac{27\!\cdots\!90}{53\!\cdots\!49}a^{11}-\frac{26\!\cdots\!24}{53\!\cdots\!49}a^{10}-\frac{10\!\cdots\!40}{53\!\cdots\!49}a^{9}+\frac{10\!\cdots\!30}{53\!\cdots\!49}a^{8}+\frac{20\!\cdots\!80}{53\!\cdots\!49}a^{7}-\frac{24\!\cdots\!60}{53\!\cdots\!49}a^{6}-\frac{15\!\cdots\!72}{53\!\cdots\!49}a^{5}+\frac{25\!\cdots\!40}{53\!\cdots\!49}a^{4}-\frac{10\!\cdots\!90}{53\!\cdots\!49}a^{3}-\frac{93\!\cdots\!70}{53\!\cdots\!49}a^{2}+\frac{42\!\cdots\!75}{53\!\cdots\!49}a-\frac{47\!\cdots\!79}{53\!\cdots\!49}$, $\frac{12\!\cdots\!66}{53\!\cdots\!49}a^{19}-\frac{11\!\cdots\!11}{53\!\cdots\!49}a^{18}-\frac{80\!\cdots\!08}{53\!\cdots\!49}a^{17}+\frac{24\!\cdots\!61}{53\!\cdots\!49}a^{16}+\frac{20\!\cdots\!29}{53\!\cdots\!49}a^{15}-\frac{93\!\cdots\!81}{53\!\cdots\!49}a^{14}-\frac{26\!\cdots\!54}{53\!\cdots\!49}a^{13}+\frac{14\!\cdots\!38}{53\!\cdots\!49}a^{12}+\frac{18\!\cdots\!38}{53\!\cdots\!49}a^{11}-\frac{11\!\cdots\!30}{53\!\cdots\!49}a^{10}-\frac{74\!\cdots\!92}{53\!\cdots\!49}a^{9}+\frac{49\!\cdots\!48}{53\!\cdots\!49}a^{8}+\frac{15\!\cdots\!94}{53\!\cdots\!49}a^{7}-\frac{11\!\cdots\!05}{53\!\cdots\!49}a^{6}-\frac{14\!\cdots\!43}{53\!\cdots\!49}a^{5}+\frac{12\!\cdots\!29}{53\!\cdots\!49}a^{4}+\frac{36\!\cdots\!78}{53\!\cdots\!49}a^{3}-\frac{48\!\cdots\!07}{53\!\cdots\!49}a^{2}+\frac{11\!\cdots\!56}{53\!\cdots\!49}a-\frac{76\!\cdots\!28}{53\!\cdots\!49}$, $\frac{59\!\cdots\!50}{53\!\cdots\!49}a^{19}-\frac{93\!\cdots\!35}{53\!\cdots\!49}a^{18}-\frac{38\!\cdots\!85}{53\!\cdots\!49}a^{17}+\frac{14\!\cdots\!65}{53\!\cdots\!49}a^{16}+\frac{96\!\cdots\!16}{53\!\cdots\!49}a^{15}-\frac{51\!\cdots\!95}{53\!\cdots\!49}a^{14}-\frac{12\!\cdots\!40}{53\!\cdots\!49}a^{13}+\frac{78\!\cdots\!40}{53\!\cdots\!49}a^{12}+\frac{89\!\cdots\!80}{53\!\cdots\!49}a^{11}-\frac{61\!\cdots\!56}{53\!\cdots\!49}a^{10}-\frac{35\!\cdots\!95}{53\!\cdots\!49}a^{9}+\frac{26\!\cdots\!60}{53\!\cdots\!49}a^{8}+\frac{72\!\cdots\!40}{53\!\cdots\!49}a^{7}-\frac{59\!\cdots\!65}{53\!\cdots\!49}a^{6}-\frac{65\!\cdots\!62}{53\!\cdots\!49}a^{5}+\frac{64\!\cdots\!75}{53\!\cdots\!49}a^{4}+\frac{13\!\cdots\!10}{53\!\cdots\!49}a^{3}-\frac{25\!\cdots\!20}{53\!\cdots\!49}a^{2}+\frac{74\!\cdots\!45}{53\!\cdots\!49}a-\frac{63\!\cdots\!96}{53\!\cdots\!49}$, $\frac{65\!\cdots\!16}{53\!\cdots\!49}a^{19}-\frac{21\!\cdots\!76}{53\!\cdots\!49}a^{18}-\frac{42\!\cdots\!23}{53\!\cdots\!49}a^{17}+\frac{10\!\cdots\!96}{53\!\cdots\!49}a^{16}+\frac{10\!\cdots\!13}{53\!\cdots\!49}a^{15}-\frac{42\!\cdots\!86}{53\!\cdots\!49}a^{14}-\frac{13\!\cdots\!14}{53\!\cdots\!49}a^{13}+\frac{67\!\cdots\!98}{53\!\cdots\!49}a^{12}+\frac{98\!\cdots\!58}{53\!\cdots\!49}a^{11}-\frac{53\!\cdots\!74}{53\!\cdots\!49}a^{10}-\frac{39\!\cdots\!97}{53\!\cdots\!49}a^{9}+\frac{23\!\cdots\!88}{53\!\cdots\!49}a^{8}+\frac{81\!\cdots\!54}{53\!\cdots\!49}a^{7}-\frac{52\!\cdots\!40}{53\!\cdots\!49}a^{6}-\frac{78\!\cdots\!81}{53\!\cdots\!49}a^{5}+\frac{56\!\cdots\!54}{53\!\cdots\!49}a^{4}+\frac{22\!\cdots\!68}{53\!\cdots\!49}a^{3}-\frac{23\!\cdots\!87}{53\!\cdots\!49}a^{2}+\frac{40\!\cdots\!11}{53\!\cdots\!49}a-\frac{12\!\cdots\!32}{53\!\cdots\!49}$, $\frac{28\!\cdots\!48}{53\!\cdots\!49}a^{19}+\frac{15\!\cdots\!39}{53\!\cdots\!49}a^{18}-\frac{18\!\cdots\!93}{53\!\cdots\!49}a^{17}-\frac{62\!\cdots\!09}{53\!\cdots\!49}a^{16}+\frac{45\!\cdots\!24}{53\!\cdots\!49}a^{15}+\frac{89\!\cdots\!07}{53\!\cdots\!49}a^{14}-\frac{59\!\cdots\!84}{53\!\cdots\!49}a^{13}-\frac{60\!\cdots\!76}{53\!\cdots\!49}a^{12}+\frac{42\!\cdots\!14}{53\!\cdots\!49}a^{11}+\frac{19\!\cdots\!60}{53\!\cdots\!49}a^{10}-\frac{17\!\cdots\!29}{53\!\cdots\!49}a^{9}-\frac{35\!\cdots\!12}{53\!\cdots\!49}a^{8}+\frac{36\!\cdots\!66}{53\!\cdots\!49}a^{7}-\frac{15\!\cdots\!23}{53\!\cdots\!49}a^{6}-\frac{38\!\cdots\!14}{53\!\cdots\!49}a^{5}+\frac{38\!\cdots\!96}{53\!\cdots\!49}a^{4}+\frac{17\!\cdots\!40}{53\!\cdots\!49}a^{3}-\frac{27\!\cdots\!72}{53\!\cdots\!49}a^{2}-\frac{18\!\cdots\!66}{53\!\cdots\!49}a+\frac{34\!\cdots\!01}{53\!\cdots\!49}$, $\frac{46\!\cdots\!78}{53\!\cdots\!49}a^{19}+\frac{84\!\cdots\!89}{53\!\cdots\!49}a^{18}-\frac{29\!\cdots\!23}{53\!\cdots\!49}a^{17}+\frac{11\!\cdots\!66}{53\!\cdots\!49}a^{16}+\frac{75\!\cdots\!24}{53\!\cdots\!49}a^{15}-\frac{14\!\cdots\!33}{53\!\cdots\!49}a^{14}-\frac{97\!\cdots\!94}{53\!\cdots\!49}a^{13}+\frac{28\!\cdots\!64}{53\!\cdots\!49}a^{12}+\frac{69\!\cdots\!04}{53\!\cdots\!49}a^{11}-\frac{24\!\cdots\!64}{53\!\cdots\!49}a^{10}-\frac{27\!\cdots\!69}{53\!\cdots\!49}a^{9}+\frac{10\!\cdots\!18}{53\!\cdots\!49}a^{8}+\frac{56\!\cdots\!46}{53\!\cdots\!49}a^{7}-\frac{25\!\cdots\!83}{53\!\cdots\!49}a^{6}-\frac{53\!\cdots\!86}{53\!\cdots\!49}a^{5}+\frac{29\!\cdots\!36}{53\!\cdots\!49}a^{4}+\frac{16\!\cdots\!50}{53\!\cdots\!49}a^{3}-\frac{12\!\cdots\!42}{53\!\cdots\!49}a^{2}+\frac{23\!\cdots\!09}{53\!\cdots\!49}a-\frac{13\!\cdots\!78}{53\!\cdots\!49}$, $\frac{71\!\cdots\!16}{53\!\cdots\!49}a^{19}-\frac{12\!\cdots\!58}{53\!\cdots\!49}a^{18}-\frac{46\!\cdots\!09}{53\!\cdots\!49}a^{17}+\frac{90\!\cdots\!48}{53\!\cdots\!49}a^{16}+\frac{11\!\cdots\!67}{53\!\cdots\!49}a^{15}-\frac{24\!\cdots\!66}{53\!\cdots\!49}a^{14}-\frac{15\!\cdots\!52}{53\!\cdots\!49}a^{13}+\frac{32\!\cdots\!08}{53\!\cdots\!49}a^{12}+\frac{10\!\cdots\!78}{53\!\cdots\!49}a^{11}-\frac{24\!\cdots\!80}{53\!\cdots\!49}a^{10}-\frac{40\!\cdots\!09}{53\!\cdots\!49}a^{9}+\frac{96\!\cdots\!89}{53\!\cdots\!49}a^{8}+\frac{77\!\cdots\!20}{53\!\cdots\!49}a^{7}-\frac{20\!\cdots\!22}{53\!\cdots\!49}a^{6}-\frac{51\!\cdots\!49}{53\!\cdots\!49}a^{5}+\frac{21\!\cdots\!70}{53\!\cdots\!49}a^{4}-\frac{26\!\cdots\!38}{53\!\cdots\!49}a^{3}-\frac{84\!\cdots\!16}{53\!\cdots\!49}a^{2}+\frac{32\!\cdots\!75}{53\!\cdots\!49}a-\frac{19\!\cdots\!49}{53\!\cdots\!49}$, $\frac{66\!\cdots\!66}{53\!\cdots\!49}a^{19}-\frac{21\!\cdots\!93}{53\!\cdots\!49}a^{18}-\frac{42\!\cdots\!94}{53\!\cdots\!49}a^{17}+\frac{23\!\cdots\!13}{53\!\cdots\!49}a^{16}+\frac{10\!\cdots\!83}{53\!\cdots\!49}a^{15}-\frac{75\!\cdots\!61}{53\!\cdots\!49}a^{14}-\frac{14\!\cdots\!92}{53\!\cdots\!49}a^{13}+\frac{11\!\cdots\!48}{53\!\cdots\!49}a^{12}+\frac{99\!\cdots\!58}{53\!\cdots\!49}a^{11}-\frac{85\!\cdots\!36}{53\!\cdots\!49}a^{10}-\frac{39\!\cdots\!04}{53\!\cdots\!49}a^{9}+\frac{35\!\cdots\!49}{53\!\cdots\!49}a^{8}+\frac{79\!\cdots\!60}{53\!\cdots\!49}a^{7}-\frac{80\!\cdots\!87}{53\!\cdots\!49}a^{6}-\frac{70\!\cdots\!11}{53\!\cdots\!49}a^{5}+\frac{86\!\cdots\!45}{53\!\cdots\!49}a^{4}+\frac{10\!\cdots\!72}{53\!\cdots\!49}a^{3}-\frac{33\!\cdots\!36}{53\!\cdots\!49}a^{2}+\frac{10\!\cdots\!20}{53\!\cdots\!49}a-\frac{78\!\cdots\!96}{53\!\cdots\!49}$, $\frac{79\!\cdots\!49}{53\!\cdots\!49}a^{19}+\frac{11\!\cdots\!47}{53\!\cdots\!49}a^{18}-\frac{50\!\cdots\!16}{53\!\cdots\!49}a^{17}+\frac{39\!\cdots\!22}{53\!\cdots\!49}a^{16}+\frac{12\!\cdots\!41}{53\!\cdots\!49}a^{15}-\frac{30\!\cdots\!60}{53\!\cdots\!49}a^{14}-\frac{16\!\cdots\!99}{53\!\cdots\!49}a^{13}+\frac{56\!\cdots\!96}{53\!\cdots\!49}a^{12}+\frac{11\!\cdots\!39}{53\!\cdots\!49}a^{11}-\frac{48\!\cdots\!52}{53\!\cdots\!49}a^{10}-\frac{47\!\cdots\!70}{53\!\cdots\!49}a^{9}+\frac{22\!\cdots\!56}{53\!\cdots\!49}a^{8}+\frac{98\!\cdots\!26}{53\!\cdots\!49}a^{7}-\frac{53\!\cdots\!18}{53\!\cdots\!49}a^{6}-\frac{92\!\cdots\!66}{53\!\cdots\!49}a^{5}+\frac{62\!\cdots\!64}{53\!\cdots\!49}a^{4}+\frac{26\!\cdots\!12}{53\!\cdots\!49}a^{3}-\frac{26\!\cdots\!62}{53\!\cdots\!49}a^{2}+\frac{53\!\cdots\!14}{53\!\cdots\!49}a-\frac{25\!\cdots\!59}{53\!\cdots\!49}$, $\frac{47\!\cdots\!06}{53\!\cdots\!49}a^{19}-\frac{26\!\cdots\!22}{53\!\cdots\!49}a^{18}-\frac{30\!\cdots\!61}{53\!\cdots\!49}a^{17}+\frac{82\!\cdots\!54}{53\!\cdots\!49}a^{16}+\frac{76\!\cdots\!76}{53\!\cdots\!49}a^{15}-\frac{32\!\cdots\!21}{53\!\cdots\!49}a^{14}-\frac{99\!\cdots\!56}{53\!\cdots\!49}a^{13}+\frac{51\!\cdots\!98}{53\!\cdots\!49}a^{12}+\frac{70\!\cdots\!10}{53\!\cdots\!49}a^{11}-\frac{40\!\cdots\!31}{53\!\cdots\!49}a^{10}-\frac{27\!\cdots\!08}{53\!\cdots\!49}a^{9}+\frac{17\!\cdots\!00}{53\!\cdots\!49}a^{8}+\frac{57\!\cdots\!72}{53\!\cdots\!49}a^{7}-\frac{39\!\cdots\!88}{53\!\cdots\!49}a^{6}-\frac{54\!\cdots\!65}{53\!\cdots\!49}a^{5}+\frac{43\!\cdots\!96}{53\!\cdots\!49}a^{4}+\frac{14\!\cdots\!27}{53\!\cdots\!49}a^{3}-\frac{17\!\cdots\!38}{53\!\cdots\!49}a^{2}+\frac{41\!\cdots\!50}{53\!\cdots\!49}a-\frac{23\!\cdots\!63}{53\!\cdots\!49}$, $\frac{10\!\cdots\!38}{53\!\cdots\!49}a^{19}-\frac{83\!\cdots\!41}{53\!\cdots\!49}a^{18}-\frac{65\!\cdots\!59}{53\!\cdots\!49}a^{17}+\frac{67\!\cdots\!74}{53\!\cdots\!49}a^{16}+\frac{16\!\cdots\!52}{53\!\cdots\!49}a^{15}-\frac{19\!\cdots\!50}{53\!\cdots\!49}a^{14}-\frac{20\!\cdots\!68}{53\!\cdots\!49}a^{13}+\frac{27\!\cdots\!89}{53\!\cdots\!49}a^{12}+\frac{14\!\cdots\!01}{53\!\cdots\!49}a^{11}-\frac{20\!\cdots\!38}{53\!\cdots\!49}a^{10}-\frac{53\!\cdots\!00}{53\!\cdots\!49}a^{9}+\frac{81\!\cdots\!76}{53\!\cdots\!49}a^{8}+\frac{96\!\cdots\!82}{53\!\cdots\!49}a^{7}-\frac{17\!\cdots\!30}{53\!\cdots\!49}a^{6}-\frac{59\!\cdots\!84}{53\!\cdots\!49}a^{5}+\frac{17\!\cdots\!80}{53\!\cdots\!49}a^{4}-\frac{20\!\cdots\!41}{53\!\cdots\!49}a^{3}-\frac{59\!\cdots\!39}{53\!\cdots\!49}a^{2}+\frac{23\!\cdots\!80}{53\!\cdots\!49}a-\frac{15\!\cdots\!70}{53\!\cdots\!49}$, $\frac{71\!\cdots\!00}{53\!\cdots\!49}a^{19}-\frac{34\!\cdots\!88}{53\!\cdots\!49}a^{18}-\frac{40\!\cdots\!27}{53\!\cdots\!49}a^{17}+\frac{22\!\cdots\!38}{53\!\cdots\!49}a^{16}+\frac{76\!\cdots\!88}{53\!\cdots\!49}a^{15}-\frac{56\!\cdots\!90}{53\!\cdots\!49}a^{14}-\frac{45\!\cdots\!32}{53\!\cdots\!49}a^{13}+\frac{68\!\cdots\!88}{53\!\cdots\!49}a^{12}-\frac{29\!\cdots\!14}{53\!\cdots\!49}a^{11}-\frac{42\!\cdots\!50}{53\!\cdots\!49}a^{10}+\frac{49\!\cdots\!32}{53\!\cdots\!49}a^{9}+\frac{12\!\cdots\!58}{53\!\cdots\!49}a^{8}-\frac{23\!\cdots\!97}{53\!\cdots\!49}a^{7}-\frac{12\!\cdots\!35}{53\!\cdots\!49}a^{6}+\frac{43\!\cdots\!05}{53\!\cdots\!49}a^{5}-\frac{13\!\cdots\!50}{53\!\cdots\!49}a^{4}-\frac{21\!\cdots\!57}{53\!\cdots\!49}a^{3}+\frac{16\!\cdots\!28}{53\!\cdots\!49}a^{2}-\frac{42\!\cdots\!63}{53\!\cdots\!49}a+\frac{33\!\cdots\!75}{53\!\cdots\!49}$, $\frac{96\!\cdots\!99}{53\!\cdots\!49}a^{19}-\frac{71\!\cdots\!10}{53\!\cdots\!49}a^{18}-\frac{61\!\cdots\!34}{53\!\cdots\!49}a^{17}+\frac{59\!\cdots\!55}{53\!\cdots\!49}a^{16}+\frac{15\!\cdots\!95}{53\!\cdots\!49}a^{15}-\frac{17\!\cdots\!16}{53\!\cdots\!49}a^{14}-\frac{19\!\cdots\!99}{53\!\cdots\!49}a^{13}+\frac{24\!\cdots\!68}{53\!\cdots\!49}a^{12}+\frac{13\!\cdots\!65}{53\!\cdots\!49}a^{11}-\frac{18\!\cdots\!85}{53\!\cdots\!49}a^{10}-\frac{52\!\cdots\!79}{53\!\cdots\!49}a^{9}+\frac{76\!\cdots\!36}{53\!\cdots\!49}a^{8}+\frac{97\!\cdots\!19}{53\!\cdots\!49}a^{7}-\frac{16\!\cdots\!15}{53\!\cdots\!49}a^{6}-\frac{62\!\cdots\!11}{53\!\cdots\!49}a^{5}+\frac{17\!\cdots\!96}{53\!\cdots\!49}a^{4}-\frac{27\!\cdots\!03}{53\!\cdots\!49}a^{3}-\frac{63\!\cdots\!48}{53\!\cdots\!49}a^{2}+\frac{32\!\cdots\!68}{53\!\cdots\!49}a-\frac{44\!\cdots\!85}{53\!\cdots\!49}$, $\frac{95\!\cdots\!65}{53\!\cdots\!49}a^{19}-\frac{89\!\cdots\!01}{53\!\cdots\!49}a^{18}-\frac{61\!\cdots\!77}{53\!\cdots\!49}a^{17}+\frac{18\!\cdots\!60}{53\!\cdots\!49}a^{16}+\frac{15\!\cdots\!64}{53\!\cdots\!49}a^{15}-\frac{71\!\cdots\!66}{53\!\cdots\!49}a^{14}-\frac{20\!\cdots\!81}{53\!\cdots\!49}a^{13}+\frac{11\!\cdots\!73}{53\!\cdots\!49}a^{12}+\frac{14\!\cdots\!72}{53\!\cdots\!49}a^{11}-\frac{87\!\cdots\!62}{53\!\cdots\!49}a^{10}-\frac{56\!\cdots\!75}{53\!\cdots\!49}a^{9}+\frac{37\!\cdots\!39}{53\!\cdots\!49}a^{8}+\frac{11\!\cdots\!61}{53\!\cdots\!49}a^{7}-\frac{84\!\cdots\!29}{53\!\cdots\!49}a^{6}-\frac{11\!\cdots\!70}{53\!\cdots\!49}a^{5}+\frac{92\!\cdots\!07}{53\!\cdots\!49}a^{4}+\frac{28\!\cdots\!71}{53\!\cdots\!49}a^{3}-\frac{37\!\cdots\!22}{53\!\cdots\!49}a^{2}+\frac{86\!\cdots\!76}{53\!\cdots\!49}a-\frac{49\!\cdots\!02}{53\!\cdots\!49}$, $\frac{32\!\cdots\!06}{53\!\cdots\!49}a^{19}+\frac{17\!\cdots\!28}{53\!\cdots\!49}a^{18}-\frac{20\!\cdots\!77}{53\!\cdots\!49}a^{17}+\frac{44\!\cdots\!59}{53\!\cdots\!49}a^{16}+\frac{52\!\cdots\!25}{53\!\cdots\!49}a^{15}-\frac{19\!\cdots\!27}{53\!\cdots\!49}a^{14}-\frac{68\!\cdots\!13}{53\!\cdots\!49}a^{13}+\frac{31\!\cdots\!76}{53\!\cdots\!49}a^{12}+\frac{48\!\cdots\!07}{53\!\cdots\!49}a^{11}-\frac{25\!\cdots\!66}{53\!\cdots\!49}a^{10}-\frac{19\!\cdots\!66}{53\!\cdots\!49}a^{9}+\frac{10\!\cdots\!70}{53\!\cdots\!49}a^{8}+\frac{39\!\cdots\!59}{53\!\cdots\!49}a^{7}-\frac{24\!\cdots\!60}{53\!\cdots\!49}a^{6}-\frac{36\!\cdots\!98}{53\!\cdots\!49}a^{5}+\frac{26\!\cdots\!35}{53\!\cdots\!49}a^{4}+\frac{96\!\cdots\!74}{53\!\cdots\!49}a^{3}-\frac{10\!\cdots\!93}{53\!\cdots\!49}a^{2}+\frac{23\!\cdots\!12}{53\!\cdots\!49}a-\frac{11\!\cdots\!34}{53\!\cdots\!49}$, $\frac{30\!\cdots\!02}{53\!\cdots\!49}a^{19}-\frac{26\!\cdots\!71}{53\!\cdots\!49}a^{18}-\frac{19\!\cdots\!53}{53\!\cdots\!49}a^{17}+\frac{21\!\cdots\!26}{53\!\cdots\!49}a^{16}+\frac{48\!\cdots\!05}{53\!\cdots\!49}a^{15}-\frac{61\!\cdots\!27}{53\!\cdots\!49}a^{14}-\frac{62\!\cdots\!19}{53\!\cdots\!49}a^{13}+\frac{85\!\cdots\!15}{53\!\cdots\!49}a^{12}+\frac{43\!\cdots\!04}{53\!\cdots\!49}a^{11}-\frac{63\!\cdots\!77}{53\!\cdots\!49}a^{10}-\frac{16\!\cdots\!64}{53\!\cdots\!49}a^{9}+\frac{26\!\cdots\!89}{53\!\cdots\!49}a^{8}+\frac{29\!\cdots\!14}{53\!\cdots\!49}a^{7}-\frac{56\!\cdots\!77}{53\!\cdots\!49}a^{6}-\frac{17\!\cdots\!73}{53\!\cdots\!49}a^{5}+\frac{58\!\cdots\!87}{53\!\cdots\!49}a^{4}-\frac{10\!\cdots\!86}{53\!\cdots\!49}a^{3}-\frac{20\!\cdots\!14}{53\!\cdots\!49}a^{2}+\frac{11\!\cdots\!40}{53\!\cdots\!49}a-\frac{15\!\cdots\!40}{53\!\cdots\!49}$, $\frac{35\!\cdots\!20}{53\!\cdots\!49}a^{19}-\frac{29\!\cdots\!14}{53\!\cdots\!49}a^{18}-\frac{24\!\cdots\!47}{53\!\cdots\!49}a^{17}+\frac{19\!\cdots\!54}{53\!\cdots\!49}a^{16}+\frac{63\!\cdots\!85}{53\!\cdots\!49}a^{15}-\frac{50\!\cdots\!60}{53\!\cdots\!49}a^{14}-\frac{81\!\cdots\!66}{53\!\cdots\!49}a^{13}+\frac{65\!\cdots\!90}{53\!\cdots\!49}a^{12}+\frac{55\!\cdots\!78}{53\!\cdots\!49}a^{11}-\frac{46\!\cdots\!34}{53\!\cdots\!49}a^{10}-\frac{19\!\cdots\!69}{53\!\cdots\!49}a^{9}+\frac{18\!\cdots\!82}{53\!\cdots\!49}a^{8}+\frac{27\!\cdots\!22}{53\!\cdots\!49}a^{7}-\frac{38\!\cdots\!38}{53\!\cdots\!49}a^{6}+\frac{16\!\cdots\!05}{53\!\cdots\!49}a^{5}+\frac{37\!\cdots\!92}{53\!\cdots\!49}a^{4}-\frac{76\!\cdots\!82}{53\!\cdots\!49}a^{3}-\frac{13\!\cdots\!22}{53\!\cdots\!49}a^{2}+\frac{46\!\cdots\!65}{53\!\cdots\!49}a-\frac{28\!\cdots\!68}{53\!\cdots\!49}$, $\frac{17\!\cdots\!60}{53\!\cdots\!49}a^{19}-\frac{15\!\cdots\!80}{53\!\cdots\!49}a^{18}-\frac{11\!\cdots\!40}{53\!\cdots\!49}a^{17}+\frac{35\!\cdots\!75}{53\!\cdots\!49}a^{16}+\frac{29\!\cdots\!20}{53\!\cdots\!49}a^{15}-\frac{13\!\cdots\!80}{53\!\cdots\!49}a^{14}-\frac{37\!\cdots\!00}{53\!\cdots\!49}a^{13}+\frac{20\!\cdots\!10}{53\!\cdots\!49}a^{12}+\frac{26\!\cdots\!20}{53\!\cdots\!49}a^{11}-\frac{16\!\cdots\!74}{53\!\cdots\!49}a^{10}-\frac{10\!\cdots\!70}{53\!\cdots\!49}a^{9}+\frac{69\!\cdots\!55}{53\!\cdots\!49}a^{8}+\frac{21\!\cdots\!60}{53\!\cdots\!49}a^{7}-\frac{15\!\cdots\!10}{53\!\cdots\!49}a^{6}-\frac{20\!\cdots\!78}{53\!\cdots\!49}a^{5}+\frac{17\!\cdots\!35}{53\!\cdots\!49}a^{4}+\frac{49\!\cdots\!90}{53\!\cdots\!49}a^{3}-\frac{69\!\cdots\!65}{53\!\cdots\!49}a^{2}+\frac{17\!\cdots\!89}{53\!\cdots\!49}a-\frac{11\!\cdots\!04}{53\!\cdots\!49}$
| 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: | \( 249459949124 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 249459949124 \cdot 1}{2\cdot\sqrt{396107830343483954099825714111328125}}\cr\approx \mathstrut & 0.207808847218498 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 64*x^18 + 78*x^17 + 1609*x^16 - 2224*x^15 - 20399*x^14 + 30774*x^13 + 139797*x^12 - 228288*x^11 - 510719*x^10 + 929822*x^9 + 878982*x^8 - 2003579*x^7 - 355510*x^6 + 2013497*x^5 - 574064*x^4 - 658628*x^3 + 441152*x^2 - 86946*x + 4621)
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 - x^19 - 64*x^18 + 78*x^17 + 1609*x^16 - 2224*x^15 - 20399*x^14 + 30774*x^13 + 139797*x^12 - 228288*x^11 - 510719*x^10 + 929822*x^9 + 878982*x^8 - 2003579*x^7 - 355510*x^6 + 2013497*x^5 - 574064*x^4 - 658628*x^3 + 441152*x^2 - 86946*x + 4621, 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 - x^19 - 64*x^18 + 78*x^17 + 1609*x^16 - 2224*x^15 - 20399*x^14 + 30774*x^13 + 139797*x^12 - 228288*x^11 - 510719*x^10 + 929822*x^9 + 878982*x^8 - 2003579*x^7 - 355510*x^6 + 2013497*x^5 - 574064*x^4 - 658628*x^3 + 441152*x^2 - 86946*x + 4621);
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 - x^19 - 64*x^18 + 78*x^17 + 1609*x^16 - 2224*x^15 - 20399*x^14 + 30774*x^13 + 139797*x^12 - 228288*x^11 - 510719*x^10 + 929822*x^9 + 878982*x^8 - 2003579*x^7 - 355510*x^6 + 2013497*x^5 - 574064*x^4 - 658628*x^3 + 441152*x^2 - 86946*x + 4621);
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
| \(\Q(\sqrt{5}) \), 4.4.6125.1, \(\Q(\zeta_{11})^+\), 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 | $20$ | $20$ | R | R | R | $20$ | $20$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $20$ | $4$ | $5$ | $15$ | |||
|
\(7\)
| 7.20.10.2 | $x^{20} + 2401 x^{12} - 16807 x^{10} + 470596 x^{8} - 823543 x^{6} + 11529602 x^{4} - 121060821 x^{2} + 847425747$ | $2$ | $10$ | $10$ | 20T1 | $[\ ]_{2}^{10}$ |
|
\(11\)
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |