Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 60*x^18 + 236*x^17 + 1435*x^16 - 5524*x^15 - 17726*x^14 + 66220*x^13 + 122474*x^12 - 437336*x^11 - 480740*x^10 + 1596028*x^9 + 1047359*x^8 - 3070872*x^7 - 1203606*x^6 + 2823768*x^5 + 614261*x^4 - 1007904*x^3 - 118608*x^2 + 70588*x + 1451)
gp: K = bnfinit(y^20 - 4*y^19 - 60*y^18 + 236*y^17 + 1435*y^16 - 5524*y^15 - 17726*y^14 + 66220*y^13 + 122474*y^12 - 437336*y^11 - 480740*y^10 + 1596028*y^9 + 1047359*y^8 - 3070872*y^7 - 1203606*y^6 + 2823768*y^5 + 614261*y^4 - 1007904*y^3 - 118608*y^2 + 70588*y + 1451, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 - 60*x^18 + 236*x^17 + 1435*x^16 - 5524*x^15 - 17726*x^14 + 66220*x^13 + 122474*x^12 - 437336*x^11 - 480740*x^10 + 1596028*x^9 + 1047359*x^8 - 3070872*x^7 - 1203606*x^6 + 2823768*x^5 + 614261*x^4 - 1007904*x^3 - 118608*x^2 + 70588*x + 1451);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 - 60*x^18 + 236*x^17 + 1435*x^16 - 5524*x^15 - 17726*x^14 + 66220*x^13 + 122474*x^12 - 437336*x^11 - 480740*x^10 + 1596028*x^9 + 1047359*x^8 - 3070872*x^7 - 1203606*x^6 + 2823768*x^5 + 614261*x^4 - 1007904*x^3 - 118608*x^2 + 70588*x + 1451)
\( x^{20} - 4 x^{19} - 60 x^{18} + 236 x^{17} + 1435 x^{16} - 5524 x^{15} - 17726 x^{14} + 66220 x^{13} + \cdots + 1451 \)
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: |
\(1505680748169532571648000000000000000\)
\(\medspace = 2^{30}\cdot 5^{15}\cdot 11^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(64.40\) | 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: | $2^{3/2}5^{3/4}11^{4/5}\approx 64.4001152950292$ | ||
| 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: | \(440=2^{3}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(67,·)$, $\chi_{440}(147,·)$, $\chi_{440}(9,·)$, $\chi_{440}(267,·)$, $\chi_{440}(81,·)$, $\chi_{440}(3,·)$, $\chi_{440}(203,·)$, $\chi_{440}(89,·)$, $\chi_{440}(27,·)$, $\chi_{440}(361,·)$, $\chi_{440}(289,·)$, $\chi_{440}(163,·)$, $\chi_{440}(401,·)$, $\chi_{440}(169,·)$, $\chi_{440}(323,·)$, $\chi_{440}(427,·)$, $\chi_{440}(49,·)$, $\chi_{440}(243,·)$, $\chi_{440}(201,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{1054855908478}a^{18}+\frac{40274534656}{527427954239}a^{17}+\frac{1600657844}{527427954239}a^{16}+\frac{39130263459}{1054855908478}a^{15}-\frac{52882400951}{1054855908478}a^{14}+\frac{151606853083}{1054855908478}a^{13}-\frac{200202658757}{1054855908478}a^{12}+\frac{179937412937}{1054855908478}a^{11}-\frac{35799794964}{527427954239}a^{10}-\frac{78309698625}{527427954239}a^{9}-\frac{56678828105}{527427954239}a^{8}+\frac{16881612043}{1054855908478}a^{7}-\frac{414260889493}{1054855908478}a^{6}+\frac{234378754158}{527427954239}a^{5}+\frac{89001208760}{527427954239}a^{4}+\frac{234198840155}{1054855908478}a^{3}+\frac{124470694961}{527427954239}a^{2}-\frac{405419500363}{1054855908478}a-\frac{238256932417}{527427954239}$, $\frac{1}{42\!\cdots\!62}a^{19}+\frac{35\!\cdots\!98}{21\!\cdots\!81}a^{18}-\frac{46\!\cdots\!65}{42\!\cdots\!62}a^{17}-\frac{10\!\cdots\!16}{21\!\cdots\!81}a^{16}-\frac{64\!\cdots\!01}{42\!\cdots\!62}a^{15}-\frac{11\!\cdots\!57}{21\!\cdots\!81}a^{14}-\frac{30\!\cdots\!28}{21\!\cdots\!81}a^{13}-\frac{13\!\cdots\!23}{21\!\cdots\!81}a^{12}-\frac{76\!\cdots\!09}{21\!\cdots\!81}a^{11}-\frac{18\!\cdots\!71}{21\!\cdots\!81}a^{10}+\frac{17\!\cdots\!99}{42\!\cdots\!62}a^{9}+\frac{90\!\cdots\!71}{21\!\cdots\!81}a^{8}-\frac{47\!\cdots\!47}{42\!\cdots\!62}a^{7}+\frac{53\!\cdots\!21}{21\!\cdots\!81}a^{6}-\frac{70\!\cdots\!73}{21\!\cdots\!81}a^{5}-\frac{11\!\cdots\!92}{21\!\cdots\!81}a^{4}+\frac{26\!\cdots\!95}{42\!\cdots\!62}a^{3}+\frac{65\!\cdots\!34}{21\!\cdots\!81}a^{2}-\frac{21\!\cdots\!27}{42\!\cdots\!62}a-\frac{17\!\cdots\!71}{21\!\cdots\!81}$
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
$C_{5}$, which has order $5$ (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: |
$\frac{137152378}{527427954239}a^{19}-\frac{767934845}{527427954239}a^{18}-\frac{7129368102}{527427954239}a^{17}+\frac{88230880555}{1054855908478}a^{16}+\frac{133864656818}{527427954239}a^{15}-\frac{990847146193}{527427954239}a^{14}-\frac{1022796009135}{527427954239}a^{13}+\frac{22258952470789}{1054855908478}a^{12}+\frac{1039272647646}{527427954239}a^{11}-\frac{66061628858913}{527427954239}a^{10}+\frac{27282365093204}{527427954239}a^{9}+\frac{400866060121759}{1054855908478}a^{8}-\frac{138344124007136}{527427954239}a^{7}-\frac{274024452425953}{527427954239}a^{6}+\frac{218437567818799}{527427954239}a^{5}+\frac{136488068467498}{527427954239}a^{4}-\frac{102551915468802}{527427954239}a^{3}-\frac{37973987743749}{1054855908478}a^{2}+\frac{8042941150016}{527427954239}a-\frac{316724530429}{1054855908478}$, $\frac{16\!\cdots\!56}{21\!\cdots\!81}a^{19}-\frac{96\!\cdots\!60}{21\!\cdots\!81}a^{18}-\frac{86\!\cdots\!24}{21\!\cdots\!81}a^{17}+\frac{55\!\cdots\!60}{21\!\cdots\!81}a^{16}+\frac{15\!\cdots\!32}{21\!\cdots\!81}a^{15}-\frac{12\!\cdots\!76}{21\!\cdots\!81}a^{14}-\frac{97\!\cdots\!20}{21\!\cdots\!81}a^{13}+\frac{13\!\cdots\!44}{21\!\cdots\!81}a^{12}-\frac{20\!\cdots\!16}{21\!\cdots\!81}a^{11}-\frac{79\!\cdots\!52}{21\!\cdots\!81}a^{10}+\frac{52\!\cdots\!48}{21\!\cdots\!81}a^{9}+\frac{23\!\cdots\!89}{21\!\cdots\!81}a^{8}-\frac{22\!\cdots\!92}{21\!\cdots\!81}a^{7}-\frac{28\!\cdots\!36}{21\!\cdots\!81}a^{6}+\frac{32\!\cdots\!96}{21\!\cdots\!81}a^{5}+\frac{10\!\cdots\!36}{21\!\cdots\!81}a^{4}-\frac{12\!\cdots\!64}{21\!\cdots\!81}a^{3}-\frac{89\!\cdots\!28}{21\!\cdots\!81}a^{2}-\frac{88\!\cdots\!32}{21\!\cdots\!81}a-\frac{62\!\cdots\!70}{21\!\cdots\!81}$, $\frac{39\!\cdots\!80}{21\!\cdots\!81}a^{19}-\frac{18\!\cdots\!40}{21\!\cdots\!81}a^{18}-\frac{21\!\cdots\!00}{21\!\cdots\!81}a^{17}+\frac{10\!\cdots\!25}{21\!\cdots\!81}a^{16}+\frac{45\!\cdots\!00}{21\!\cdots\!81}a^{15}-\frac{23\!\cdots\!40}{21\!\cdots\!81}a^{14}-\frac{45\!\cdots\!40}{21\!\cdots\!81}a^{13}+\frac{26\!\cdots\!95}{21\!\cdots\!81}a^{12}+\frac{20\!\cdots\!40}{21\!\cdots\!81}a^{11}-\frac{16\!\cdots\!84}{21\!\cdots\!81}a^{10}-\frac{26\!\cdots\!20}{21\!\cdots\!81}a^{9}+\frac{49\!\cdots\!60}{21\!\cdots\!81}a^{8}-\frac{74\!\cdots\!60}{21\!\cdots\!81}a^{7}-\frac{69\!\cdots\!10}{21\!\cdots\!81}a^{6}+\frac{17\!\cdots\!24}{21\!\cdots\!81}a^{5}+\frac{37\!\cdots\!45}{21\!\cdots\!81}a^{4}-\frac{58\!\cdots\!60}{21\!\cdots\!81}a^{3}-\frac{62\!\cdots\!60}{21\!\cdots\!81}a^{2}-\frac{27\!\cdots\!60}{21\!\cdots\!81}a+\frac{16\!\cdots\!19}{21\!\cdots\!81}$, $\frac{56\!\cdots\!36}{21\!\cdots\!81}a^{19}-\frac{28\!\cdots\!00}{21\!\cdots\!81}a^{18}-\frac{30\!\cdots\!24}{21\!\cdots\!81}a^{17}+\frac{16\!\cdots\!85}{21\!\cdots\!81}a^{16}+\frac{60\!\cdots\!32}{21\!\cdots\!81}a^{15}-\frac{36\!\cdots\!16}{21\!\cdots\!81}a^{14}-\frac{54\!\cdots\!60}{21\!\cdots\!81}a^{13}+\frac{40\!\cdots\!39}{21\!\cdots\!81}a^{12}+\frac{18\!\cdots\!24}{21\!\cdots\!81}a^{11}-\frac{23\!\cdots\!36}{21\!\cdots\!81}a^{10}+\frac{25\!\cdots\!28}{21\!\cdots\!81}a^{9}+\frac{72\!\cdots\!49}{21\!\cdots\!81}a^{8}-\frac{29\!\cdots\!52}{21\!\cdots\!81}a^{7}-\frac{98\!\cdots\!46}{21\!\cdots\!81}a^{6}+\frac{49\!\cdots\!20}{21\!\cdots\!81}a^{5}+\frac{48\!\cdots\!81}{21\!\cdots\!81}a^{4}-\frac{18\!\cdots\!24}{21\!\cdots\!81}a^{3}-\frac{71\!\cdots\!88}{21\!\cdots\!81}a^{2}-\frac{36\!\cdots\!92}{21\!\cdots\!81}a-\frac{11\!\cdots\!32}{21\!\cdots\!81}$, $\frac{41\!\cdots\!20}{21\!\cdots\!81}a^{19}-\frac{21\!\cdots\!90}{21\!\cdots\!81}a^{18}-\frac{22\!\cdots\!60}{21\!\cdots\!81}a^{17}+\frac{12\!\cdots\!85}{21\!\cdots\!81}a^{16}+\frac{43\!\cdots\!56}{21\!\cdots\!81}a^{15}-\frac{28\!\cdots\!30}{21\!\cdots\!81}a^{14}-\frac{36\!\cdots\!20}{21\!\cdots\!81}a^{13}+\frac{31\!\cdots\!90}{21\!\cdots\!81}a^{12}+\frac{89\!\cdots\!40}{21\!\cdots\!81}a^{11}-\frac{19\!\cdots\!70}{21\!\cdots\!81}a^{10}+\frac{49\!\cdots\!20}{21\!\cdots\!81}a^{9}+\frac{59\!\cdots\!10}{21\!\cdots\!81}a^{8}-\frac{31\!\cdots\!80}{21\!\cdots\!81}a^{7}-\frac{88\!\cdots\!20}{21\!\cdots\!81}a^{6}+\frac{51\!\cdots\!28}{21\!\cdots\!81}a^{5}+\frac{53\!\cdots\!45}{21\!\cdots\!81}a^{4}-\frac{22\!\cdots\!60}{21\!\cdots\!81}a^{3}-\frac{13\!\cdots\!60}{21\!\cdots\!81}a^{2}+\frac{63\!\cdots\!40}{21\!\cdots\!81}a+\frac{70\!\cdots\!70}{21\!\cdots\!81}$, $\frac{25\!\cdots\!50}{21\!\cdots\!81}a^{19}-\frac{16\!\cdots\!04}{21\!\cdots\!81}a^{18}-\frac{12\!\cdots\!07}{21\!\cdots\!81}a^{17}+\frac{19\!\cdots\!19}{42\!\cdots\!62}a^{16}+\frac{20\!\cdots\!24}{21\!\cdots\!81}a^{15}-\frac{21\!\cdots\!10}{21\!\cdots\!81}a^{14}-\frac{97\!\cdots\!79}{21\!\cdots\!81}a^{13}+\frac{49\!\cdots\!37}{42\!\cdots\!62}a^{12}-\frac{81\!\cdots\!02}{21\!\cdots\!81}a^{11}-\frac{29\!\cdots\!63}{42\!\cdots\!62}a^{10}+\frac{10\!\cdots\!29}{21\!\cdots\!81}a^{9}+\frac{44\!\cdots\!28}{21\!\cdots\!81}a^{8}-\frac{43\!\cdots\!62}{21\!\cdots\!81}a^{7}-\frac{60\!\cdots\!53}{21\!\cdots\!81}a^{6}+\frac{66\!\cdots\!47}{21\!\cdots\!81}a^{5}+\frac{58\!\cdots\!43}{42\!\cdots\!62}a^{4}-\frac{35\!\cdots\!62}{21\!\cdots\!81}a^{3}-\frac{37\!\cdots\!03}{21\!\cdots\!81}a^{2}+\frac{51\!\cdots\!19}{21\!\cdots\!81}a+\frac{39\!\cdots\!94}{21\!\cdots\!81}$, $\frac{13\!\cdots\!78}{21\!\cdots\!81}a^{19}+\frac{20\!\cdots\!75}{21\!\cdots\!81}a^{18}-\frac{13\!\cdots\!22}{21\!\cdots\!81}a^{17}-\frac{25\!\cdots\!75}{42\!\cdots\!62}a^{16}+\frac{52\!\cdots\!98}{21\!\cdots\!81}a^{15}+\frac{31\!\cdots\!67}{21\!\cdots\!81}a^{14}-\frac{95\!\cdots\!35}{21\!\cdots\!81}a^{13}-\frac{80\!\cdots\!71}{42\!\cdots\!62}a^{12}+\frac{94\!\cdots\!46}{21\!\cdots\!81}a^{11}+\frac{27\!\cdots\!47}{21\!\cdots\!81}a^{10}-\frac{51\!\cdots\!16}{21\!\cdots\!81}a^{9}-\frac{21\!\cdots\!21}{42\!\cdots\!62}a^{8}+\frac{15\!\cdots\!84}{21\!\cdots\!81}a^{7}+\frac{21\!\cdots\!07}{21\!\cdots\!81}a^{6}-\frac{21\!\cdots\!97}{21\!\cdots\!81}a^{5}-\frac{21\!\cdots\!22}{21\!\cdots\!81}a^{4}+\frac{10\!\cdots\!58}{21\!\cdots\!81}a^{3}+\frac{15\!\cdots\!51}{42\!\cdots\!62}a^{2}-\frac{86\!\cdots\!44}{21\!\cdots\!81}a-\frac{16\!\cdots\!87}{42\!\cdots\!62}$, $\frac{13\!\cdots\!38}{21\!\cdots\!81}a^{19}-\frac{84\!\cdots\!85}{21\!\cdots\!81}a^{18}-\frac{68\!\cdots\!42}{21\!\cdots\!81}a^{17}+\frac{97\!\cdots\!75}{42\!\cdots\!62}a^{16}+\frac{12\!\cdots\!78}{21\!\cdots\!81}a^{15}-\frac{11\!\cdots\!53}{21\!\cdots\!81}a^{14}-\frac{90\!\cdots\!05}{21\!\cdots\!81}a^{13}+\frac{25\!\cdots\!69}{42\!\cdots\!62}a^{12}+\frac{34\!\cdots\!38}{21\!\cdots\!81}a^{11}-\frac{75\!\cdots\!13}{21\!\cdots\!81}a^{10}+\frac{28\!\cdots\!24}{21\!\cdots\!81}a^{9}+\frac{46\!\cdots\!99}{42\!\cdots\!62}a^{8}-\frac{13\!\cdots\!56}{21\!\cdots\!81}a^{7}-\frac{31\!\cdots\!33}{21\!\cdots\!81}a^{6}+\frac{18\!\cdots\!89}{21\!\cdots\!81}a^{5}+\frac{15\!\cdots\!58}{21\!\cdots\!81}a^{4}-\frac{68\!\cdots\!22}{21\!\cdots\!81}a^{3}-\frac{33\!\cdots\!07}{42\!\cdots\!62}a^{2}-\frac{13\!\cdots\!88}{21\!\cdots\!81}a-\frac{10\!\cdots\!55}{42\!\cdots\!62}$, $\frac{11\!\cdots\!54}{21\!\cdots\!81}a^{19}+\frac{64\!\cdots\!43}{42\!\cdots\!62}a^{18}-\frac{19\!\cdots\!36}{21\!\cdots\!81}a^{17}-\frac{18\!\cdots\!07}{21\!\cdots\!81}a^{16}+\frac{88\!\cdots\!30}{21\!\cdots\!81}a^{15}+\frac{83\!\cdots\!17}{42\!\cdots\!62}a^{14}-\frac{17\!\cdots\!31}{21\!\cdots\!81}a^{13}-\frac{46\!\cdots\!18}{21\!\cdots\!81}a^{12}+\frac{18\!\cdots\!28}{21\!\cdots\!81}a^{11}+\frac{51\!\cdots\!95}{42\!\cdots\!62}a^{10}-\frac{10\!\cdots\!97}{21\!\cdots\!81}a^{9}-\frac{64\!\cdots\!72}{21\!\cdots\!81}a^{8}+\frac{32\!\cdots\!68}{21\!\cdots\!81}a^{7}+\frac{42\!\cdots\!70}{21\!\cdots\!81}a^{6}-\frac{43\!\cdots\!35}{21\!\cdots\!81}a^{5}+\frac{63\!\cdots\!75}{42\!\cdots\!62}a^{4}+\frac{22\!\cdots\!36}{21\!\cdots\!81}a^{3}-\frac{25\!\cdots\!77}{42\!\cdots\!62}a^{2}-\frac{28\!\cdots\!08}{21\!\cdots\!81}a-\frac{21\!\cdots\!91}{21\!\cdots\!81}$, $\frac{27\!\cdots\!71}{42\!\cdots\!62}a^{19}-\frac{14\!\cdots\!11}{42\!\cdots\!62}a^{18}-\frac{71\!\cdots\!73}{21\!\cdots\!81}a^{17}+\frac{84\!\cdots\!17}{42\!\cdots\!62}a^{16}+\frac{27\!\cdots\!53}{42\!\cdots\!62}a^{15}-\frac{18\!\cdots\!55}{42\!\cdots\!62}a^{14}-\frac{11\!\cdots\!12}{21\!\cdots\!81}a^{13}+\frac{10\!\cdots\!78}{21\!\cdots\!81}a^{12}+\frac{47\!\cdots\!39}{42\!\cdots\!62}a^{11}-\frac{62\!\cdots\!26}{21\!\cdots\!81}a^{10}+\frac{19\!\cdots\!17}{21\!\cdots\!81}a^{9}+\frac{38\!\cdots\!61}{42\!\cdots\!62}a^{8}-\frac{11\!\cdots\!18}{21\!\cdots\!81}a^{7}-\frac{51\!\cdots\!47}{42\!\cdots\!62}a^{6}+\frac{36\!\cdots\!51}{42\!\cdots\!62}a^{5}+\frac{12\!\cdots\!29}{21\!\cdots\!81}a^{4}-\frac{17\!\cdots\!33}{42\!\cdots\!62}a^{3}-\frac{33\!\cdots\!35}{42\!\cdots\!62}a^{2}+\frac{68\!\cdots\!36}{21\!\cdots\!81}a+\frac{14\!\cdots\!35}{42\!\cdots\!62}$, $\frac{15\!\cdots\!47}{42\!\cdots\!62}a^{19}-\frac{58\!\cdots\!15}{42\!\cdots\!62}a^{18}-\frac{46\!\cdots\!00}{21\!\cdots\!81}a^{17}+\frac{17\!\cdots\!70}{21\!\cdots\!81}a^{16}+\frac{10\!\cdots\!21}{21\!\cdots\!81}a^{15}-\frac{38\!\cdots\!86}{21\!\cdots\!81}a^{14}-\frac{12\!\cdots\!95}{21\!\cdots\!81}a^{13}+\frac{45\!\cdots\!53}{21\!\cdots\!81}a^{12}+\frac{80\!\cdots\!99}{21\!\cdots\!81}a^{11}-\frac{28\!\cdots\!45}{21\!\cdots\!81}a^{10}-\frac{54\!\cdots\!29}{42\!\cdots\!62}a^{9}+\frac{19\!\cdots\!67}{42\!\cdots\!62}a^{8}+\frac{46\!\cdots\!22}{21\!\cdots\!81}a^{7}-\frac{17\!\cdots\!62}{21\!\cdots\!81}a^{6}-\frac{77\!\cdots\!41}{42\!\cdots\!62}a^{5}+\frac{29\!\cdots\!47}{42\!\cdots\!62}a^{4}+\frac{14\!\cdots\!61}{21\!\cdots\!81}a^{3}-\frac{47\!\cdots\!82}{21\!\cdots\!81}a^{2}-\frac{30\!\cdots\!89}{21\!\cdots\!81}a+\frac{33\!\cdots\!55}{21\!\cdots\!81}$, $\frac{43\!\cdots\!90}{21\!\cdots\!81}a^{19}-\frac{51\!\cdots\!91}{42\!\cdots\!62}a^{18}-\frac{20\!\cdots\!80}{19\!\cdots\!09}a^{17}+\frac{14\!\cdots\!84}{21\!\cdots\!81}a^{16}+\frac{41\!\cdots\!38}{21\!\cdots\!81}a^{15}-\frac{66\!\cdots\!25}{42\!\cdots\!62}a^{14}-\frac{28\!\cdots\!70}{21\!\cdots\!81}a^{13}+\frac{37\!\cdots\!53}{21\!\cdots\!81}a^{12}-\frac{72\!\cdots\!88}{21\!\cdots\!81}a^{11}-\frac{44\!\cdots\!27}{42\!\cdots\!62}a^{10}+\frac{10\!\cdots\!18}{21\!\cdots\!81}a^{9}+\frac{67\!\cdots\!76}{21\!\cdots\!81}a^{8}-\frac{50\!\cdots\!82}{21\!\cdots\!81}a^{7}-\frac{93\!\cdots\!08}{21\!\cdots\!81}a^{6}+\frac{77\!\cdots\!92}{21\!\cdots\!81}a^{5}+\frac{94\!\cdots\!55}{42\!\cdots\!62}a^{4}-\frac{36\!\cdots\!40}{21\!\cdots\!81}a^{3}-\frac{15\!\cdots\!31}{42\!\cdots\!62}a^{2}+\frac{28\!\cdots\!38}{21\!\cdots\!81}a+\frac{16\!\cdots\!33}{21\!\cdots\!81}$, $\frac{57\!\cdots\!70}{21\!\cdots\!81}a^{19}-\frac{49\!\cdots\!99}{42\!\cdots\!62}a^{18}-\frac{31\!\cdots\!89}{21\!\cdots\!81}a^{17}+\frac{27\!\cdots\!11}{42\!\cdots\!62}a^{16}+\frac{64\!\cdots\!36}{21\!\cdots\!81}a^{15}-\frac{30\!\cdots\!90}{21\!\cdots\!81}a^{14}-\frac{12\!\cdots\!65}{42\!\cdots\!62}a^{13}+\frac{32\!\cdots\!51}{21\!\cdots\!81}a^{12}+\frac{21\!\cdots\!32}{21\!\cdots\!81}a^{11}-\frac{37\!\cdots\!23}{42\!\cdots\!62}a^{10}+\frac{54\!\cdots\!23}{42\!\cdots\!62}a^{9}+\frac{51\!\cdots\!52}{21\!\cdots\!81}a^{8}-\frac{72\!\cdots\!23}{42\!\cdots\!62}a^{7}-\frac{11\!\cdots\!19}{42\!\cdots\!62}a^{6}+\frac{69\!\cdots\!87}{21\!\cdots\!81}a^{5}+\frac{13\!\cdots\!45}{21\!\cdots\!81}a^{4}-\frac{38\!\cdots\!10}{21\!\cdots\!81}a^{3}+\frac{35\!\cdots\!69}{21\!\cdots\!81}a^{2}+\frac{10\!\cdots\!85}{42\!\cdots\!62}a+\frac{41\!\cdots\!35}{42\!\cdots\!62}$, $\frac{88\!\cdots\!97}{21\!\cdots\!81}a^{19}-\frac{84\!\cdots\!17}{21\!\cdots\!81}a^{18}-\frac{76\!\cdots\!83}{42\!\cdots\!62}a^{17}+\frac{10\!\cdots\!03}{42\!\cdots\!62}a^{16}+\frac{90\!\cdots\!19}{42\!\cdots\!62}a^{15}-\frac{11\!\cdots\!87}{21\!\cdots\!81}a^{14}+\frac{19\!\cdots\!79}{21\!\cdots\!81}a^{13}+\frac{14\!\cdots\!14}{21\!\cdots\!81}a^{12}-\frac{80\!\cdots\!73}{21\!\cdots\!81}a^{11}-\frac{18\!\cdots\!99}{42\!\cdots\!62}a^{10}+\frac{62\!\cdots\!89}{21\!\cdots\!81}a^{9}+\frac{63\!\cdots\!47}{42\!\cdots\!62}a^{8}-\frac{39\!\cdots\!87}{42\!\cdots\!62}a^{7}-\frac{10\!\cdots\!93}{42\!\cdots\!62}a^{6}+\frac{48\!\cdots\!69}{42\!\cdots\!62}a^{5}+\frac{80\!\cdots\!55}{42\!\cdots\!62}a^{4}-\frac{15\!\cdots\!93}{42\!\cdots\!62}a^{3}-\frac{10\!\cdots\!86}{21\!\cdots\!81}a^{2}-\frac{18\!\cdots\!39}{42\!\cdots\!62}a-\frac{96\!\cdots\!39}{42\!\cdots\!62}$, $\frac{22\!\cdots\!78}{21\!\cdots\!81}a^{19}-\frac{83\!\cdots\!26}{21\!\cdots\!81}a^{18}-\frac{27\!\cdots\!93}{42\!\cdots\!62}a^{17}+\frac{96\!\cdots\!17}{42\!\cdots\!62}a^{16}+\frac{33\!\cdots\!06}{21\!\cdots\!81}a^{15}-\frac{10\!\cdots\!06}{21\!\cdots\!81}a^{14}-\frac{85\!\cdots\!19}{42\!\cdots\!62}a^{13}+\frac{25\!\cdots\!59}{42\!\cdots\!62}a^{12}+\frac{62\!\cdots\!21}{42\!\cdots\!62}a^{11}-\frac{15\!\cdots\!59}{42\!\cdots\!62}a^{10}-\frac{13\!\cdots\!87}{21\!\cdots\!81}a^{9}+\frac{23\!\cdots\!59}{21\!\cdots\!81}a^{8}+\frac{34\!\cdots\!99}{21\!\cdots\!81}a^{7}-\frac{34\!\cdots\!61}{21\!\cdots\!81}a^{6}-\frac{10\!\cdots\!65}{42\!\cdots\!62}a^{5}+\frac{29\!\cdots\!91}{42\!\cdots\!62}a^{4}+\frac{33\!\cdots\!95}{21\!\cdots\!81}a^{3}+\frac{15\!\cdots\!83}{21\!\cdots\!81}a^{2}-\frac{70\!\cdots\!72}{21\!\cdots\!81}a+\frac{20\!\cdots\!74}{21\!\cdots\!81}$, $\frac{17\!\cdots\!52}{21\!\cdots\!81}a^{19}-\frac{10\!\cdots\!69}{21\!\cdots\!81}a^{18}-\frac{97\!\cdots\!08}{21\!\cdots\!81}a^{17}+\frac{58\!\cdots\!77}{21\!\cdots\!81}a^{16}+\frac{20\!\cdots\!48}{21\!\cdots\!81}a^{15}-\frac{26\!\cdots\!19}{42\!\cdots\!62}a^{14}-\frac{20\!\cdots\!86}{21\!\cdots\!81}a^{13}+\frac{29\!\cdots\!43}{42\!\cdots\!62}a^{12}+\frac{10\!\cdots\!16}{21\!\cdots\!81}a^{11}-\frac{86\!\cdots\!08}{21\!\cdots\!81}a^{10}-\frac{28\!\cdots\!35}{21\!\cdots\!81}a^{9}+\frac{50\!\cdots\!27}{42\!\cdots\!62}a^{8}+\frac{46\!\cdots\!96}{21\!\cdots\!81}a^{7}-\frac{56\!\cdots\!87}{42\!\cdots\!62}a^{6}-\frac{99\!\cdots\!76}{21\!\cdots\!81}a^{5}+\frac{14\!\cdots\!71}{42\!\cdots\!62}a^{4}+\frac{11\!\cdots\!94}{21\!\cdots\!81}a^{3}+\frac{81\!\cdots\!97}{21\!\cdots\!81}a^{2}-\frac{21\!\cdots\!24}{21\!\cdots\!81}a-\frac{43\!\cdots\!86}{21\!\cdots\!81}$, $\frac{19\!\cdots\!62}{19\!\cdots\!09}a^{19}-\frac{24\!\cdots\!55}{42\!\cdots\!62}a^{18}-\frac{11\!\cdots\!62}{21\!\cdots\!81}a^{17}+\frac{14\!\cdots\!67}{42\!\cdots\!62}a^{16}+\frac{25\!\cdots\!88}{21\!\cdots\!81}a^{15}-\frac{33\!\cdots\!31}{42\!\cdots\!62}a^{14}-\frac{55\!\cdots\!01}{42\!\cdots\!62}a^{13}+\frac{20\!\cdots\!17}{21\!\cdots\!81}a^{12}+\frac{15\!\cdots\!30}{21\!\cdots\!81}a^{11}-\frac{26\!\cdots\!97}{42\!\cdots\!62}a^{10}-\frac{91\!\cdots\!13}{42\!\cdots\!62}a^{9}+\frac{48\!\cdots\!30}{21\!\cdots\!81}a^{8}+\frac{10\!\cdots\!69}{42\!\cdots\!62}a^{7}-\frac{18\!\cdots\!39}{42\!\cdots\!62}a^{6}-\frac{61\!\cdots\!92}{21\!\cdots\!81}a^{5}+\frac{16\!\cdots\!47}{42\!\cdots\!62}a^{4}-\frac{50\!\cdots\!06}{21\!\cdots\!81}a^{3}-\frac{28\!\cdots\!35}{21\!\cdots\!81}a^{2}+\frac{78\!\cdots\!87}{42\!\cdots\!62}a+\frac{23\!\cdots\!17}{21\!\cdots\!81}$, $\frac{37\!\cdots\!22}{21\!\cdots\!81}a^{19}-\frac{43\!\cdots\!75}{42\!\cdots\!62}a^{18}-\frac{18\!\cdots\!58}{21\!\cdots\!81}a^{17}+\frac{24\!\cdots\!95}{42\!\cdots\!62}a^{16}+\frac{31\!\cdots\!00}{21\!\cdots\!81}a^{15}-\frac{53\!\cdots\!09}{42\!\cdots\!62}a^{14}-\frac{14\!\cdots\!30}{21\!\cdots\!81}a^{13}+\frac{29\!\cdots\!41}{21\!\cdots\!81}a^{12}-\frac{13\!\cdots\!22}{21\!\cdots\!81}a^{11}-\frac{16\!\cdots\!52}{21\!\cdots\!81}a^{10}+\frac{17\!\cdots\!46}{21\!\cdots\!81}a^{9}+\frac{93\!\cdots\!61}{42\!\cdots\!62}a^{8}-\frac{71\!\cdots\!12}{21\!\cdots\!81}a^{7}-\frac{10\!\cdots\!99}{42\!\cdots\!62}a^{6}+\frac{11\!\cdots\!04}{21\!\cdots\!81}a^{5}+\frac{11\!\cdots\!62}{21\!\cdots\!81}a^{4}-\frac{56\!\cdots\!48}{21\!\cdots\!81}a^{3}+\frac{60\!\cdots\!07}{42\!\cdots\!62}a^{2}+\frac{63\!\cdots\!87}{21\!\cdots\!81}a-\frac{31\!\cdots\!13}{42\!\cdots\!62}$, $\frac{63\!\cdots\!96}{21\!\cdots\!81}a^{19}-\frac{73\!\cdots\!95}{42\!\cdots\!62}a^{18}-\frac{63\!\cdots\!03}{42\!\cdots\!62}a^{17}+\frac{21\!\cdots\!93}{21\!\cdots\!81}a^{16}+\frac{10\!\cdots\!53}{42\!\cdots\!62}a^{15}-\frac{46\!\cdots\!43}{21\!\cdots\!81}a^{14}-\frac{31\!\cdots\!85}{21\!\cdots\!81}a^{13}+\frac{51\!\cdots\!67}{21\!\cdots\!81}a^{12}-\frac{27\!\cdots\!01}{42\!\cdots\!62}a^{11}-\frac{29\!\cdots\!51}{21\!\cdots\!81}a^{10}+\frac{47\!\cdots\!05}{42\!\cdots\!62}a^{9}+\frac{17\!\cdots\!63}{42\!\cdots\!62}a^{8}-\frac{19\!\cdots\!13}{42\!\cdots\!62}a^{7}-\frac{10\!\cdots\!10}{21\!\cdots\!81}a^{6}+\frac{15\!\cdots\!56}{21\!\cdots\!81}a^{5}+\frac{73\!\cdots\!95}{42\!\cdots\!62}a^{4}-\frac{79\!\cdots\!77}{21\!\cdots\!81}a^{3}-\frac{45\!\cdots\!79}{21\!\cdots\!81}a^{2}+\frac{12\!\cdots\!34}{21\!\cdots\!81}a+\frac{17\!\cdots\!85}{21\!\cdots\!81}$
| 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: | \( 107373870215 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 107373870215 \cdot 5}{2\cdot\sqrt{1505680748169532571648000000000000000}}\cr\approx \mathstrut & 0.229388731839267 \end{aligned}\] (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 - 60*x^18 + 236*x^17 + 1435*x^16 - 5524*x^15 - 17726*x^14 + 66220*x^13 + 122474*x^12 - 437336*x^11 - 480740*x^10 + 1596028*x^9 + 1047359*x^8 - 3070872*x^7 - 1203606*x^6 + 2823768*x^5 + 614261*x^4 - 1007904*x^3 - 118608*x^2 + 70588*x + 1451)
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 - 60*x^18 + 236*x^17 + 1435*x^16 - 5524*x^15 - 17726*x^14 + 66220*x^13 + 122474*x^12 - 437336*x^11 - 480740*x^10 + 1596028*x^9 + 1047359*x^8 - 3070872*x^7 - 1203606*x^6 + 2823768*x^5 + 614261*x^4 - 1007904*x^3 - 118608*x^2 + 70588*x + 1451, 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 - 60*x^18 + 236*x^17 + 1435*x^16 - 5524*x^15 - 17726*x^14 + 66220*x^13 + 122474*x^12 - 437336*x^11 - 480740*x^10 + 1596028*x^9 + 1047359*x^8 - 3070872*x^7 - 1203606*x^6 + 2823768*x^5 + 614261*x^4 - 1007904*x^3 - 118608*x^2 + 70588*x + 1451);
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 - 60*x^18 + 236*x^17 + 1435*x^16 - 5524*x^15 - 17726*x^14 + 66220*x^13 + 122474*x^12 - 437336*x^11 - 480740*x^10 + 1596028*x^9 + 1047359*x^8 - 3070872*x^7 - 1203606*x^6 + 2823768*x^5 + 614261*x^4 - 1007904*x^3 - 118608*x^2 + 70588*x + 1451);
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
| \(\Q(\sqrt{5}) \), 4.4.8000.1, \(\Q(\zeta_{11})^+\), 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 | R | $20$ | R | $20$ | R | $20$ | $20$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $20$ | $2$ | $10$ | $30$ | |||
|
\(5\)
| Deg $20$ | $4$ | $5$ | $15$ | |||
|
\(11\)
| 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 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |