Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 40*x^18 - 124*x^17 + 815*x^16 - 2164*x^15 + 10834*x^14 - 24620*x^13 + 101774*x^12 - 198776*x^11 + 710860*x^10 - 1188452*x^9 + 3826719*x^8 - 5429272*x^7 + 15991074*x^6 - 18262392*x^5 + 47151361*x^4 - 38741504*x^3 + 92992732*x^2 - 43101732*x + 90594151)
gp: K = bnfinit(y^20 - 4*y^19 + 40*y^18 - 124*y^17 + 815*y^16 - 2164*y^15 + 10834*y^14 - 24620*y^13 + 101774*y^12 - 198776*y^11 + 710860*y^10 - 1188452*y^9 + 3826719*y^8 - 5429272*y^7 + 15991074*y^6 - 18262392*y^5 + 47151361*y^4 - 38741504*y^3 + 92992732*y^2 - 43101732*y + 90594151, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 + 40*x^18 - 124*x^17 + 815*x^16 - 2164*x^15 + 10834*x^14 - 24620*x^13 + 101774*x^12 - 198776*x^11 + 710860*x^10 - 1188452*x^9 + 3826719*x^8 - 5429272*x^7 + 15991074*x^6 - 18262392*x^5 + 47151361*x^4 - 38741504*x^3 + 92992732*x^2 - 43101732*x + 90594151);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 + 40*x^18 - 124*x^17 + 815*x^16 - 2164*x^15 + 10834*x^14 - 24620*x^13 + 101774*x^12 - 198776*x^11 + 710860*x^10 - 1188452*x^9 + 3826719*x^8 - 5429272*x^7 + 15991074*x^6 - 18262392*x^5 + 47151361*x^4 - 38741504*x^3 + 92992732*x^2 - 43101732*x + 90594151)
\( x^{20} - 4 x^{19} + 40 x^{18} - 124 x^{17} + 815 x^{16} - 2164 x^{15} + 10834 x^{14} - 24620 x^{13} + \cdots + 90594151 \)
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: | $[0, 10]$ | 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$$\chi_{440}(1,·)$, $\chi_{440}(133,·)$, $\chi_{440}(201,·)$, $\chi_{440}(397,·)$, $\chi_{440}(333,·)$, $\chi_{440}(81,·)$, $\chi_{440}(37,·)$, $\chi_{440}(213,·)$, $\chi_{440}(89,·)$, $\chi_{440}(93,·)$, $\chi_{440}(361,·)$, $\chi_{440}(289,·)$, $\chi_{440}(357,·)$, $\chi_{440}(401,·)$, $\chi_{440}(169,·)$, $\chi_{440}(157,·)$, $\chi_{440}(49,·)$, $\chi_{440}(53,·)$, $\chi_{440}(9,·)$, $\chi_{440}(317,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{12\!\cdots\!82}a^{18}-\frac{43\!\cdots\!93}{12\!\cdots\!82}a^{17}-\frac{57\!\cdots\!87}{12\!\cdots\!82}a^{16}+\frac{53\!\cdots\!17}{64\!\cdots\!41}a^{15}-\frac{21\!\cdots\!39}{12\!\cdots\!82}a^{14}-\frac{32\!\cdots\!89}{12\!\cdots\!82}a^{13}-\frac{62\!\cdots\!49}{64\!\cdots\!41}a^{12}-\frac{12\!\cdots\!33}{12\!\cdots\!82}a^{11}+\frac{17\!\cdots\!87}{64\!\cdots\!41}a^{10}-\frac{70\!\cdots\!39}{64\!\cdots\!41}a^{9}-\frac{51\!\cdots\!25}{12\!\cdots\!82}a^{8}-\frac{18\!\cdots\!31}{64\!\cdots\!41}a^{7}-\frac{55\!\cdots\!83}{12\!\cdots\!82}a^{6}-\frac{21\!\cdots\!83}{12\!\cdots\!82}a^{5}+\frac{52\!\cdots\!09}{64\!\cdots\!41}a^{4}-\frac{30\!\cdots\!11}{64\!\cdots\!41}a^{3}-\frac{54\!\cdots\!23}{12\!\cdots\!82}a^{2}+\frac{17\!\cdots\!70}{64\!\cdots\!41}a-\frac{49\!\cdots\!53}{11\!\cdots\!98}$, $\frac{1}{11\!\cdots\!42}a^{19}+\frac{24\!\cdots\!41}{11\!\cdots\!42}a^{18}+\frac{18\!\cdots\!86}{59\!\cdots\!21}a^{17}+\frac{15\!\cdots\!95}{11\!\cdots\!42}a^{16}+\frac{10\!\cdots\!02}{59\!\cdots\!21}a^{15}+\frac{12\!\cdots\!43}{11\!\cdots\!42}a^{14}+\frac{88\!\cdots\!83}{59\!\cdots\!21}a^{13}+\frac{24\!\cdots\!39}{11\!\cdots\!42}a^{12}-\frac{97\!\cdots\!71}{11\!\cdots\!42}a^{11}-\frac{36\!\cdots\!64}{59\!\cdots\!21}a^{10}+\frac{16\!\cdots\!96}{59\!\cdots\!21}a^{9}+\frac{10\!\cdots\!57}{59\!\cdots\!21}a^{8}-\frac{28\!\cdots\!06}{59\!\cdots\!21}a^{7}-\frac{28\!\cdots\!39}{59\!\cdots\!21}a^{6}+\frac{16\!\cdots\!76}{59\!\cdots\!21}a^{5}+\frac{19\!\cdots\!55}{59\!\cdots\!21}a^{4}+\frac{47\!\cdots\!19}{11\!\cdots\!42}a^{3}-\frac{40\!\cdots\!99}{11\!\cdots\!42}a^{2}+\frac{22\!\cdots\!01}{11\!\cdots\!42}a+\frac{24\!\cdots\!91}{54\!\cdots\!69}$
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
$C_{22082}$, which has order $22082$ (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: | $9$ | 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: |
$\frac{404924790182}{64\!\cdots\!41}a^{19}+\frac{1764075181645}{64\!\cdots\!41}a^{18}+\frac{1618421850242}{64\!\cdots\!41}a^{17}+\frac{107520892245065}{12\!\cdots\!82}a^{16}-\frac{11594532537538}{64\!\cdots\!41}a^{15}+\frac{911820723687953}{64\!\cdots\!41}a^{14}-\frac{875504250517845}{64\!\cdots\!41}a^{13}+\frac{20\!\cdots\!91}{12\!\cdots\!82}a^{12}-\frac{94\!\cdots\!06}{64\!\cdots\!41}a^{11}+\frac{79\!\cdots\!03}{64\!\cdots\!41}a^{10}-\frac{73\!\cdots\!84}{64\!\cdots\!41}a^{9}+\frac{94\!\cdots\!61}{12\!\cdots\!82}a^{8}-\frac{36\!\cdots\!84}{64\!\cdots\!41}a^{7}+\frac{21\!\cdots\!53}{64\!\cdots\!41}a^{6}-\frac{14\!\cdots\!39}{64\!\cdots\!41}a^{5}+\frac{71\!\cdots\!12}{64\!\cdots\!41}a^{4}-\frac{33\!\cdots\!38}{64\!\cdots\!41}a^{3}+\frac{31\!\cdots\!29}{12\!\cdots\!82}a^{2}-\frac{41\!\cdots\!36}{64\!\cdots\!41}a+\frac{10\!\cdots\!53}{11\!\cdots\!98}$, $\frac{32\!\cdots\!80}{59\!\cdots\!21}a^{19}+\frac{11\!\cdots\!90}{59\!\cdots\!21}a^{18}-\frac{20\!\cdots\!20}{59\!\cdots\!21}a^{17}+\frac{38\!\cdots\!90}{59\!\cdots\!21}a^{16}-\frac{66\!\cdots\!76}{59\!\cdots\!21}a^{15}+\frac{72\!\cdots\!70}{59\!\cdots\!21}a^{14}-\frac{11\!\cdots\!40}{59\!\cdots\!21}a^{13}+\frac{89\!\cdots\!90}{59\!\cdots\!21}a^{12}-\frac{11\!\cdots\!00}{59\!\cdots\!21}a^{11}+\frac{77\!\cdots\!30}{59\!\cdots\!21}a^{10}-\frac{83\!\cdots\!20}{59\!\cdots\!21}a^{9}+\frac{50\!\cdots\!90}{59\!\cdots\!21}a^{8}-\frac{43\!\cdots\!60}{59\!\cdots\!21}a^{7}+\frac{25\!\cdots\!40}{59\!\cdots\!21}a^{6}-\frac{16\!\cdots\!12}{59\!\cdots\!21}a^{5}+\frac{98\!\cdots\!75}{59\!\cdots\!21}a^{4}-\frac{42\!\cdots\!40}{59\!\cdots\!21}a^{3}+\frac{23\!\cdots\!20}{59\!\cdots\!21}a^{2}-\frac{52\!\cdots\!20}{59\!\cdots\!21}a+\frac{31\!\cdots\!60}{54\!\cdots\!69}$, $\frac{31\!\cdots\!04}{59\!\cdots\!21}a^{19}+\frac{72\!\cdots\!60}{59\!\cdots\!21}a^{18}-\frac{51\!\cdots\!16}{59\!\cdots\!21}a^{17}+\frac{24\!\cdots\!75}{59\!\cdots\!21}a^{16}-\frac{22\!\cdots\!52}{59\!\cdots\!21}a^{15}+\frac{46\!\cdots\!96}{59\!\cdots\!21}a^{14}-\frac{41\!\cdots\!80}{59\!\cdots\!21}a^{13}+\frac{58\!\cdots\!96}{59\!\cdots\!21}a^{12}-\frac{43\!\cdots\!44}{59\!\cdots\!21}a^{11}+\frac{51\!\cdots\!12}{59\!\cdots\!21}a^{10}-\frac{29\!\cdots\!08}{59\!\cdots\!21}a^{9}+\frac{33\!\cdots\!31}{59\!\cdots\!21}a^{8}-\frac{13\!\cdots\!88}{59\!\cdots\!21}a^{7}+\frac{16\!\cdots\!56}{59\!\cdots\!21}a^{6}-\frac{35\!\cdots\!40}{59\!\cdots\!21}a^{5}+\frac{61\!\cdots\!24}{59\!\cdots\!21}a^{4}-\frac{10\!\cdots\!16}{59\!\cdots\!21}a^{3}+\frac{14\!\cdots\!08}{59\!\cdots\!21}a^{2}+\frac{60\!\cdots\!72}{59\!\cdots\!21}a+\frac{12\!\cdots\!31}{54\!\cdots\!69}$, $\frac{13\!\cdots\!96}{59\!\cdots\!21}a^{19}+\frac{21\!\cdots\!80}{59\!\cdots\!21}a^{18}-\frac{12\!\cdots\!04}{59\!\cdots\!21}a^{17}+\frac{10\!\cdots\!00}{59\!\cdots\!21}a^{16}-\frac{46\!\cdots\!08}{59\!\cdots\!21}a^{15}+\frac{21\!\cdots\!24}{59\!\cdots\!21}a^{14}-\frac{85\!\cdots\!80}{59\!\cdots\!21}a^{13}+\frac{29\!\cdots\!79}{59\!\cdots\!21}a^{12}-\frac{98\!\cdots\!76}{59\!\cdots\!21}a^{11}+\frac{26\!\cdots\!24}{59\!\cdots\!21}a^{10}-\frac{78\!\cdots\!32}{59\!\cdots\!21}a^{9}+\frac{17\!\cdots\!89}{59\!\cdots\!21}a^{8}-\frac{44\!\cdots\!12}{59\!\cdots\!21}a^{7}+\frac{86\!\cdots\!74}{59\!\cdots\!21}a^{6}-\frac{18\!\cdots\!24}{59\!\cdots\!21}a^{5}+\frac{32\!\cdots\!01}{59\!\cdots\!21}a^{4}-\frac{53\!\cdots\!84}{59\!\cdots\!21}a^{3}+\frac{75\!\cdots\!92}{59\!\cdots\!21}a^{2}-\frac{69\!\cdots\!92}{59\!\cdots\!21}a+\frac{85\!\cdots\!02}{54\!\cdots\!69}$, $\frac{81\!\cdots\!76}{59\!\cdots\!21}a^{19}+\frac{37\!\cdots\!30}{59\!\cdots\!21}a^{18}-\frac{15\!\cdots\!04}{59\!\cdots\!21}a^{17}+\frac{13\!\cdots\!15}{59\!\cdots\!21}a^{16}-\frac{43\!\cdots\!24}{59\!\cdots\!21}a^{15}+\frac{25\!\cdots\!74}{59\!\cdots\!21}a^{14}-\frac{69\!\cdots\!60}{59\!\cdots\!21}a^{13}+\frac{31\!\cdots\!94}{59\!\cdots\!21}a^{12}-\frac{72\!\cdots\!56}{59\!\cdots\!21}a^{11}+\frac{26\!\cdots\!18}{59\!\cdots\!21}a^{10}-\frac{53\!\cdots\!12}{59\!\cdots\!21}a^{9}+\frac{17\!\cdots\!59}{59\!\cdots\!21}a^{8}-\frac{30\!\cdots\!72}{59\!\cdots\!21}a^{7}+\frac{88\!\cdots\!84}{59\!\cdots\!21}a^{6}-\frac{13\!\cdots\!72}{59\!\cdots\!21}a^{5}+\frac{36\!\cdots\!51}{59\!\cdots\!21}a^{4}-\frac{42\!\cdots\!24}{59\!\cdots\!21}a^{3}+\frac{91\!\cdots\!12}{59\!\cdots\!21}a^{2}-\frac{58\!\cdots\!92}{59\!\cdots\!21}a+\frac{14\!\cdots\!60}{54\!\cdots\!69}$, $\frac{51\!\cdots\!82}{59\!\cdots\!21}a^{19}+\frac{11\!\cdots\!65}{59\!\cdots\!21}a^{18}-\frac{31\!\cdots\!38}{59\!\cdots\!21}a^{17}+\frac{86\!\cdots\!75}{11\!\cdots\!42}a^{16}-\frac{10\!\cdots\!38}{59\!\cdots\!21}a^{15}+\frac{84\!\cdots\!93}{59\!\cdots\!21}a^{14}-\frac{18\!\cdots\!85}{59\!\cdots\!21}a^{13}+\frac{21\!\cdots\!91}{11\!\cdots\!42}a^{12}-\frac{20\!\cdots\!86}{59\!\cdots\!21}a^{11}+\frac{85\!\cdots\!07}{54\!\cdots\!69}a^{10}-\frac{15\!\cdots\!04}{59\!\cdots\!21}a^{9}+\frac{12\!\cdots\!81}{11\!\cdots\!42}a^{8}-\frac{82\!\cdots\!44}{59\!\cdots\!21}a^{7}+\frac{29\!\cdots\!13}{59\!\cdots\!21}a^{6}-\frac{33\!\cdots\!03}{59\!\cdots\!21}a^{5}+\frac{10\!\cdots\!72}{59\!\cdots\!21}a^{4}-\frac{91\!\cdots\!98}{59\!\cdots\!21}a^{3}+\frac{51\!\cdots\!89}{11\!\cdots\!42}a^{2}-\frac{11\!\cdots\!16}{59\!\cdots\!21}a+\frac{66\!\cdots\!73}{10\!\cdots\!38}$, $\frac{19\!\cdots\!78}{59\!\cdots\!21}a^{19}+\frac{44\!\cdots\!05}{59\!\cdots\!21}a^{18}-\frac{26\!\cdots\!22}{59\!\cdots\!21}a^{17}+\frac{37\!\cdots\!25}{11\!\cdots\!42}a^{16}-\frac{82\!\cdots\!86}{59\!\cdots\!21}a^{15}+\frac{37\!\cdots\!97}{59\!\cdots\!21}a^{14}-\frac{14\!\cdots\!05}{59\!\cdots\!21}a^{13}+\frac{95\!\cdots\!99}{11\!\cdots\!42}a^{12}-\frac{15\!\cdots\!42}{59\!\cdots\!21}a^{11}+\frac{41\!\cdots\!51}{59\!\cdots\!21}a^{10}-\frac{12\!\cdots\!96}{59\!\cdots\!21}a^{9}+\frac{54\!\cdots\!19}{11\!\cdots\!42}a^{8}-\frac{69\!\cdots\!56}{59\!\cdots\!21}a^{7}+\frac{13\!\cdots\!57}{59\!\cdots\!21}a^{6}-\frac{29\!\cdots\!63}{59\!\cdots\!21}a^{5}+\frac{47\!\cdots\!48}{59\!\cdots\!21}a^{4}-\frac{90\!\cdots\!82}{59\!\cdots\!21}a^{3}+\frac{21\!\cdots\!73}{11\!\cdots\!42}a^{2}-\frac{12\!\cdots\!88}{59\!\cdots\!21}a+\frac{41\!\cdots\!11}{10\!\cdots\!38}$, $\frac{66\!\cdots\!82}{59\!\cdots\!21}a^{19}+\frac{22\!\cdots\!25}{59\!\cdots\!21}a^{18}-\frac{13\!\cdots\!18}{59\!\cdots\!21}a^{17}+\frac{17\!\cdots\!25}{11\!\cdots\!42}a^{16}-\frac{35\!\cdots\!78}{59\!\cdots\!21}a^{15}+\frac{15\!\cdots\!73}{59\!\cdots\!21}a^{14}-\frac{58\!\cdots\!25}{59\!\cdots\!21}a^{13}+\frac{36\!\cdots\!41}{11\!\cdots\!42}a^{12}-\frac{59\!\cdots\!66}{59\!\cdots\!21}a^{11}+\frac{15\!\cdots\!27}{59\!\cdots\!21}a^{10}-\frac{44\!\cdots\!64}{59\!\cdots\!21}a^{9}+\frac{18\!\cdots\!41}{11\!\cdots\!42}a^{8}-\frac{25\!\cdots\!44}{59\!\cdots\!21}a^{7}+\frac{43\!\cdots\!83}{59\!\cdots\!21}a^{6}-\frac{11\!\cdots\!39}{59\!\cdots\!21}a^{5}+\frac{15\!\cdots\!47}{59\!\cdots\!21}a^{4}-\frac{37\!\cdots\!98}{59\!\cdots\!21}a^{3}+\frac{66\!\cdots\!89}{11\!\cdots\!42}a^{2}-\frac{53\!\cdots\!96}{59\!\cdots\!21}a+\frac{13\!\cdots\!69}{10\!\cdots\!38}$, $\frac{75\!\cdots\!30}{59\!\cdots\!21}a^{19}+\frac{58\!\cdots\!86}{59\!\cdots\!21}a^{18}-\frac{15\!\cdots\!65}{59\!\cdots\!21}a^{17}+\frac{23\!\cdots\!01}{59\!\cdots\!21}a^{16}-\frac{30\!\cdots\!90}{59\!\cdots\!21}a^{15}+\frac{97\!\cdots\!25}{11\!\cdots\!42}a^{14}-\frac{70\!\cdots\!01}{59\!\cdots\!21}a^{13}+\frac{12\!\cdots\!77}{11\!\cdots\!42}a^{12}-\frac{84\!\cdots\!66}{59\!\cdots\!21}a^{11}+\frac{57\!\cdots\!14}{59\!\cdots\!21}a^{10}-\frac{67\!\cdots\!61}{59\!\cdots\!21}a^{9}+\frac{76\!\cdots\!51}{11\!\cdots\!42}a^{8}-\frac{37\!\cdots\!86}{59\!\cdots\!21}a^{7}+\frac{38\!\cdots\!41}{11\!\cdots\!42}a^{6}-\frac{16\!\cdots\!00}{59\!\cdots\!21}a^{5}+\frac{14\!\cdots\!87}{11\!\cdots\!42}a^{4}-\frac{49\!\cdots\!18}{59\!\cdots\!21}a^{3}+\frac{17\!\cdots\!24}{59\!\cdots\!21}a^{2}-\frac{66\!\cdots\!03}{59\!\cdots\!21}a+\frac{20\!\cdots\!67}{54\!\cdots\!69}$
| 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: | \( 140644.599182 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 140644.599182 \cdot 22082}{2\cdot\sqrt{1505680748169532571648000000000000000}}\cr\approx \mathstrut & 0.121356684930 \end{aligned}\] (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 + 40*x^18 - 124*x^17 + 815*x^16 - 2164*x^15 + 10834*x^14 - 24620*x^13 + 101774*x^12 - 198776*x^11 + 710860*x^10 - 1188452*x^9 + 3826719*x^8 - 5429272*x^7 + 15991074*x^6 - 18262392*x^5 + 47151361*x^4 - 38741504*x^3 + 92992732*x^2 - 43101732*x + 90594151)
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 + 40*x^18 - 124*x^17 + 815*x^16 - 2164*x^15 + 10834*x^14 - 24620*x^13 + 101774*x^12 - 198776*x^11 + 710860*x^10 - 1188452*x^9 + 3826719*x^8 - 5429272*x^7 + 15991074*x^6 - 18262392*x^5 + 47151361*x^4 - 38741504*x^3 + 92992732*x^2 - 43101732*x + 90594151, 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 + 40*x^18 - 124*x^17 + 815*x^16 - 2164*x^15 + 10834*x^14 - 24620*x^13 + 101774*x^12 - 198776*x^11 + 710860*x^10 - 1188452*x^9 + 3826719*x^8 - 5429272*x^7 + 15991074*x^6 - 18262392*x^5 + 47151361*x^4 - 38741504*x^3 + 92992732*x^2 - 43101732*x + 90594151);
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 + 40*x^18 - 124*x^17 + 815*x^16 - 2164*x^15 + 10834*x^14 - 24620*x^13 + 101774*x^12 - 198776*x^11 + 710860*x^10 - 1188452*x^9 + 3826719*x^8 - 5429272*x^7 + 15991074*x^6 - 18262392*x^5 + 47151361*x^4 - 38741504*x^3 + 92992732*x^2 - 43101732*x + 90594151);
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
| \(\Q(\sqrt{5}) \), 4.0.8000.2, \(\Q(\zeta_{11})^+\), 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 | $20$ | $20$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 $20$ | $2$ | $10$ | $30$ | |||
|
\(5\)
| 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 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |