Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 65*x^18 - 118*x^17 + 2550*x^16 - 4024*x^15 + 72661*x^14 - 96362*x^13 + 1573687*x^12 - 1687702*x^11 + 26178289*x^10 - 22501622*x^9 + 332045847*x^8 - 225430718*x^7 + 3115413077*x^6 - 1566380644*x^5 + 20342958970*x^4 - 6430154154*x^3 + 82545355413*x^2 - 10933110054*x + 156709628451)
gp: K = bnfinit(y^20 - 2*y^19 + 65*y^18 - 118*y^17 + 2550*y^16 - 4024*y^15 + 72661*y^14 - 96362*y^13 + 1573687*y^12 - 1687702*y^11 + 26178289*y^10 - 22501622*y^9 + 332045847*y^8 - 225430718*y^7 + 3115413077*y^6 - 1566380644*y^5 + 20342958970*y^4 - 6430154154*y^3 + 82545355413*y^2 - 10933110054*y + 156709628451, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 65*x^18 - 118*x^17 + 2550*x^16 - 4024*x^15 + 72661*x^14 - 96362*x^13 + 1573687*x^12 - 1687702*x^11 + 26178289*x^10 - 22501622*x^9 + 332045847*x^8 - 225430718*x^7 + 3115413077*x^6 - 1566380644*x^5 + 20342958970*x^4 - 6430154154*x^3 + 82545355413*x^2 - 10933110054*x + 156709628451);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 + 65*x^18 - 118*x^17 + 2550*x^16 - 4024*x^15 + 72661*x^14 - 96362*x^13 + 1573687*x^12 - 1687702*x^11 + 26178289*x^10 - 22501622*x^9 + 332045847*x^8 - 225430718*x^7 + 3115413077*x^6 - 1566380644*x^5 + 20342958970*x^4 - 6430154154*x^3 + 82545355413*x^2 - 10933110054*x + 156709628451)
\( x^{20} - 2 x^{19} + 65 x^{18} - 118 x^{17} + 2550 x^{16} - 4024 x^{15} + 72661 x^{14} + \cdots + 156709628451 \)
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: |
\(1123852547182106083132474859412520960000000000\)
\(\medspace = 2^{30}\cdot 5^{10}\cdot 41^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(178.87\) | 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: | $2^{3/2}5^{1/2}41^{9/10}\approx 178.86915968604137$ | ||
| Ramified primes: |
\(2\), \(5\), \(41\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1640=2^{3}\cdot 5\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1640}(1,·)$, $\chi_{1640}(939,·)$, $\chi_{1640}(1521,·)$, $\chi_{1640}(961,·)$, $\chi_{1640}(201,·)$, $\chi_{1640}(139,·)$, $\chi_{1640}(1281,·)$, $\chi_{1640}(81,·)$, $\chi_{1640}(441,·)$, $\chi_{1640}(1179,·)$, $\chi_{1640}(1499,·)$, $\chi_{1640}(739,·)$, $\chi_{1640}(1041,·)$, $\chi_{1640}(681,·)$, $\chi_{1640}(619,·)$, $\chi_{1640}(59,·)$, $\chi_{1640}(819,·)$, $\chi_{1640}(761,·)$, $\chi_{1640}(379,·)$, $\chi_{1640}(1419,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{14}+\frac{1}{9}a^{13}-\frac{1}{9}a^{12}+\frac{1}{9}a^{10}-\frac{4}{9}a^{7}+\frac{2}{9}a^{6}+\frac{2}{9}a^{5}+\frac{1}{9}a^{4}+\frac{1}{3}a^{3}-\frac{4}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{1}{9}a^{10}-\frac{4}{9}a^{8}-\frac{1}{3}a^{7}-\frac{4}{9}a^{5}+\frac{2}{9}a^{4}+\frac{2}{9}a^{3}+\frac{1}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{14}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}-\frac{1}{9}a^{11}-\frac{1}{9}a^{9}-\frac{1}{3}a^{8}-\frac{4}{9}a^{6}+\frac{2}{9}a^{5}+\frac{2}{9}a^{4}+\frac{1}{9}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{2241}a^{18}+\frac{43}{2241}a^{17}-\frac{31}{2241}a^{16}+\frac{47}{2241}a^{15}+\frac{49}{747}a^{14}+\frac{11}{2241}a^{13}-\frac{305}{2241}a^{12}-\frac{209}{2241}a^{11}-\frac{242}{2241}a^{10}+\frac{20}{2241}a^{9}+\frac{1048}{2241}a^{8}+\frac{76}{2241}a^{7}-\frac{18}{83}a^{6}+\frac{253}{2241}a^{5}+\frac{485}{2241}a^{4}+\frac{176}{2241}a^{3}-\frac{437}{2241}a^{2}-\frac{316}{747}a-\frac{67}{249}$, $\frac{1}{11\!\cdots\!81}a^{19}-\frac{19\!\cdots\!26}{11\!\cdots\!81}a^{18}+\frac{51\!\cdots\!61}{11\!\cdots\!81}a^{17}-\frac{38\!\cdots\!21}{11\!\cdots\!81}a^{16}+\frac{43\!\cdots\!29}{36\!\cdots\!27}a^{15}-\frac{15\!\cdots\!91}{11\!\cdots\!81}a^{14}+\frac{14\!\cdots\!54}{11\!\cdots\!81}a^{13}-\frac{15\!\cdots\!34}{11\!\cdots\!81}a^{12}-\frac{43\!\cdots\!51}{11\!\cdots\!81}a^{11}+\frac{24\!\cdots\!78}{11\!\cdots\!81}a^{10}+\frac{51\!\cdots\!24}{11\!\cdots\!81}a^{9}+\frac{53\!\cdots\!21}{11\!\cdots\!81}a^{8}+\frac{47\!\cdots\!80}{13\!\cdots\!01}a^{7}-\frac{44\!\cdots\!88}{11\!\cdots\!81}a^{6}-\frac{42\!\cdots\!69}{11\!\cdots\!81}a^{5}-\frac{53\!\cdots\!83}{11\!\cdots\!81}a^{4}+\frac{37\!\cdots\!11}{11\!\cdots\!81}a^{3}-\frac{89\!\cdots\!70}{36\!\cdots\!27}a^{2}+\frac{11\!\cdots\!30}{40\!\cdots\!03}a-\frac{74\!\cdots\!95}{40\!\cdots\!03}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{2}\times C_{2}\times C_{15053420}$, which has order $60213680$ (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: |
$\frac{13\!\cdots\!26}{11\!\cdots\!81}a^{19}+\frac{44\!\cdots\!36}{11\!\cdots\!81}a^{18}+\frac{51\!\cdots\!18}{11\!\cdots\!81}a^{17}+\frac{30\!\cdots\!24}{11\!\cdots\!81}a^{16}+\frac{50\!\cdots\!74}{36\!\cdots\!27}a^{15}+\frac{12\!\cdots\!16}{11\!\cdots\!81}a^{14}+\frac{32\!\cdots\!42}{11\!\cdots\!81}a^{13}+\frac{34\!\cdots\!76}{11\!\cdots\!81}a^{12}+\frac{51\!\cdots\!06}{11\!\cdots\!81}a^{11}+\frac{74\!\cdots\!96}{11\!\cdots\!81}a^{10}+\frac{59\!\cdots\!92}{11\!\cdots\!81}a^{9}+\frac{11\!\cdots\!22}{11\!\cdots\!81}a^{8}+\frac{13\!\cdots\!52}{36\!\cdots\!27}a^{7}+\frac{13\!\cdots\!48}{11\!\cdots\!81}a^{6}+\frac{67\!\cdots\!14}{11\!\cdots\!81}a^{5}+\frac{11\!\cdots\!88}{11\!\cdots\!81}a^{4}-\frac{10\!\cdots\!50}{11\!\cdots\!81}a^{3}+\frac{61\!\cdots\!17}{12\!\cdots\!09}a^{2}-\frac{51\!\cdots\!56}{12\!\cdots\!09}a+\frac{47\!\cdots\!70}{40\!\cdots\!03}$, $\frac{21\!\cdots\!48}{13\!\cdots\!07}a^{19}-\frac{41\!\cdots\!39}{13\!\cdots\!07}a^{18}+\frac{14\!\cdots\!98}{13\!\cdots\!07}a^{17}-\frac{24\!\cdots\!65}{13\!\cdots\!07}a^{16}+\frac{18\!\cdots\!48}{44\!\cdots\!69}a^{15}-\frac{84\!\cdots\!97}{13\!\cdots\!07}a^{14}+\frac{16\!\cdots\!80}{13\!\cdots\!07}a^{13}-\frac{20\!\cdots\!49}{13\!\cdots\!07}a^{12}+\frac{35\!\cdots\!24}{13\!\cdots\!07}a^{11}-\frac{35\!\cdots\!77}{13\!\cdots\!07}a^{10}+\frac{60\!\cdots\!82}{13\!\cdots\!07}a^{9}-\frac{47\!\cdots\!50}{13\!\cdots\!07}a^{8}+\frac{87\!\cdots\!72}{14\!\cdots\!23}a^{7}-\frac{47\!\cdots\!72}{13\!\cdots\!07}a^{6}+\frac{77\!\cdots\!10}{13\!\cdots\!07}a^{5}-\frac{33\!\cdots\!83}{13\!\cdots\!07}a^{4}+\frac{56\!\cdots\!52}{13\!\cdots\!07}a^{3}-\frac{46\!\cdots\!41}{44\!\cdots\!69}a^{2}+\frac{25\!\cdots\!10}{16\!\cdots\!47}a-\frac{89\!\cdots\!98}{49\!\cdots\!41}$, $\frac{78\!\cdots\!76}{11\!\cdots\!81}a^{19}+\frac{53\!\cdots\!40}{11\!\cdots\!81}a^{18}+\frac{19\!\cdots\!32}{11\!\cdots\!81}a^{17}+\frac{34\!\cdots\!53}{11\!\cdots\!81}a^{16}+\frac{11\!\cdots\!32}{36\!\cdots\!27}a^{15}+\frac{13\!\cdots\!56}{11\!\cdots\!81}a^{14}+\frac{20\!\cdots\!44}{11\!\cdots\!81}a^{13}+\frac{36\!\cdots\!80}{11\!\cdots\!81}a^{12}-\frac{89\!\cdots\!00}{11\!\cdots\!81}a^{11}+\frac{76\!\cdots\!32}{11\!\cdots\!81}a^{10}-\frac{30\!\cdots\!76}{11\!\cdots\!81}a^{9}+\frac{11\!\cdots\!72}{11\!\cdots\!81}a^{8}-\frac{19\!\cdots\!08}{36\!\cdots\!27}a^{7}+\frac{13\!\cdots\!28}{11\!\cdots\!81}a^{6}-\frac{71\!\cdots\!92}{11\!\cdots\!81}a^{5}+\frac{10\!\cdots\!12}{11\!\cdots\!81}a^{4}-\frac{48\!\cdots\!24}{11\!\cdots\!81}a^{3}+\frac{59\!\cdots\!30}{12\!\cdots\!09}a^{2}-\frac{15\!\cdots\!20}{12\!\cdots\!09}a+\frac{44\!\cdots\!76}{40\!\cdots\!03}$, $\frac{52\!\cdots\!00}{11\!\cdots\!81}a^{19}+\frac{51\!\cdots\!60}{11\!\cdots\!81}a^{18}+\frac{63\!\cdots\!00}{11\!\cdots\!81}a^{17}+\frac{33\!\cdots\!30}{11\!\cdots\!81}a^{16}-\frac{29\!\cdots\!40}{36\!\cdots\!27}a^{15}+\frac{12\!\cdots\!40}{11\!\cdots\!81}a^{14}-\frac{94\!\cdots\!60}{11\!\cdots\!81}a^{13}+\frac{34\!\cdots\!00}{11\!\cdots\!81}a^{12}-\frac{31\!\cdots\!40}{11\!\cdots\!81}a^{11}+\frac{71\!\cdots\!00}{11\!\cdots\!81}a^{10}-\frac{63\!\cdots\!40}{11\!\cdots\!81}a^{9}+\frac{10\!\cdots\!60}{11\!\cdots\!81}a^{8}-\frac{31\!\cdots\!80}{36\!\cdots\!27}a^{7}+\frac{12\!\cdots\!80}{11\!\cdots\!81}a^{6}-\frac{97\!\cdots\!76}{11\!\cdots\!81}a^{5}+\frac{99\!\cdots\!45}{11\!\cdots\!81}a^{4}-\frac{60\!\cdots\!60}{11\!\cdots\!81}a^{3}+\frac{59\!\cdots\!80}{13\!\cdots\!01}a^{2}-\frac{17\!\cdots\!00}{12\!\cdots\!09}a+\frac{40\!\cdots\!60}{40\!\cdots\!03}$, $\frac{33\!\cdots\!78}{36\!\cdots\!27}a^{19}+\frac{32\!\cdots\!41}{13\!\cdots\!01}a^{18}+\frac{12\!\cdots\!30}{36\!\cdots\!27}a^{17}+\frac{23\!\cdots\!56}{13\!\cdots\!01}a^{16}+\frac{39\!\cdots\!12}{36\!\cdots\!27}a^{15}+\frac{25\!\cdots\!32}{36\!\cdots\!27}a^{14}+\frac{86\!\cdots\!56}{36\!\cdots\!27}a^{13}+\frac{27\!\cdots\!89}{13\!\cdots\!01}a^{12}+\frac{47\!\cdots\!18}{12\!\cdots\!09}a^{11}+\frac{15\!\cdots\!33}{36\!\cdots\!27}a^{10}+\frac{17\!\cdots\!56}{36\!\cdots\!27}a^{9}+\frac{84\!\cdots\!82}{12\!\cdots\!09}a^{8}+\frac{14\!\cdots\!58}{36\!\cdots\!27}a^{7}+\frac{29\!\cdots\!06}{36\!\cdots\!27}a^{6}+\frac{56\!\cdots\!36}{36\!\cdots\!27}a^{5}+\frac{82\!\cdots\!55}{12\!\cdots\!09}a^{4}-\frac{39\!\cdots\!90}{36\!\cdots\!27}a^{3}+\frac{12\!\cdots\!14}{36\!\cdots\!27}a^{2}-\frac{20\!\cdots\!68}{12\!\cdots\!09}a+\frac{33\!\cdots\!38}{40\!\cdots\!03}$, $\frac{15\!\cdots\!84}{11\!\cdots\!81}a^{19}+\frac{86\!\cdots\!45}{11\!\cdots\!81}a^{18}+\frac{42\!\cdots\!64}{11\!\cdots\!81}a^{17}+\frac{57\!\cdots\!84}{11\!\cdots\!81}a^{16}+\frac{10\!\cdots\!44}{12\!\cdots\!09}a^{15}+\frac{22\!\cdots\!08}{11\!\cdots\!81}a^{14}+\frac{12\!\cdots\!00}{11\!\cdots\!81}a^{13}+\frac{61\!\cdots\!55}{11\!\cdots\!81}a^{12}+\frac{22\!\cdots\!32}{11\!\cdots\!81}a^{11}+\frac{12\!\cdots\!74}{11\!\cdots\!81}a^{10}-\frac{27\!\cdots\!62}{11\!\cdots\!81}a^{9}+\frac{19\!\cdots\!42}{11\!\cdots\!81}a^{8}-\frac{24\!\cdots\!00}{36\!\cdots\!27}a^{7}+\frac{22\!\cdots\!05}{11\!\cdots\!81}a^{6}-\frac{10\!\cdots\!26}{11\!\cdots\!81}a^{5}+\frac{18\!\cdots\!07}{11\!\cdots\!81}a^{4}-\frac{74\!\cdots\!86}{11\!\cdots\!81}a^{3}+\frac{30\!\cdots\!59}{36\!\cdots\!27}a^{2}-\frac{23\!\cdots\!26}{12\!\cdots\!09}a+\frac{25\!\cdots\!61}{13\!\cdots\!01}$, $\frac{21\!\cdots\!76}{36\!\cdots\!27}a^{19}+\frac{31\!\cdots\!52}{36\!\cdots\!27}a^{18}+\frac{32\!\cdots\!14}{12\!\cdots\!09}a^{17}+\frac{24\!\cdots\!44}{36\!\cdots\!27}a^{16}+\frac{31\!\cdots\!74}{36\!\cdots\!27}a^{15}+\frac{10\!\cdots\!39}{36\!\cdots\!27}a^{14}+\frac{25\!\cdots\!42}{12\!\cdots\!09}a^{13}+\frac{31\!\cdots\!79}{36\!\cdots\!27}a^{12}+\frac{13\!\cdots\!52}{36\!\cdots\!27}a^{11}+\frac{23\!\cdots\!50}{12\!\cdots\!09}a^{10}+\frac{64\!\cdots\!34}{12\!\cdots\!09}a^{9}+\frac{11\!\cdots\!97}{36\!\cdots\!27}a^{8}+\frac{19\!\cdots\!78}{36\!\cdots\!27}a^{7}+\frac{14\!\cdots\!56}{36\!\cdots\!27}a^{6}+\frac{42\!\cdots\!72}{12\!\cdots\!09}a^{5}+\frac{11\!\cdots\!18}{36\!\cdots\!27}a^{4}+\frac{54\!\cdots\!28}{40\!\cdots\!03}a^{3}+\frac{62\!\cdots\!30}{36\!\cdots\!27}a^{2}+\frac{31\!\cdots\!42}{12\!\cdots\!09}a+\frac{17\!\cdots\!31}{40\!\cdots\!03}$, $\frac{14\!\cdots\!62}{36\!\cdots\!27}a^{19}+\frac{79\!\cdots\!37}{36\!\cdots\!27}a^{18}+\frac{48\!\cdots\!50}{36\!\cdots\!27}a^{17}+\frac{51\!\cdots\!02}{36\!\cdots\!27}a^{16}+\frac{42\!\cdots\!66}{12\!\cdots\!09}a^{15}+\frac{19\!\cdots\!43}{36\!\cdots\!27}a^{14}+\frac{23\!\cdots\!84}{36\!\cdots\!27}a^{13}+\frac{55\!\cdots\!88}{36\!\cdots\!27}a^{12}+\frac{29\!\cdots\!26}{36\!\cdots\!27}a^{11}+\frac{11\!\cdots\!86}{36\!\cdots\!27}a^{10}+\frac{21\!\cdots\!44}{36\!\cdots\!27}a^{9}+\frac{17\!\cdots\!31}{36\!\cdots\!27}a^{8}-\frac{28\!\cdots\!56}{12\!\cdots\!09}a^{7}+\frac{20\!\cdots\!06}{36\!\cdots\!27}a^{6}-\frac{41\!\cdots\!04}{36\!\cdots\!27}a^{5}+\frac{16\!\cdots\!40}{36\!\cdots\!27}a^{4}-\frac{38\!\cdots\!96}{36\!\cdots\!27}a^{3}+\frac{26\!\cdots\!91}{12\!\cdots\!09}a^{2}-\frac{12\!\cdots\!40}{40\!\cdots\!03}a+\frac{66\!\cdots\!38}{13\!\cdots\!01}$, $\frac{10\!\cdots\!06}{11\!\cdots\!81}a^{19}+\frac{51\!\cdots\!86}{11\!\cdots\!81}a^{18}+\frac{30\!\cdots\!08}{11\!\cdots\!81}a^{17}+\frac{34\!\cdots\!62}{11\!\cdots\!81}a^{16}+\frac{24\!\cdots\!50}{36\!\cdots\!27}a^{15}+\frac{13\!\cdots\!18}{11\!\cdots\!81}a^{14}+\frac{11\!\cdots\!12}{11\!\cdots\!81}a^{13}+\frac{37\!\cdots\!03}{11\!\cdots\!81}a^{12}+\frac{83\!\cdots\!90}{11\!\cdots\!81}a^{11}+\frac{79\!\cdots\!01}{11\!\cdots\!81}a^{10}-\frac{67\!\cdots\!54}{11\!\cdots\!81}a^{9}+\frac{12\!\cdots\!54}{11\!\cdots\!81}a^{8}-\frac{38\!\cdots\!16}{12\!\cdots\!09}a^{7}+\frac{14\!\cdots\!46}{11\!\cdots\!81}a^{6}-\frac{54\!\cdots\!50}{11\!\cdots\!81}a^{5}+\frac{11\!\cdots\!61}{11\!\cdots\!81}a^{4}-\frac{41\!\cdots\!72}{11\!\cdots\!81}a^{3}+\frac{18\!\cdots\!80}{36\!\cdots\!27}a^{2}-\frac{15\!\cdots\!90}{13\!\cdots\!01}a+\frac{48\!\cdots\!84}{40\!\cdots\!03}$
| 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: | \( 5104264.636551031 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 5104264.636551031 \cdot 60213680}{2\cdot\sqrt{1123852547182106083132474859412520960000000000}}\cr\approx \mathstrut & 0.439584503310514 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 65*x^18 - 118*x^17 + 2550*x^16 - 4024*x^15 + 72661*x^14 - 96362*x^13 + 1573687*x^12 - 1687702*x^11 + 26178289*x^10 - 22501622*x^9 + 332045847*x^8 - 225430718*x^7 + 3115413077*x^6 - 1566380644*x^5 + 20342958970*x^4 - 6430154154*x^3 + 82545355413*x^2 - 10933110054*x + 156709628451)
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 - 2*x^19 + 65*x^18 - 118*x^17 + 2550*x^16 - 4024*x^15 + 72661*x^14 - 96362*x^13 + 1573687*x^12 - 1687702*x^11 + 26178289*x^10 - 22501622*x^9 + 332045847*x^8 - 225430718*x^7 + 3115413077*x^6 - 1566380644*x^5 + 20342958970*x^4 - 6430154154*x^3 + 82545355413*x^2 - 10933110054*x + 156709628451, 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 - 2*x^19 + 65*x^18 - 118*x^17 + 2550*x^16 - 4024*x^15 + 72661*x^14 - 96362*x^13 + 1573687*x^12 - 1687702*x^11 + 26178289*x^10 - 22501622*x^9 + 332045847*x^8 - 225430718*x^7 + 3115413077*x^6 - 1566380644*x^5 + 20342958970*x^4 - 6430154154*x^3 + 82545355413*x^2 - 10933110054*x + 156709628451);
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 - 2*x^19 + 65*x^18 - 118*x^17 + 2550*x^16 - 4024*x^15 + 72661*x^14 - 96362*x^13 + 1573687*x^12 - 1687702*x^11 + 26178289*x^10 - 22501622*x^9 + 332045847*x^8 - 225430718*x^7 + 3115413077*x^6 - 1566380644*x^5 + 20342958970*x^4 - 6430154154*x^3 + 82545355413*x^2 - 10933110054*x + 156709628451);
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
$C_2\times C_{10}$ (as 20T3):
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)
| An abelian group of order 20 |
| The 20 conjugacy class representatives for $C_2\times C_{10}$ |
| Character table for $C_2\times C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-410}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-10}, \sqrt{41})\), 5.5.2825761.1, 10.10.327381934393961.1, 10.0.33523910081941606400000.1, 10.0.817656343461990400000.2 |
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 | ${\href{/padicField/3.2.0.1}{2} }^{10}$ | R | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{4}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/43.10.0.1}{10} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.15.13 | $x^{10} - 12 x^{9} + 174 x^{8} - 640 x^{7} - 40 x^{6} + 11424 x^{5} - 7984 x^{4} - 79360 x^{3} + 84112 x^{2} + 955968 x + 1404384$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
| 2.10.15.13 | $x^{10} - 12 x^{9} + 174 x^{8} - 640 x^{7} - 40 x^{6} + 11424 x^{5} - 7984 x^{4} - 79360 x^{3} + 84112 x^{2} + 955968 x + 1404384$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ | |
|
\(5\)
| 5.10.5.2 | $x^{10} + 2500 x^{2} - 9375$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 5.10.5.2 | $x^{10} + 2500 x^{2} - 9375$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(41\)
| 41.10.9.1 | $x^{10} + 41$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 41.10.9.1 | $x^{10} + 41$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |