Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 42*x^18 - 108*x^17 + 585*x^16 - 1163*x^15 + 4645*x^14 - 10617*x^13 + 31833*x^12 - 75285*x^11 + 168816*x^10 - 330321*x^9 + 666734*x^8 - 1317875*x^7 + 2500617*x^6 - 4132047*x^5 + 5546802*x^4 - 5570593*x^3 + 3798320*x^2 - 1539672*x + 282861)
gp: K = bnfinit(y^20 - 4*y^19 + 42*y^18 - 108*y^17 + 585*y^16 - 1163*y^15 + 4645*y^14 - 10617*y^13 + 31833*y^12 - 75285*y^11 + 168816*y^10 - 330321*y^9 + 666734*y^8 - 1317875*y^7 + 2500617*y^6 - 4132047*y^5 + 5546802*y^4 - 5570593*y^3 + 3798320*y^2 - 1539672*y + 282861, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 + 42*x^18 - 108*x^17 + 585*x^16 - 1163*x^15 + 4645*x^14 - 10617*x^13 + 31833*x^12 - 75285*x^11 + 168816*x^10 - 330321*x^9 + 666734*x^8 - 1317875*x^7 + 2500617*x^6 - 4132047*x^5 + 5546802*x^4 - 5570593*x^3 + 3798320*x^2 - 1539672*x + 282861);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 4*x^19 + 42*x^18 - 108*x^17 + 585*x^16 - 1163*x^15 + 4645*x^14 - 10617*x^13 + 31833*x^12 - 75285*x^11 + 168816*x^10 - 330321*x^9 + 666734*x^8 - 1317875*x^7 + 2500617*x^6 - 4132047*x^5 + 5546802*x^4 - 5570593*x^3 + 3798320*x^2 - 1539672*x + 282861)
\( x^{20} - 4 x^{19} + 42 x^{18} - 108 x^{17} + 585 x^{16} - 1163 x^{15} + 4645 x^{14} - 10617 x^{13} + \cdots + 282861 \)
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: |
\(3280906026328914297094848663330078125\)
\(\medspace = 5^{15}\cdot 401^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(66.96\) | 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: | $5^{3/4}401^{1/2}\approx 66.95757085563274$ | ||
| Ramified primes: |
\(5\), \(401\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{13}-\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{1}{9}a^{10}-\frac{1}{9}a^{9}-\frac{1}{9}a^{8}+\frac{1}{9}a^{7}+\frac{2}{9}a^{6}-\frac{4}{9}a^{5}+\frac{2}{9}a^{4}-\frac{2}{9}a^{3}-\frac{2}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{13}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{4}{9}a^{6}-\frac{2}{9}a^{5}-\frac{1}{9}a^{3}+\frac{4}{9}a^{2}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{2}{9}a^{6}-\frac{2}{9}a^{5}-\frac{1}{3}a^{4}-\frac{1}{9}a^{2}$, $\frac{1}{621}a^{17}+\frac{1}{207}a^{16}+\frac{8}{621}a^{15}+\frac{1}{23}a^{14}+\frac{100}{621}a^{13}-\frac{65}{621}a^{12}+\frac{103}{621}a^{11}+\frac{70}{621}a^{10}+\frac{103}{621}a^{9}-\frac{83}{621}a^{8}+\frac{43}{621}a^{7}+\frac{34}{621}a^{6}+\frac{11}{207}a^{5}+\frac{2}{27}a^{4}+\frac{305}{621}a^{3}+\frac{85}{621}a^{2}+\frac{83}{207}a+\frac{14}{69}$, $\frac{1}{169968321}a^{18}-\frac{35659}{56656107}a^{17}-\frac{7163593}{169968321}a^{16}+\frac{1452103}{56656107}a^{15}-\frac{5788499}{169968321}a^{14}-\frac{25547555}{169968321}a^{13}-\frac{22708262}{169968321}a^{12}+\frac{25093246}{169968321}a^{11}+\frac{339622}{169968321}a^{10}-\frac{21016640}{169968321}a^{9}+\frac{11496070}{169968321}a^{8}+\frac{13642942}{169968321}a^{7}+\frac{5233985}{18885369}a^{6}+\frac{18744838}{169968321}a^{5}-\frac{75006145}{169968321}a^{4}-\frac{19572014}{169968321}a^{3}-\frac{26257567}{56656107}a^{2}+\frac{106705}{320091}a+\frac{87352}{6295123}$, $\frac{1}{51\!\cdots\!77}a^{19}-\frac{84\!\cdots\!57}{51\!\cdots\!77}a^{18}+\frac{12\!\cdots\!52}{18\!\cdots\!51}a^{17}+\frac{27\!\cdots\!68}{51\!\cdots\!77}a^{16}-\frac{91\!\cdots\!96}{17\!\cdots\!59}a^{15}-\frac{87\!\cdots\!86}{51\!\cdots\!77}a^{14}+\frac{95\!\cdots\!39}{17\!\cdots\!59}a^{13}+\frac{31\!\cdots\!46}{18\!\cdots\!51}a^{12}-\frac{11\!\cdots\!14}{17\!\cdots\!59}a^{11}-\frac{10\!\cdots\!14}{13\!\cdots\!43}a^{10}-\frac{79\!\cdots\!08}{56\!\cdots\!53}a^{9}+\frac{27\!\cdots\!30}{17\!\cdots\!59}a^{8}-\frac{65\!\cdots\!12}{39\!\cdots\!29}a^{7}+\frac{26\!\cdots\!18}{74\!\cdots\!93}a^{6}+\frac{12\!\cdots\!57}{24\!\cdots\!11}a^{5}-\frac{29\!\cdots\!51}{22\!\cdots\!99}a^{4}-\frac{71\!\cdots\!02}{17\!\cdots\!59}a^{3}-\frac{12\!\cdots\!64}{51\!\cdots\!77}a^{2}+\frac{46\!\cdots\!47}{17\!\cdots\!59}a-\frac{42\!\cdots\!98}{10\!\cdots\!01}$
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
$C_{2}\times C_{808}$, which has order $1616$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $404$ (assuming GRH)
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: |
$\frac{85\!\cdots\!61}{13\!\cdots\!43}a^{19}-\frac{83\!\cdots\!99}{39\!\cdots\!29}a^{18}+\frac{10\!\cdots\!86}{39\!\cdots\!29}a^{17}-\frac{19\!\cdots\!25}{39\!\cdots\!29}a^{16}+\frac{13\!\cdots\!08}{39\!\cdots\!29}a^{15}-\frac{19\!\cdots\!15}{39\!\cdots\!29}a^{14}+\frac{10\!\cdots\!29}{39\!\cdots\!29}a^{13}-\frac{19\!\cdots\!56}{39\!\cdots\!29}a^{12}+\frac{66\!\cdots\!05}{39\!\cdots\!29}a^{11}-\frac{14\!\cdots\!36}{39\!\cdots\!29}a^{10}+\frac{32\!\cdots\!97}{39\!\cdots\!29}a^{9}-\frac{59\!\cdots\!13}{39\!\cdots\!29}a^{8}+\frac{12\!\cdots\!65}{39\!\cdots\!29}a^{7}-\frac{45\!\cdots\!73}{74\!\cdots\!93}a^{6}+\frac{45\!\cdots\!09}{39\!\cdots\!29}a^{5}-\frac{78\!\cdots\!91}{43\!\cdots\!81}a^{4}+\frac{28\!\cdots\!77}{13\!\cdots\!43}a^{3}-\frac{74\!\cdots\!50}{39\!\cdots\!29}a^{2}+\frac{12\!\cdots\!83}{13\!\cdots\!43}a-\frac{18\!\cdots\!62}{82\!\cdots\!77}$, $\frac{24\!\cdots\!38}{13\!\cdots\!43}a^{19}-\frac{10\!\cdots\!54}{17\!\cdots\!23}a^{18}+\frac{96\!\cdots\!46}{13\!\cdots\!43}a^{17}-\frac{60\!\cdots\!85}{39\!\cdots\!29}a^{16}+\frac{12\!\cdots\!43}{13\!\cdots\!43}a^{15}-\frac{60\!\cdots\!45}{39\!\cdots\!29}a^{14}+\frac{29\!\cdots\!62}{39\!\cdots\!29}a^{13}-\frac{58\!\cdots\!98}{39\!\cdots\!29}a^{12}+\frac{18\!\cdots\!25}{39\!\cdots\!29}a^{11}-\frac{42\!\cdots\!44}{39\!\cdots\!29}a^{10}+\frac{93\!\cdots\!66}{39\!\cdots\!29}a^{9}-\frac{17\!\cdots\!59}{39\!\cdots\!29}a^{8}+\frac{36\!\cdots\!95}{39\!\cdots\!29}a^{7}-\frac{44\!\cdots\!68}{24\!\cdots\!31}a^{6}+\frac{13\!\cdots\!71}{39\!\cdots\!29}a^{5}-\frac{20\!\cdots\!92}{39\!\cdots\!29}a^{4}+\frac{43\!\cdots\!67}{66\!\cdots\!31}a^{3}-\frac{73\!\cdots\!05}{13\!\cdots\!43}a^{2}+\frac{41\!\cdots\!56}{14\!\cdots\!27}a-\frac{18\!\cdots\!62}{27\!\cdots\!59}$, $\frac{61\!\cdots\!38}{39\!\cdots\!29}a^{19}-\frac{34\!\cdots\!71}{66\!\cdots\!31}a^{18}+\frac{81\!\cdots\!77}{13\!\cdots\!43}a^{17}-\frac{49\!\cdots\!40}{39\!\cdots\!29}a^{16}+\frac{10\!\cdots\!04}{13\!\cdots\!43}a^{15}-\frac{49\!\cdots\!45}{39\!\cdots\!29}a^{14}+\frac{83\!\cdots\!28}{13\!\cdots\!43}a^{13}-\frac{16\!\cdots\!92}{13\!\cdots\!43}a^{12}+\frac{53\!\cdots\!55}{13\!\cdots\!43}a^{11}-\frac{11\!\cdots\!20}{13\!\cdots\!43}a^{10}+\frac{87\!\cdots\!53}{43\!\cdots\!81}a^{9}-\frac{49\!\cdots\!66}{13\!\cdots\!43}a^{8}+\frac{30\!\cdots\!50}{39\!\cdots\!29}a^{7}-\frac{11\!\cdots\!43}{74\!\cdots\!93}a^{6}+\frac{37\!\cdots\!66}{13\!\cdots\!43}a^{5}-\frac{17\!\cdots\!59}{39\!\cdots\!29}a^{4}+\frac{31\!\cdots\!24}{56\!\cdots\!41}a^{3}-\frac{18\!\cdots\!70}{39\!\cdots\!29}a^{2}+\frac{31\!\cdots\!43}{13\!\cdots\!43}a-\frac{42\!\cdots\!21}{82\!\cdots\!77}$, $\frac{60\!\cdots\!08}{13\!\cdots\!43}a^{19}-\frac{59\!\cdots\!77}{39\!\cdots\!29}a^{18}+\frac{71\!\cdots\!98}{39\!\cdots\!29}a^{17}-\frac{61\!\cdots\!50}{17\!\cdots\!23}a^{16}+\frac{94\!\cdots\!69}{39\!\cdots\!29}a^{15}-\frac{13\!\cdots\!40}{39\!\cdots\!29}a^{14}+\frac{73\!\cdots\!87}{39\!\cdots\!29}a^{13}-\frac{13\!\cdots\!33}{39\!\cdots\!29}a^{12}+\frac{46\!\cdots\!90}{39\!\cdots\!29}a^{11}-\frac{10\!\cdots\!45}{39\!\cdots\!29}a^{10}+\frac{22\!\cdots\!96}{39\!\cdots\!29}a^{9}-\frac{42\!\cdots\!14}{39\!\cdots\!29}a^{8}+\frac{87\!\cdots\!25}{39\!\cdots\!29}a^{7}-\frac{32\!\cdots\!19}{74\!\cdots\!93}a^{6}+\frac{31\!\cdots\!07}{39\!\cdots\!29}a^{5}-\frac{55\!\cdots\!63}{43\!\cdots\!81}a^{4}+\frac{20\!\cdots\!61}{13\!\cdots\!43}a^{3}-\frac{52\!\cdots\!50}{39\!\cdots\!29}a^{2}+\frac{88\!\cdots\!04}{13\!\cdots\!43}a-\frac{12\!\cdots\!77}{82\!\cdots\!77}$, $\frac{10\!\cdots\!54}{39\!\cdots\!29}a^{19}+\frac{70\!\cdots\!73}{39\!\cdots\!29}a^{18}+\frac{78\!\cdots\!71}{39\!\cdots\!29}a^{17}+\frac{30\!\cdots\!36}{39\!\cdots\!29}a^{16}-\frac{17\!\cdots\!49}{39\!\cdots\!29}a^{15}+\frac{12\!\cdots\!45}{13\!\cdots\!43}a^{14}-\frac{28\!\cdots\!92}{43\!\cdots\!81}a^{13}+\frac{73\!\cdots\!41}{13\!\cdots\!43}a^{12}-\frac{14\!\cdots\!27}{13\!\cdots\!43}a^{11}+\frac{13\!\cdots\!28}{43\!\cdots\!81}a^{10}-\frac{11\!\cdots\!98}{13\!\cdots\!43}a^{9}+\frac{65\!\cdots\!39}{43\!\cdots\!81}a^{8}-\frac{11\!\cdots\!11}{39\!\cdots\!29}a^{7}+\frac{46\!\cdots\!77}{74\!\cdots\!93}a^{6}-\frac{47\!\cdots\!13}{39\!\cdots\!29}a^{5}+\frac{39\!\cdots\!01}{17\!\cdots\!23}a^{4}-\frac{13\!\cdots\!17}{39\!\cdots\!29}a^{3}+\frac{14\!\cdots\!18}{43\!\cdots\!81}a^{2}-\frac{73\!\cdots\!41}{43\!\cdots\!81}a+\frac{57\!\cdots\!31}{27\!\cdots\!59}$, $\frac{23\!\cdots\!29}{22\!\cdots\!99}a^{19}-\frac{73\!\cdots\!84}{20\!\cdots\!91}a^{18}+\frac{27\!\cdots\!59}{68\!\cdots\!97}a^{17}-\frac{18\!\cdots\!71}{20\!\cdots\!91}a^{16}+\frac{37\!\cdots\!17}{68\!\cdots\!97}a^{15}-\frac{18\!\cdots\!35}{20\!\cdots\!91}a^{14}+\frac{86\!\cdots\!89}{20\!\cdots\!91}a^{13}-\frac{17\!\cdots\!74}{20\!\cdots\!91}a^{12}+\frac{56\!\cdots\!15}{20\!\cdots\!91}a^{11}-\frac{12\!\cdots\!57}{20\!\cdots\!91}a^{10}+\frac{28\!\cdots\!97}{20\!\cdots\!91}a^{9}-\frac{53\!\cdots\!65}{20\!\cdots\!91}a^{8}+\frac{10\!\cdots\!08}{20\!\cdots\!91}a^{7}-\frac{15\!\cdots\!69}{14\!\cdots\!61}a^{6}+\frac{40\!\cdots\!25}{20\!\cdots\!91}a^{5}-\frac{27\!\cdots\!33}{89\!\cdots\!17}a^{4}+\frac{79\!\cdots\!86}{20\!\cdots\!91}a^{3}-\frac{75\!\cdots\!59}{22\!\cdots\!99}a^{2}+\frac{38\!\cdots\!00}{22\!\cdots\!99}a-\frac{52\!\cdots\!55}{14\!\cdots\!61}$, $\frac{20\!\cdots\!27}{39\!\cdots\!29}a^{19}-\frac{68\!\cdots\!18}{39\!\cdots\!29}a^{18}+\frac{35\!\cdots\!35}{17\!\cdots\!23}a^{17}-\frac{16\!\cdots\!34}{39\!\cdots\!29}a^{16}+\frac{10\!\cdots\!36}{39\!\cdots\!29}a^{15}-\frac{54\!\cdots\!45}{13\!\cdots\!43}a^{14}+\frac{83\!\cdots\!48}{39\!\cdots\!29}a^{13}-\frac{15\!\cdots\!07}{39\!\cdots\!29}a^{12}+\frac{23\!\cdots\!59}{17\!\cdots\!23}a^{11}-\frac{11\!\cdots\!58}{39\!\cdots\!29}a^{10}+\frac{26\!\cdots\!34}{39\!\cdots\!29}a^{9}-\frac{49\!\cdots\!80}{39\!\cdots\!29}a^{8}+\frac{10\!\cdots\!23}{39\!\cdots\!29}a^{7}-\frac{41\!\cdots\!44}{82\!\cdots\!77}a^{6}+\frac{36\!\cdots\!94}{39\!\cdots\!29}a^{5}-\frac{57\!\cdots\!08}{39\!\cdots\!29}a^{4}+\frac{23\!\cdots\!84}{13\!\cdots\!43}a^{3}-\frac{10\!\cdots\!49}{66\!\cdots\!31}a^{2}+\frac{10\!\cdots\!17}{13\!\cdots\!43}a-\frac{24\!\cdots\!63}{13\!\cdots\!03}$, $\frac{80\!\cdots\!19}{13\!\cdots\!43}a^{19}-\frac{26\!\cdots\!87}{13\!\cdots\!43}a^{18}+\frac{94\!\cdots\!28}{39\!\cdots\!29}a^{17}-\frac{63\!\cdots\!85}{13\!\cdots\!43}a^{16}+\frac{12\!\cdots\!72}{39\!\cdots\!29}a^{15}-\frac{62\!\cdots\!86}{13\!\cdots\!43}a^{14}+\frac{96\!\cdots\!51}{39\!\cdots\!29}a^{13}-\frac{18\!\cdots\!11}{39\!\cdots\!29}a^{12}+\frac{61\!\cdots\!38}{39\!\cdots\!29}a^{11}-\frac{13\!\cdots\!62}{39\!\cdots\!29}a^{10}+\frac{29\!\cdots\!18}{39\!\cdots\!29}a^{9}-\frac{56\!\cdots\!82}{39\!\cdots\!29}a^{8}+\frac{11\!\cdots\!61}{39\!\cdots\!29}a^{7}-\frac{18\!\cdots\!99}{32\!\cdots\!91}a^{6}+\frac{14\!\cdots\!97}{13\!\cdots\!43}a^{5}-\frac{66\!\cdots\!16}{39\!\cdots\!29}a^{4}+\frac{81\!\cdots\!81}{39\!\cdots\!29}a^{3}-\frac{69\!\cdots\!30}{39\!\cdots\!29}a^{2}+\frac{51\!\cdots\!44}{56\!\cdots\!41}a-\frac{72\!\cdots\!53}{35\!\cdots\!99}$, $\frac{12\!\cdots\!45}{39\!\cdots\!29}a^{19}-\frac{43\!\cdots\!77}{39\!\cdots\!29}a^{18}+\frac{55\!\cdots\!53}{43\!\cdots\!81}a^{17}-\frac{10\!\cdots\!35}{39\!\cdots\!29}a^{16}+\frac{22\!\cdots\!99}{13\!\cdots\!43}a^{15}-\frac{36\!\cdots\!77}{13\!\cdots\!43}a^{14}+\frac{51\!\cdots\!67}{39\!\cdots\!29}a^{13}-\frac{10\!\cdots\!44}{39\!\cdots\!29}a^{12}+\frac{33\!\cdots\!09}{39\!\cdots\!29}a^{11}-\frac{74\!\cdots\!50}{39\!\cdots\!29}a^{10}+\frac{16\!\cdots\!31}{39\!\cdots\!29}a^{9}-\frac{31\!\cdots\!53}{39\!\cdots\!29}a^{8}+\frac{23\!\cdots\!08}{14\!\cdots\!27}a^{7}-\frac{23\!\cdots\!84}{74\!\cdots\!93}a^{6}+\frac{23\!\cdots\!26}{39\!\cdots\!29}a^{5}-\frac{13\!\cdots\!56}{14\!\cdots\!27}a^{4}+\frac{46\!\cdots\!37}{39\!\cdots\!29}a^{3}-\frac{39\!\cdots\!13}{39\!\cdots\!29}a^{2}+\frac{67\!\cdots\!31}{13\!\cdots\!43}a-\frac{42\!\cdots\!27}{35\!\cdots\!99}$
| 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: | \( 31495162.1453 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 31495162.1453 \cdot 1616}{2\cdot\sqrt{3280906026328914297094848663330078125}}\cr\approx \mathstrut & 1.34727731484 \end{aligned}\] (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 + 42*x^18 - 108*x^17 + 585*x^16 - 1163*x^15 + 4645*x^14 - 10617*x^13 + 31833*x^12 - 75285*x^11 + 168816*x^10 - 330321*x^9 + 666734*x^8 - 1317875*x^7 + 2500617*x^6 - 4132047*x^5 + 5546802*x^4 - 5570593*x^3 + 3798320*x^2 - 1539672*x + 282861)
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 + 42*x^18 - 108*x^17 + 585*x^16 - 1163*x^15 + 4645*x^14 - 10617*x^13 + 31833*x^12 - 75285*x^11 + 168816*x^10 - 330321*x^9 + 666734*x^8 - 1317875*x^7 + 2500617*x^6 - 4132047*x^5 + 5546802*x^4 - 5570593*x^3 + 3798320*x^2 - 1539672*x + 282861, 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 + 42*x^18 - 108*x^17 + 585*x^16 - 1163*x^15 + 4645*x^14 - 10617*x^13 + 31833*x^12 - 75285*x^11 + 168816*x^10 - 330321*x^9 + 666734*x^8 - 1317875*x^7 + 2500617*x^6 - 4132047*x^5 + 5546802*x^4 - 5570593*x^3 + 3798320*x^2 - 1539672*x + 282861);
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 + 42*x^18 - 108*x^17 + 585*x^16 - 1163*x^15 + 4645*x^14 - 10617*x^13 + 31833*x^12 - 75285*x^11 + 168816*x^10 - 330321*x^9 + 666734*x^8 - 1317875*x^7 + 2500617*x^6 - 4132047*x^5 + 5546802*x^4 - 5570593*x^3 + 3798320*x^2 - 1539672*x + 282861);
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 solvable group of order 40 |
| The 13 conjugacy class representatives for $D_{20}$ |
| Character table for $D_{20}$ |
Intermediate fields
| \(\Q(\sqrt{2005}) \), 4.0.20100125.1, 5.5.160801.1, 10.10.32402005006253125.1 |
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]
Sibling fields
| Galois closure: | deg 40 |
| Degree 20 sibling: | 20.0.8181810539473601738391143798828125.1 |
| Minimal sibling: | 20.0.8181810539473601738391143798828125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $20$ | ${\href{/padicField/3.2.0.1}{2} }^{9}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | $20$ | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{9}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{10}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{10}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/padicField/53.2.0.1}{2} }^{9}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}$ |
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 $20$ | $4$ | $5$ | $15$ | |||
|
\(401\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |