Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 23*x^18 - 38*x^17 + 157*x^16 + 126*x^15 - 239*x^14 + 5543*x^13 - 9767*x^12 + 48731*x^11 - 52647*x^10 + 197277*x^9 + 60753*x^8 + 239947*x^7 + 1556367*x^6 - 309932*x^5 + 5752551*x^4 + 1073140*x^3 + 5045800*x^2 + 3532675*x - 2727775)
gp: K = bnfinit(y^20 - 4*y^19 + 23*y^18 - 38*y^17 + 157*y^16 + 126*y^15 - 239*y^14 + 5543*y^13 - 9767*y^12 + 48731*y^11 - 52647*y^10 + 197277*y^9 + 60753*y^8 + 239947*y^7 + 1556367*y^6 - 309932*y^5 + 5752551*y^4 + 1073140*y^3 + 5045800*y^2 + 3532675*y - 2727775, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 + 23*x^18 - 38*x^17 + 157*x^16 + 126*x^15 - 239*x^14 + 5543*x^13 - 9767*x^12 + 48731*x^11 - 52647*x^10 + 197277*x^9 + 60753*x^8 + 239947*x^7 + 1556367*x^6 - 309932*x^5 + 5752551*x^4 + 1073140*x^3 + 5045800*x^2 + 3532675*x - 2727775);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 + 23*x^18 - 38*x^17 + 157*x^16 + 126*x^15 - 239*x^14 + 5543*x^13 - 9767*x^12 + 48731*x^11 - 52647*x^10 + 197277*x^9 + 60753*x^8 + 239947*x^7 + 1556367*x^6 - 309932*x^5 + 5752551*x^4 + 1073140*x^3 + 5045800*x^2 + 3532675*x - 2727775)
\( x^{20} - 4 x^{19} + 23 x^{18} - 38 x^{17} + 157 x^{16} + 126 x^{15} - 239 x^{14} + 5543 x^{13} + \cdots - 2727775 \)
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: | $[4, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1969799822613689762148183291015625\)
\(\medspace = 5^{10}\cdot 61^{6}\cdot 397^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(46.21\) | 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: | $5^{1/2}61^{1/2}397^{1/2}\approx 347.9727000786125$ | ||
| Ramified primes: |
\(5\), \(61\), \(397\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13}a^{16}+\frac{3}{13}a^{15}-\frac{6}{13}a^{14}-\frac{2}{13}a^{13}-\frac{5}{13}a^{12}+\frac{3}{13}a^{11}-\frac{2}{13}a^{10}+\frac{3}{13}a^{9}-\frac{4}{13}a^{7}-\frac{2}{13}a^{6}+\frac{3}{13}a^{5}+\frac{6}{13}a^{4}+\frac{5}{13}a^{3}-\frac{1}{13}a^{2}+\frac{4}{13}a+\frac{2}{13}$, $\frac{1}{13}a^{17}-\frac{2}{13}a^{15}+\frac{3}{13}a^{14}+\frac{1}{13}a^{13}+\frac{5}{13}a^{12}+\frac{2}{13}a^{11}-\frac{4}{13}a^{10}+\frac{4}{13}a^{9}-\frac{4}{13}a^{8}-\frac{3}{13}a^{7}-\frac{4}{13}a^{6}-\frac{3}{13}a^{5}-\frac{3}{13}a^{3}-\frac{6}{13}a^{2}+\frac{3}{13}a-\frac{6}{13}$, $\frac{1}{2023757255}a^{18}+\frac{34688361}{2023757255}a^{17}+\frac{4622583}{2023757255}a^{16}-\frac{875663403}{2023757255}a^{15}-\frac{952657138}{2023757255}a^{14}-\frac{57288609}{2023757255}a^{13}-\frac{984295349}{2023757255}a^{12}+\frac{64292786}{155673635}a^{11}+\frac{242075633}{2023757255}a^{10}+\frac{546141946}{2023757255}a^{9}+\frac{611856803}{2023757255}a^{8}-\frac{694960663}{2023757255}a^{7}+\frac{77571781}{155673635}a^{6}+\frac{2873229}{155673635}a^{5}+\frac{900648717}{2023757255}a^{4}-\frac{424063127}{2023757255}a^{3}+\frac{12439897}{155673635}a^{2}-\frac{87511968}{404751451}a+\frac{13138271}{404751451}$, $\frac{1}{14\!\cdots\!55}a^{19}-\frac{11\!\cdots\!17}{11\!\cdots\!35}a^{18}-\frac{74\!\cdots\!64}{14\!\cdots\!55}a^{17}+\frac{29\!\cdots\!16}{14\!\cdots\!55}a^{16}+\frac{46\!\cdots\!83}{14\!\cdots\!55}a^{15}+\frac{54\!\cdots\!97}{14\!\cdots\!55}a^{14}+\frac{16\!\cdots\!39}{14\!\cdots\!55}a^{13}+\frac{48\!\cdots\!31}{14\!\cdots\!55}a^{12}-\frac{68\!\cdots\!63}{14\!\cdots\!55}a^{11}+\frac{10\!\cdots\!15}{29\!\cdots\!71}a^{10}+\frac{34\!\cdots\!71}{14\!\cdots\!55}a^{9}+\frac{64\!\cdots\!86}{14\!\cdots\!55}a^{8}+\frac{50\!\cdots\!89}{14\!\cdots\!55}a^{7}-\frac{13\!\cdots\!98}{11\!\cdots\!35}a^{6}-\frac{43\!\cdots\!12}{14\!\cdots\!55}a^{5}-\frac{36\!\cdots\!16}{14\!\cdots\!55}a^{4}-\frac{38\!\cdots\!91}{29\!\cdots\!71}a^{3}-\frac{17\!\cdots\!07}{14\!\cdots\!55}a^{2}+\frac{13\!\cdots\!05}{29\!\cdots\!71}a+\frac{20\!\cdots\!95}{29\!\cdots\!71}$
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
$C_{2}$, which has order $2$ (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: | $11$ | 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: |
$\frac{49\!\cdots\!74}{36\!\cdots\!05}a^{19}-\frac{94\!\cdots\!41}{36\!\cdots\!05}a^{18}+\frac{68\!\cdots\!87}{36\!\cdots\!05}a^{17}+\frac{47\!\cdots\!58}{36\!\cdots\!05}a^{16}+\frac{33\!\cdots\!88}{36\!\cdots\!05}a^{15}+\frac{21\!\cdots\!74}{36\!\cdots\!05}a^{14}-\frac{47\!\cdots\!56}{36\!\cdots\!05}a^{13}+\frac{23\!\cdots\!47}{36\!\cdots\!05}a^{12}+\frac{47\!\cdots\!52}{36\!\cdots\!05}a^{11}+\frac{12\!\cdots\!49}{36\!\cdots\!05}a^{10}+\frac{21\!\cdots\!92}{36\!\cdots\!05}a^{9}+\frac{27\!\cdots\!18}{36\!\cdots\!05}a^{8}+\frac{21\!\cdots\!82}{36\!\cdots\!05}a^{7}+\frac{89\!\cdots\!08}{36\!\cdots\!05}a^{6}+\frac{89\!\cdots\!13}{36\!\cdots\!05}a^{5}+\frac{95\!\cdots\!67}{36\!\cdots\!05}a^{4}+\frac{16\!\cdots\!34}{36\!\cdots\!05}a^{3}+\frac{97\!\cdots\!89}{72\!\cdots\!21}a^{2}+\frac{21\!\cdots\!48}{72\!\cdots\!21}a+\frac{87\!\cdots\!18}{72\!\cdots\!21}$, $\frac{48\!\cdots\!56}{17\!\cdots\!81}a^{19}-\frac{90\!\cdots\!18}{17\!\cdots\!81}a^{18}+\frac{60\!\cdots\!56}{17\!\cdots\!81}a^{17}+\frac{79\!\cdots\!03}{17\!\cdots\!81}a^{16}+\frac{22\!\cdots\!06}{17\!\cdots\!81}a^{15}+\frac{21\!\cdots\!15}{17\!\cdots\!81}a^{14}-\frac{27\!\cdots\!31}{17\!\cdots\!81}a^{13}+\frac{20\!\cdots\!97}{17\!\cdots\!81}a^{12}+\frac{12\!\cdots\!65}{17\!\cdots\!81}a^{11}+\frac{92\!\cdots\!63}{17\!\cdots\!81}a^{10}+\frac{24\!\cdots\!81}{17\!\cdots\!81}a^{9}+\frac{16\!\cdots\!95}{13\!\cdots\!37}a^{8}+\frac{20\!\cdots\!13}{17\!\cdots\!81}a^{7}+\frac{10\!\cdots\!26}{13\!\cdots\!37}a^{6}+\frac{66\!\cdots\!56}{17\!\cdots\!81}a^{5}+\frac{13\!\cdots\!92}{17\!\cdots\!81}a^{4}+\frac{13\!\cdots\!30}{17\!\cdots\!81}a^{3}+\frac{46\!\cdots\!75}{17\!\cdots\!81}a^{2}+\frac{13\!\cdots\!00}{17\!\cdots\!81}a+\frac{75\!\cdots\!28}{17\!\cdots\!81}$, $\frac{18\!\cdots\!88}{17\!\cdots\!81}a^{19}+\frac{57\!\cdots\!29}{17\!\cdots\!81}a^{18}+\frac{50\!\cdots\!19}{17\!\cdots\!81}a^{17}+\frac{53\!\cdots\!16}{17\!\cdots\!81}a^{16}+\frac{72\!\cdots\!72}{17\!\cdots\!81}a^{15}+\frac{24\!\cdots\!48}{17\!\cdots\!81}a^{14}+\frac{39\!\cdots\!08}{17\!\cdots\!81}a^{13}+\frac{30\!\cdots\!14}{17\!\cdots\!81}a^{12}+\frac{37\!\cdots\!70}{17\!\cdots\!81}a^{11}+\frac{21\!\cdots\!12}{17\!\cdots\!81}a^{10}+\frac{45\!\cdots\!20}{17\!\cdots\!81}a^{9}+\frac{71\!\cdots\!93}{13\!\cdots\!37}a^{8}+\frac{43\!\cdots\!26}{17\!\cdots\!81}a^{7}+\frac{26\!\cdots\!11}{13\!\cdots\!37}a^{6}+\frac{23\!\cdots\!23}{17\!\cdots\!81}a^{5}+\frac{30\!\cdots\!87}{17\!\cdots\!81}a^{4}+\frac{58\!\cdots\!30}{17\!\cdots\!81}a^{3}+\frac{20\!\cdots\!50}{17\!\cdots\!81}a^{2}+\frac{52\!\cdots\!50}{17\!\cdots\!81}a+\frac{23\!\cdots\!21}{17\!\cdots\!81}$, $\frac{46\!\cdots\!68}{17\!\cdots\!81}a^{19}-\frac{95\!\cdots\!47}{17\!\cdots\!81}a^{18}+\frac{55\!\cdots\!37}{17\!\cdots\!81}a^{17}+\frac{74\!\cdots\!87}{17\!\cdots\!81}a^{16}+\frac{15\!\cdots\!34}{17\!\cdots\!81}a^{15}+\frac{19\!\cdots\!67}{17\!\cdots\!81}a^{14}-\frac{66\!\cdots\!39}{17\!\cdots\!81}a^{13}+\frac{17\!\cdots\!83}{17\!\cdots\!81}a^{12}+\frac{86\!\cdots\!95}{17\!\cdots\!81}a^{11}+\frac{70\!\cdots\!51}{17\!\cdots\!81}a^{10}+\frac{20\!\cdots\!61}{17\!\cdots\!81}a^{9}+\frac{97\!\cdots\!02}{13\!\cdots\!37}a^{8}+\frac{15\!\cdots\!87}{17\!\cdots\!81}a^{7}+\frac{83\!\cdots\!15}{13\!\cdots\!37}a^{6}+\frac{43\!\cdots\!33}{17\!\cdots\!81}a^{5}+\frac{10\!\cdots\!05}{17\!\cdots\!81}a^{4}+\frac{76\!\cdots\!00}{17\!\cdots\!81}a^{3}+\frac{25\!\cdots\!25}{17\!\cdots\!81}a^{2}+\frac{78\!\cdots\!50}{17\!\cdots\!81}a-\frac{15\!\cdots\!93}{17\!\cdots\!81}$, $\frac{16\!\cdots\!82}{17\!\cdots\!81}a^{19}+\frac{19\!\cdots\!02}{17\!\cdots\!81}a^{18}-\frac{44\!\cdots\!20}{17\!\cdots\!81}a^{17}+\frac{49\!\cdots\!51}{17\!\cdots\!81}a^{16}-\frac{35\!\cdots\!06}{17\!\cdots\!81}a^{15}+\frac{37\!\cdots\!90}{17\!\cdots\!81}a^{14}+\frac{64\!\cdots\!90}{17\!\cdots\!81}a^{13}+\frac{61\!\cdots\!34}{17\!\cdots\!81}a^{12}+\frac{12\!\cdots\!70}{17\!\cdots\!81}a^{11}-\frac{82\!\cdots\!85}{17\!\cdots\!81}a^{10}+\frac{10\!\cdots\!67}{17\!\cdots\!81}a^{9}-\frac{80\!\cdots\!78}{13\!\cdots\!37}a^{8}+\frac{46\!\cdots\!96}{17\!\cdots\!81}a^{7}+\frac{35\!\cdots\!74}{13\!\cdots\!37}a^{6}+\frac{89\!\cdots\!82}{17\!\cdots\!81}a^{5}+\frac{44\!\cdots\!62}{17\!\cdots\!81}a^{4}+\frac{30\!\cdots\!05}{17\!\cdots\!81}a^{3}+\frac{14\!\cdots\!75}{17\!\cdots\!81}a^{2}+\frac{50\!\cdots\!25}{17\!\cdots\!81}a+\frac{23\!\cdots\!29}{17\!\cdots\!81}$, $\frac{21\!\cdots\!14}{14\!\cdots\!55}a^{19}-\frac{33\!\cdots\!96}{14\!\cdots\!55}a^{18}+\frac{32\!\cdots\!82}{14\!\cdots\!55}a^{17}+\frac{23\!\cdots\!38}{14\!\cdots\!55}a^{16}+\frac{19\!\cdots\!48}{14\!\cdots\!55}a^{15}+\frac{10\!\cdots\!14}{14\!\cdots\!55}a^{14}+\frac{13\!\cdots\!84}{14\!\cdots\!55}a^{13}+\frac{11\!\cdots\!67}{14\!\cdots\!55}a^{12}+\frac{50\!\cdots\!02}{14\!\cdots\!55}a^{11}+\frac{68\!\cdots\!59}{14\!\cdots\!55}a^{10}+\frac{12\!\cdots\!92}{14\!\cdots\!55}a^{9}+\frac{14\!\cdots\!01}{11\!\cdots\!35}a^{8}+\frac{12\!\cdots\!12}{14\!\cdots\!55}a^{7}+\frac{50\!\cdots\!21}{11\!\cdots\!35}a^{6}+\frac{55\!\cdots\!28}{14\!\cdots\!55}a^{5}+\frac{64\!\cdots\!02}{14\!\cdots\!55}a^{4}+\frac{11\!\cdots\!84}{14\!\cdots\!55}a^{3}+\frac{74\!\cdots\!89}{29\!\cdots\!71}a^{2}+\frac{17\!\cdots\!98}{29\!\cdots\!71}a+\frac{74\!\cdots\!29}{29\!\cdots\!71}$, $\frac{55\!\cdots\!38}{14\!\cdots\!55}a^{19}-\frac{11\!\cdots\!92}{14\!\cdots\!55}a^{18}+\frac{68\!\cdots\!59}{14\!\cdots\!55}a^{17}+\frac{71\!\cdots\!46}{14\!\cdots\!55}a^{16}+\frac{26\!\cdots\!21}{14\!\cdots\!55}a^{15}+\frac{21\!\cdots\!23}{14\!\cdots\!55}a^{14}-\frac{48\!\cdots\!07}{14\!\cdots\!55}a^{13}+\frac{21\!\cdots\!74}{14\!\cdots\!55}a^{12}+\frac{89\!\cdots\!99}{14\!\cdots\!55}a^{11}+\frac{10\!\cdots\!33}{14\!\cdots\!55}a^{10}+\frac{21\!\cdots\!84}{14\!\cdots\!55}a^{9}+\frac{26\!\cdots\!51}{14\!\cdots\!55}a^{8}+\frac{17\!\cdots\!74}{14\!\cdots\!55}a^{7}+\frac{12\!\cdots\!82}{11\!\cdots\!35}a^{6}+\frac{55\!\cdots\!91}{14\!\cdots\!55}a^{5}+\frac{12\!\cdots\!74}{14\!\cdots\!55}a^{4}+\frac{10\!\cdots\!83}{14\!\cdots\!55}a^{3}+\frac{72\!\cdots\!68}{29\!\cdots\!71}a^{2}+\frac{21\!\cdots\!51}{29\!\cdots\!71}a-\frac{79\!\cdots\!32}{29\!\cdots\!71}$, $\frac{16\!\cdots\!54}{14\!\cdots\!55}a^{19}-\frac{58\!\cdots\!01}{14\!\cdots\!55}a^{18}+\frac{23\!\cdots\!32}{14\!\cdots\!55}a^{17}+\frac{77\!\cdots\!66}{11\!\cdots\!35}a^{16}-\frac{46\!\cdots\!17}{14\!\cdots\!55}a^{15}+\frac{63\!\cdots\!79}{14\!\cdots\!55}a^{14}-\frac{82\!\cdots\!16}{14\!\cdots\!55}a^{13}+\frac{66\!\cdots\!52}{14\!\cdots\!55}a^{12}-\frac{31\!\cdots\!98}{14\!\cdots\!55}a^{11}+\frac{22\!\cdots\!14}{14\!\cdots\!55}a^{10}+\frac{60\!\cdots\!92}{14\!\cdots\!55}a^{9}-\frac{57\!\cdots\!62}{14\!\cdots\!55}a^{8}+\frac{57\!\cdots\!82}{14\!\cdots\!55}a^{7}-\frac{14\!\cdots\!49}{11\!\cdots\!35}a^{6}+\frac{17\!\cdots\!08}{14\!\cdots\!55}a^{5}+\frac{24\!\cdots\!14}{11\!\cdots\!35}a^{4}+\frac{30\!\cdots\!74}{14\!\cdots\!55}a^{3}+\frac{21\!\cdots\!54}{29\!\cdots\!71}a^{2}+\frac{57\!\cdots\!03}{29\!\cdots\!71}a+\frac{13\!\cdots\!19}{29\!\cdots\!71}$, $\frac{38\!\cdots\!94}{14\!\cdots\!55}a^{19}+\frac{23\!\cdots\!04}{14\!\cdots\!55}a^{18}-\frac{10\!\cdots\!83}{14\!\cdots\!55}a^{17}+\frac{51\!\cdots\!68}{14\!\cdots\!55}a^{16}+\frac{15\!\cdots\!88}{14\!\cdots\!55}a^{15}+\frac{23\!\cdots\!94}{14\!\cdots\!55}a^{14}+\frac{30\!\cdots\!44}{14\!\cdots\!55}a^{13}+\frac{57\!\cdots\!22}{14\!\cdots\!55}a^{12}+\frac{16\!\cdots\!17}{14\!\cdots\!55}a^{11}+\frac{56\!\cdots\!34}{14\!\cdots\!55}a^{10}+\frac{47\!\cdots\!87}{14\!\cdots\!55}a^{9}+\frac{47\!\cdots\!08}{14\!\cdots\!55}a^{8}+\frac{24\!\cdots\!96}{31\!\cdots\!65}a^{7}+\frac{13\!\cdots\!46}{11\!\cdots\!35}a^{6}+\frac{29\!\cdots\!48}{14\!\cdots\!55}a^{5}+\frac{60\!\cdots\!97}{14\!\cdots\!55}a^{4}+\frac{98\!\cdots\!44}{14\!\cdots\!55}a^{3}+\frac{65\!\cdots\!74}{29\!\cdots\!71}a^{2}+\frac{13\!\cdots\!11}{22\!\cdots\!67}a+\frac{22\!\cdots\!57}{29\!\cdots\!71}$, $\frac{11\!\cdots\!22}{14\!\cdots\!55}a^{19}-\frac{11\!\cdots\!19}{29\!\cdots\!71}a^{18}+\frac{31\!\cdots\!64}{14\!\cdots\!55}a^{17}-\frac{70\!\cdots\!22}{14\!\cdots\!55}a^{16}+\frac{37\!\cdots\!31}{22\!\cdots\!67}a^{15}-\frac{42\!\cdots\!62}{14\!\cdots\!55}a^{14}-\frac{48\!\cdots\!32}{29\!\cdots\!71}a^{13}+\frac{68\!\cdots\!14}{14\!\cdots\!55}a^{12}-\frac{34\!\cdots\!36}{29\!\cdots\!71}a^{11}+\frac{71\!\cdots\!31}{14\!\cdots\!55}a^{10}-\frac{92\!\cdots\!52}{11\!\cdots\!35}a^{9}+\frac{33\!\cdots\!68}{14\!\cdots\!55}a^{8}-\frac{19\!\cdots\!53}{14\!\cdots\!55}a^{7}+\frac{34\!\cdots\!11}{11\!\cdots\!35}a^{6}+\frac{30\!\cdots\!26}{29\!\cdots\!71}a^{5}-\frac{15\!\cdots\!28}{14\!\cdots\!55}a^{4}+\frac{81\!\cdots\!61}{14\!\cdots\!55}a^{3}-\frac{52\!\cdots\!97}{14\!\cdots\!55}a^{2}+\frac{20\!\cdots\!42}{29\!\cdots\!71}a-\frac{81\!\cdots\!20}{29\!\cdots\!71}$, $\frac{14\!\cdots\!44}{87\!\cdots\!05}a^{19}-\frac{69\!\cdots\!49}{87\!\cdots\!05}a^{18}+\frac{39\!\cdots\!84}{87\!\cdots\!05}a^{17}-\frac{84\!\cdots\!11}{87\!\cdots\!05}a^{16}+\frac{22\!\cdots\!49}{67\!\cdots\!85}a^{15}-\frac{22\!\cdots\!02}{87\!\cdots\!05}a^{14}-\frac{37\!\cdots\!99}{87\!\cdots\!05}a^{13}+\frac{85\!\cdots\!09}{87\!\cdots\!05}a^{12}-\frac{20\!\cdots\!97}{87\!\cdots\!05}a^{11}+\frac{17\!\cdots\!04}{17\!\cdots\!81}a^{10}-\frac{10\!\cdots\!32}{67\!\cdots\!85}a^{9}+\frac{38\!\cdots\!54}{87\!\cdots\!05}a^{8}-\frac{17\!\cdots\!99}{87\!\cdots\!05}a^{7}+\frac{32\!\cdots\!73}{67\!\cdots\!85}a^{6}+\frac{21\!\cdots\!52}{87\!\cdots\!05}a^{5}-\frac{23\!\cdots\!69}{87\!\cdots\!05}a^{4}+\frac{21\!\cdots\!23}{17\!\cdots\!81}a^{3}-\frac{69\!\cdots\!88}{87\!\cdots\!05}a^{2}+\frac{26\!\cdots\!93}{17\!\cdots\!81}a-\frac{10\!\cdots\!15}{17\!\cdots\!81}$
| 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: | \( 390465777.545 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 390465777.545 \cdot 2}{2\cdot\sqrt{1969799822613689762148183291015625}}\cr\approx \mathstrut & 0.341925002343 \end{aligned}\] (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 + 23*x^18 - 38*x^17 + 157*x^16 + 126*x^15 - 239*x^14 + 5543*x^13 - 9767*x^12 + 48731*x^11 - 52647*x^10 + 197277*x^9 + 60753*x^8 + 239947*x^7 + 1556367*x^6 - 309932*x^5 + 5752551*x^4 + 1073140*x^3 + 5045800*x^2 + 3532675*x - 2727775)
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 + 23*x^18 - 38*x^17 + 157*x^16 + 126*x^15 - 239*x^14 + 5543*x^13 - 9767*x^12 + 48731*x^11 - 52647*x^10 + 197277*x^9 + 60753*x^8 + 239947*x^7 + 1556367*x^6 - 309932*x^5 + 5752551*x^4 + 1073140*x^3 + 5045800*x^2 + 3532675*x - 2727775, 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 + 23*x^18 - 38*x^17 + 157*x^16 + 126*x^15 - 239*x^14 + 5543*x^13 - 9767*x^12 + 48731*x^11 - 52647*x^10 + 197277*x^9 + 60753*x^8 + 239947*x^7 + 1556367*x^6 - 309932*x^5 + 5752551*x^4 + 1073140*x^3 + 5045800*x^2 + 3532675*x - 2727775);
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 + 23*x^18 - 38*x^17 + 157*x^16 + 126*x^15 - 239*x^14 + 5543*x^13 - 9767*x^12 + 48731*x^11 - 52647*x^10 + 197277*x^9 + 60753*x^8 + 239947*x^7 + 1556367*x^6 - 309932*x^5 + 5752551*x^4 + 1073140*x^3 + 5045800*x^2 + 3532675*x - 2727775);
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
$C_2\wr S_5$ (as 20T288):
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 non-solvable group of order 3840 |
| The 36 conjugacy class representatives for $C_2\wr S_5$ |
| Character table for $C_2\wr S_5$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 5.5.24217.1, 10.10.1832697153125.1, 10.2.71011883131565.1, 10.2.8876485391445625.1 |
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
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.2.71011883131565.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{10}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ | |
|
\(61\)
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(397\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |