Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 16*x^18 + 121*x^17 + 95*x^16 - 933*x^15 - 226*x^14 + 3630*x^13 - 25*x^12 - 7849*x^11 + 1238*x^10 + 9696*x^9 - 2643*x^8 - 6638*x^7 + 2522*x^6 + 2214*x^5 - 1128*x^4 - 209*x^3 + 182*x^2 - 26*x + 1)
gp: K = bnfinit(y^20 - 6*y^19 - 16*y^18 + 121*y^17 + 95*y^16 - 933*y^15 - 226*y^14 + 3630*y^13 - 25*y^12 - 7849*y^11 + 1238*y^10 + 9696*y^9 - 2643*y^8 - 6638*y^7 + 2522*y^6 + 2214*y^5 - 1128*y^4 - 209*y^3 + 182*y^2 - 26*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^19 - 16*x^18 + 121*x^17 + 95*x^16 - 933*x^15 - 226*x^14 + 3630*x^13 - 25*x^12 - 7849*x^11 + 1238*x^10 + 9696*x^9 - 2643*x^8 - 6638*x^7 + 2522*x^6 + 2214*x^5 - 1128*x^4 - 209*x^3 + 182*x^2 - 26*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 6*x^19 - 16*x^18 + 121*x^17 + 95*x^16 - 933*x^15 - 226*x^14 + 3630*x^13 - 25*x^12 - 7849*x^11 + 1238*x^10 + 9696*x^9 - 2643*x^8 - 6638*x^7 + 2522*x^6 + 2214*x^5 - 1128*x^4 - 209*x^3 + 182*x^2 - 26*x + 1)
\( x^{20} - 6 x^{19} - 16 x^{18} + 121 x^{17} + 95 x^{16} - 933 x^{15} - 226 x^{14} + 3630 x^{13} + \cdots + 1 \)
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: |
\(333025103817911062616373171479401\)
\(\medspace = 19^{10}\cdot 293^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(42.28\) | 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: | $19^{1/2}293^{1/2}\approx 74.61233142048303$ | ||
| Ramified primes: |
\(19\), \(293\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{33\!\cdots\!23}a^{19}+\frac{564382434816272}{33\!\cdots\!23}a^{18}-\frac{206660134027447}{33\!\cdots\!23}a^{17}+\frac{12\!\cdots\!70}{33\!\cdots\!23}a^{16}+\frac{252417889324493}{33\!\cdots\!23}a^{15}-\frac{15\!\cdots\!61}{33\!\cdots\!23}a^{14}+\frac{579973081206849}{33\!\cdots\!23}a^{13}+\frac{457139677050149}{33\!\cdots\!23}a^{12}-\frac{16\!\cdots\!03}{33\!\cdots\!23}a^{11}-\frac{358664398820179}{33\!\cdots\!23}a^{10}+\frac{830079917363802}{33\!\cdots\!23}a^{9}-\frac{707081182098395}{33\!\cdots\!23}a^{8}-\frac{621163320277074}{33\!\cdots\!23}a^{7}+\frac{791364865198392}{33\!\cdots\!23}a^{6}-\frac{191856093941960}{33\!\cdots\!23}a^{5}+\frac{440441417632769}{33\!\cdots\!23}a^{4}-\frac{10\!\cdots\!78}{33\!\cdots\!23}a^{3}-\frac{989459496811115}{33\!\cdots\!23}a^{2}-\frac{923113311719802}{33\!\cdots\!23}a+\frac{771527516705096}{33\!\cdots\!23}$
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
Trivial group, which has order $1$ (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: |
$a$, $\frac{12\!\cdots\!78}{33\!\cdots\!23}a^{19}-\frac{69\!\cdots\!36}{33\!\cdots\!23}a^{18}-\frac{23\!\cdots\!47}{33\!\cdots\!23}a^{17}+\frac{14\!\cdots\!08}{33\!\cdots\!23}a^{16}+\frac{18\!\cdots\!37}{33\!\cdots\!23}a^{15}-\frac{10\!\cdots\!39}{33\!\cdots\!23}a^{14}-\frac{81\!\cdots\!34}{33\!\cdots\!23}a^{13}+\frac{41\!\cdots\!21}{33\!\cdots\!23}a^{12}+\frac{19\!\cdots\!40}{33\!\cdots\!23}a^{11}-\frac{87\!\cdots\!29}{33\!\cdots\!23}a^{10}-\frac{27\!\cdots\!52}{33\!\cdots\!23}a^{9}+\frac{10\!\cdots\!46}{33\!\cdots\!23}a^{8}+\frac{19\!\cdots\!19}{33\!\cdots\!23}a^{7}-\frac{70\!\cdots\!99}{33\!\cdots\!23}a^{6}-\frac{40\!\cdots\!50}{33\!\cdots\!23}a^{5}+\frac{24\!\cdots\!56}{33\!\cdots\!23}a^{4}-\frac{17\!\cdots\!71}{33\!\cdots\!23}a^{3}-\frac{31\!\cdots\!70}{33\!\cdots\!23}a^{2}+\frac{63\!\cdots\!62}{33\!\cdots\!23}a-\frac{26\!\cdots\!93}{33\!\cdots\!23}$, $\frac{82\!\cdots\!11}{33\!\cdots\!23}a^{19}-\frac{44\!\cdots\!08}{33\!\cdots\!23}a^{18}-\frac{15\!\cdots\!02}{33\!\cdots\!23}a^{17}+\frac{90\!\cdots\!53}{33\!\cdots\!23}a^{16}+\frac{12\!\cdots\!33}{33\!\cdots\!23}a^{15}-\frac{69\!\cdots\!74}{33\!\cdots\!23}a^{14}-\frac{57\!\cdots\!85}{33\!\cdots\!23}a^{13}+\frac{26\!\cdots\!58}{33\!\cdots\!23}a^{12}+\frac{14\!\cdots\!20}{33\!\cdots\!23}a^{11}-\frac{57\!\cdots\!87}{33\!\cdots\!23}a^{10}-\frac{20\!\cdots\!98}{33\!\cdots\!23}a^{9}+\frac{69\!\cdots\!18}{33\!\cdots\!23}a^{8}+\frac{15\!\cdots\!09}{33\!\cdots\!23}a^{7}-\frac{47\!\cdots\!89}{33\!\cdots\!23}a^{6}-\frac{44\!\cdots\!83}{33\!\cdots\!23}a^{5}+\frac{16\!\cdots\!08}{33\!\cdots\!23}a^{4}-\frac{52\!\cdots\!26}{33\!\cdots\!23}a^{3}-\frac{21\!\cdots\!72}{33\!\cdots\!23}a^{2}+\frac{32\!\cdots\!26}{33\!\cdots\!23}a-\frac{99\!\cdots\!52}{33\!\cdots\!23}$, $\frac{86\!\cdots\!83}{33\!\cdots\!23}a^{19}-\frac{47\!\cdots\!35}{33\!\cdots\!23}a^{18}-\frac{16\!\cdots\!13}{33\!\cdots\!23}a^{17}+\frac{96\!\cdots\!65}{33\!\cdots\!23}a^{16}+\frac{13\!\cdots\!77}{33\!\cdots\!23}a^{15}-\frac{74\!\cdots\!59}{33\!\cdots\!23}a^{14}-\frac{58\!\cdots\!95}{33\!\cdots\!23}a^{13}+\frac{28\!\cdots\!49}{33\!\cdots\!23}a^{12}+\frac{14\!\cdots\!65}{33\!\cdots\!23}a^{11}-\frac{60\!\cdots\!71}{33\!\cdots\!23}a^{10}-\frac{20\!\cdots\!42}{33\!\cdots\!23}a^{9}+\frac{73\!\cdots\!64}{33\!\cdots\!23}a^{8}+\frac{15\!\cdots\!19}{33\!\cdots\!23}a^{7}-\frac{50\!\cdots\!00}{33\!\cdots\!23}a^{6}-\frac{38\!\cdots\!41}{33\!\cdots\!23}a^{5}+\frac{17\!\cdots\!10}{33\!\cdots\!23}a^{4}-\frac{99\!\cdots\!77}{33\!\cdots\!23}a^{3}-\frac{22\!\cdots\!93}{33\!\cdots\!23}a^{2}+\frac{41\!\cdots\!94}{33\!\cdots\!23}a-\frac{18\!\cdots\!03}{33\!\cdots\!23}$, $\frac{32\!\cdots\!30}{33\!\cdots\!23}a^{19}-\frac{19\!\cdots\!93}{33\!\cdots\!23}a^{18}-\frac{51\!\cdots\!99}{33\!\cdots\!23}a^{17}+\frac{37\!\cdots\!70}{33\!\cdots\!23}a^{16}+\frac{30\!\cdots\!67}{33\!\cdots\!23}a^{15}-\frac{28\!\cdots\!08}{33\!\cdots\!23}a^{14}-\frac{83\!\cdots\!53}{33\!\cdots\!23}a^{13}+\frac{10\!\cdots\!34}{33\!\cdots\!23}a^{12}+\frac{76\!\cdots\!48}{33\!\cdots\!23}a^{11}-\frac{20\!\cdots\!93}{33\!\cdots\!23}a^{10}+\frac{96\!\cdots\!94}{33\!\cdots\!23}a^{9}+\frac{23\!\cdots\!23}{33\!\cdots\!23}a^{8}-\frac{29\!\cdots\!98}{33\!\cdots\!23}a^{7}-\frac{14\!\cdots\!20}{33\!\cdots\!23}a^{6}+\frac{28\!\cdots\!31}{33\!\cdots\!23}a^{5}+\frac{43\!\cdots\!23}{33\!\cdots\!23}a^{4}-\frac{11\!\cdots\!85}{33\!\cdots\!23}a^{3}-\frac{47\!\cdots\!17}{33\!\cdots\!23}a^{2}+\frac{16\!\cdots\!04}{33\!\cdots\!23}a-\frac{56\!\cdots\!24}{33\!\cdots\!23}$, $\frac{71\!\cdots\!53}{33\!\cdots\!23}a^{19}-\frac{37\!\cdots\!98}{33\!\cdots\!23}a^{18}-\frac{14\!\cdots\!20}{33\!\cdots\!23}a^{17}+\frac{76\!\cdots\!74}{33\!\cdots\!23}a^{16}+\frac{13\!\cdots\!78}{33\!\cdots\!23}a^{15}-\frac{59\!\cdots\!05}{33\!\cdots\!23}a^{14}-\frac{65\!\cdots\!74}{33\!\cdots\!23}a^{13}+\frac{23\!\cdots\!23}{33\!\cdots\!23}a^{12}+\frac{18\!\cdots\!12}{33\!\cdots\!23}a^{11}-\frac{49\!\cdots\!74}{33\!\cdots\!23}a^{10}-\frac{29\!\cdots\!76}{33\!\cdots\!23}a^{9}+\frac{61\!\cdots\!69}{33\!\cdots\!23}a^{8}+\frac{25\!\cdots\!99}{33\!\cdots\!23}a^{7}-\frac{42\!\cdots\!25}{33\!\cdots\!23}a^{6}-\frac{10\!\cdots\!00}{33\!\cdots\!23}a^{5}+\frac{15\!\cdots\!95}{33\!\cdots\!23}a^{4}+\frac{96\!\cdots\!80}{33\!\cdots\!23}a^{3}-\frac{22\!\cdots\!17}{33\!\cdots\!23}a^{2}+\frac{22\!\cdots\!53}{33\!\cdots\!23}a+\frac{731619851770462}{33\!\cdots\!23}$, $\frac{32\!\cdots\!30}{33\!\cdots\!23}a^{19}-\frac{19\!\cdots\!93}{33\!\cdots\!23}a^{18}-\frac{51\!\cdots\!99}{33\!\cdots\!23}a^{17}+\frac{37\!\cdots\!70}{33\!\cdots\!23}a^{16}+\frac{30\!\cdots\!67}{33\!\cdots\!23}a^{15}-\frac{28\!\cdots\!08}{33\!\cdots\!23}a^{14}-\frac{83\!\cdots\!53}{33\!\cdots\!23}a^{13}+\frac{10\!\cdots\!34}{33\!\cdots\!23}a^{12}+\frac{76\!\cdots\!48}{33\!\cdots\!23}a^{11}-\frac{20\!\cdots\!93}{33\!\cdots\!23}a^{10}+\frac{96\!\cdots\!94}{33\!\cdots\!23}a^{9}+\frac{23\!\cdots\!23}{33\!\cdots\!23}a^{8}-\frac{29\!\cdots\!98}{33\!\cdots\!23}a^{7}-\frac{14\!\cdots\!20}{33\!\cdots\!23}a^{6}+\frac{28\!\cdots\!31}{33\!\cdots\!23}a^{5}+\frac{43\!\cdots\!23}{33\!\cdots\!23}a^{4}-\frac{11\!\cdots\!85}{33\!\cdots\!23}a^{3}-\frac{47\!\cdots\!17}{33\!\cdots\!23}a^{2}+\frac{16\!\cdots\!27}{33\!\cdots\!23}a-\frac{56\!\cdots\!24}{33\!\cdots\!23}$, $\frac{16\!\cdots\!49}{33\!\cdots\!23}a^{19}-\frac{88\!\cdots\!75}{33\!\cdots\!23}a^{18}-\frac{34\!\cdots\!23}{33\!\cdots\!23}a^{17}+\frac{18\!\cdots\!52}{33\!\cdots\!23}a^{16}+\frac{30\!\cdots\!24}{33\!\cdots\!23}a^{15}-\frac{15\!\cdots\!00}{33\!\cdots\!23}a^{14}-\frac{14\!\cdots\!70}{33\!\cdots\!23}a^{13}+\frac{61\!\cdots\!21}{33\!\cdots\!23}a^{12}+\frac{39\!\cdots\!00}{33\!\cdots\!23}a^{11}-\frac{14\!\cdots\!85}{33\!\cdots\!23}a^{10}-\frac{58\!\cdots\!59}{33\!\cdots\!23}a^{9}+\frac{19\!\cdots\!39}{33\!\cdots\!23}a^{8}+\frac{44\!\cdots\!49}{33\!\cdots\!23}a^{7}-\frac{14\!\cdots\!59}{33\!\cdots\!23}a^{6}-\frac{12\!\cdots\!16}{33\!\cdots\!23}a^{5}+\frac{56\!\cdots\!87}{33\!\cdots\!23}a^{4}-\frac{19\!\cdots\!64}{33\!\cdots\!23}a^{3}-\frac{84\!\cdots\!08}{33\!\cdots\!23}a^{2}+\frac{10\!\cdots\!45}{33\!\cdots\!23}a-\frac{18\!\cdots\!58}{33\!\cdots\!23}$, $\frac{11\!\cdots\!86}{33\!\cdots\!23}a^{19}-\frac{66\!\cdots\!04}{33\!\cdots\!23}a^{18}-\frac{19\!\cdots\!21}{33\!\cdots\!23}a^{17}+\frac{12\!\cdots\!62}{33\!\cdots\!23}a^{16}+\frac{14\!\cdots\!58}{33\!\cdots\!23}a^{15}-\frac{90\!\cdots\!70}{33\!\cdots\!23}a^{14}-\frac{51\!\cdots\!69}{33\!\cdots\!23}a^{13}+\frac{31\!\cdots\!87}{33\!\cdots\!23}a^{12}+\frac{10\!\cdots\!17}{33\!\cdots\!23}a^{11}-\frac{55\!\cdots\!88}{33\!\cdots\!23}a^{10}-\frac{10\!\cdots\!05}{33\!\cdots\!23}a^{9}+\frac{51\!\cdots\!57}{33\!\cdots\!23}a^{8}+\frac{33\!\cdots\!89}{33\!\cdots\!23}a^{7}-\frac{23\!\cdots\!85}{33\!\cdots\!23}a^{6}+\frac{20\!\cdots\!22}{33\!\cdots\!23}a^{5}+\frac{36\!\cdots\!07}{33\!\cdots\!23}a^{4}-\frac{18\!\cdots\!92}{33\!\cdots\!23}a^{3}+\frac{20\!\cdots\!95}{33\!\cdots\!23}a^{2}+\frac{41\!\cdots\!01}{33\!\cdots\!23}a-\frac{32\!\cdots\!87}{33\!\cdots\!23}$, $\frac{14\!\cdots\!92}{33\!\cdots\!23}a^{19}-\frac{82\!\cdots\!09}{33\!\cdots\!23}a^{18}-\frac{27\!\cdots\!95}{33\!\cdots\!23}a^{17}+\frac{16\!\cdots\!09}{33\!\cdots\!23}a^{16}+\frac{21\!\cdots\!81}{33\!\cdots\!23}a^{15}-\frac{12\!\cdots\!90}{33\!\cdots\!23}a^{14}-\frac{88\!\cdots\!83}{33\!\cdots\!23}a^{13}+\frac{49\!\cdots\!39}{33\!\cdots\!23}a^{12}+\frac{21\!\cdots\!32}{33\!\cdots\!23}a^{11}-\frac{10\!\cdots\!37}{33\!\cdots\!23}a^{10}-\frac{28\!\cdots\!11}{33\!\cdots\!23}a^{9}+\frac{12\!\cdots\!71}{33\!\cdots\!23}a^{8}+\frac{18\!\cdots\!81}{33\!\cdots\!23}a^{7}-\frac{86\!\cdots\!07}{33\!\cdots\!23}a^{6}-\frac{22\!\cdots\!23}{33\!\cdots\!23}a^{5}+\frac{30\!\cdots\!32}{33\!\cdots\!23}a^{4}-\frac{27\!\cdots\!04}{33\!\cdots\!23}a^{3}-\frac{40\!\cdots\!85}{33\!\cdots\!23}a^{2}+\frac{83\!\cdots\!14}{33\!\cdots\!23}a-\frac{32\!\cdots\!14}{33\!\cdots\!23}$, $\frac{84\!\cdots\!70}{33\!\cdots\!23}a^{19}-\frac{44\!\cdots\!54}{33\!\cdots\!23}a^{18}-\frac{16\!\cdots\!79}{33\!\cdots\!23}a^{17}+\frac{91\!\cdots\!68}{33\!\cdots\!23}a^{16}+\frac{14\!\cdots\!15}{33\!\cdots\!23}a^{15}-\frac{70\!\cdots\!50}{33\!\cdots\!23}a^{14}-\frac{69\!\cdots\!53}{33\!\cdots\!23}a^{13}+\frac{27\!\cdots\!41}{33\!\cdots\!23}a^{12}+\frac{18\!\cdots\!93}{33\!\cdots\!23}a^{11}-\frac{58\!\cdots\!37}{33\!\cdots\!23}a^{10}-\frac{29\!\cdots\!27}{33\!\cdots\!23}a^{9}+\frac{71\!\cdots\!75}{33\!\cdots\!23}a^{8}+\frac{25\!\cdots\!77}{33\!\cdots\!23}a^{7}-\frac{49\!\cdots\!57}{33\!\cdots\!23}a^{6}-\frac{10\!\cdots\!03}{33\!\cdots\!23}a^{5}+\frac{17\!\cdots\!74}{33\!\cdots\!23}a^{4}+\frac{13\!\cdots\!64}{33\!\cdots\!23}a^{3}-\frac{24\!\cdots\!78}{33\!\cdots\!23}a^{2}+\frac{12\!\cdots\!09}{33\!\cdots\!23}a+\frac{19\!\cdots\!42}{33\!\cdots\!23}$, $\frac{71\!\cdots\!53}{33\!\cdots\!23}a^{19}-\frac{37\!\cdots\!98}{33\!\cdots\!23}a^{18}-\frac{14\!\cdots\!20}{33\!\cdots\!23}a^{17}+\frac{76\!\cdots\!74}{33\!\cdots\!23}a^{16}+\frac{13\!\cdots\!78}{33\!\cdots\!23}a^{15}-\frac{59\!\cdots\!05}{33\!\cdots\!23}a^{14}-\frac{65\!\cdots\!74}{33\!\cdots\!23}a^{13}+\frac{23\!\cdots\!23}{33\!\cdots\!23}a^{12}+\frac{18\!\cdots\!12}{33\!\cdots\!23}a^{11}-\frac{49\!\cdots\!74}{33\!\cdots\!23}a^{10}-\frac{29\!\cdots\!76}{33\!\cdots\!23}a^{9}+\frac{61\!\cdots\!69}{33\!\cdots\!23}a^{8}+\frac{25\!\cdots\!99}{33\!\cdots\!23}a^{7}-\frac{42\!\cdots\!25}{33\!\cdots\!23}a^{6}-\frac{10\!\cdots\!00}{33\!\cdots\!23}a^{5}+\frac{15\!\cdots\!95}{33\!\cdots\!23}a^{4}+\frac{96\!\cdots\!80}{33\!\cdots\!23}a^{3}-\frac{22\!\cdots\!17}{33\!\cdots\!23}a^{2}+\frac{22\!\cdots\!53}{33\!\cdots\!23}a+\frac{40\!\cdots\!85}{33\!\cdots\!23}$, $\frac{60\!\cdots\!25}{33\!\cdots\!23}a^{19}-\frac{33\!\cdots\!99}{33\!\cdots\!23}a^{18}-\frac{11\!\cdots\!36}{33\!\cdots\!23}a^{17}+\frac{68\!\cdots\!42}{33\!\cdots\!23}a^{16}+\frac{85\!\cdots\!13}{33\!\cdots\!23}a^{15}-\frac{52\!\cdots\!83}{33\!\cdots\!23}a^{14}-\frac{35\!\cdots\!03}{33\!\cdots\!23}a^{13}+\frac{20\!\cdots\!74}{33\!\cdots\!23}a^{12}+\frac{82\!\cdots\!34}{33\!\cdots\!23}a^{11}-\frac{44\!\cdots\!53}{33\!\cdots\!23}a^{10}-\frac{10\!\cdots\!05}{33\!\cdots\!23}a^{9}+\frac{55\!\cdots\!00}{33\!\cdots\!23}a^{8}+\frac{53\!\cdots\!16}{33\!\cdots\!23}a^{7}-\frac{38\!\cdots\!96}{33\!\cdots\!23}a^{6}+\frac{97\!\cdots\!16}{33\!\cdots\!23}a^{5}+\frac{13\!\cdots\!45}{33\!\cdots\!23}a^{4}-\frac{20\!\cdots\!03}{33\!\cdots\!23}a^{3}-\frac{17\!\cdots\!34}{33\!\cdots\!23}a^{2}+\frac{49\!\cdots\!91}{33\!\cdots\!23}a-\frac{38\!\cdots\!68}{33\!\cdots\!23}$, $\frac{18\!\cdots\!82}{33\!\cdots\!23}a^{19}-\frac{10\!\cdots\!80}{33\!\cdots\!23}a^{18}-\frac{35\!\cdots\!52}{33\!\cdots\!23}a^{17}+\frac{20\!\cdots\!42}{33\!\cdots\!23}a^{16}+\frac{28\!\cdots\!40}{33\!\cdots\!23}a^{15}-\frac{15\!\cdots\!42}{33\!\cdots\!23}a^{14}-\frac{12\!\cdots\!60}{33\!\cdots\!23}a^{13}+\frac{60\!\cdots\!57}{33\!\cdots\!23}a^{12}+\frac{29\!\cdots\!83}{33\!\cdots\!23}a^{11}-\frac{12\!\cdots\!48}{33\!\cdots\!23}a^{10}-\frac{40\!\cdots\!73}{33\!\cdots\!23}a^{9}+\frac{15\!\cdots\!09}{33\!\cdots\!23}a^{8}+\frac{25\!\cdots\!08}{33\!\cdots\!23}a^{7}-\frac{10\!\cdots\!52}{33\!\cdots\!23}a^{6}-\frac{24\!\cdots\!15}{33\!\cdots\!23}a^{5}+\frac{35\!\cdots\!64}{33\!\cdots\!23}a^{4}-\frac{42\!\cdots\!35}{33\!\cdots\!23}a^{3}-\frac{46\!\cdots\!07}{33\!\cdots\!23}a^{2}+\frac{12\!\cdots\!47}{33\!\cdots\!23}a-\frac{69\!\cdots\!37}{33\!\cdots\!23}$, $\frac{89\!\cdots\!24}{33\!\cdots\!23}a^{19}-\frac{47\!\cdots\!78}{33\!\cdots\!23}a^{18}-\frac{18\!\cdots\!94}{33\!\cdots\!23}a^{17}+\frac{97\!\cdots\!60}{33\!\cdots\!23}a^{16}+\frac{16\!\cdots\!83}{33\!\cdots\!23}a^{15}-\frac{76\!\cdots\!11}{33\!\cdots\!23}a^{14}-\frac{78\!\cdots\!61}{33\!\cdots\!23}a^{13}+\frac{30\!\cdots\!30}{33\!\cdots\!23}a^{12}+\frac{21\!\cdots\!07}{33\!\cdots\!23}a^{11}-\frac{66\!\cdots\!07}{33\!\cdots\!23}a^{10}-\frac{32\!\cdots\!12}{33\!\cdots\!23}a^{9}+\frac{84\!\cdots\!13}{33\!\cdots\!23}a^{8}+\frac{25\!\cdots\!73}{33\!\cdots\!23}a^{7}-\frac{60\!\cdots\!86}{33\!\cdots\!23}a^{6}-\frac{83\!\cdots\!19}{33\!\cdots\!23}a^{5}+\frac{22\!\cdots\!50}{33\!\cdots\!23}a^{4}-\frac{42\!\cdots\!87}{33\!\cdots\!23}a^{3}-\frac{31\!\cdots\!01}{33\!\cdots\!23}a^{2}+\frac{48\!\cdots\!80}{33\!\cdots\!23}a-\frac{14\!\cdots\!17}{33\!\cdots\!23}$, $\frac{17\!\cdots\!45}{33\!\cdots\!23}a^{19}-\frac{97\!\cdots\!74}{33\!\cdots\!23}a^{18}-\frac{32\!\cdots\!09}{33\!\cdots\!23}a^{17}+\frac{19\!\cdots\!56}{33\!\cdots\!23}a^{16}+\frac{25\!\cdots\!13}{33\!\cdots\!23}a^{15}-\frac{14\!\cdots\!31}{33\!\cdots\!23}a^{14}-\frac{10\!\cdots\!15}{33\!\cdots\!23}a^{13}+\frac{57\!\cdots\!24}{33\!\cdots\!23}a^{12}+\frac{26\!\cdots\!62}{33\!\cdots\!23}a^{11}-\frac{12\!\cdots\!12}{33\!\cdots\!23}a^{10}-\frac{36\!\cdots\!78}{33\!\cdots\!23}a^{9}+\frac{14\!\cdots\!57}{33\!\cdots\!23}a^{8}+\frac{26\!\cdots\!98}{33\!\cdots\!23}a^{7}-\frac{95\!\cdots\!39}{33\!\cdots\!23}a^{6}-\frac{62\!\cdots\!98}{33\!\cdots\!23}a^{5}+\frac{32\!\cdots\!85}{33\!\cdots\!23}a^{4}-\frac{17\!\cdots\!63}{33\!\cdots\!23}a^{3}-\frac{41\!\cdots\!17}{33\!\cdots\!23}a^{2}+\frac{71\!\cdots\!47}{33\!\cdots\!23}a-\frac{27\!\cdots\!02}{33\!\cdots\!23}$, $\frac{15\!\cdots\!17}{33\!\cdots\!23}a^{19}-\frac{83\!\cdots\!25}{33\!\cdots\!23}a^{18}-\frac{28\!\cdots\!53}{33\!\cdots\!23}a^{17}+\frac{16\!\cdots\!98}{33\!\cdots\!23}a^{16}+\frac{23\!\cdots\!66}{33\!\cdots\!23}a^{15}-\frac{12\!\cdots\!17}{33\!\cdots\!23}a^{14}-\frac{10\!\cdots\!24}{33\!\cdots\!23}a^{13}+\frac{49\!\cdots\!00}{33\!\cdots\!23}a^{12}+\frac{25\!\cdots\!67}{33\!\cdots\!23}a^{11}-\frac{10\!\cdots\!49}{33\!\cdots\!23}a^{10}-\frac{35\!\cdots\!34}{33\!\cdots\!23}a^{9}+\frac{12\!\cdots\!25}{33\!\cdots\!23}a^{8}+\frac{25\!\cdots\!82}{33\!\cdots\!23}a^{7}-\frac{86\!\cdots\!92}{33\!\cdots\!23}a^{6}-\frac{57\!\cdots\!64}{33\!\cdots\!23}a^{5}+\frac{30\!\cdots\!80}{33\!\cdots\!23}a^{4}-\frac{20\!\cdots\!06}{33\!\cdots\!23}a^{3}-\frac{40\!\cdots\!89}{33\!\cdots\!23}a^{2}+\frac{81\!\cdots\!30}{33\!\cdots\!23}a-\frac{38\!\cdots\!80}{33\!\cdots\!23}$, $\frac{51\!\cdots\!01}{33\!\cdots\!23}a^{19}-\frac{28\!\cdots\!59}{33\!\cdots\!23}a^{18}-\frac{91\!\cdots\!87}{33\!\cdots\!23}a^{17}+\frac{57\!\cdots\!64}{33\!\cdots\!23}a^{16}+\frac{68\!\cdots\!72}{33\!\cdots\!23}a^{15}-\frac{42\!\cdots\!70}{33\!\cdots\!23}a^{14}-\frac{27\!\cdots\!06}{33\!\cdots\!23}a^{13}+\frac{15\!\cdots\!04}{33\!\cdots\!23}a^{12}+\frac{60\!\cdots\!82}{33\!\cdots\!23}a^{11}-\frac{31\!\cdots\!64}{33\!\cdots\!23}a^{10}-\frac{71\!\cdots\!53}{33\!\cdots\!23}a^{9}+\frac{33\!\cdots\!90}{33\!\cdots\!23}a^{8}+\frac{36\!\cdots\!03}{33\!\cdots\!23}a^{7}-\frac{19\!\cdots\!72}{33\!\cdots\!23}a^{6}+\frac{48\!\cdots\!33}{33\!\cdots\!23}a^{5}+\frac{52\!\cdots\!21}{33\!\cdots\!23}a^{4}-\frac{10\!\cdots\!84}{33\!\cdots\!23}a^{3}-\frac{46\!\cdots\!16}{33\!\cdots\!23}a^{2}+\frac{20\!\cdots\!65}{33\!\cdots\!23}a-\frac{20\!\cdots\!92}{33\!\cdots\!23}$, $\frac{90\!\cdots\!82}{33\!\cdots\!23}a^{19}-\frac{49\!\cdots\!27}{33\!\cdots\!23}a^{18}-\frac{17\!\cdots\!40}{33\!\cdots\!23}a^{17}+\frac{99\!\cdots\!18}{33\!\cdots\!23}a^{16}+\frac{14\!\cdots\!92}{33\!\cdots\!23}a^{15}-\frac{76\!\cdots\!95}{33\!\cdots\!23}a^{14}-\frac{62\!\cdots\!29}{33\!\cdots\!23}a^{13}+\frac{29\!\cdots\!88}{33\!\cdots\!23}a^{12}+\frac{15\!\cdots\!84}{33\!\cdots\!23}a^{11}-\frac{61\!\cdots\!28}{33\!\cdots\!23}a^{10}-\frac{21\!\cdots\!09}{33\!\cdots\!23}a^{9}+\frac{74\!\cdots\!06}{33\!\cdots\!23}a^{8}+\frac{15\!\cdots\!04}{33\!\cdots\!23}a^{7}-\frac{50\!\cdots\!70}{33\!\cdots\!23}a^{6}-\frac{34\!\cdots\!80}{33\!\cdots\!23}a^{5}+\frac{17\!\cdots\!62}{33\!\cdots\!23}a^{4}-\frac{12\!\cdots\!15}{33\!\cdots\!23}a^{3}-\frac{22\!\cdots\!93}{33\!\cdots\!23}a^{2}+\frac{47\!\cdots\!12}{33\!\cdots\!23}a-\frac{25\!\cdots\!29}{33\!\cdots\!23}$
| 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: | \( 6609693766.39 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 6609693766.39 \cdot 1}{2\cdot\sqrt{333025103817911062616373171479401}}\cr\approx \mathstrut & 0.189894667835 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 16*x^18 + 121*x^17 + 95*x^16 - 933*x^15 - 226*x^14 + 3630*x^13 - 25*x^12 - 7849*x^11 + 1238*x^10 + 9696*x^9 - 2643*x^8 - 6638*x^7 + 2522*x^6 + 2214*x^5 - 1128*x^4 - 209*x^3 + 182*x^2 - 26*x + 1)
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 - 6*x^19 - 16*x^18 + 121*x^17 + 95*x^16 - 933*x^15 - 226*x^14 + 3630*x^13 - 25*x^12 - 7849*x^11 + 1238*x^10 + 9696*x^9 - 2643*x^8 - 6638*x^7 + 2522*x^6 + 2214*x^5 - 1128*x^4 - 209*x^3 + 182*x^2 - 26*x + 1, 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 - 6*x^19 - 16*x^18 + 121*x^17 + 95*x^16 - 933*x^15 - 226*x^14 + 3630*x^13 - 25*x^12 - 7849*x^11 + 1238*x^10 + 9696*x^9 - 2643*x^8 - 6638*x^7 + 2522*x^6 + 2214*x^5 - 1128*x^4 - 209*x^3 + 182*x^2 - 26*x + 1);
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 - 6*x^19 - 16*x^18 + 121*x^17 + 95*x^16 - 933*x^15 - 226*x^14 + 3630*x^13 - 25*x^12 - 7849*x^11 + 1238*x^10 + 9696*x^9 - 2643*x^8 - 6638*x^7 + 2522*x^6 + 2214*x^5 - 1128*x^4 - 209*x^3 + 182*x^2 - 26*x + 1);
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
$C_2\times A_5$ (as 20T36):
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 120 |
| The 10 conjugacy class representatives for $C_2\times A_5$ |
| Character table for $C_2\times A_5$ |
Intermediate fields
| 10.10.960472390437121.1 |
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 sibling: | data not computed |
| Degree 12 siblings: | data not computed |
| Degree 20 sibling: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 40 sibling: | data not computed |
| Minimal sibling: | 10.10.281418410398076453.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.3.0.1}{3} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(293\)
| 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 $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |