Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 54*x^18 + 142*x^17 + 1215*x^16 - 1658*x^15 - 14226*x^14 + 5956*x^13 + 91351*x^12 + 20666*x^11 - 319628*x^10 - 208840*x^9 + 584299*x^8 + 561074*x^7 - 505908*x^6 - 639678*x^5 + 156250*x^4 + 296450*x^3 - 2608*x^2 - 43810*x - 263)
gp: K = bnfinit(y^20 - 4*y^19 - 54*y^18 + 142*y^17 + 1215*y^16 - 1658*y^15 - 14226*y^14 + 5956*y^13 + 91351*y^12 + 20666*y^11 - 319628*y^10 - 208840*y^9 + 584299*y^8 + 561074*y^7 - 505908*y^6 - 639678*y^5 + 156250*y^4 + 296450*y^3 - 2608*y^2 - 43810*y - 263, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 - 54*x^18 + 142*x^17 + 1215*x^16 - 1658*x^15 - 14226*x^14 + 5956*x^13 + 91351*x^12 + 20666*x^11 - 319628*x^10 - 208840*x^9 + 584299*x^8 + 561074*x^7 - 505908*x^6 - 639678*x^5 + 156250*x^4 + 296450*x^3 - 2608*x^2 - 43810*x - 263);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 - 54*x^18 + 142*x^17 + 1215*x^16 - 1658*x^15 - 14226*x^14 + 5956*x^13 + 91351*x^12 + 20666*x^11 - 319628*x^10 - 208840*x^9 + 584299*x^8 + 561074*x^7 - 505908*x^6 - 639678*x^5 + 156250*x^4 + 296450*x^3 - 2608*x^2 - 43810*x - 263)
\( x^{20} - 4 x^{19} - 54 x^{18} + 142 x^{17} + 1215 x^{16} - 1658 x^{15} - 14226 x^{14} + 5956 x^{13} + \cdots - 263 \)
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: |
\(7303816416639774784171798530359296\)
\(\medspace = 2^{30}\cdot 11^{16}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(49.34\) | 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: | $2^{31/16}11^{4/5}23^{1/2}\approx 125.0903140152616$ | ||
| Ramified primes: |
\(2\), \(11\), \(23\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{77}a^{15}+\frac{25}{77}a^{14}+\frac{9}{77}a^{13}-\frac{16}{77}a^{12}+\frac{13}{77}a^{11}+\frac{1}{7}a^{10}+\frac{16}{77}a^{9}-\frac{24}{77}a^{8}+\frac{36}{77}a^{7}-\frac{32}{77}a^{6}+\frac{3}{7}a^{5}-\frac{12}{77}a^{4}-\frac{26}{77}a^{3}-\frac{19}{77}a^{2}-\frac{6}{77}a+\frac{1}{77}$, $\frac{1}{77}a^{16}-\frac{10}{77}a^{13}+\frac{4}{11}a^{12}-\frac{6}{77}a^{11}-\frac{4}{11}a^{10}+\frac{38}{77}a^{9}+\frac{20}{77}a^{8}-\frac{8}{77}a^{7}-\frac{2}{11}a^{6}+\frac{10}{77}a^{5}-\frac{34}{77}a^{4}+\frac{15}{77}a^{3}+\frac{1}{11}a^{2}-\frac{3}{77}a-\frac{25}{77}$, $\frac{1}{77}a^{17}-\frac{10}{77}a^{14}+\frac{4}{11}a^{13}-\frac{6}{77}a^{12}-\frac{4}{11}a^{11}+\frac{38}{77}a^{10}+\frac{20}{77}a^{9}-\frac{8}{77}a^{8}-\frac{2}{11}a^{7}+\frac{10}{77}a^{6}-\frac{34}{77}a^{5}+\frac{15}{77}a^{4}+\frac{1}{11}a^{3}-\frac{3}{77}a^{2}-\frac{25}{77}a$, $\frac{1}{77}a^{18}-\frac{30}{77}a^{14}+\frac{1}{11}a^{13}-\frac{34}{77}a^{12}+\frac{2}{11}a^{11}-\frac{24}{77}a^{10}-\frac{2}{77}a^{9}-\frac{23}{77}a^{8}-\frac{15}{77}a^{7}+\frac{31}{77}a^{6}+\frac{37}{77}a^{5}-\frac{36}{77}a^{4}-\frac{32}{77}a^{3}+\frac{16}{77}a^{2}+\frac{17}{77}a+\frac{10}{77}$, $\frac{1}{56\!\cdots\!59}a^{19}+\frac{20\!\cdots\!15}{56\!\cdots\!59}a^{18}-\frac{15\!\cdots\!74}{46\!\cdots\!79}a^{17}+\frac{50\!\cdots\!85}{81\!\cdots\!37}a^{16}+\frac{15\!\cdots\!18}{56\!\cdots\!59}a^{15}+\frac{32\!\cdots\!22}{56\!\cdots\!59}a^{14}+\frac{15\!\cdots\!97}{56\!\cdots\!59}a^{13}+\frac{16\!\cdots\!47}{56\!\cdots\!59}a^{12}-\frac{13\!\cdots\!98}{56\!\cdots\!59}a^{11}+\frac{18\!\cdots\!05}{56\!\cdots\!59}a^{10}+\frac{76\!\cdots\!50}{51\!\cdots\!69}a^{9}-\frac{17\!\cdots\!89}{56\!\cdots\!59}a^{8}-\frac{12\!\cdots\!21}{56\!\cdots\!59}a^{7}-\frac{13\!\cdots\!92}{56\!\cdots\!59}a^{6}+\frac{22\!\cdots\!12}{56\!\cdots\!59}a^{5}-\frac{25\!\cdots\!38}{56\!\cdots\!59}a^{4}+\frac{27\!\cdots\!04}{56\!\cdots\!59}a^{3}-\frac{86\!\cdots\!35}{56\!\cdots\!59}a^{2}-\frac{51\!\cdots\!21}{56\!\cdots\!59}a+\frac{59\!\cdots\!84}{56\!\cdots\!59}$
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: |
$\frac{18\!\cdots\!34}{51\!\cdots\!69}a^{19}-\frac{99\!\cdots\!14}{51\!\cdots\!69}a^{18}-\frac{11\!\cdots\!44}{73\!\cdots\!67}a^{17}+\frac{38\!\cdots\!34}{51\!\cdots\!69}a^{16}+\frac{16\!\cdots\!76}{51\!\cdots\!69}a^{15}-\frac{54\!\cdots\!72}{51\!\cdots\!69}a^{14}-\frac{17\!\cdots\!38}{51\!\cdots\!69}a^{13}+\frac{37\!\cdots\!66}{51\!\cdots\!69}a^{12}+\frac{98\!\cdots\!84}{46\!\cdots\!79}a^{11}-\frac{12\!\cdots\!34}{51\!\cdots\!69}a^{10}-\frac{38\!\cdots\!52}{51\!\cdots\!69}a^{9}+\frac{20\!\cdots\!23}{51\!\cdots\!69}a^{8}+\frac{74\!\cdots\!80}{51\!\cdots\!69}a^{7}-\frac{10\!\cdots\!24}{51\!\cdots\!69}a^{6}-\frac{73\!\cdots\!08}{51\!\cdots\!69}a^{5}-\frac{55\!\cdots\!49}{73\!\cdots\!67}a^{4}+\frac{33\!\cdots\!10}{51\!\cdots\!69}a^{3}+\frac{33\!\cdots\!66}{51\!\cdots\!69}a^{2}-\frac{52\!\cdots\!20}{51\!\cdots\!69}a-\frac{55\!\cdots\!05}{73\!\cdots\!67}$, $\frac{56\!\cdots\!00}{73\!\cdots\!67}a^{19}-\frac{32\!\cdots\!74}{73\!\cdots\!67}a^{18}-\frac{17\!\cdots\!06}{51\!\cdots\!69}a^{17}+\frac{86\!\cdots\!13}{51\!\cdots\!69}a^{16}+\frac{33\!\cdots\!60}{51\!\cdots\!69}a^{15}-\frac{12\!\cdots\!68}{51\!\cdots\!69}a^{14}-\frac{35\!\cdots\!82}{51\!\cdots\!69}a^{13}+\frac{12\!\cdots\!95}{73\!\cdots\!67}a^{12}+\frac{21\!\cdots\!84}{51\!\cdots\!69}a^{11}-\frac{30\!\cdots\!56}{51\!\cdots\!69}a^{10}-\frac{75\!\cdots\!78}{51\!\cdots\!69}a^{9}+\frac{48\!\cdots\!78}{46\!\cdots\!79}a^{8}+\frac{14\!\cdots\!38}{51\!\cdots\!69}a^{7}-\frac{39\!\cdots\!86}{51\!\cdots\!69}a^{6}-\frac{14\!\cdots\!00}{51\!\cdots\!69}a^{5}+\frac{55\!\cdots\!30}{51\!\cdots\!69}a^{4}+\frac{60\!\cdots\!70}{51\!\cdots\!69}a^{3}+\frac{29\!\cdots\!78}{51\!\cdots\!69}a^{2}-\frac{86\!\cdots\!22}{51\!\cdots\!69}a-\frac{11\!\cdots\!39}{51\!\cdots\!69}$, $\frac{67\!\cdots\!36}{51\!\cdots\!69}a^{19}-\frac{38\!\cdots\!60}{51\!\cdots\!69}a^{18}-\frac{30\!\cdots\!04}{51\!\cdots\!69}a^{17}+\frac{20\!\cdots\!65}{73\!\cdots\!67}a^{16}+\frac{57\!\cdots\!48}{51\!\cdots\!69}a^{15}-\frac{20\!\cdots\!84}{51\!\cdots\!69}a^{14}-\frac{61\!\cdots\!00}{51\!\cdots\!69}a^{13}+\frac{14\!\cdots\!42}{51\!\cdots\!69}a^{12}+\frac{37\!\cdots\!96}{51\!\cdots\!69}a^{11}-\frac{50\!\cdots\!88}{51\!\cdots\!69}a^{10}-\frac{13\!\cdots\!64}{51\!\cdots\!69}a^{9}+\frac{84\!\cdots\!49}{51\!\cdots\!69}a^{8}+\frac{25\!\cdots\!48}{51\!\cdots\!69}a^{7}-\frac{56\!\cdots\!76}{51\!\cdots\!69}a^{6}-\frac{25\!\cdots\!64}{51\!\cdots\!69}a^{5}-\frac{10\!\cdots\!22}{51\!\cdots\!69}a^{4}+\frac{11\!\cdots\!36}{51\!\cdots\!69}a^{3}+\frac{83\!\cdots\!52}{51\!\cdots\!69}a^{2}-\frac{15\!\cdots\!72}{46\!\cdots\!79}a-\frac{85\!\cdots\!79}{51\!\cdots\!69}$, $\frac{10\!\cdots\!44}{51\!\cdots\!69}a^{19}-\frac{41\!\cdots\!90}{51\!\cdots\!69}a^{18}-\frac{56\!\cdots\!78}{46\!\cdots\!79}a^{17}+\frac{15\!\cdots\!49}{51\!\cdots\!69}a^{16}+\frac{18\!\cdots\!88}{66\!\cdots\!97}a^{15}-\frac{20\!\cdots\!28}{51\!\cdots\!69}a^{14}-\frac{17\!\cdots\!02}{51\!\cdots\!69}a^{13}+\frac{10\!\cdots\!93}{51\!\cdots\!69}a^{12}+\frac{11\!\cdots\!24}{51\!\cdots\!69}a^{11}-\frac{86\!\cdots\!24}{46\!\cdots\!79}a^{10}-\frac{42\!\cdots\!98}{51\!\cdots\!69}a^{9}-\frac{14\!\cdots\!39}{73\!\cdots\!67}a^{8}+\frac{86\!\cdots\!94}{51\!\cdots\!69}a^{7}+\frac{32\!\cdots\!06}{51\!\cdots\!69}a^{6}-\frac{92\!\cdots\!60}{51\!\cdots\!69}a^{5}-\frac{35\!\cdots\!14}{51\!\cdots\!69}a^{4}+\frac{48\!\cdots\!26}{51\!\cdots\!69}a^{3}+\frac{12\!\cdots\!26}{51\!\cdots\!69}a^{2}-\frac{97\!\cdots\!02}{51\!\cdots\!69}a-\frac{30\!\cdots\!67}{51\!\cdots\!69}$, $\frac{27\!\cdots\!53}{81\!\cdots\!37}a^{19}-\frac{10\!\cdots\!96}{56\!\cdots\!59}a^{18}-\frac{76\!\cdots\!08}{51\!\cdots\!69}a^{17}+\frac{41\!\cdots\!08}{56\!\cdots\!59}a^{16}+\frac{15\!\cdots\!82}{56\!\cdots\!59}a^{15}-\frac{85\!\cdots\!70}{81\!\cdots\!37}a^{14}-\frac{24\!\cdots\!84}{81\!\cdots\!37}a^{13}+\frac{41\!\cdots\!81}{56\!\cdots\!59}a^{12}+\frac{10\!\cdots\!09}{56\!\cdots\!59}a^{11}-\frac{14\!\cdots\!30}{56\!\cdots\!59}a^{10}-\frac{33\!\cdots\!49}{51\!\cdots\!69}a^{9}+\frac{35\!\cdots\!62}{81\!\cdots\!37}a^{8}+\frac{99\!\cdots\!15}{81\!\cdots\!37}a^{7}-\frac{17\!\cdots\!56}{56\!\cdots\!59}a^{6}-\frac{68\!\cdots\!50}{56\!\cdots\!59}a^{5}+\frac{86\!\cdots\!87}{56\!\cdots\!59}a^{4}+\frac{30\!\cdots\!70}{56\!\cdots\!59}a^{3}+\frac{20\!\cdots\!94}{56\!\cdots\!59}a^{2}-\frac{45\!\cdots\!96}{56\!\cdots\!59}a-\frac{34\!\cdots\!79}{56\!\cdots\!59}$, $\frac{18\!\cdots\!70}{56\!\cdots\!59}a^{19}-\frac{10\!\cdots\!94}{56\!\cdots\!59}a^{18}-\frac{77\!\cdots\!57}{51\!\cdots\!69}a^{17}+\frac{40\!\cdots\!64}{56\!\cdots\!59}a^{16}+\frac{16\!\cdots\!79}{56\!\cdots\!59}a^{15}-\frac{57\!\cdots\!36}{56\!\cdots\!59}a^{14}-\frac{17\!\cdots\!49}{56\!\cdots\!59}a^{13}+\frac{56\!\cdots\!64}{81\!\cdots\!37}a^{12}+\frac{10\!\cdots\!67}{56\!\cdots\!59}a^{11}-\frac{13\!\cdots\!24}{56\!\cdots\!59}a^{10}-\frac{35\!\cdots\!86}{51\!\cdots\!69}a^{9}+\frac{22\!\cdots\!65}{56\!\cdots\!59}a^{8}+\frac{74\!\cdots\!77}{56\!\cdots\!59}a^{7}-\frac{13\!\cdots\!74}{56\!\cdots\!59}a^{6}-\frac{74\!\cdots\!14}{56\!\cdots\!59}a^{5}-\frac{24\!\cdots\!75}{56\!\cdots\!59}a^{4}+\frac{33\!\cdots\!20}{56\!\cdots\!59}a^{3}+\frac{29\!\cdots\!66}{56\!\cdots\!59}a^{2}-\frac{51\!\cdots\!24}{56\!\cdots\!59}a-\frac{42\!\cdots\!00}{81\!\cdots\!37}$, $\frac{24\!\cdots\!28}{56\!\cdots\!59}a^{19}-\frac{13\!\cdots\!98}{56\!\cdots\!59}a^{18}-\frac{90\!\cdots\!07}{46\!\cdots\!79}a^{17}+\frac{51\!\cdots\!61}{56\!\cdots\!59}a^{16}+\frac{30\!\cdots\!18}{81\!\cdots\!37}a^{15}-\frac{73\!\cdots\!72}{56\!\cdots\!59}a^{14}-\frac{23\!\cdots\!98}{56\!\cdots\!59}a^{13}+\frac{50\!\cdots\!99}{56\!\cdots\!59}a^{12}+\frac{14\!\cdots\!87}{56\!\cdots\!59}a^{11}-\frac{17\!\cdots\!46}{56\!\cdots\!59}a^{10}-\frac{45\!\cdots\!70}{51\!\cdots\!69}a^{9}+\frac{28\!\cdots\!70}{56\!\cdots\!59}a^{8}+\frac{97\!\cdots\!76}{56\!\cdots\!59}a^{7}-\frac{17\!\cdots\!04}{56\!\cdots\!59}a^{6}-\frac{97\!\cdots\!56}{56\!\cdots\!59}a^{5}-\frac{30\!\cdots\!23}{56\!\cdots\!59}a^{4}+\frac{43\!\cdots\!94}{56\!\cdots\!59}a^{3}+\frac{37\!\cdots\!48}{56\!\cdots\!59}a^{2}-\frac{96\!\cdots\!79}{81\!\cdots\!37}a-\frac{35\!\cdots\!94}{56\!\cdots\!59}$, $\frac{42\!\cdots\!87}{56\!\cdots\!59}a^{19}-\frac{26\!\cdots\!36}{56\!\cdots\!59}a^{18}-\frac{15\!\cdots\!50}{51\!\cdots\!69}a^{17}+\frac{10\!\cdots\!62}{56\!\cdots\!59}a^{16}+\frac{30\!\cdots\!17}{56\!\cdots\!59}a^{15}-\frac{14\!\cdots\!74}{56\!\cdots\!59}a^{14}-\frac{29\!\cdots\!04}{56\!\cdots\!59}a^{13}+\frac{10\!\cdots\!27}{56\!\cdots\!59}a^{12}+\frac{17\!\cdots\!54}{56\!\cdots\!59}a^{11}-\frac{37\!\cdots\!42}{56\!\cdots\!59}a^{10}-\frac{55\!\cdots\!24}{51\!\cdots\!69}a^{9}+\frac{73\!\cdots\!33}{56\!\cdots\!59}a^{8}+\frac{11\!\cdots\!05}{56\!\cdots\!59}a^{7}-\frac{72\!\cdots\!34}{56\!\cdots\!59}a^{6}-\frac{10\!\cdots\!82}{56\!\cdots\!59}a^{5}+\frac{33\!\cdots\!58}{56\!\cdots\!59}a^{4}+\frac{63\!\cdots\!01}{81\!\cdots\!37}a^{3}-\frac{67\!\cdots\!26}{56\!\cdots\!59}a^{2}-\frac{55\!\cdots\!31}{56\!\cdots\!59}a+\frac{65\!\cdots\!26}{56\!\cdots\!59}$, $\frac{28\!\cdots\!60}{81\!\cdots\!37}a^{19}-\frac{10\!\cdots\!08}{56\!\cdots\!59}a^{18}-\frac{82\!\cdots\!45}{51\!\cdots\!69}a^{17}+\frac{41\!\cdots\!30}{56\!\cdots\!59}a^{16}+\frac{17\!\cdots\!06}{56\!\cdots\!59}a^{15}-\frac{59\!\cdots\!60}{56\!\cdots\!59}a^{14}-\frac{19\!\cdots\!40}{56\!\cdots\!59}a^{13}+\frac{40\!\cdots\!68}{56\!\cdots\!59}a^{12}+\frac{11\!\cdots\!65}{56\!\cdots\!59}a^{11}-\frac{13\!\cdots\!18}{56\!\cdots\!59}a^{10}-\frac{54\!\cdots\!16}{73\!\cdots\!67}a^{9}+\frac{21\!\cdots\!74}{56\!\cdots\!59}a^{8}+\frac{81\!\cdots\!02}{56\!\cdots\!59}a^{7}-\frac{16\!\cdots\!10}{81\!\cdots\!37}a^{6}-\frac{81\!\cdots\!20}{56\!\cdots\!59}a^{5}-\frac{47\!\cdots\!93}{56\!\cdots\!59}a^{4}+\frac{36\!\cdots\!96}{56\!\cdots\!59}a^{3}+\frac{38\!\cdots\!62}{56\!\cdots\!59}a^{2}-\frac{58\!\cdots\!29}{56\!\cdots\!59}a-\frac{62\!\cdots\!05}{56\!\cdots\!59}$, $\frac{19\!\cdots\!98}{56\!\cdots\!59}a^{19}-\frac{11\!\cdots\!04}{56\!\cdots\!59}a^{18}-\frac{79\!\cdots\!45}{51\!\cdots\!69}a^{17}+\frac{42\!\cdots\!17}{56\!\cdots\!59}a^{16}+\frac{16\!\cdots\!78}{56\!\cdots\!59}a^{15}-\frac{60\!\cdots\!06}{56\!\cdots\!59}a^{14}-\frac{17\!\cdots\!03}{56\!\cdots\!59}a^{13}+\frac{42\!\cdots\!75}{56\!\cdots\!59}a^{12}+\frac{10\!\cdots\!86}{56\!\cdots\!59}a^{11}-\frac{14\!\cdots\!39}{56\!\cdots\!59}a^{10}-\frac{35\!\cdots\!56}{51\!\cdots\!69}a^{9}+\frac{25\!\cdots\!95}{56\!\cdots\!59}a^{8}+\frac{74\!\cdots\!02}{56\!\cdots\!59}a^{7}-\frac{17\!\cdots\!02}{56\!\cdots\!59}a^{6}-\frac{74\!\cdots\!44}{56\!\cdots\!59}a^{5}+\frac{76\!\cdots\!07}{56\!\cdots\!59}a^{4}+\frac{33\!\cdots\!15}{56\!\cdots\!59}a^{3}+\frac{30\!\cdots\!24}{81\!\cdots\!37}a^{2}-\frac{50\!\cdots\!18}{56\!\cdots\!59}a-\frac{37\!\cdots\!89}{56\!\cdots\!59}$, $\frac{72\!\cdots\!09}{56\!\cdots\!59}a^{19}-\frac{45\!\cdots\!41}{56\!\cdots\!59}a^{18}-\frac{27\!\cdots\!51}{51\!\cdots\!69}a^{17}+\frac{17\!\cdots\!33}{56\!\cdots\!59}a^{16}+\frac{52\!\cdots\!58}{56\!\cdots\!59}a^{15}-\frac{25\!\cdots\!41}{56\!\cdots\!59}a^{14}-\frac{52\!\cdots\!45}{56\!\cdots\!59}a^{13}+\frac{17\!\cdots\!44}{56\!\cdots\!59}a^{12}+\frac{30\!\cdots\!36}{56\!\cdots\!59}a^{11}-\frac{67\!\cdots\!31}{56\!\cdots\!59}a^{10}-\frac{94\!\cdots\!53}{51\!\cdots\!69}a^{9}+\frac{13\!\cdots\!56}{56\!\cdots\!59}a^{8}+\frac{19\!\cdots\!58}{56\!\cdots\!59}a^{7}-\frac{13\!\cdots\!79}{56\!\cdots\!59}a^{6}-\frac{17\!\cdots\!00}{56\!\cdots\!59}a^{5}+\frac{67\!\cdots\!04}{56\!\cdots\!59}a^{4}+\frac{60\!\cdots\!92}{56\!\cdots\!59}a^{3}-\frac{13\!\cdots\!93}{56\!\cdots\!59}a^{2}-\frac{41\!\cdots\!50}{56\!\cdots\!59}a-\frac{11\!\cdots\!15}{56\!\cdots\!59}$, $\frac{15\!\cdots\!45}{56\!\cdots\!59}a^{19}+\frac{50\!\cdots\!63}{56\!\cdots\!59}a^{18}-\frac{29\!\cdots\!26}{51\!\cdots\!69}a^{17}-\frac{20\!\cdots\!59}{56\!\cdots\!59}a^{16}+\frac{17\!\cdots\!58}{81\!\cdots\!37}a^{15}+\frac{34\!\cdots\!23}{56\!\cdots\!59}a^{14}-\frac{18\!\cdots\!64}{56\!\cdots\!59}a^{13}-\frac{33\!\cdots\!21}{56\!\cdots\!59}a^{12}+\frac{18\!\cdots\!08}{81\!\cdots\!37}a^{11}+\frac{19\!\cdots\!80}{56\!\cdots\!59}a^{10}-\frac{43\!\cdots\!46}{46\!\cdots\!79}a^{9}-\frac{63\!\cdots\!42}{56\!\cdots\!59}a^{8}+\frac{11\!\cdots\!64}{56\!\cdots\!59}a^{7}+\frac{11\!\cdots\!78}{56\!\cdots\!59}a^{6}-\frac{13\!\cdots\!68}{56\!\cdots\!59}a^{5}-\frac{14\!\cdots\!65}{81\!\cdots\!37}a^{4}+\frac{75\!\cdots\!96}{56\!\cdots\!59}a^{3}+\frac{32\!\cdots\!73}{56\!\cdots\!59}a^{2}-\frac{17\!\cdots\!85}{56\!\cdots\!59}a-\frac{25\!\cdots\!68}{56\!\cdots\!59}$, $\frac{98\!\cdots\!30}{56\!\cdots\!59}a^{19}-\frac{52\!\cdots\!47}{56\!\cdots\!59}a^{18}-\frac{41\!\cdots\!64}{51\!\cdots\!69}a^{17}+\frac{20\!\cdots\!94}{56\!\cdots\!59}a^{16}+\frac{12\!\cdots\!62}{81\!\cdots\!37}a^{15}-\frac{28\!\cdots\!46}{56\!\cdots\!59}a^{14}-\frac{99\!\cdots\!62}{56\!\cdots\!59}a^{13}+\frac{19\!\cdots\!49}{56\!\cdots\!59}a^{12}+\frac{88\!\cdots\!24}{81\!\cdots\!37}a^{11}-\frac{62\!\cdots\!62}{56\!\cdots\!59}a^{10}-\frac{28\!\cdots\!74}{73\!\cdots\!67}a^{9}+\frac{90\!\cdots\!44}{56\!\cdots\!59}a^{8}+\frac{43\!\cdots\!25}{56\!\cdots\!59}a^{7}-\frac{33\!\cdots\!80}{81\!\cdots\!37}a^{6}-\frac{43\!\cdots\!83}{56\!\cdots\!59}a^{5}-\frac{51\!\cdots\!56}{56\!\cdots\!59}a^{4}+\frac{28\!\cdots\!87}{81\!\cdots\!37}a^{3}+\frac{28\!\cdots\!31}{56\!\cdots\!59}a^{2}-\frac{31\!\cdots\!90}{56\!\cdots\!59}a-\frac{71\!\cdots\!95}{56\!\cdots\!59}$, $\frac{16\!\cdots\!57}{56\!\cdots\!59}a^{19}-\frac{90\!\cdots\!79}{56\!\cdots\!59}a^{18}-\frac{98\!\cdots\!04}{73\!\cdots\!67}a^{17}+\frac{34\!\cdots\!97}{56\!\cdots\!59}a^{16}+\frac{14\!\cdots\!49}{56\!\cdots\!59}a^{15}-\frac{49\!\cdots\!52}{56\!\cdots\!59}a^{14}-\frac{16\!\cdots\!55}{56\!\cdots\!59}a^{13}+\frac{33\!\cdots\!90}{56\!\cdots\!59}a^{12}+\frac{14\!\cdots\!71}{81\!\cdots\!37}a^{11}-\frac{11\!\cdots\!20}{56\!\cdots\!59}a^{10}-\frac{45\!\cdots\!71}{73\!\cdots\!67}a^{9}+\frac{18\!\cdots\!00}{56\!\cdots\!59}a^{8}+\frac{68\!\cdots\!94}{56\!\cdots\!59}a^{7}-\frac{97\!\cdots\!00}{56\!\cdots\!59}a^{6}-\frac{97\!\cdots\!06}{81\!\cdots\!37}a^{5}-\frac{37\!\cdots\!23}{56\!\cdots\!59}a^{4}+\frac{30\!\cdots\!02}{56\!\cdots\!59}a^{3}+\frac{30\!\cdots\!67}{56\!\cdots\!59}a^{2}-\frac{48\!\cdots\!17}{56\!\cdots\!59}a-\frac{29\!\cdots\!65}{56\!\cdots\!59}$, $\frac{56\!\cdots\!23}{81\!\cdots\!37}a^{19}-\frac{24\!\cdots\!52}{56\!\cdots\!59}a^{18}-\frac{15\!\cdots\!13}{51\!\cdots\!69}a^{17}+\frac{91\!\cdots\!44}{56\!\cdots\!59}a^{16}+\frac{30\!\cdots\!78}{56\!\cdots\!59}a^{15}-\frac{13\!\cdots\!62}{56\!\cdots\!59}a^{14}-\frac{31\!\cdots\!38}{56\!\cdots\!59}a^{13}+\frac{92\!\cdots\!66}{56\!\cdots\!59}a^{12}+\frac{26\!\cdots\!26}{81\!\cdots\!37}a^{11}-\frac{33\!\cdots\!88}{56\!\cdots\!59}a^{10}-\frac{59\!\cdots\!36}{51\!\cdots\!69}a^{9}+\frac{63\!\cdots\!50}{56\!\cdots\!59}a^{8}+\frac{12\!\cdots\!41}{56\!\cdots\!59}a^{7}-\frac{56\!\cdots\!74}{56\!\cdots\!59}a^{6}-\frac{17\!\cdots\!55}{81\!\cdots\!37}a^{5}+\frac{18\!\cdots\!22}{56\!\cdots\!59}a^{4}+\frac{74\!\cdots\!87}{81\!\cdots\!37}a^{3}-\frac{10\!\cdots\!02}{56\!\cdots\!59}a^{2}-\frac{71\!\cdots\!83}{56\!\cdots\!59}a+\frac{71\!\cdots\!47}{56\!\cdots\!59}$, $\frac{25\!\cdots\!19}{56\!\cdots\!59}a^{19}-\frac{13\!\cdots\!93}{56\!\cdots\!59}a^{18}-\frac{10\!\cdots\!89}{51\!\cdots\!69}a^{17}+\frac{53\!\cdots\!36}{56\!\cdots\!59}a^{16}+\frac{21\!\cdots\!52}{56\!\cdots\!59}a^{15}-\frac{75\!\cdots\!90}{56\!\cdots\!59}a^{14}-\frac{23\!\cdots\!86}{56\!\cdots\!59}a^{13}+\frac{51\!\cdots\!89}{56\!\cdots\!59}a^{12}+\frac{14\!\cdots\!54}{56\!\cdots\!59}a^{11}-\frac{17\!\cdots\!29}{56\!\cdots\!59}a^{10}-\frac{46\!\cdots\!52}{51\!\cdots\!69}a^{9}+\frac{28\!\cdots\!18}{56\!\cdots\!59}a^{8}+\frac{98\!\cdots\!10}{56\!\cdots\!59}a^{7}-\frac{14\!\cdots\!65}{56\!\cdots\!59}a^{6}-\frac{97\!\cdots\!35}{56\!\cdots\!59}a^{5}-\frac{55\!\cdots\!30}{56\!\cdots\!59}a^{4}+\frac{42\!\cdots\!81}{56\!\cdots\!59}a^{3}+\frac{46\!\cdots\!28}{56\!\cdots\!59}a^{2}-\frac{64\!\cdots\!13}{56\!\cdots\!59}a-\frac{57\!\cdots\!02}{56\!\cdots\!59}$, $\frac{15\!\cdots\!35}{56\!\cdots\!59}a^{19}-\frac{86\!\cdots\!86}{56\!\cdots\!59}a^{18}-\frac{62\!\cdots\!66}{51\!\cdots\!69}a^{17}+\frac{33\!\cdots\!03}{56\!\cdots\!59}a^{16}+\frac{13\!\cdots\!29}{56\!\cdots\!59}a^{15}-\frac{47\!\cdots\!74}{56\!\cdots\!59}a^{14}-\frac{14\!\cdots\!87}{56\!\cdots\!59}a^{13}+\frac{32\!\cdots\!52}{56\!\cdots\!59}a^{12}+\frac{86\!\cdots\!93}{56\!\cdots\!59}a^{11}-\frac{11\!\cdots\!44}{56\!\cdots\!59}a^{10}-\frac{27\!\cdots\!58}{51\!\cdots\!69}a^{9}+\frac{19\!\cdots\!76}{56\!\cdots\!59}a^{8}+\frac{59\!\cdots\!90}{56\!\cdots\!59}a^{7}-\frac{12\!\cdots\!12}{56\!\cdots\!59}a^{6}-\frac{84\!\cdots\!85}{81\!\cdots\!37}a^{5}-\frac{40\!\cdots\!11}{56\!\cdots\!59}a^{4}+\frac{37\!\cdots\!19}{81\!\cdots\!37}a^{3}+\frac{19\!\cdots\!92}{56\!\cdots\!59}a^{2}-\frac{40\!\cdots\!54}{56\!\cdots\!59}a-\frac{18\!\cdots\!43}{56\!\cdots\!59}$, $\frac{17\!\cdots\!87}{56\!\cdots\!59}a^{19}-\frac{14\!\cdots\!15}{81\!\cdots\!37}a^{18}-\frac{68\!\cdots\!34}{51\!\cdots\!69}a^{17}+\frac{38\!\cdots\!52}{56\!\cdots\!59}a^{16}+\frac{14\!\cdots\!77}{56\!\cdots\!59}a^{15}-\frac{55\!\cdots\!38}{56\!\cdots\!59}a^{14}-\frac{14\!\cdots\!86}{56\!\cdots\!59}a^{13}+\frac{38\!\cdots\!48}{56\!\cdots\!59}a^{12}+\frac{90\!\cdots\!49}{56\!\cdots\!59}a^{11}-\frac{13\!\cdots\!19}{56\!\cdots\!59}a^{10}-\frac{28\!\cdots\!71}{51\!\cdots\!69}a^{9}+\frac{24\!\cdots\!97}{56\!\cdots\!59}a^{8}+\frac{60\!\cdots\!20}{56\!\cdots\!59}a^{7}-\frac{19\!\cdots\!94}{56\!\cdots\!59}a^{6}-\frac{58\!\cdots\!21}{56\!\cdots\!59}a^{5}+\frac{39\!\cdots\!36}{56\!\cdots\!59}a^{4}+\frac{25\!\cdots\!32}{56\!\cdots\!59}a^{3}+\frac{83\!\cdots\!99}{56\!\cdots\!59}a^{2}-\frac{35\!\cdots\!80}{56\!\cdots\!59}a-\frac{43\!\cdots\!61}{56\!\cdots\!59}$, $\frac{93\!\cdots\!13}{56\!\cdots\!59}a^{19}-\frac{54\!\cdots\!29}{56\!\cdots\!59}a^{18}-\frac{36\!\cdots\!87}{51\!\cdots\!69}a^{17}+\frac{20\!\cdots\!95}{56\!\cdots\!59}a^{16}+\frac{74\!\cdots\!20}{56\!\cdots\!59}a^{15}-\frac{29\!\cdots\!92}{56\!\cdots\!59}a^{14}-\frac{11\!\cdots\!57}{81\!\cdots\!37}a^{13}+\frac{20\!\cdots\!34}{56\!\cdots\!59}a^{12}+\frac{46\!\cdots\!28}{56\!\cdots\!59}a^{11}-\frac{73\!\cdots\!50}{56\!\cdots\!59}a^{10}-\frac{14\!\cdots\!91}{51\!\cdots\!69}a^{9}+\frac{13\!\cdots\!46}{56\!\cdots\!59}a^{8}+\frac{30\!\cdots\!47}{56\!\cdots\!59}a^{7}-\frac{15\!\cdots\!18}{81\!\cdots\!37}a^{6}-\frac{42\!\cdots\!89}{81\!\cdots\!37}a^{5}+\frac{22\!\cdots\!06}{56\!\cdots\!59}a^{4}+\frac{18\!\cdots\!34}{81\!\cdots\!37}a^{3}+\frac{70\!\cdots\!87}{81\!\cdots\!37}a^{2}-\frac{17\!\cdots\!12}{56\!\cdots\!59}a-\frac{53\!\cdots\!78}{56\!\cdots\!59}$
| 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: | \( 36744656987.7 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 36744656987.7 \cdot 1}{2\cdot\sqrt{7303816416639774784171798530359296}}\cr\approx \mathstrut & 0.225418310749 \end{aligned}\] (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 - 54*x^18 + 142*x^17 + 1215*x^16 - 1658*x^15 - 14226*x^14 + 5956*x^13 + 91351*x^12 + 20666*x^11 - 319628*x^10 - 208840*x^9 + 584299*x^8 + 561074*x^7 - 505908*x^6 - 639678*x^5 + 156250*x^4 + 296450*x^3 - 2608*x^2 - 43810*x - 263)
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 - 54*x^18 + 142*x^17 + 1215*x^16 - 1658*x^15 - 14226*x^14 + 5956*x^13 + 91351*x^12 + 20666*x^11 - 319628*x^10 - 208840*x^9 + 584299*x^8 + 561074*x^7 - 505908*x^6 - 639678*x^5 + 156250*x^4 + 296450*x^3 - 2608*x^2 - 43810*x - 263, 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 - 54*x^18 + 142*x^17 + 1215*x^16 - 1658*x^15 - 14226*x^14 + 5956*x^13 + 91351*x^12 + 20666*x^11 - 319628*x^10 - 208840*x^9 + 584299*x^8 + 561074*x^7 - 505908*x^6 - 639678*x^5 + 156250*x^4 + 296450*x^3 - 2608*x^2 - 43810*x - 263);
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 - 54*x^18 + 142*x^17 + 1215*x^16 - 1658*x^15 - 14226*x^14 + 5956*x^13 + 91351*x^12 + 20666*x^11 - 319628*x^10 - 208840*x^9 + 584299*x^8 + 561074*x^7 - 505908*x^6 - 639678*x^5 + 156250*x^4 + 296450*x^3 - 2608*x^2 - 43810*x - 263);
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
$C_2\wr C_5$ (as 20T46):
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 160 |
| The 16 conjugacy class representatives for $C_2\wr C_5$ |
| Character table for $C_2\wr C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.10.2670699013250048.2, 10.10.5048580365312.1, 10.10.116117348402176.1 |
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 32 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.10.5048580365312.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $20$ | $4$ | $5$ | $30$ | |||
|
\(11\)
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |