Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 33*x^18 + 54*x^17 + 399*x^16 - 470*x^15 - 2298*x^14 + 1592*x^13 + 6862*x^12 - 2024*x^11 - 10603*x^10 + 385*x^9 + 8089*x^8 + 480*x^7 - 3142*x^6 - 173*x^5 + 625*x^4 + 2*x^3 - 55*x^2 + 3*x + 1)
gp: K = bnfinit(y^20 - 2*y^19 - 33*y^18 + 54*y^17 + 399*y^16 - 470*y^15 - 2298*y^14 + 1592*y^13 + 6862*y^12 - 2024*y^11 - 10603*y^10 + 385*y^9 + 8089*y^8 + 480*y^7 - 3142*y^6 - 173*y^5 + 625*y^4 + 2*y^3 - 55*y^2 + 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 - 33*x^18 + 54*x^17 + 399*x^16 - 470*x^15 - 2298*x^14 + 1592*x^13 + 6862*x^12 - 2024*x^11 - 10603*x^10 + 385*x^9 + 8089*x^8 + 480*x^7 - 3142*x^6 - 173*x^5 + 625*x^4 + 2*x^3 - 55*x^2 + 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 - 33*x^18 + 54*x^17 + 399*x^16 - 470*x^15 - 2298*x^14 + 1592*x^13 + 6862*x^12 - 2024*x^11 - 10603*x^10 + 385*x^9 + 8089*x^8 + 480*x^7 - 3142*x^6 - 173*x^5 + 625*x^4 + 2*x^3 - 55*x^2 + 3*x + 1)
\( x^{20} - 2 x^{19} - 33 x^{18} + 54 x^{17} + 399 x^{16} - 470 x^{15} - 2298 x^{14} + 1592 x^{13} + \cdots + 1 \)
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: |
\(3165599696740582502041142585281\)
\(\medspace = 11^{16}\cdot 43^{4}\cdot 67^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.50\) | 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: | $11^{4/5}43^{1/2}67^{1/2}\approx 365.49865006155005$ | ||
| Ramified primes: |
\(11\), \(43\), \(67\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11}a^{15}-\frac{5}{11}a^{14}-\frac{4}{11}a^{13}+\frac{3}{11}a^{11}+\frac{4}{11}a^{10}-\frac{4}{11}a^{9}+\frac{4}{11}a^{8}-\frac{2}{11}a^{7}-\frac{5}{11}a^{6}+\frac{4}{11}a^{5}-\frac{2}{11}a^{4}+\frac{2}{11}a^{2}+\frac{2}{11}a+\frac{1}{11}$, $\frac{1}{11}a^{16}+\frac{4}{11}a^{14}+\frac{2}{11}a^{13}+\frac{3}{11}a^{12}-\frac{3}{11}a^{11}+\frac{5}{11}a^{10}-\frac{5}{11}a^{9}-\frac{4}{11}a^{8}-\frac{4}{11}a^{7}+\frac{1}{11}a^{6}-\frac{4}{11}a^{5}+\frac{1}{11}a^{4}+\frac{2}{11}a^{3}+\frac{1}{11}a^{2}+\frac{5}{11}$, $\frac{1}{11}a^{17}-\frac{3}{11}a^{13}-\frac{3}{11}a^{12}+\frac{4}{11}a^{11}+\frac{1}{11}a^{10}+\frac{1}{11}a^{9}+\frac{2}{11}a^{8}-\frac{2}{11}a^{7}+\frac{5}{11}a^{6}-\frac{4}{11}a^{5}-\frac{1}{11}a^{4}+\frac{1}{11}a^{3}+\frac{3}{11}a^{2}-\frac{3}{11}a-\frac{4}{11}$, $\frac{1}{11}a^{18}-\frac{3}{11}a^{14}-\frac{3}{11}a^{13}+\frac{4}{11}a^{12}+\frac{1}{11}a^{11}+\frac{1}{11}a^{10}+\frac{2}{11}a^{9}-\frac{2}{11}a^{8}+\frac{5}{11}a^{7}-\frac{4}{11}a^{6}-\frac{1}{11}a^{5}+\frac{1}{11}a^{4}+\frac{3}{11}a^{3}-\frac{3}{11}a^{2}-\frac{4}{11}a$, $\frac{1}{599891102963471}a^{19}-\frac{11953905355039}{599891102963471}a^{18}-\frac{10702609768043}{599891102963471}a^{17}-\frac{1837671988623}{54535554814861}a^{16}+\frac{121813648770}{54535554814861}a^{15}+\frac{130915099735718}{599891102963471}a^{14}+\frac{83998874557407}{599891102963471}a^{13}+\frac{256678055273520}{599891102963471}a^{12}-\frac{121289523549966}{599891102963471}a^{11}-\frac{103954886844602}{599891102963471}a^{10}+\frac{283123979059233}{599891102963471}a^{9}+\frac{161477848611038}{599891102963471}a^{8}+\frac{273201634975738}{599891102963471}a^{7}-\frac{178400656581219}{599891102963471}a^{6}-\frac{1172042131983}{26082221867977}a^{5}+\frac{233838651208639}{599891102963471}a^{4}-\frac{85132545774436}{599891102963471}a^{3}+\frac{93402393057284}{599891102963471}a^{2}-\frac{190462426347679}{599891102963471}a-\frac{146367845238720}{599891102963471}$
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: |
$\frac{643469054477520}{26082221867977}a^{19}-\frac{985841694586439}{26082221867977}a^{18}-\frac{21\!\cdots\!64}{26082221867977}a^{17}+\frac{24\!\cdots\!25}{26082221867977}a^{16}+\frac{24\!\cdots\!89}{2371111078907}a^{15}-\frac{17\!\cdots\!23}{26082221867977}a^{14}-\frac{15\!\cdots\!90}{26082221867977}a^{13}+\frac{30\!\cdots\!88}{26082221867977}a^{12}+\frac{46\!\cdots\!35}{26082221867977}a^{11}+\frac{79\!\cdots\!58}{26082221867977}a^{10}-\frac{66\!\cdots\!31}{26082221867977}a^{9}-\frac{27\!\cdots\!04}{26082221867977}a^{8}+\frac{42\!\cdots\!01}{26082221867977}a^{7}+\frac{22\!\cdots\!27}{26082221867977}a^{6}-\frac{11\!\cdots\!25}{26082221867977}a^{5}-\frac{65\!\cdots\!93}{26082221867977}a^{4}+\frac{15\!\cdots\!03}{26082221867977}a^{3}+\frac{65\!\cdots\!99}{2371111078907}a^{2}-\frac{92\!\cdots\!33}{26082221867977}a-\frac{17\!\cdots\!01}{26082221867977}$, $\frac{286077894050551}{26082221867977}a^{19}-\frac{471224059663139}{26082221867977}a^{18}-\frac{95\!\cdots\!04}{26082221867977}a^{17}+\frac{12\!\cdots\!07}{26082221867977}a^{16}+\frac{11\!\cdots\!72}{26082221867977}a^{15}-\frac{83\!\cdots\!89}{2371111078907}a^{14}-\frac{68\!\cdots\!32}{26082221867977}a^{13}+\frac{20\!\cdots\!78}{26082221867977}a^{12}+\frac{20\!\cdots\!20}{26082221867977}a^{11}+\frac{15\!\cdots\!53}{26082221867977}a^{10}-\frac{29\!\cdots\!77}{26082221867977}a^{9}-\frac{87\!\cdots\!35}{2371111078907}a^{8}+\frac{18\!\cdots\!86}{26082221867977}a^{7}+\frac{83\!\cdots\!73}{26082221867977}a^{6}-\frac{54\!\cdots\!68}{26082221867977}a^{5}-\frac{25\!\cdots\!27}{26082221867977}a^{4}+\frac{70\!\cdots\!53}{26082221867977}a^{3}+\frac{28\!\cdots\!06}{26082221867977}a^{2}-\frac{36\!\cdots\!06}{26082221867977}a-\frac{680423178131644}{26082221867977}$, $\frac{471359615788958}{26082221867977}a^{19}-\frac{733079818972404}{26082221867977}a^{18}-\frac{14\!\cdots\!91}{2371111078907}a^{17}+\frac{18\!\cdots\!54}{26082221867977}a^{16}+\frac{19\!\cdots\!92}{26082221867977}a^{15}-\frac{13\!\cdots\!88}{26082221867977}a^{14}-\frac{11\!\cdots\!19}{26082221867977}a^{13}+\frac{24\!\cdots\!15}{26082221867977}a^{12}+\frac{30\!\cdots\!53}{2371111078907}a^{11}+\frac{52\!\cdots\!97}{26082221867977}a^{10}-\frac{48\!\cdots\!44}{26082221867977}a^{9}-\frac{19\!\cdots\!99}{26082221867977}a^{8}+\frac{27\!\cdots\!80}{2371111078907}a^{7}+\frac{15\!\cdots\!30}{26082221867977}a^{6}-\frac{84\!\cdots\!46}{26082221867977}a^{5}-\frac{45\!\cdots\!14}{26082221867977}a^{4}+\frac{11\!\cdots\!16}{26082221867977}a^{3}+\frac{49\!\cdots\!28}{26082221867977}a^{2}-\frac{62\!\cdots\!74}{26082221867977}a-\frac{12\!\cdots\!56}{26082221867977}$, $\frac{644529984344056}{26082221867977}a^{19}-\frac{10\!\cdots\!43}{26082221867977}a^{18}-\frac{21\!\cdots\!13}{26082221867977}a^{17}+\frac{25\!\cdots\!13}{26082221867977}a^{16}+\frac{26\!\cdots\!16}{26082221867977}a^{15}-\frac{18\!\cdots\!65}{26082221867977}a^{14}-\frac{15\!\cdots\!16}{26082221867977}a^{13}+\frac{36\!\cdots\!13}{26082221867977}a^{12}+\frac{41\!\cdots\!48}{2371111078907}a^{11}+\frac{62\!\cdots\!49}{26082221867977}a^{10}-\frac{65\!\cdots\!19}{26082221867977}a^{9}-\frac{25\!\cdots\!66}{26082221867977}a^{8}+\frac{41\!\cdots\!62}{26082221867977}a^{7}+\frac{20\!\cdots\!37}{26082221867977}a^{6}-\frac{11\!\cdots\!73}{26082221867977}a^{5}-\frac{59\!\cdots\!43}{26082221867977}a^{4}+\frac{14\!\cdots\!24}{26082221867977}a^{3}+\frac{63\!\cdots\!79}{26082221867977}a^{2}-\frac{79\!\cdots\!42}{26082221867977}a-\frac{138399029298957}{2371111078907}$, $\frac{12\!\cdots\!91}{599891102963471}a^{19}-\frac{20\!\cdots\!49}{599891102963471}a^{18}-\frac{41\!\cdots\!62}{599891102963471}a^{17}+\frac{53\!\cdots\!63}{599891102963471}a^{16}+\frac{46\!\cdots\!84}{54535554814861}a^{15}-\frac{41\!\cdots\!88}{599891102963471}a^{14}-\frac{26\!\cdots\!03}{54535554814861}a^{13}+\frac{96\!\cdots\!23}{599891102963471}a^{12}+\frac{85\!\cdots\!15}{599891102963471}a^{11}+\frac{41\!\cdots\!76}{54535554814861}a^{10}-\frac{12\!\cdots\!45}{599891102963471}a^{9}-\frac{37\!\cdots\!26}{599891102963471}a^{8}+\frac{78\!\cdots\!90}{599891102963471}a^{7}+\frac{32\!\cdots\!23}{599891102963471}a^{6}-\frac{96\!\cdots\!77}{26082221867977}a^{5}-\frac{96\!\cdots\!33}{599891102963471}a^{4}+\frac{28\!\cdots\!06}{599891102963471}a^{3}+\frac{10\!\cdots\!52}{599891102963471}a^{2}-\frac{14\!\cdots\!73}{599891102963471}a-\frac{26\!\cdots\!78}{599891102963471}$, $\frac{20\!\cdots\!67}{599891102963471}a^{19}-\frac{33\!\cdots\!78}{599891102963471}a^{18}-\frac{68\!\cdots\!43}{599891102963471}a^{17}+\frac{84\!\cdots\!89}{599891102963471}a^{16}+\frac{83\!\cdots\!07}{599891102963471}a^{15}-\frac{63\!\cdots\!24}{599891102963471}a^{14}-\frac{48\!\cdots\!87}{599891102963471}a^{13}+\frac{13\!\cdots\!57}{599891102963471}a^{12}+\frac{14\!\cdots\!78}{599891102963471}a^{11}+\frac{13\!\cdots\!68}{599891102963471}a^{10}-\frac{20\!\cdots\!24}{599891102963471}a^{9}-\frac{68\!\cdots\!54}{599891102963471}a^{8}+\frac{12\!\cdots\!07}{599891102963471}a^{7}+\frac{55\!\cdots\!13}{599891102963471}a^{6}-\frac{14\!\cdots\!88}{26082221867977}a^{5}-\frac{15\!\cdots\!86}{599891102963471}a^{4}+\frac{41\!\cdots\!91}{599891102963471}a^{3}+\frac{14\!\cdots\!75}{599891102963471}a^{2}-\frac{19\!\cdots\!75}{599891102963471}a-\frac{32\!\cdots\!22}{599891102963471}$, $\frac{33\!\cdots\!51}{599891102963471}a^{19}-\frac{45\!\cdots\!59}{599891102963471}a^{18}-\frac{11\!\cdots\!76}{599891102963471}a^{17}+\frac{10\!\cdots\!68}{599891102963471}a^{16}+\frac{14\!\cdots\!99}{599891102963471}a^{15}-\frac{67\!\cdots\!12}{599891102963471}a^{14}-\frac{85\!\cdots\!26}{599891102963471}a^{13}+\frac{12\!\cdots\!09}{599891102963471}a^{12}+\frac{25\!\cdots\!75}{599891102963471}a^{11}+\frac{84\!\cdots\!69}{599891102963471}a^{10}-\frac{35\!\cdots\!04}{599891102963471}a^{9}-\frac{20\!\cdots\!32}{599891102963471}a^{8}+\frac{21\!\cdots\!74}{599891102963471}a^{7}+\frac{15\!\cdots\!01}{599891102963471}a^{6}-\frac{23\!\cdots\!65}{26082221867977}a^{5}-\frac{44\!\cdots\!07}{599891102963471}a^{4}+\frac{59\!\cdots\!67}{599891102963471}a^{3}+\frac{47\!\cdots\!46}{599891102963471}a^{2}-\frac{33\!\cdots\!18}{599891102963471}a-\frac{93\!\cdots\!15}{599891102963471}$, $\frac{10\!\cdots\!88}{599891102963471}a^{19}-\frac{15\!\cdots\!47}{599891102963471}a^{18}-\frac{33\!\cdots\!27}{599891102963471}a^{17}+\frac{39\!\cdots\!31}{599891102963471}a^{16}+\frac{41\!\cdots\!69}{599891102963471}a^{15}-\frac{29\!\cdots\!72}{599891102963471}a^{14}-\frac{24\!\cdots\!85}{599891102963471}a^{13}+\frac{55\!\cdots\!87}{599891102963471}a^{12}+\frac{72\!\cdots\!92}{599891102963471}a^{11}+\frac{10\!\cdots\!53}{599891102963471}a^{10}-\frac{10\!\cdots\!80}{599891102963471}a^{9}-\frac{40\!\cdots\!00}{599891102963471}a^{8}+\frac{60\!\cdots\!03}{54535554814861}a^{7}+\frac{33\!\cdots\!68}{599891102963471}a^{6}-\frac{82\!\cdots\!03}{26082221867977}a^{5}-\frac{98\!\cdots\!58}{599891102963471}a^{4}+\frac{25\!\cdots\!39}{599891102963471}a^{3}+\frac{10\!\cdots\!92}{599891102963471}a^{2}-\frac{13\!\cdots\!58}{599891102963471}a-\frac{22\!\cdots\!37}{54535554814861}$, $\frac{13\!\cdots\!87}{599891102963471}a^{19}-\frac{20\!\cdots\!19}{599891102963471}a^{18}-\frac{45\!\cdots\!59}{599891102963471}a^{17}+\frac{52\!\cdots\!22}{599891102963471}a^{16}+\frac{56\!\cdots\!29}{599891102963471}a^{15}-\frac{38\!\cdots\!03}{599891102963471}a^{14}-\frac{32\!\cdots\!83}{599891102963471}a^{13}+\frac{69\!\cdots\!68}{599891102963471}a^{12}+\frac{96\!\cdots\!08}{599891102963471}a^{11}+\frac{15\!\cdots\!84}{599891102963471}a^{10}-\frac{14\!\cdots\!36}{599891102963471}a^{9}-\frac{50\!\cdots\!84}{54535554814861}a^{8}+\frac{89\!\cdots\!52}{599891102963471}a^{7}+\frac{45\!\cdots\!50}{599891102963471}a^{6}-\frac{11\!\cdots\!19}{26082221867977}a^{5}-\frac{13\!\cdots\!30}{599891102963471}a^{4}+\frac{34\!\cdots\!86}{599891102963471}a^{3}+\frac{15\!\cdots\!43}{599891102963471}a^{2}-\frac{19\!\cdots\!25}{599891102963471}a-\frac{36\!\cdots\!46}{599891102963471}$, $\frac{12\!\cdots\!55}{599891102963471}a^{19}-\frac{20\!\cdots\!57}{599891102963471}a^{18}-\frac{42\!\cdots\!54}{599891102963471}a^{17}+\frac{48\!\cdots\!42}{54535554814861}a^{16}+\frac{46\!\cdots\!97}{54535554814861}a^{15}-\frac{40\!\cdots\!97}{599891102963471}a^{14}-\frac{29\!\cdots\!47}{599891102963471}a^{13}+\frac{89\!\cdots\!02}{599891102963471}a^{12}+\frac{87\!\cdots\!66}{599891102963471}a^{11}+\frac{69\!\cdots\!62}{599891102963471}a^{10}-\frac{12\!\cdots\!94}{599891102963471}a^{9}-\frac{41\!\cdots\!03}{599891102963471}a^{8}+\frac{78\!\cdots\!00}{599891102963471}a^{7}+\frac{34\!\cdots\!95}{599891102963471}a^{6}-\frac{93\!\cdots\!14}{26082221867977}a^{5}-\frac{97\!\cdots\!25}{599891102963471}a^{4}+\frac{26\!\cdots\!48}{599891102963471}a^{3}+\frac{10\!\cdots\!68}{599891102963471}a^{2}-\frac{13\!\cdots\!65}{599891102963471}a-\frac{24\!\cdots\!78}{599891102963471}$, $\frac{223048460985727}{54535554814861}a^{19}-\frac{26\!\cdots\!40}{599891102963471}a^{18}-\frac{85\!\cdots\!37}{599891102963471}a^{17}+\frac{51\!\cdots\!76}{54535554814861}a^{16}+\frac{10\!\cdots\!44}{599891102963471}a^{15}-\frac{24\!\cdots\!07}{599891102963471}a^{14}-\frac{65\!\cdots\!35}{599891102963471}a^{13}-\frac{12\!\cdots\!24}{599891102963471}a^{12}+\frac{19\!\cdots\!29}{599891102963471}a^{11}+\frac{98\!\cdots\!91}{599891102963471}a^{10}-\frac{25\!\cdots\!72}{54535554814861}a^{9}-\frac{19\!\cdots\!92}{599891102963471}a^{8}+\frac{16\!\cdots\!36}{599891102963471}a^{7}+\frac{14\!\cdots\!28}{599891102963471}a^{6}-\frac{17\!\cdots\!96}{26082221867977}a^{5}-\frac{39\!\cdots\!56}{599891102963471}a^{4}+\frac{50\!\cdots\!46}{599891102963471}a^{3}+\frac{42\!\cdots\!65}{599891102963471}a^{2}-\frac{36\!\cdots\!93}{599891102963471}a-\frac{89\!\cdots\!43}{599891102963471}$, $\frac{42\!\cdots\!30}{599891102963471}a^{19}-\frac{60\!\cdots\!11}{599891102963471}a^{18}-\frac{13\!\cdots\!45}{54535554814861}a^{17}+\frac{14\!\cdots\!87}{599891102963471}a^{16}+\frac{18\!\cdots\!40}{599891102963471}a^{15}-\frac{99\!\cdots\!34}{599891102963471}a^{14}-\frac{10\!\cdots\!68}{599891102963471}a^{13}+\frac{99\!\cdots\!41}{599891102963471}a^{12}+\frac{31\!\cdots\!66}{599891102963471}a^{11}+\frac{81\!\cdots\!74}{599891102963471}a^{10}-\frac{44\!\cdots\!75}{599891102963471}a^{9}-\frac{22\!\cdots\!57}{599891102963471}a^{8}+\frac{28\!\cdots\!14}{599891102963471}a^{7}+\frac{17\!\cdots\!10}{599891102963471}a^{6}-\frac{33\!\cdots\!65}{26082221867977}a^{5}-\frac{51\!\cdots\!45}{599891102963471}a^{4}+\frac{10\!\cdots\!13}{599891102963471}a^{3}+\frac{57\!\cdots\!98}{599891102963471}a^{2}-\frac{70\!\cdots\!86}{599891102963471}a-\frac{15\!\cdots\!89}{599891102963471}$, $\frac{14\!\cdots\!17}{599891102963471}a^{19}-\frac{15\!\cdots\!88}{599891102963471}a^{18}-\frac{49\!\cdots\!56}{599891102963471}a^{17}+\frac{35\!\cdots\!87}{599891102963471}a^{16}+\frac{57\!\cdots\!26}{54535554814861}a^{15}-\frac{18\!\cdots\!58}{599891102963471}a^{14}-\frac{35\!\cdots\!39}{54535554814861}a^{13}-\frac{47\!\cdots\!74}{599891102963471}a^{12}+\frac{11\!\cdots\!71}{599891102963471}a^{11}+\frac{50\!\cdots\!28}{599891102963471}a^{10}-\frac{17\!\cdots\!19}{599891102963471}a^{9}-\frac{11\!\cdots\!26}{599891102963471}a^{8}+\frac{11\!\cdots\!00}{599891102963471}a^{7}+\frac{88\!\cdots\!75}{599891102963471}a^{6}-\frac{16\!\cdots\!34}{26082221867977}a^{5}-\frac{28\!\cdots\!95}{599891102963471}a^{4}+\frac{60\!\cdots\!93}{599891102963471}a^{3}+\frac{36\!\cdots\!71}{599891102963471}a^{2}-\frac{45\!\cdots\!67}{599891102963471}a-\frac{91\!\cdots\!86}{599891102963471}$, $\frac{19\!\cdots\!98}{599891102963471}a^{19}-\frac{31\!\cdots\!14}{599891102963471}a^{18}-\frac{60\!\cdots\!54}{54535554814861}a^{17}+\frac{78\!\cdots\!25}{599891102963471}a^{16}+\frac{81\!\cdots\!56}{599891102963471}a^{15}-\frac{58\!\cdots\!17}{599891102963471}a^{14}-\frac{47\!\cdots\!52}{599891102963471}a^{13}+\frac{11\!\cdots\!98}{599891102963471}a^{12}+\frac{13\!\cdots\!96}{599891102963471}a^{11}+\frac{18\!\cdots\!49}{599891102963471}a^{10}-\frac{19\!\cdots\!58}{599891102963471}a^{9}-\frac{76\!\cdots\!61}{599891102963471}a^{8}+\frac{12\!\cdots\!28}{599891102963471}a^{7}+\frac{62\!\cdots\!96}{599891102963471}a^{6}-\frac{15\!\cdots\!39}{26082221867977}a^{5}-\frac{18\!\cdots\!52}{599891102963471}a^{4}+\frac{45\!\cdots\!92}{599891102963471}a^{3}+\frac{20\!\cdots\!75}{599891102963471}a^{2}-\frac{22\!\cdots\!78}{54535554814861}a-\frac{49\!\cdots\!23}{599891102963471}$, $\frac{10\!\cdots\!32}{54535554814861}a^{19}-\frac{16\!\cdots\!14}{599891102963471}a^{18}-\frac{37\!\cdots\!32}{599891102963471}a^{17}+\frac{41\!\cdots\!18}{599891102963471}a^{16}+\frac{46\!\cdots\!84}{599891102963471}a^{15}-\frac{29\!\cdots\!50}{599891102963471}a^{14}-\frac{27\!\cdots\!75}{599891102963471}a^{13}+\frac{46\!\cdots\!39}{599891102963471}a^{12}+\frac{80\!\cdots\!75}{599891102963471}a^{11}+\frac{15\!\cdots\!78}{599891102963471}a^{10}-\frac{11\!\cdots\!23}{599891102963471}a^{9}-\frac{46\!\cdots\!34}{54535554814861}a^{8}+\frac{74\!\cdots\!70}{599891102963471}a^{7}+\frac{41\!\cdots\!53}{599891102963471}a^{6}-\frac{93\!\cdots\!00}{26082221867977}a^{5}-\frac{12\!\cdots\!99}{599891102963471}a^{4}+\frac{29\!\cdots\!20}{599891102963471}a^{3}+\frac{13\!\cdots\!12}{599891102963471}a^{2}-\frac{17\!\cdots\!49}{599891102963471}a-\frac{31\!\cdots\!66}{599891102963471}$, $\frac{11\!\cdots\!53}{54535554814861}a^{19}-\frac{18\!\cdots\!15}{599891102963471}a^{18}-\frac{40\!\cdots\!44}{599891102963471}a^{17}+\frac{46\!\cdots\!13}{599891102963471}a^{16}+\frac{50\!\cdots\!26}{599891102963471}a^{15}-\frac{33\!\cdots\!16}{599891102963471}a^{14}-\frac{29\!\cdots\!60}{599891102963471}a^{13}+\frac{59\!\cdots\!99}{599891102963471}a^{12}+\frac{86\!\cdots\!41}{599891102963471}a^{11}+\frac{14\!\cdots\!63}{599891102963471}a^{10}-\frac{12\!\cdots\!11}{599891102963471}a^{9}-\frac{46\!\cdots\!54}{54535554814861}a^{8}+\frac{71\!\cdots\!73}{54535554814861}a^{7}+\frac{41\!\cdots\!39}{599891102963471}a^{6}-\frac{86\!\cdots\!78}{2371111078907}a^{5}-\frac{11\!\cdots\!22}{599891102963471}a^{4}+\frac{28\!\cdots\!22}{599891102963471}a^{3}+\frac{12\!\cdots\!26}{599891102963471}a^{2}-\frac{15\!\cdots\!24}{599891102963471}a-\frac{30\!\cdots\!51}{599891102963471}$, $\frac{68\!\cdots\!89}{599891102963471}a^{19}-\frac{10\!\cdots\!95}{54535554814861}a^{18}-\frac{22\!\cdots\!79}{599891102963471}a^{17}+\frac{30\!\cdots\!85}{599891102963471}a^{16}+\frac{27\!\cdots\!65}{599891102963471}a^{15}-\frac{23\!\cdots\!83}{599891102963471}a^{14}-\frac{16\!\cdots\!15}{599891102963471}a^{13}+\frac{59\!\cdots\!24}{599891102963471}a^{12}+\frac{47\!\cdots\!71}{599891102963471}a^{11}+\frac{88\!\cdots\!56}{599891102963471}a^{10}-\frac{68\!\cdots\!17}{599891102963471}a^{9}-\frac{18\!\cdots\!52}{599891102963471}a^{8}+\frac{44\!\cdots\!14}{599891102963471}a^{7}+\frac{17\!\cdots\!74}{599891102963471}a^{6}-\frac{55\!\cdots\!70}{26082221867977}a^{5}-\frac{52\!\cdots\!77}{599891102963471}a^{4}+\frac{16\!\cdots\!17}{599891102963471}a^{3}+\frac{58\!\cdots\!20}{599891102963471}a^{2}-\frac{89\!\cdots\!49}{599891102963471}a-\frac{13\!\cdots\!48}{599891102963471}$, $\frac{15\!\cdots\!78}{599891102963471}a^{19}-\frac{24\!\cdots\!44}{599891102963471}a^{18}-\frac{52\!\cdots\!38}{599891102963471}a^{17}+\frac{60\!\cdots\!44}{599891102963471}a^{16}+\frac{65\!\cdots\!13}{599891102963471}a^{15}-\frac{44\!\cdots\!79}{599891102963471}a^{14}-\frac{38\!\cdots\!62}{599891102963471}a^{13}+\frac{79\!\cdots\!27}{599891102963471}a^{12}+\frac{11\!\cdots\!35}{599891102963471}a^{11}+\frac{18\!\cdots\!83}{599891102963471}a^{10}-\frac{16\!\cdots\!11}{599891102963471}a^{9}-\frac{65\!\cdots\!53}{599891102963471}a^{8}+\frac{10\!\cdots\!62}{599891102963471}a^{7}+\frac{53\!\cdots\!77}{599891102963471}a^{6}-\frac{12\!\cdots\!44}{26082221867977}a^{5}-\frac{15\!\cdots\!39}{599891102963471}a^{4}+\frac{37\!\cdots\!91}{599891102963471}a^{3}+\frac{17\!\cdots\!36}{599891102963471}a^{2}-\frac{21\!\cdots\!43}{599891102963471}a-\frac{42\!\cdots\!40}{599891102963471}$, $\frac{935025250852951}{599891102963471}a^{19}-\frac{10\!\cdots\!70}{599891102963471}a^{18}-\frac{32\!\cdots\!09}{599891102963471}a^{17}+\frac{22\!\cdots\!26}{599891102963471}a^{16}+\frac{40\!\cdots\!26}{599891102963471}a^{15}-\frac{88\!\cdots\!70}{599891102963471}a^{14}-\frac{21\!\cdots\!60}{54535554814861}a^{13}-\frac{48\!\cdots\!02}{54535554814861}a^{12}+\frac{64\!\cdots\!04}{599891102963471}a^{11}+\frac{34\!\cdots\!04}{54535554814861}a^{10}-\frac{77\!\cdots\!67}{599891102963471}a^{9}-\frac{71\!\cdots\!88}{599891102963471}a^{8}+\frac{25\!\cdots\!66}{599891102963471}a^{7}+\frac{40\!\cdots\!19}{599891102963471}a^{6}+\frac{17\!\cdots\!46}{26082221867977}a^{5}-\frac{63\!\cdots\!84}{599891102963471}a^{4}-\frac{18\!\cdots\!50}{599891102963471}a^{3}+\frac{882148667687853}{54535554814861}a^{2}+\frac{10\!\cdots\!57}{599891102963471}a+\frac{19119982110520}{54535554814861}$
| 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: | \( 500606346.877 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 500606346.877 \cdot 1}{2\cdot\sqrt{3165599696740582502041142585281}}\cr\approx \mathstrut & 0.147515709101 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 33*x^18 + 54*x^17 + 399*x^16 - 470*x^15 - 2298*x^14 + 1592*x^13 + 6862*x^12 - 2024*x^11 - 10603*x^10 + 385*x^9 + 8089*x^8 + 480*x^7 - 3142*x^6 - 173*x^5 + 625*x^4 + 2*x^3 - 55*x^2 + 3*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 - 2*x^19 - 33*x^18 + 54*x^17 + 399*x^16 - 470*x^15 - 2298*x^14 + 1592*x^13 + 6862*x^12 - 2024*x^11 - 10603*x^10 + 385*x^9 + 8089*x^8 + 480*x^7 - 3142*x^6 - 173*x^5 + 625*x^4 + 2*x^3 - 55*x^2 + 3*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 - 2*x^19 - 33*x^18 + 54*x^17 + 399*x^16 - 470*x^15 - 2298*x^14 + 1592*x^13 + 6862*x^12 - 2024*x^11 - 10603*x^10 + 385*x^9 + 8089*x^8 + 480*x^7 - 3142*x^6 - 173*x^5 + 625*x^4 + 2*x^3 - 55*x^2 + 3*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 - 2*x^19 - 33*x^18 + 54*x^17 + 399*x^16 - 470*x^15 - 2298*x^14 + 1592*x^13 + 6862*x^12 - 2024*x^11 - 10603*x^10 + 385*x^9 + 8089*x^8 + 480*x^7 - 3142*x^6 - 173*x^5 + 625*x^4 + 2*x^3 - 55*x^2 + 3*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
$C_2\wr C_5$ (as 20T41):
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
| \(\Q(\zeta_{11})^+\), 10.10.617567936161.1 x2, 10.10.1779213224079841.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
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: | 10.10.617567936161.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{4}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
|
\(43\)
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(67\)
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.4.2.1 | $x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 67.4.2.1 | $x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |