Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429)
gp: K = bnfinit(y^20 - 8*y^19 - 20*y^18 + 284*y^17 - 21*y^16 - 4198*y^15 + 3737*y^14 + 33772*y^13 - 42360*y^12 - 162403*y^11 + 230822*y^10 + 483926*y^9 - 710001*y^8 - 898678*y^7 + 1266895*y^6 + 1017051*y^5 - 1261123*y^4 - 651709*y^3 + 616317*y^2 + 185386*y - 101429, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 8*x^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 8*x^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429)
\( x^{20} - 8 x^{19} - 20 x^{18} + 284 x^{17} - 21 x^{16} - 4198 x^{15} + 3737 x^{14} + 33772 x^{13} + \cdots - 101429 \)
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: |
\(19184293930982457734868438720703125\)
\(\medspace = 3^{10}\cdot 5^{15}\cdot 239^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(51.78\) | 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: | $3^{1/2}5^{3/4}239^{1/2}\approx 89.53381316204927$ | ||
| Ramified primes: |
\(3\), \(5\), \(239\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{239}a^{17}-\frac{22}{239}a^{16}-\frac{74}{239}a^{15}-\frac{68}{239}a^{14}-\frac{40}{239}a^{13}+\frac{89}{239}a^{12}-\frac{41}{239}a^{11}+\frac{22}{239}a^{10}+\frac{7}{239}a^{9}+\frac{18}{239}a^{8}+\frac{65}{239}a^{7}-\frac{44}{239}a^{6}+\frac{18}{239}a^{5}+\frac{62}{239}a^{3}+\frac{114}{239}a^{2}-\frac{60}{239}a-\frac{3}{239}$, $\frac{1}{207691}a^{18}-\frac{94}{207691}a^{17}-\frac{7333}{207691}a^{16}+\frac{26531}{207691}a^{15}-\frac{558}{18881}a^{14}-\frac{64668}{207691}a^{13}-\frac{87231}{207691}a^{12}-\frac{95972}{207691}a^{11}+\frac{27581}{207691}a^{10}-\frac{77444}{207691}a^{9}+\frac{31990}{207691}a^{8}+\frac{37340}{207691}a^{7}-\frac{51545}{207691}a^{6}-\frac{55310}{207691}a^{5}-\frac{18819}{207691}a^{4}+\frac{4428}{18881}a^{3}-\frac{2077}{18881}a^{2}-\frac{16954}{207691}a-\frac{90604}{207691}$, $\frac{1}{11\!\cdots\!59}a^{19}-\frac{17\!\cdots\!60}{11\!\cdots\!59}a^{18}-\frac{56\!\cdots\!88}{10\!\cdots\!69}a^{17}+\frac{23\!\cdots\!83}{11\!\cdots\!59}a^{16}+\frac{43\!\cdots\!09}{11\!\cdots\!59}a^{15}+\frac{50\!\cdots\!20}{11\!\cdots\!59}a^{14}-\frac{78\!\cdots\!89}{62\!\cdots\!61}a^{13}-\frac{51\!\cdots\!84}{11\!\cdots\!59}a^{12}-\frac{33\!\cdots\!64}{11\!\cdots\!59}a^{11}+\frac{26\!\cdots\!91}{11\!\cdots\!59}a^{10}+\frac{20\!\cdots\!86}{11\!\cdots\!59}a^{9}-\frac{12\!\cdots\!21}{10\!\cdots\!69}a^{8}+\frac{54\!\cdots\!27}{11\!\cdots\!59}a^{7}-\frac{46\!\cdots\!96}{11\!\cdots\!59}a^{6}-\frac{51\!\cdots\!89}{11\!\cdots\!59}a^{5}+\frac{96\!\cdots\!23}{11\!\cdots\!59}a^{4}+\frac{41\!\cdots\!47}{10\!\cdots\!69}a^{3}-\frac{26\!\cdots\!05}{11\!\cdots\!59}a^{2}-\frac{35\!\cdots\!86}{11\!\cdots\!59}a+\frac{37\!\cdots\!91}{11\!\cdots\!59}$
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: |
$\frac{18\!\cdots\!31}{20\!\cdots\!01}a^{19}-\frac{11\!\cdots\!13}{20\!\cdots\!01}a^{18}-\frac{56\!\cdots\!50}{20\!\cdots\!01}a^{17}+\frac{42\!\cdots\!15}{20\!\cdots\!01}a^{16}+\frac{67\!\cdots\!73}{20\!\cdots\!01}a^{15}-\frac{65\!\cdots\!79}{20\!\cdots\!01}a^{14}-\frac{41\!\cdots\!68}{20\!\cdots\!01}a^{13}+\frac{55\!\cdots\!45}{20\!\cdots\!01}a^{12}+\frac{14\!\cdots\!57}{20\!\cdots\!01}a^{11}-\frac{27\!\cdots\!02}{20\!\cdots\!01}a^{10}-\frac{34\!\cdots\!90}{20\!\cdots\!01}a^{9}+\frac{83\!\cdots\!09}{20\!\cdots\!01}a^{8}+\frac{85\!\cdots\!62}{20\!\cdots\!01}a^{7}-\frac{15\!\cdots\!48}{20\!\cdots\!01}a^{6}-\frac{19\!\cdots\!83}{20\!\cdots\!01}a^{5}+\frac{15\!\cdots\!47}{20\!\cdots\!01}a^{4}+\frac{26\!\cdots\!81}{20\!\cdots\!01}a^{3}-\frac{76\!\cdots\!01}{20\!\cdots\!01}a^{2}-\frac{13\!\cdots\!44}{20\!\cdots\!01}a+\frac{11\!\cdots\!53}{20\!\cdots\!01}$, $\frac{64\!\cdots\!11}{10\!\cdots\!69}a^{19}-\frac{45\!\cdots\!75}{11\!\cdots\!59}a^{18}-\frac{21\!\cdots\!63}{11\!\cdots\!59}a^{17}+\frac{16\!\cdots\!14}{11\!\cdots\!59}a^{16}+\frac{26\!\cdots\!44}{11\!\cdots\!59}a^{15}-\frac{23\!\cdots\!67}{10\!\cdots\!69}a^{14}-\frac{86\!\cdots\!29}{62\!\cdots\!61}a^{13}+\frac{21\!\cdots\!37}{11\!\cdots\!59}a^{12}+\frac{57\!\cdots\!51}{11\!\cdots\!59}a^{11}-\frac{10\!\cdots\!19}{11\!\cdots\!59}a^{10}-\frac{14\!\cdots\!46}{11\!\cdots\!59}a^{9}+\frac{32\!\cdots\!82}{11\!\cdots\!59}a^{8}+\frac{35\!\cdots\!75}{11\!\cdots\!59}a^{7}-\frac{58\!\cdots\!71}{11\!\cdots\!59}a^{6}-\frac{78\!\cdots\!86}{11\!\cdots\!59}a^{5}+\frac{60\!\cdots\!31}{11\!\cdots\!59}a^{4}+\frac{94\!\cdots\!01}{10\!\cdots\!69}a^{3}-\frac{26\!\cdots\!89}{10\!\cdots\!69}a^{2}-\frac{53\!\cdots\!41}{11\!\cdots\!59}a+\frac{43\!\cdots\!25}{11\!\cdots\!59}$, $\frac{22\!\cdots\!49}{10\!\cdots\!69}a^{19}-\frac{16\!\cdots\!29}{11\!\cdots\!59}a^{18}-\frac{77\!\cdots\!58}{11\!\cdots\!59}a^{17}+\frac{59\!\cdots\!62}{11\!\cdots\!59}a^{16}+\frac{92\!\cdots\!88}{11\!\cdots\!59}a^{15}-\frac{82\!\cdots\!11}{10\!\cdots\!69}a^{14}-\frac{29\!\cdots\!90}{62\!\cdots\!61}a^{13}+\frac{76\!\cdots\!15}{11\!\cdots\!59}a^{12}+\frac{18\!\cdots\!07}{11\!\cdots\!59}a^{11}-\frac{38\!\cdots\!39}{11\!\cdots\!59}a^{10}-\frac{43\!\cdots\!03}{11\!\cdots\!59}a^{9}+\frac{11\!\cdots\!68}{11\!\cdots\!59}a^{8}+\frac{10\!\cdots\!14}{11\!\cdots\!59}a^{7}-\frac{21\!\cdots\!22}{11\!\cdots\!59}a^{6}-\frac{24\!\cdots\!41}{11\!\cdots\!59}a^{5}+\frac{21\!\cdots\!77}{11\!\cdots\!59}a^{4}+\frac{31\!\cdots\!23}{10\!\cdots\!69}a^{3}-\frac{96\!\cdots\!57}{10\!\cdots\!69}a^{2}-\frac{18\!\cdots\!19}{11\!\cdots\!59}a+\frac{15\!\cdots\!14}{11\!\cdots\!59}$, $\frac{23\!\cdots\!03}{11\!\cdots\!59}a^{19}-\frac{14\!\cdots\!61}{11\!\cdots\!59}a^{18}-\frac{71\!\cdots\!54}{11\!\cdots\!59}a^{17}+\frac{54\!\cdots\!66}{11\!\cdots\!59}a^{16}+\frac{86\!\cdots\!19}{11\!\cdots\!59}a^{15}-\frac{83\!\cdots\!70}{11\!\cdots\!59}a^{14}-\frac{28\!\cdots\!22}{62\!\cdots\!61}a^{13}+\frac{69\!\cdots\!12}{11\!\cdots\!59}a^{12}+\frac{19\!\cdots\!34}{11\!\cdots\!59}a^{11}-\frac{34\!\cdots\!40}{11\!\cdots\!59}a^{10}-\frac{49\!\cdots\!62}{11\!\cdots\!59}a^{9}+\frac{10\!\cdots\!51}{11\!\cdots\!59}a^{8}+\frac{12\!\cdots\!93}{11\!\cdots\!59}a^{7}-\frac{18\!\cdots\!80}{11\!\cdots\!59}a^{6}-\frac{27\!\cdots\!39}{11\!\cdots\!59}a^{5}+\frac{19\!\cdots\!68}{11\!\cdots\!59}a^{4}+\frac{31\!\cdots\!96}{10\!\cdots\!69}a^{3}-\frac{95\!\cdots\!26}{11\!\cdots\!59}a^{2}-\frac{17\!\cdots\!42}{11\!\cdots\!59}a+\frac{14\!\cdots\!73}{11\!\cdots\!59}$, $\frac{84\!\cdots\!95}{11\!\cdots\!59}a^{19}-\frac{53\!\cdots\!18}{11\!\cdots\!59}a^{18}-\frac{25\!\cdots\!04}{11\!\cdots\!59}a^{17}+\frac{19\!\cdots\!52}{11\!\cdots\!59}a^{16}+\frac{30\!\cdots\!30}{11\!\cdots\!59}a^{15}-\frac{30\!\cdots\!97}{11\!\cdots\!59}a^{14}-\frac{98\!\cdots\!16}{62\!\cdots\!61}a^{13}+\frac{25\!\cdots\!74}{11\!\cdots\!59}a^{12}+\frac{63\!\cdots\!00}{11\!\cdots\!59}a^{11}-\frac{12\!\cdots\!22}{11\!\cdots\!59}a^{10}-\frac{14\!\cdots\!96}{11\!\cdots\!59}a^{9}+\frac{38\!\cdots\!34}{11\!\cdots\!59}a^{8}+\frac{36\!\cdots\!57}{11\!\cdots\!59}a^{7}-\frac{69\!\cdots\!41}{11\!\cdots\!59}a^{6}-\frac{87\!\cdots\!38}{11\!\cdots\!59}a^{5}+\frac{71\!\cdots\!15}{11\!\cdots\!59}a^{4}+\frac{10\!\cdots\!73}{10\!\cdots\!69}a^{3}-\frac{35\!\cdots\!30}{11\!\cdots\!59}a^{2}-\frac{63\!\cdots\!65}{11\!\cdots\!59}a+\frac{51\!\cdots\!98}{11\!\cdots\!59}$, $\frac{15\!\cdots\!10}{11\!\cdots\!59}a^{19}-\frac{10\!\cdots\!83}{11\!\cdots\!59}a^{18}-\frac{49\!\cdots\!65}{11\!\cdots\!59}a^{17}+\frac{37\!\cdots\!86}{11\!\cdots\!59}a^{16}+\frac{59\!\cdots\!03}{11\!\cdots\!59}a^{15}-\frac{57\!\cdots\!45}{11\!\cdots\!59}a^{14}-\frac{17\!\cdots\!16}{57\!\cdots\!51}a^{13}+\frac{47\!\cdots\!74}{11\!\cdots\!59}a^{12}+\frac{13\!\cdots\!43}{11\!\cdots\!59}a^{11}-\frac{21\!\cdots\!56}{10\!\cdots\!69}a^{10}-\frac{33\!\cdots\!37}{11\!\cdots\!59}a^{9}+\frac{72\!\cdots\!04}{11\!\cdots\!59}a^{8}+\frac{82\!\cdots\!35}{11\!\cdots\!59}a^{7}-\frac{13\!\cdots\!21}{11\!\cdots\!59}a^{6}-\frac{16\!\cdots\!65}{10\!\cdots\!69}a^{5}+\frac{13\!\cdots\!21}{11\!\cdots\!59}a^{4}+\frac{21\!\cdots\!61}{10\!\cdots\!69}a^{3}-\frac{66\!\cdots\!79}{11\!\cdots\!59}a^{2}-\frac{11\!\cdots\!10}{10\!\cdots\!69}a+\frac{98\!\cdots\!42}{11\!\cdots\!59}$, $\frac{38\!\cdots\!32}{10\!\cdots\!69}a^{19}-\frac{27\!\cdots\!49}{11\!\cdots\!59}a^{18}-\frac{13\!\cdots\!91}{11\!\cdots\!59}a^{17}+\frac{99\!\cdots\!75}{11\!\cdots\!59}a^{16}+\frac{15\!\cdots\!88}{11\!\cdots\!59}a^{15}-\frac{13\!\cdots\!02}{10\!\cdots\!69}a^{14}-\frac{51\!\cdots\!58}{62\!\cdots\!61}a^{13}+\frac{12\!\cdots\!86}{11\!\cdots\!59}a^{12}+\frac{34\!\cdots\!19}{11\!\cdots\!59}a^{11}-\frac{63\!\cdots\!98}{11\!\cdots\!59}a^{10}-\frac{87\!\cdots\!38}{11\!\cdots\!59}a^{9}+\frac{19\!\cdots\!83}{11\!\cdots\!59}a^{8}+\frac{21\!\cdots\!82}{11\!\cdots\!59}a^{7}-\frac{35\!\cdots\!44}{11\!\cdots\!59}a^{6}-\frac{47\!\cdots\!32}{11\!\cdots\!59}a^{5}+\frac{36\!\cdots\!14}{11\!\cdots\!59}a^{4}+\frac{56\!\cdots\!71}{10\!\cdots\!69}a^{3}-\frac{16\!\cdots\!51}{10\!\cdots\!69}a^{2}-\frac{32\!\cdots\!64}{11\!\cdots\!59}a+\frac{26\!\cdots\!39}{11\!\cdots\!59}$, $\frac{19\!\cdots\!63}{11\!\cdots\!59}a^{19}-\frac{12\!\cdots\!27}{11\!\cdots\!59}a^{18}-\frac{59\!\cdots\!90}{11\!\cdots\!59}a^{17}+\frac{44\!\cdots\!86}{11\!\cdots\!59}a^{16}+\frac{73\!\cdots\!13}{11\!\cdots\!59}a^{15}-\frac{68\!\cdots\!94}{11\!\cdots\!59}a^{14}-\frac{24\!\cdots\!82}{62\!\cdots\!61}a^{13}+\frac{57\!\cdots\!61}{11\!\cdots\!59}a^{12}+\frac{17\!\cdots\!85}{11\!\cdots\!59}a^{11}-\frac{28\!\cdots\!63}{11\!\cdots\!59}a^{10}-\frac{48\!\cdots\!04}{11\!\cdots\!59}a^{9}+\frac{87\!\cdots\!00}{11\!\cdots\!59}a^{8}+\frac{12\!\cdots\!57}{11\!\cdots\!59}a^{7}-\frac{14\!\cdots\!65}{10\!\cdots\!69}a^{6}-\frac{25\!\cdots\!62}{11\!\cdots\!59}a^{5}+\frac{16\!\cdots\!07}{11\!\cdots\!59}a^{4}+\frac{28\!\cdots\!07}{10\!\cdots\!69}a^{3}-\frac{81\!\cdots\!38}{11\!\cdots\!59}a^{2}-\frac{15\!\cdots\!60}{11\!\cdots\!59}a+\frac{12\!\cdots\!52}{11\!\cdots\!59}$, $\frac{12\!\cdots\!34}{11\!\cdots\!59}a^{19}-\frac{76\!\cdots\!14}{11\!\cdots\!59}a^{18}-\frac{37\!\cdots\!34}{11\!\cdots\!59}a^{17}+\frac{28\!\cdots\!73}{11\!\cdots\!59}a^{16}+\frac{44\!\cdots\!81}{11\!\cdots\!59}a^{15}-\frac{43\!\cdots\!54}{11\!\cdots\!59}a^{14}-\frac{14\!\cdots\!76}{62\!\cdots\!61}a^{13}+\frac{36\!\cdots\!61}{11\!\cdots\!59}a^{12}+\frac{95\!\cdots\!01}{11\!\cdots\!59}a^{11}-\frac{18\!\cdots\!27}{11\!\cdots\!59}a^{10}-\frac{23\!\cdots\!49}{11\!\cdots\!59}a^{9}+\frac{54\!\cdots\!99}{11\!\cdots\!59}a^{8}+\frac{56\!\cdots\!43}{11\!\cdots\!59}a^{7}-\frac{90\!\cdots\!07}{10\!\cdots\!69}a^{6}-\frac{12\!\cdots\!36}{11\!\cdots\!59}a^{5}+\frac{10\!\cdots\!09}{11\!\cdots\!59}a^{4}+\frac{15\!\cdots\!58}{10\!\cdots\!69}a^{3}-\frac{49\!\cdots\!14}{11\!\cdots\!59}a^{2}-\frac{90\!\cdots\!96}{11\!\cdots\!59}a+\frac{73\!\cdots\!20}{11\!\cdots\!59}$, $\frac{40\!\cdots\!16}{11\!\cdots\!59}a^{19}-\frac{21\!\cdots\!98}{10\!\cdots\!69}a^{18}-\frac{13\!\cdots\!10}{11\!\cdots\!59}a^{17}+\frac{85\!\cdots\!74}{11\!\cdots\!59}a^{16}+\frac{19\!\cdots\!67}{11\!\cdots\!59}a^{15}-\frac{12\!\cdots\!38}{11\!\cdots\!59}a^{14}-\frac{82\!\cdots\!39}{62\!\cdots\!61}a^{13}+\frac{10\!\cdots\!01}{11\!\cdots\!59}a^{12}+\frac{75\!\cdots\!50}{10\!\cdots\!69}a^{11}-\frac{50\!\cdots\!62}{11\!\cdots\!59}a^{10}-\frac{27\!\cdots\!11}{10\!\cdots\!69}a^{9}+\frac{14\!\cdots\!48}{11\!\cdots\!59}a^{8}+\frac{75\!\cdots\!78}{11\!\cdots\!59}a^{7}-\frac{24\!\cdots\!81}{11\!\cdots\!59}a^{6}-\frac{11\!\cdots\!00}{11\!\cdots\!59}a^{5}+\frac{22\!\cdots\!19}{11\!\cdots\!59}a^{4}+\frac{82\!\cdots\!75}{10\!\cdots\!69}a^{3}-\frac{10\!\cdots\!43}{11\!\cdots\!59}a^{2}-\frac{29\!\cdots\!36}{11\!\cdots\!59}a+\frac{17\!\cdots\!90}{11\!\cdots\!59}$, $\frac{30\!\cdots\!68}{11\!\cdots\!59}a^{19}-\frac{19\!\cdots\!14}{11\!\cdots\!59}a^{18}-\frac{94\!\cdots\!78}{11\!\cdots\!59}a^{17}+\frac{70\!\cdots\!63}{11\!\cdots\!59}a^{16}+\frac{11\!\cdots\!10}{11\!\cdots\!59}a^{15}-\frac{10\!\cdots\!70}{11\!\cdots\!59}a^{14}-\frac{38\!\cdots\!19}{62\!\cdots\!61}a^{13}+\frac{91\!\cdots\!62}{11\!\cdots\!59}a^{12}+\frac{26\!\cdots\!71}{11\!\cdots\!59}a^{11}-\frac{45\!\cdots\!74}{11\!\cdots\!59}a^{10}-\frac{71\!\cdots\!30}{11\!\cdots\!59}a^{9}+\frac{13\!\cdots\!65}{11\!\cdots\!59}a^{8}+\frac{17\!\cdots\!09}{11\!\cdots\!59}a^{7}-\frac{24\!\cdots\!54}{11\!\cdots\!59}a^{6}-\frac{37\!\cdots\!49}{11\!\cdots\!59}a^{5}+\frac{25\!\cdots\!12}{11\!\cdots\!59}a^{4}+\frac{42\!\cdots\!72}{10\!\cdots\!69}a^{3}-\frac{12\!\cdots\!47}{11\!\cdots\!59}a^{2}-\frac{23\!\cdots\!20}{11\!\cdots\!59}a+\frac{18\!\cdots\!46}{11\!\cdots\!59}$, $\frac{49\!\cdots\!37}{11\!\cdots\!59}a^{19}-\frac{31\!\cdots\!11}{11\!\cdots\!59}a^{18}-\frac{15\!\cdots\!72}{11\!\cdots\!59}a^{17}+\frac{10\!\cdots\!18}{10\!\cdots\!69}a^{16}+\frac{18\!\cdots\!45}{11\!\cdots\!59}a^{15}-\frac{17\!\cdots\!95}{11\!\cdots\!59}a^{14}-\frac{58\!\cdots\!85}{62\!\cdots\!61}a^{13}+\frac{13\!\cdots\!52}{10\!\cdots\!69}a^{12}+\frac{38\!\cdots\!99}{11\!\cdots\!59}a^{11}-\frac{72\!\cdots\!14}{11\!\cdots\!59}a^{10}-\frac{93\!\cdots\!68}{11\!\cdots\!59}a^{9}+\frac{22\!\cdots\!33}{11\!\cdots\!59}a^{8}+\frac{22\!\cdots\!41}{11\!\cdots\!59}a^{7}-\frac{39\!\cdots\!45}{11\!\cdots\!59}a^{6}-\frac{51\!\cdots\!77}{11\!\cdots\!59}a^{5}+\frac{40\!\cdots\!37}{11\!\cdots\!59}a^{4}+\frac{62\!\cdots\!45}{10\!\cdots\!69}a^{3}-\frac{19\!\cdots\!72}{11\!\cdots\!59}a^{2}-\frac{35\!\cdots\!40}{11\!\cdots\!59}a+\frac{29\!\cdots\!38}{11\!\cdots\!59}$, $\frac{10\!\cdots\!24}{11\!\cdots\!59}a^{19}-\frac{75\!\cdots\!03}{11\!\cdots\!59}a^{18}-\frac{26\!\cdots\!66}{11\!\cdots\!59}a^{17}+\frac{26\!\cdots\!02}{11\!\cdots\!59}a^{16}+\frac{16\!\cdots\!18}{10\!\cdots\!69}a^{15}-\frac{37\!\cdots\!35}{11\!\cdots\!59}a^{14}+\frac{25\!\cdots\!30}{62\!\cdots\!61}a^{13}+\frac{29\!\cdots\!93}{11\!\cdots\!59}a^{12}-\frac{12\!\cdots\!62}{11\!\cdots\!59}a^{11}-\frac{13\!\cdots\!74}{11\!\cdots\!59}a^{10}+\frac{64\!\cdots\!93}{11\!\cdots\!59}a^{9}+\frac{36\!\cdots\!92}{11\!\cdots\!59}a^{8}-\frac{15\!\cdots\!95}{11\!\cdots\!59}a^{7}-\frac{60\!\cdots\!13}{11\!\cdots\!59}a^{6}+\frac{16\!\cdots\!35}{11\!\cdots\!59}a^{5}+\frac{51\!\cdots\!16}{10\!\cdots\!69}a^{4}-\frac{48\!\cdots\!66}{10\!\cdots\!69}a^{3}-\frac{25\!\cdots\!83}{11\!\cdots\!59}a^{2}-\frac{14\!\cdots\!16}{11\!\cdots\!59}a+\frac{27\!\cdots\!37}{10\!\cdots\!69}$, $\frac{63\!\cdots\!37}{11\!\cdots\!59}a^{19}-\frac{45\!\cdots\!69}{11\!\cdots\!59}a^{18}-\frac{17\!\cdots\!67}{11\!\cdots\!59}a^{17}+\frac{16\!\cdots\!46}{11\!\cdots\!59}a^{16}+\frac{15\!\cdots\!39}{11\!\cdots\!59}a^{15}-\frac{26\!\cdots\!41}{11\!\cdots\!59}a^{14}-\frac{98\!\cdots\!95}{62\!\cdots\!61}a^{13}+\frac{22\!\cdots\!22}{11\!\cdots\!59}a^{12}-\frac{47\!\cdots\!82}{11\!\cdots\!59}a^{11}-\frac{11\!\cdots\!58}{11\!\cdots\!59}a^{10}+\frac{29\!\cdots\!21}{11\!\cdots\!59}a^{9}+\frac{38\!\cdots\!09}{11\!\cdots\!59}a^{8}-\frac{65\!\cdots\!86}{11\!\cdots\!59}a^{7}-\frac{76\!\cdots\!25}{11\!\cdots\!59}a^{6}+\frac{29\!\cdots\!82}{11\!\cdots\!59}a^{5}+\frac{85\!\cdots\!11}{11\!\cdots\!59}a^{4}+\frac{73\!\cdots\!66}{10\!\cdots\!69}a^{3}-\frac{46\!\cdots\!74}{11\!\cdots\!59}a^{2}-\frac{77\!\cdots\!35}{11\!\cdots\!59}a+\frac{69\!\cdots\!41}{11\!\cdots\!59}$, $\frac{88\!\cdots\!15}{11\!\cdots\!59}a^{19}-\frac{55\!\cdots\!99}{11\!\cdots\!59}a^{18}-\frac{27\!\cdots\!21}{11\!\cdots\!59}a^{17}+\frac{20\!\cdots\!82}{11\!\cdots\!59}a^{16}+\frac{31\!\cdots\!55}{10\!\cdots\!69}a^{15}-\frac{31\!\cdots\!10}{11\!\cdots\!59}a^{14}-\frac{11\!\cdots\!47}{62\!\cdots\!61}a^{13}+\frac{26\!\cdots\!77}{11\!\cdots\!59}a^{12}+\frac{85\!\cdots\!69}{11\!\cdots\!59}a^{11}-\frac{13\!\cdots\!79}{11\!\cdots\!59}a^{10}-\frac{23\!\cdots\!28}{11\!\cdots\!59}a^{9}+\frac{40\!\cdots\!01}{11\!\cdots\!59}a^{8}+\frac{54\!\cdots\!37}{11\!\cdots\!59}a^{7}-\frac{74\!\cdots\!75}{11\!\cdots\!59}a^{6}-\frac{11\!\cdots\!75}{11\!\cdots\!59}a^{5}+\frac{69\!\cdots\!60}{10\!\cdots\!69}a^{4}+\frac{12\!\cdots\!89}{10\!\cdots\!69}a^{3}-\frac{37\!\cdots\!27}{11\!\cdots\!59}a^{2}-\frac{68\!\cdots\!34}{11\!\cdots\!59}a+\frac{50\!\cdots\!72}{10\!\cdots\!69}$, $\frac{18\!\cdots\!32}{11\!\cdots\!59}a^{19}-\frac{14\!\cdots\!38}{11\!\cdots\!59}a^{18}-\frac{33\!\cdots\!03}{11\!\cdots\!59}a^{17}+\frac{50\!\cdots\!19}{11\!\cdots\!59}a^{16}-\frac{10\!\cdots\!49}{11\!\cdots\!59}a^{15}-\frac{71\!\cdots\!07}{11\!\cdots\!59}a^{14}+\frac{37\!\cdots\!71}{62\!\cdots\!61}a^{13}+\frac{54\!\cdots\!02}{11\!\cdots\!59}a^{12}-\frac{71\!\cdots\!55}{11\!\cdots\!59}a^{11}-\frac{23\!\cdots\!32}{11\!\cdots\!59}a^{10}+\frac{34\!\cdots\!38}{11\!\cdots\!59}a^{9}+\frac{63\!\cdots\!45}{11\!\cdots\!59}a^{8}-\frac{80\!\cdots\!91}{10\!\cdots\!69}a^{7}-\frac{98\!\cdots\!13}{11\!\cdots\!59}a^{6}+\frac{12\!\cdots\!73}{11\!\cdots\!59}a^{5}+\frac{81\!\cdots\!79}{11\!\cdots\!59}a^{4}-\frac{69\!\cdots\!40}{10\!\cdots\!69}a^{3}-\frac{26\!\cdots\!74}{11\!\cdots\!59}a^{2}+\frac{14\!\cdots\!27}{11\!\cdots\!59}a-\frac{19\!\cdots\!91}{11\!\cdots\!59}$, $\frac{45\!\cdots\!61}{11\!\cdots\!59}a^{19}-\frac{30\!\cdots\!49}{11\!\cdots\!59}a^{18}-\frac{13\!\cdots\!90}{11\!\cdots\!59}a^{17}+\frac{11\!\cdots\!32}{11\!\cdots\!59}a^{16}+\frac{13\!\cdots\!96}{11\!\cdots\!59}a^{15}-\frac{17\!\cdots\!07}{11\!\cdots\!59}a^{14}-\frac{27\!\cdots\!00}{57\!\cdots\!51}a^{13}+\frac{14\!\cdots\!20}{11\!\cdots\!59}a^{12}-\frac{43\!\cdots\!44}{11\!\cdots\!59}a^{11}-\frac{64\!\cdots\!77}{10\!\cdots\!69}a^{10}+\frac{76\!\cdots\!77}{11\!\cdots\!59}a^{9}+\frac{21\!\cdots\!41}{11\!\cdots\!59}a^{8}-\frac{20\!\cdots\!76}{11\!\cdots\!59}a^{7}-\frac{39\!\cdots\!32}{11\!\cdots\!59}a^{6}+\frac{95\!\cdots\!60}{10\!\cdots\!69}a^{5}+\frac{40\!\cdots\!35}{11\!\cdots\!59}a^{4}+\frac{23\!\cdots\!96}{10\!\cdots\!69}a^{3}-\frac{19\!\cdots\!62}{11\!\cdots\!59}a^{2}-\frac{23\!\cdots\!02}{10\!\cdots\!69}a+\frac{27\!\cdots\!79}{11\!\cdots\!59}$, $\frac{27\!\cdots\!50}{11\!\cdots\!59}a^{19}-\frac{17\!\cdots\!71}{11\!\cdots\!59}a^{18}-\frac{84\!\cdots\!22}{11\!\cdots\!59}a^{17}+\frac{64\!\cdots\!16}{11\!\cdots\!59}a^{16}+\frac{10\!\cdots\!06}{11\!\cdots\!59}a^{15}-\frac{98\!\cdots\!26}{11\!\cdots\!59}a^{14}-\frac{33\!\cdots\!35}{62\!\cdots\!61}a^{13}+\frac{82\!\cdots\!82}{11\!\cdots\!59}a^{12}+\frac{22\!\cdots\!82}{11\!\cdots\!59}a^{11}-\frac{41\!\cdots\!05}{11\!\cdots\!59}a^{10}-\frac{56\!\cdots\!80}{11\!\cdots\!59}a^{9}+\frac{12\!\cdots\!88}{11\!\cdots\!59}a^{8}+\frac{13\!\cdots\!98}{11\!\cdots\!59}a^{7}-\frac{22\!\cdots\!88}{11\!\cdots\!59}a^{6}-\frac{30\!\cdots\!90}{11\!\cdots\!59}a^{5}+\frac{23\!\cdots\!19}{11\!\cdots\!59}a^{4}+\frac{36\!\cdots\!80}{10\!\cdots\!69}a^{3}-\frac{11\!\cdots\!16}{11\!\cdots\!59}a^{2}-\frac{20\!\cdots\!80}{11\!\cdots\!59}a+\frac{16\!\cdots\!77}{11\!\cdots\!59}$, $\frac{10\!\cdots\!27}{11\!\cdots\!59}a^{19}-\frac{65\!\cdots\!59}{11\!\cdots\!59}a^{18}-\frac{35\!\cdots\!10}{11\!\cdots\!59}a^{17}+\frac{24\!\cdots\!96}{11\!\cdots\!59}a^{16}+\frac{49\!\cdots\!56}{11\!\cdots\!59}a^{15}-\frac{39\!\cdots\!46}{11\!\cdots\!59}a^{14}-\frac{19\!\cdots\!99}{62\!\cdots\!61}a^{13}+\frac{33\!\cdots\!77}{11\!\cdots\!59}a^{12}+\frac{18\!\cdots\!73}{11\!\cdots\!59}a^{11}-\frac{17\!\cdots\!59}{11\!\cdots\!59}a^{10}-\frac{67\!\cdots\!75}{11\!\cdots\!59}a^{9}+\frac{55\!\cdots\!17}{11\!\cdots\!59}a^{8}+\frac{18\!\cdots\!85}{11\!\cdots\!59}a^{7}-\frac{10\!\cdots\!15}{11\!\cdots\!59}a^{6}-\frac{32\!\cdots\!82}{11\!\cdots\!59}a^{5}+\frac{11\!\cdots\!26}{11\!\cdots\!59}a^{4}+\frac{31\!\cdots\!63}{10\!\cdots\!69}a^{3}-\frac{61\!\cdots\!24}{11\!\cdots\!59}a^{2}-\frac{14\!\cdots\!89}{11\!\cdots\!59}a+\frac{99\!\cdots\!35}{11\!\cdots\!59}$
| 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: | \( 62134578326.1 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 62134578326.1 \cdot 1}{2\cdot\sqrt{19184293930982457734868438720703125}}\cr\approx \mathstrut & 0.235196233782 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429)
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 - 8*x^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429, 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 - 8*x^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429);
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 - 8*x^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429);
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
$C_4\times D_5$ (as 20T6):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 40 |
| The 16 conjugacy class representatives for $C_4\times D_5$ |
| Character table for $C_4\times D_5$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 5.5.12852225.1, 10.10.825898437253125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Galois closure: | deg 40 |
| Degree 20 sibling: | deg 20 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $20$ | R | R | $20$ | ${\href{/padicField/11.2.0.1}{2} }^{10}$ | $20$ | $20$ | ${\href{/padicField/19.2.0.1}{2} }^{10}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/47.4.0.1}{4} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(239\)
| $\Q_{239}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{239}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{239}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{239}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |