Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 - x^20 + 18*x^19 - 17*x^18 - 44*x^17 + 110*x^16 - 3*x^15 - 254*x^14 + 59*x^13 + 240*x^12 - 102*x^11 - 552*x^10 - 230*x^9 + 307*x^8 + 320*x^7 - 3*x^6 - 154*x^5 - 62*x^4 + 18*x^3 + 19*x^2 + 2*x - 1)
gp: K = bnfinit(y^22 - 2*y^21 - y^20 + 18*y^19 - 17*y^18 - 44*y^17 + 110*y^16 - 3*y^15 - 254*y^14 + 59*y^13 + 240*y^12 - 102*y^11 - 552*y^10 - 230*y^9 + 307*y^8 + 320*y^7 - 3*y^6 - 154*y^5 - 62*y^4 + 18*y^3 + 19*y^2 + 2*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^21 - x^20 + 18*x^19 - 17*x^18 - 44*x^17 + 110*x^16 - 3*x^15 - 254*x^14 + 59*x^13 + 240*x^12 - 102*x^11 - 552*x^10 - 230*x^9 + 307*x^8 + 320*x^7 - 3*x^6 - 154*x^5 - 62*x^4 + 18*x^3 + 19*x^2 + 2*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 2*x^21 - x^20 + 18*x^19 - 17*x^18 - 44*x^17 + 110*x^16 - 3*x^15 - 254*x^14 + 59*x^13 + 240*x^12 - 102*x^11 - 552*x^10 - 230*x^9 + 307*x^8 + 320*x^7 - 3*x^6 - 154*x^5 - 62*x^4 + 18*x^3 + 19*x^2 + 2*x - 1)
\( x^{22} - 2 x^{21} - x^{20} + 18 x^{19} - 17 x^{18} - 44 x^{17} + 110 x^{16} - 3 x^{15} - 254 x^{14} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[6, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1447402975463751668017578125\)
\(\medspace = 5^{11}\cdot 7^{4}\cdot 83^{4}\cdot 127^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.16\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $5^{1/2}7^{1/2}83^{2/3}127^{2/3}\approx 2844.1679033165196$ | ||
| Ramified primes: |
\(5\), \(7\), \(83\), \(127\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}+\frac{2}{5}a^{17}-\frac{1}{5}a^{14}+\frac{2}{5}a^{12}-\frac{1}{5}a^{11}-\frac{1}{5}a^{10}-\frac{1}{5}a^{9}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{19}+\frac{1}{5}a^{17}-\frac{1}{5}a^{15}+\frac{2}{5}a^{14}+\frac{2}{5}a^{13}+\frac{1}{5}a^{11}+\frac{1}{5}a^{10}+\frac{2}{5}a^{9}-\frac{2}{5}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{20}-\frac{2}{5}a^{17}-\frac{1}{5}a^{16}+\frac{2}{5}a^{15}-\frac{2}{5}a^{14}-\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{2}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{56\!\cdots\!45}a^{21}-\frac{13\!\cdots\!66}{56\!\cdots\!45}a^{20}-\frac{35\!\cdots\!29}{56\!\cdots\!45}a^{19}-\frac{49\!\cdots\!99}{56\!\cdots\!45}a^{18}-\frac{18\!\cdots\!22}{56\!\cdots\!45}a^{17}-\frac{54\!\cdots\!22}{56\!\cdots\!45}a^{16}+\frac{29\!\cdots\!90}{11\!\cdots\!29}a^{15}-\frac{39\!\cdots\!54}{56\!\cdots\!45}a^{14}+\frac{26\!\cdots\!76}{56\!\cdots\!45}a^{13}+\frac{12\!\cdots\!79}{56\!\cdots\!45}a^{12}+\frac{18\!\cdots\!74}{56\!\cdots\!45}a^{11}+\frac{24\!\cdots\!71}{56\!\cdots\!45}a^{10}+\frac{23\!\cdots\!26}{56\!\cdots\!45}a^{9}-\frac{21\!\cdots\!61}{56\!\cdots\!45}a^{8}+\frac{43\!\cdots\!48}{11\!\cdots\!29}a^{7}-\frac{23\!\cdots\!48}{56\!\cdots\!45}a^{6}-\frac{20\!\cdots\!28}{56\!\cdots\!45}a^{5}-\frac{13\!\cdots\!51}{56\!\cdots\!45}a^{4}-\frac{86\!\cdots\!52}{56\!\cdots\!45}a^{3}+\frac{27\!\cdots\!19}{56\!\cdots\!45}a^{2}+\frac{58\!\cdots\!05}{11\!\cdots\!29}a+\frac{10\!\cdots\!28}{56\!\cdots\!45}$
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: | $13$ | 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: |
$\frac{38\!\cdots\!38}{56\!\cdots\!45}a^{21}-\frac{53\!\cdots\!88}{56\!\cdots\!45}a^{20}-\frac{14\!\cdots\!42}{11\!\cdots\!29}a^{19}+\frac{67\!\cdots\!54}{56\!\cdots\!45}a^{18}-\frac{32\!\cdots\!42}{56\!\cdots\!45}a^{17}-\frac{17\!\cdots\!46}{56\!\cdots\!45}a^{16}+\frac{34\!\cdots\!13}{56\!\cdots\!45}a^{15}+\frac{82\!\cdots\!81}{56\!\cdots\!45}a^{14}-\frac{85\!\cdots\!68}{56\!\cdots\!45}a^{13}+\frac{10\!\cdots\!19}{56\!\cdots\!45}a^{12}+\frac{20\!\cdots\!38}{56\!\cdots\!45}a^{11}-\frac{11\!\cdots\!51}{56\!\cdots\!45}a^{10}-\frac{99\!\cdots\!69}{56\!\cdots\!45}a^{9}-\frac{27\!\cdots\!58}{56\!\cdots\!45}a^{8}-\frac{95\!\cdots\!39}{56\!\cdots\!45}a^{7}+\frac{14\!\cdots\!53}{56\!\cdots\!45}a^{6}+\frac{16\!\cdots\!88}{56\!\cdots\!45}a^{5}+\frac{85\!\cdots\!66}{56\!\cdots\!45}a^{4}-\frac{74\!\cdots\!34}{56\!\cdots\!45}a^{3}-\frac{23\!\cdots\!22}{56\!\cdots\!45}a^{2}+\frac{51\!\cdots\!73}{11\!\cdots\!29}a+\frac{50\!\cdots\!52}{56\!\cdots\!45}$, $\frac{40\!\cdots\!77}{56\!\cdots\!45}a^{21}-\frac{18\!\cdots\!16}{56\!\cdots\!45}a^{20}+\frac{57\!\cdots\!40}{11\!\cdots\!29}a^{19}+\frac{52\!\cdots\!46}{56\!\cdots\!45}a^{18}-\frac{50\!\cdots\!42}{11\!\cdots\!29}a^{17}+\frac{44\!\cdots\!40}{11\!\cdots\!29}a^{16}+\frac{56\!\cdots\!14}{56\!\cdots\!45}a^{15}-\frac{15\!\cdots\!33}{56\!\cdots\!45}a^{14}+\frac{65\!\cdots\!68}{56\!\cdots\!45}a^{13}+\frac{39\!\cdots\!10}{11\!\cdots\!29}a^{12}-\frac{26\!\cdots\!96}{56\!\cdots\!45}a^{11}-\frac{27\!\cdots\!61}{56\!\cdots\!45}a^{10}+\frac{14\!\cdots\!04}{11\!\cdots\!29}a^{9}+\frac{15\!\cdots\!16}{56\!\cdots\!45}a^{8}-\frac{16\!\cdots\!79}{56\!\cdots\!45}a^{7}-\frac{15\!\cdots\!89}{56\!\cdots\!45}a^{6}+\frac{25\!\cdots\!12}{56\!\cdots\!45}a^{5}+\frac{14\!\cdots\!62}{11\!\cdots\!29}a^{4}+\frac{43\!\cdots\!87}{56\!\cdots\!45}a^{3}-\frac{26\!\cdots\!08}{56\!\cdots\!45}a^{2}-\frac{11\!\cdots\!19}{56\!\cdots\!45}a-\frac{22\!\cdots\!91}{56\!\cdots\!45}$, $\frac{17\!\cdots\!43}{56\!\cdots\!45}a^{21}-\frac{43\!\cdots\!21}{56\!\cdots\!45}a^{20}+\frac{61\!\cdots\!14}{11\!\cdots\!29}a^{19}+\frac{31\!\cdots\!44}{56\!\cdots\!45}a^{18}-\frac{44\!\cdots\!66}{56\!\cdots\!45}a^{17}-\frac{55\!\cdots\!53}{56\!\cdots\!45}a^{16}+\frac{21\!\cdots\!47}{56\!\cdots\!45}a^{15}-\frac{10\!\cdots\!98}{56\!\cdots\!45}a^{14}-\frac{39\!\cdots\!03}{56\!\cdots\!45}a^{13}+\frac{28\!\cdots\!47}{56\!\cdots\!45}a^{12}+\frac{28\!\cdots\!62}{56\!\cdots\!45}a^{11}-\frac{62\!\cdots\!60}{11\!\cdots\!29}a^{10}-\frac{81\!\cdots\!82}{56\!\cdots\!45}a^{9}-\frac{14\!\cdots\!62}{56\!\cdots\!45}a^{8}+\frac{10\!\cdots\!97}{11\!\cdots\!29}a^{7}+\frac{29\!\cdots\!46}{56\!\cdots\!45}a^{6}-\frac{14\!\cdots\!12}{56\!\cdots\!45}a^{5}-\frac{19\!\cdots\!62}{56\!\cdots\!45}a^{4}-\frac{13\!\cdots\!53}{56\!\cdots\!45}a^{3}+\frac{37\!\cdots\!98}{56\!\cdots\!45}a^{2}+\frac{58\!\cdots\!82}{56\!\cdots\!45}a-\frac{36\!\cdots\!91}{56\!\cdots\!45}$, $\frac{36\!\cdots\!91}{56\!\cdots\!45}a^{21}-\frac{18\!\cdots\!85}{11\!\cdots\!29}a^{20}+\frac{12\!\cdots\!26}{11\!\cdots\!29}a^{19}+\frac{65\!\cdots\!68}{56\!\cdots\!45}a^{18}-\frac{93\!\cdots\!91}{56\!\cdots\!45}a^{17}-\frac{11\!\cdots\!38}{56\!\cdots\!45}a^{16}+\frac{45\!\cdots\!63}{56\!\cdots\!45}a^{15}-\frac{45\!\cdots\!84}{11\!\cdots\!29}a^{14}-\frac{82\!\cdots\!16}{56\!\cdots\!45}a^{13}+\frac{60\!\cdots\!72}{56\!\cdots\!45}a^{12}+\frac{59\!\cdots\!93}{56\!\cdots\!45}a^{11}-\frac{65\!\cdots\!44}{56\!\cdots\!45}a^{10}-\frac{17\!\cdots\!32}{56\!\cdots\!45}a^{9}-\frac{29\!\cdots\!48}{56\!\cdots\!45}a^{8}+\frac{11\!\cdots\!99}{56\!\cdots\!45}a^{7}+\frac{12\!\cdots\!27}{11\!\cdots\!29}a^{6}-\frac{31\!\cdots\!19}{56\!\cdots\!45}a^{5}-\frac{41\!\cdots\!02}{56\!\cdots\!45}a^{4}-\frac{58\!\cdots\!76}{11\!\cdots\!29}a^{3}+\frac{79\!\cdots\!91}{56\!\cdots\!45}a^{2}+\frac{31\!\cdots\!31}{56\!\cdots\!45}a-\frac{84\!\cdots\!49}{11\!\cdots\!29}$, $\frac{11\!\cdots\!63}{56\!\cdots\!45}a^{21}-\frac{37\!\cdots\!46}{56\!\cdots\!45}a^{20}+\frac{60\!\cdots\!38}{11\!\cdots\!29}a^{19}+\frac{19\!\cdots\!34}{56\!\cdots\!45}a^{18}-\frac{45\!\cdots\!56}{56\!\cdots\!45}a^{17}-\frac{26\!\cdots\!38}{56\!\cdots\!45}a^{16}+\frac{15\!\cdots\!42}{56\!\cdots\!45}a^{15}-\frac{20\!\cdots\!93}{56\!\cdots\!45}a^{14}-\frac{11\!\cdots\!18}{56\!\cdots\!45}a^{13}+\frac{32\!\cdots\!22}{56\!\cdots\!45}a^{12}-\frac{13\!\cdots\!28}{56\!\cdots\!45}a^{11}-\frac{33\!\cdots\!69}{11\!\cdots\!29}a^{10}-\frac{29\!\cdots\!12}{56\!\cdots\!45}a^{9}+\frac{15\!\cdots\!78}{56\!\cdots\!45}a^{8}+\frac{16\!\cdots\!31}{11\!\cdots\!29}a^{7}-\frac{23\!\cdots\!79}{56\!\cdots\!45}a^{6}-\frac{41\!\cdots\!42}{56\!\cdots\!45}a^{5}-\frac{18\!\cdots\!82}{56\!\cdots\!45}a^{4}+\frac{12\!\cdots\!32}{56\!\cdots\!45}a^{3}-\frac{92\!\cdots\!32}{56\!\cdots\!45}a^{2}-\frac{47\!\cdots\!93}{56\!\cdots\!45}a-\frac{38\!\cdots\!61}{56\!\cdots\!45}$, $\frac{22\!\cdots\!93}{56\!\cdots\!45}a^{21}-\frac{11\!\cdots\!86}{11\!\cdots\!29}a^{20}+\frac{16\!\cdots\!14}{11\!\cdots\!29}a^{19}+\frac{41\!\cdots\!09}{56\!\cdots\!45}a^{18}-\frac{64\!\cdots\!08}{56\!\cdots\!45}a^{17}-\frac{69\!\cdots\!44}{56\!\cdots\!45}a^{16}+\frac{31\!\cdots\!74}{56\!\cdots\!45}a^{15}-\frac{36\!\cdots\!09}{11\!\cdots\!29}a^{14}-\frac{55\!\cdots\!78}{56\!\cdots\!45}a^{13}+\frac{55\!\cdots\!06}{56\!\cdots\!45}a^{12}+\frac{37\!\cdots\!64}{56\!\cdots\!45}a^{11}-\frac{65\!\cdots\!87}{56\!\cdots\!45}a^{10}-\frac{93\!\cdots\!41}{56\!\cdots\!45}a^{9}+\frac{33\!\cdots\!06}{56\!\cdots\!45}a^{8}+\frac{78\!\cdots\!67}{56\!\cdots\!45}a^{7}+\frac{15\!\cdots\!41}{11\!\cdots\!29}a^{6}-\frac{33\!\cdots\!92}{56\!\cdots\!45}a^{5}-\frac{17\!\cdots\!26}{56\!\cdots\!45}a^{4}+\frac{15\!\cdots\!98}{11\!\cdots\!29}a^{3}+\frac{50\!\cdots\!68}{56\!\cdots\!45}a^{2}-\frac{24\!\cdots\!67}{56\!\cdots\!45}a-\frac{12\!\cdots\!60}{11\!\cdots\!29}$, $\frac{14\!\cdots\!21}{11\!\cdots\!29}a^{21}-\frac{25\!\cdots\!20}{11\!\cdots\!29}a^{20}-\frac{14\!\cdots\!70}{11\!\cdots\!29}a^{19}+\frac{25\!\cdots\!79}{11\!\cdots\!29}a^{18}-\frac{22\!\cdots\!34}{11\!\cdots\!29}a^{17}-\frac{57\!\cdots\!12}{11\!\cdots\!29}a^{16}+\frac{14\!\cdots\!80}{11\!\cdots\!29}a^{15}-\frac{21\!\cdots\!93}{11\!\cdots\!29}a^{14}-\frac{31\!\cdots\!70}{11\!\cdots\!29}a^{13}+\frac{11\!\cdots\!88}{11\!\cdots\!29}a^{12}+\frac{13\!\cdots\!44}{11\!\cdots\!29}a^{11}-\frac{15\!\cdots\!55}{11\!\cdots\!29}a^{10}-\frac{51\!\cdots\!94}{11\!\cdots\!29}a^{9}-\frac{54\!\cdots\!34}{11\!\cdots\!29}a^{8}+\frac{27\!\cdots\!25}{11\!\cdots\!29}a^{7}+\frac{20\!\cdots\!01}{11\!\cdots\!29}a^{6}+\frac{27\!\cdots\!30}{11\!\cdots\!29}a^{5}+\frac{15\!\cdots\!88}{11\!\cdots\!29}a^{4}-\frac{10\!\cdots\!86}{11\!\cdots\!29}a^{3}-\frac{62\!\cdots\!64}{11\!\cdots\!29}a^{2}-\frac{70\!\cdots\!01}{11\!\cdots\!29}a+\frac{19\!\cdots\!06}{11\!\cdots\!29}$, $\frac{28\!\cdots\!21}{11\!\cdots\!29}a^{21}-\frac{18\!\cdots\!89}{56\!\cdots\!45}a^{20}-\frac{35\!\cdots\!22}{56\!\cdots\!45}a^{19}+\frac{25\!\cdots\!54}{56\!\cdots\!45}a^{18}-\frac{66\!\cdots\!71}{56\!\cdots\!45}a^{17}-\frac{84\!\cdots\!06}{56\!\cdots\!45}a^{16}+\frac{12\!\cdots\!09}{56\!\cdots\!45}a^{15}+\frac{23\!\cdots\!08}{11\!\cdots\!29}a^{14}-\frac{40\!\cdots\!84}{56\!\cdots\!45}a^{13}-\frac{15\!\cdots\!63}{56\!\cdots\!45}a^{12}+\frac{49\!\cdots\!36}{56\!\cdots\!45}a^{11}+\frac{27\!\cdots\!82}{56\!\cdots\!45}a^{10}-\frac{98\!\cdots\!82}{56\!\cdots\!45}a^{9}-\frac{75\!\cdots\!32}{56\!\cdots\!45}a^{8}+\frac{35\!\cdots\!06}{56\!\cdots\!45}a^{7}+\frac{70\!\cdots\!22}{56\!\cdots\!45}a^{6}+\frac{14\!\cdots\!98}{56\!\cdots\!45}a^{5}-\frac{24\!\cdots\!93}{56\!\cdots\!45}a^{4}-\frac{16\!\cdots\!79}{56\!\cdots\!45}a^{3}+\frac{24\!\cdots\!14}{56\!\cdots\!45}a^{2}+\frac{27\!\cdots\!81}{56\!\cdots\!45}a+\frac{28\!\cdots\!23}{56\!\cdots\!45}$, $\frac{16\!\cdots\!73}{56\!\cdots\!45}a^{21}-\frac{22\!\cdots\!01}{56\!\cdots\!45}a^{20}-\frac{10\!\cdots\!08}{11\!\cdots\!29}a^{19}+\frac{32\!\cdots\!17}{56\!\cdots\!45}a^{18}-\frac{17\!\cdots\!03}{11\!\cdots\!29}a^{17}-\frac{11\!\cdots\!13}{56\!\cdots\!45}a^{16}+\frac{16\!\cdots\!82}{56\!\cdots\!45}a^{15}+\frac{15\!\cdots\!29}{56\!\cdots\!45}a^{14}-\frac{57\!\cdots\!03}{56\!\cdots\!45}a^{13}-\frac{99\!\cdots\!52}{56\!\cdots\!45}a^{12}+\frac{71\!\cdots\!14}{56\!\cdots\!45}a^{11}-\frac{10\!\cdots\!18}{56\!\cdots\!45}a^{10}-\frac{23\!\cdots\!70}{11\!\cdots\!29}a^{9}-\frac{82\!\cdots\!42}{56\!\cdots\!45}a^{8}+\frac{15\!\cdots\!31}{11\!\cdots\!29}a^{7}+\frac{18\!\cdots\!64}{11\!\cdots\!29}a^{6}+\frac{38\!\cdots\!37}{56\!\cdots\!45}a^{5}-\frac{39\!\cdots\!08}{56\!\cdots\!45}a^{4}-\frac{19\!\cdots\!51}{56\!\cdots\!45}a^{3}+\frac{66\!\cdots\!64}{56\!\cdots\!45}a^{2}+\frac{95\!\cdots\!60}{11\!\cdots\!29}a+\frac{62\!\cdots\!26}{56\!\cdots\!45}$, $\frac{38\!\cdots\!46}{11\!\cdots\!29}a^{21}-\frac{48\!\cdots\!22}{56\!\cdots\!45}a^{20}+\frac{16\!\cdots\!52}{56\!\cdots\!45}a^{19}+\frac{35\!\cdots\!03}{56\!\cdots\!45}a^{18}-\frac{51\!\cdots\!68}{56\!\cdots\!45}a^{17}-\frac{66\!\cdots\!38}{56\!\cdots\!45}a^{16}+\frac{25\!\cdots\!24}{56\!\cdots\!45}a^{15}-\frac{24\!\cdots\!91}{11\!\cdots\!29}a^{14}-\frac{48\!\cdots\!01}{56\!\cdots\!45}a^{13}+\frac{37\!\cdots\!33}{56\!\cdots\!45}a^{12}+\frac{76\!\cdots\!85}{11\!\cdots\!29}a^{11}-\frac{44\!\cdots\!42}{56\!\cdots\!45}a^{10}-\frac{95\!\cdots\!71}{56\!\cdots\!45}a^{9}+\frac{12\!\cdots\!89}{56\!\cdots\!45}a^{8}+\frac{74\!\cdots\!67}{56\!\cdots\!45}a^{7}+\frac{44\!\cdots\!11}{11\!\cdots\!29}a^{6}-\frac{27\!\cdots\!12}{56\!\cdots\!45}a^{5}-\frac{21\!\cdots\!67}{56\!\cdots\!45}a^{4}+\frac{54\!\cdots\!29}{56\!\cdots\!45}a^{3}+\frac{11\!\cdots\!39}{11\!\cdots\!29}a^{2}+\frac{19\!\cdots\!74}{11\!\cdots\!29}a-\frac{10\!\cdots\!11}{56\!\cdots\!45}$, $\frac{50\!\cdots\!94}{56\!\cdots\!45}a^{21}-\frac{18\!\cdots\!91}{11\!\cdots\!29}a^{20}-\frac{25\!\cdots\!78}{56\!\cdots\!45}a^{19}+\frac{79\!\cdots\!53}{56\!\cdots\!45}a^{18}-\frac{14\!\cdots\!39}{11\!\cdots\!29}a^{17}-\frac{16\!\cdots\!57}{56\!\cdots\!45}a^{16}+\frac{83\!\cdots\!54}{11\!\cdots\!29}a^{15}-\frac{33\!\cdots\!57}{56\!\cdots\!45}a^{14}-\frac{16\!\cdots\!66}{11\!\cdots\!29}a^{13}-\frac{40\!\cdots\!58}{11\!\cdots\!29}a^{12}+\frac{55\!\cdots\!98}{56\!\cdots\!45}a^{11}+\frac{77\!\cdots\!78}{11\!\cdots\!29}a^{10}-\frac{50\!\cdots\!47}{11\!\cdots\!29}a^{9}-\frac{22\!\cdots\!87}{56\!\cdots\!45}a^{8}+\frac{61\!\cdots\!52}{56\!\cdots\!45}a^{7}+\frac{21\!\cdots\!42}{56\!\cdots\!45}a^{6}+\frac{67\!\cdots\!06}{56\!\cdots\!45}a^{5}-\frac{41\!\cdots\!89}{56\!\cdots\!45}a^{4}-\frac{72\!\cdots\!78}{56\!\cdots\!45}a^{3}-\frac{93\!\cdots\!65}{11\!\cdots\!29}a^{2}+\frac{11\!\cdots\!34}{56\!\cdots\!45}a+\frac{28\!\cdots\!83}{56\!\cdots\!45}$, $\frac{42\!\cdots\!36}{56\!\cdots\!45}a^{21}-\frac{16\!\cdots\!34}{11\!\cdots\!29}a^{20}-\frac{49\!\cdots\!96}{56\!\cdots\!45}a^{19}+\frac{15\!\cdots\!01}{11\!\cdots\!29}a^{18}-\frac{69\!\cdots\!93}{56\!\cdots\!45}a^{17}-\frac{19\!\cdots\!58}{56\!\cdots\!45}a^{16}+\frac{47\!\cdots\!29}{56\!\cdots\!45}a^{15}+\frac{15\!\cdots\!11}{56\!\cdots\!45}a^{14}-\frac{11\!\cdots\!88}{56\!\cdots\!45}a^{13}+\frac{25\!\cdots\!56}{56\!\cdots\!45}a^{12}+\frac{22\!\cdots\!82}{11\!\cdots\!29}a^{11}-\frac{57\!\cdots\!47}{56\!\cdots\!45}a^{10}-\frac{24\!\cdots\!26}{56\!\cdots\!45}a^{9}-\frac{88\!\cdots\!98}{56\!\cdots\!45}a^{8}+\frac{12\!\cdots\!11}{56\!\cdots\!45}a^{7}+\frac{11\!\cdots\!84}{56\!\cdots\!45}a^{6}-\frac{30\!\cdots\!92}{11\!\cdots\!29}a^{5}-\frac{54\!\cdots\!84}{56\!\cdots\!45}a^{4}-\frac{16\!\cdots\!51}{56\!\cdots\!45}a^{3}+\frac{79\!\cdots\!78}{56\!\cdots\!45}a^{2}+\frac{39\!\cdots\!56}{56\!\cdots\!45}a-\frac{68\!\cdots\!99}{56\!\cdots\!45}$, $\frac{37\!\cdots\!31}{56\!\cdots\!45}a^{21}-\frac{82\!\cdots\!78}{56\!\cdots\!45}a^{20}-\frac{24\!\cdots\!06}{56\!\cdots\!45}a^{19}+\frac{69\!\cdots\!41}{56\!\cdots\!45}a^{18}-\frac{15\!\cdots\!66}{11\!\cdots\!29}a^{17}-\frac{31\!\cdots\!65}{11\!\cdots\!29}a^{16}+\frac{46\!\cdots\!08}{56\!\cdots\!45}a^{15}-\frac{10\!\cdots\!34}{56\!\cdots\!45}a^{14}-\frac{10\!\cdots\!13}{56\!\cdots\!45}a^{13}+\frac{51\!\cdots\!51}{56\!\cdots\!45}a^{12}+\frac{87\!\cdots\!43}{56\!\cdots\!45}a^{11}-\frac{74\!\cdots\!87}{56\!\cdots\!45}a^{10}-\frac{38\!\cdots\!37}{11\!\cdots\!29}a^{9}-\frac{31\!\cdots\!77}{56\!\cdots\!45}a^{8}+\frac{26\!\cdots\!22}{11\!\cdots\!29}a^{7}+\frac{14\!\cdots\!18}{11\!\cdots\!29}a^{6}-\frac{32\!\cdots\!12}{56\!\cdots\!45}a^{5}-\frac{47\!\cdots\!49}{56\!\cdots\!45}a^{4}-\frac{45\!\cdots\!71}{56\!\cdots\!45}a^{3}+\frac{20\!\cdots\!68}{11\!\cdots\!29}a^{2}+\frac{30\!\cdots\!19}{56\!\cdots\!45}a-\frac{44\!\cdots\!48}{56\!\cdots\!45}$
| 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: | \( 95631.3345105 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 95631.3345105 \cdot 1}{2\cdot\sqrt{1447402975463751668017578125}}\cr\approx \mathstrut & 0.195386435924 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 - x^20 + 18*x^19 - 17*x^18 - 44*x^17 + 110*x^16 - 3*x^15 - 254*x^14 + 59*x^13 + 240*x^12 - 102*x^11 - 552*x^10 - 230*x^9 + 307*x^8 + 320*x^7 - 3*x^6 - 154*x^5 - 62*x^4 + 18*x^3 + 19*x^2 + 2*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^22 - 2*x^21 - x^20 + 18*x^19 - 17*x^18 - 44*x^17 + 110*x^16 - 3*x^15 - 254*x^14 + 59*x^13 + 240*x^12 - 102*x^11 - 552*x^10 - 230*x^9 + 307*x^8 + 320*x^7 - 3*x^6 - 154*x^5 - 62*x^4 + 18*x^3 + 19*x^2 + 2*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^22 - 2*x^21 - x^20 + 18*x^19 - 17*x^18 - 44*x^17 + 110*x^16 - 3*x^15 - 254*x^14 + 59*x^13 + 240*x^12 - 102*x^11 - 552*x^10 - 230*x^9 + 307*x^8 + 320*x^7 - 3*x^6 - 154*x^5 - 62*x^4 + 18*x^3 + 19*x^2 + 2*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^22 - 2*x^21 - x^20 + 18*x^19 - 17*x^18 - 44*x^17 + 110*x^16 - 3*x^15 - 254*x^14 + 59*x^13 + 240*x^12 - 102*x^11 - 552*x^10 - 230*x^9 + 307*x^8 + 320*x^7 - 3*x^6 - 154*x^5 - 62*x^4 + 18*x^3 + 19*x^2 + 2*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
$C_2\times A_{11}$ (as 22T46):
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 39916800 |
| The 62 conjugacy class representatives for $C_2\times A_{11}$ are not computed |
| Character table for $C_2\times A_{11}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 11.3.136113034225.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $22$ | $22$ | R | R | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.7.0.1}{7} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.14.0.1 | $x^{14} + 5 x^{7} + 6 x^{5} + 2 x^{4} + 3 x^{2} + 6 x + 3$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(83\)
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 83.6.4.1 | $x^{6} + 246 x^{5} + 20178 x^{4} + 552518 x^{3} + 60774 x^{2} + 1674264 x + 45729605$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 83.10.0.1 | $x^{10} + 7 x^{5} + 73 x^{3} + 53 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(127\)
| 127.2.0.1 | $x^{2} + 126 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 127.6.4.1 | $x^{6} + 378 x^{5} + 47637 x^{4} + 2002898 x^{3} + 190917 x^{2} + 6049872 x + 253919890$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 127.14.0.1 | $x^{14} - x + 29$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |