Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 - 4*x^20 + 14*x^18 + 27*x^17 - 40*x^16 - 66*x^15 + 4*x^14 + 154*x^13 + 292*x^12 - 44*x^11 - 237*x^10 + 44*x^9 + 22*x^8 - 93*x^7 - 3*x^6 + 33*x^5 - 4*x^4 - 2*x^3 + 7*x^2 - x - 1)
gp: K = bnfinit(y^22 - 2*y^21 - 4*y^20 + 14*y^18 + 27*y^17 - 40*y^16 - 66*y^15 + 4*y^14 + 154*y^13 + 292*y^12 - 44*y^11 - 237*y^10 + 44*y^9 + 22*y^8 - 93*y^7 - 3*y^6 + 33*y^5 - 4*y^4 - 2*y^3 + 7*y^2 - y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^21 - 4*x^20 + 14*x^18 + 27*x^17 - 40*x^16 - 66*x^15 + 4*x^14 + 154*x^13 + 292*x^12 - 44*x^11 - 237*x^10 + 44*x^9 + 22*x^8 - 93*x^7 - 3*x^6 + 33*x^5 - 4*x^4 - 2*x^3 + 7*x^2 - x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 2*x^21 - 4*x^20 + 14*x^18 + 27*x^17 - 40*x^16 - 66*x^15 + 4*x^14 + 154*x^13 + 292*x^12 - 44*x^11 - 237*x^10 + 44*x^9 + 22*x^8 - 93*x^7 - 3*x^6 + 33*x^5 - 4*x^4 - 2*x^3 + 7*x^2 - x - 1)
\( x^{22} - 2 x^{21} - 4 x^{20} + 14 x^{18} + 27 x^{17} - 40 x^{16} - 66 x^{15} + 4 x^{14} + 154 x^{13} + \cdots - 1 \)
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: | $[10, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(854993079356720164347705078125\)
\(\medspace = 5^{11}\cdot 211441^{2}\cdot 625831^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.94\) | 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: | $5^{1/2}211441^{1/2}625831^{1/2}\approx 813407.4393285323$ | ||
| Ramified primes: |
\(5\), \(211441\), \(625831\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{14\!\cdots\!39}a^{21}-\frac{45\!\cdots\!77}{14\!\cdots\!39}a^{20}-\frac{13\!\cdots\!76}{14\!\cdots\!39}a^{19}-\frac{46\!\cdots\!74}{14\!\cdots\!39}a^{18}+\frac{66\!\cdots\!79}{14\!\cdots\!39}a^{17}+\frac{61\!\cdots\!18}{14\!\cdots\!39}a^{16}+\frac{66\!\cdots\!80}{14\!\cdots\!39}a^{15}+\frac{18\!\cdots\!83}{14\!\cdots\!39}a^{14}+\frac{40\!\cdots\!79}{14\!\cdots\!39}a^{13}+\frac{70\!\cdots\!95}{14\!\cdots\!39}a^{12}+\frac{53\!\cdots\!62}{14\!\cdots\!39}a^{11}+\frac{43\!\cdots\!63}{14\!\cdots\!39}a^{10}-\frac{13\!\cdots\!68}{14\!\cdots\!39}a^{9}-\frac{50\!\cdots\!50}{14\!\cdots\!39}a^{8}-\frac{69\!\cdots\!18}{14\!\cdots\!39}a^{7}-\frac{39\!\cdots\!62}{14\!\cdots\!39}a^{6}+\frac{16\!\cdots\!63}{14\!\cdots\!39}a^{5}-\frac{87\!\cdots\!35}{14\!\cdots\!39}a^{4}+\frac{15\!\cdots\!11}{14\!\cdots\!39}a^{3}+\frac{39\!\cdots\!48}{14\!\cdots\!39}a^{2}+\frac{47\!\cdots\!42}{14\!\cdots\!39}a+\frac{53\!\cdots\!93}{14\!\cdots\!39}$
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: | $15$ | 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: |
$\frac{86\!\cdots\!84}{14\!\cdots\!39}a^{21}-\frac{17\!\cdots\!52}{14\!\cdots\!39}a^{20}-\frac{34\!\cdots\!06}{14\!\cdots\!39}a^{19}+\frac{13\!\cdots\!10}{14\!\cdots\!39}a^{18}+\frac{12\!\cdots\!32}{14\!\cdots\!39}a^{17}+\frac{23\!\cdots\!20}{14\!\cdots\!39}a^{16}-\frac{35\!\cdots\!01}{14\!\cdots\!39}a^{15}-\frac{56\!\cdots\!04}{14\!\cdots\!39}a^{14}+\frac{54\!\cdots\!04}{14\!\cdots\!39}a^{13}+\frac{13\!\cdots\!66}{14\!\cdots\!39}a^{12}+\frac{24\!\cdots\!08}{14\!\cdots\!39}a^{11}-\frac{48\!\cdots\!02}{14\!\cdots\!39}a^{10}-\frac{21\!\cdots\!93}{14\!\cdots\!39}a^{9}+\frac{38\!\cdots\!18}{14\!\cdots\!39}a^{8}+\frac{17\!\cdots\!34}{14\!\cdots\!39}a^{7}-\frac{81\!\cdots\!10}{14\!\cdots\!39}a^{6}-\frac{13\!\cdots\!34}{14\!\cdots\!39}a^{5}+\frac{29\!\cdots\!24}{14\!\cdots\!39}a^{4}-\frac{34\!\cdots\!60}{14\!\cdots\!39}a^{3}-\frac{16\!\cdots\!30}{14\!\cdots\!39}a^{2}+\frac{61\!\cdots\!76}{14\!\cdots\!39}a-\frac{93\!\cdots\!57}{14\!\cdots\!39}$, $\frac{71\!\cdots\!65}{14\!\cdots\!39}a^{21}-\frac{47\!\cdots\!74}{14\!\cdots\!39}a^{20}-\frac{45\!\cdots\!18}{14\!\cdots\!39}a^{19}-\frac{38\!\cdots\!77}{14\!\cdots\!39}a^{18}+\frac{84\!\cdots\!11}{14\!\cdots\!39}a^{17}+\frac{31\!\cdots\!22}{14\!\cdots\!39}a^{16}-\frac{42\!\cdots\!77}{14\!\cdots\!39}a^{15}-\frac{74\!\cdots\!20}{14\!\cdots\!39}a^{14}-\frac{58\!\cdots\!66}{14\!\cdots\!39}a^{13}+\frac{88\!\cdots\!68}{14\!\cdots\!39}a^{12}+\frac{33\!\cdots\!33}{14\!\cdots\!39}a^{11}+\frac{27\!\cdots\!12}{14\!\cdots\!39}a^{10}-\frac{10\!\cdots\!07}{14\!\cdots\!39}a^{9}-\frac{10\!\cdots\!72}{14\!\cdots\!39}a^{8}+\frac{17\!\cdots\!46}{14\!\cdots\!39}a^{7}-\frac{87\!\cdots\!48}{14\!\cdots\!39}a^{6}-\frac{85\!\cdots\!40}{14\!\cdots\!39}a^{5}-\frac{29\!\cdots\!14}{14\!\cdots\!39}a^{4}+\frac{34\!\cdots\!54}{14\!\cdots\!39}a^{3}-\frac{53\!\cdots\!57}{14\!\cdots\!39}a^{2}+\frac{70\!\cdots\!51}{14\!\cdots\!39}a+\frac{40\!\cdots\!22}{14\!\cdots\!39}$, $\frac{30\!\cdots\!84}{14\!\cdots\!39}a^{21}-\frac{43\!\cdots\!30}{14\!\cdots\!39}a^{20}-\frac{13\!\cdots\!10}{14\!\cdots\!39}a^{19}-\frac{11\!\cdots\!56}{14\!\cdots\!39}a^{18}+\frac{33\!\cdots\!48}{14\!\cdots\!39}a^{17}+\frac{10\!\cdots\!41}{14\!\cdots\!39}a^{16}-\frac{44\!\cdots\!40}{14\!\cdots\!39}a^{15}-\frac{20\!\cdots\!68}{14\!\cdots\!39}a^{14}-\frac{15\!\cdots\!30}{14\!\cdots\!39}a^{13}+\frac{33\!\cdots\!20}{14\!\cdots\!39}a^{12}+\frac{10\!\cdots\!06}{14\!\cdots\!39}a^{11}+\frac{69\!\cdots\!85}{14\!\cdots\!39}a^{10}-\frac{16\!\cdots\!22}{14\!\cdots\!39}a^{9}+\frac{14\!\cdots\!14}{14\!\cdots\!39}a^{8}+\frac{14\!\cdots\!98}{14\!\cdots\!39}a^{7}-\frac{12\!\cdots\!18}{14\!\cdots\!39}a^{6}-\frac{95\!\cdots\!52}{14\!\cdots\!39}a^{5}+\frac{12\!\cdots\!24}{14\!\cdots\!39}a^{4}-\frac{12\!\cdots\!38}{14\!\cdots\!39}a^{3}-\frac{12\!\cdots\!88}{14\!\cdots\!39}a^{2}+\frac{68\!\cdots\!73}{14\!\cdots\!39}a-\frac{86\!\cdots\!84}{14\!\cdots\!39}$, $\frac{72\!\cdots\!03}{14\!\cdots\!39}a^{21}-\frac{48\!\cdots\!48}{14\!\cdots\!39}a^{20}-\frac{46\!\cdots\!74}{14\!\cdots\!39}a^{19}-\frac{39\!\cdots\!05}{14\!\cdots\!39}a^{18}+\frac{86\!\cdots\!84}{14\!\cdots\!39}a^{17}+\frac{31\!\cdots\!42}{14\!\cdots\!39}a^{16}-\frac{40\!\cdots\!01}{14\!\cdots\!39}a^{15}-\frac{76\!\cdots\!86}{14\!\cdots\!39}a^{14}-\frac{59\!\cdots\!82}{14\!\cdots\!39}a^{13}+\frac{90\!\cdots\!46}{14\!\cdots\!39}a^{12}+\frac{34\!\cdots\!14}{14\!\cdots\!39}a^{11}+\frac{27\!\cdots\!98}{14\!\cdots\!39}a^{10}-\frac{10\!\cdots\!89}{14\!\cdots\!39}a^{9}-\frac{10\!\cdots\!22}{14\!\cdots\!39}a^{8}+\frac{19\!\cdots\!40}{14\!\cdots\!39}a^{7}-\frac{88\!\cdots\!48}{14\!\cdots\!39}a^{6}-\frac{86\!\cdots\!88}{14\!\cdots\!39}a^{5}-\frac{29\!\cdots\!16}{14\!\cdots\!39}a^{4}+\frac{33\!\cdots\!34}{14\!\cdots\!39}a^{3}-\frac{54\!\cdots\!72}{14\!\cdots\!39}a^{2}+\frac{47\!\cdots\!12}{14\!\cdots\!39}a+\frac{40\!\cdots\!06}{14\!\cdots\!39}$, $\frac{72\!\cdots\!96}{14\!\cdots\!39}a^{21}-\frac{56\!\cdots\!14}{14\!\cdots\!39}a^{20}-\frac{44\!\cdots\!74}{14\!\cdots\!39}a^{19}-\frac{37\!\cdots\!24}{14\!\cdots\!39}a^{18}+\frac{87\!\cdots\!54}{14\!\cdots\!39}a^{17}+\frac{30\!\cdots\!00}{14\!\cdots\!39}a^{16}-\frac{21\!\cdots\!00}{14\!\cdots\!39}a^{15}-\frac{73\!\cdots\!52}{14\!\cdots\!39}a^{14}-\frac{56\!\cdots\!14}{14\!\cdots\!39}a^{13}+\frac{91\!\cdots\!49}{14\!\cdots\!39}a^{12}+\frac{32\!\cdots\!30}{14\!\cdots\!39}a^{11}+\frac{25\!\cdots\!28}{14\!\cdots\!39}a^{10}-\frac{10\!\cdots\!87}{14\!\cdots\!39}a^{9}-\frac{10\!\cdots\!50}{14\!\cdots\!39}a^{8}+\frac{14\!\cdots\!80}{14\!\cdots\!39}a^{7}-\frac{86\!\cdots\!59}{14\!\cdots\!39}a^{6}-\frac{67\!\cdots\!78}{14\!\cdots\!39}a^{5}-\frac{24\!\cdots\!50}{14\!\cdots\!39}a^{4}-\frac{28\!\cdots\!68}{14\!\cdots\!39}a^{3}-\frac{12\!\cdots\!22}{14\!\cdots\!39}a^{2}+\frac{46\!\cdots\!70}{14\!\cdots\!39}a+\frac{25\!\cdots\!33}{14\!\cdots\!39}$, $\frac{46\!\cdots\!42}{14\!\cdots\!39}a^{21}-\frac{27\!\cdots\!90}{14\!\cdots\!39}a^{20}-\frac{30\!\cdots\!42}{14\!\cdots\!39}a^{19}-\frac{26\!\cdots\!50}{14\!\cdots\!39}a^{18}+\frac{60\!\cdots\!08}{14\!\cdots\!39}a^{17}+\frac{21\!\cdots\!00}{14\!\cdots\!39}a^{16}-\frac{20\!\cdots\!78}{14\!\cdots\!39}a^{15}-\frac{53\!\cdots\!48}{14\!\cdots\!39}a^{14}-\frac{41\!\cdots\!64}{14\!\cdots\!39}a^{13}+\frac{67\!\cdots\!22}{14\!\cdots\!39}a^{12}+\frac{22\!\cdots\!08}{14\!\cdots\!39}a^{11}+\frac{17\!\cdots\!04}{14\!\cdots\!39}a^{10}-\frac{10\!\cdots\!23}{14\!\cdots\!39}a^{9}-\frac{11\!\cdots\!24}{14\!\cdots\!39}a^{8}+\frac{38\!\cdots\!86}{14\!\cdots\!39}a^{7}-\frac{24\!\cdots\!62}{14\!\cdots\!39}a^{6}-\frac{56\!\cdots\!34}{14\!\cdots\!39}a^{5}-\frac{26\!\cdots\!70}{14\!\cdots\!39}a^{4}+\frac{41\!\cdots\!73}{14\!\cdots\!39}a^{3}-\frac{34\!\cdots\!10}{14\!\cdots\!39}a^{2}+\frac{15\!\cdots\!20}{14\!\cdots\!39}a+\frac{22\!\cdots\!91}{14\!\cdots\!39}$, $\frac{14\!\cdots\!18}{14\!\cdots\!39}a^{21}-\frac{20\!\cdots\!75}{14\!\cdots\!39}a^{20}-\frac{75\!\cdots\!08}{14\!\cdots\!39}a^{19}-\frac{34\!\cdots\!99}{14\!\cdots\!39}a^{18}+\frac{19\!\cdots\!34}{14\!\cdots\!39}a^{17}+\frac{50\!\cdots\!88}{14\!\cdots\!39}a^{16}-\frac{33\!\cdots\!93}{14\!\cdots\!39}a^{15}-\frac{12\!\cdots\!33}{14\!\cdots\!39}a^{14}-\frac{48\!\cdots\!20}{14\!\cdots\!39}a^{13}+\frac{20\!\cdots\!04}{14\!\cdots\!39}a^{12}+\frac{54\!\cdots\!76}{14\!\cdots\!39}a^{11}+\frac{20\!\cdots\!04}{14\!\cdots\!39}a^{10}-\frac{31\!\cdots\!40}{14\!\cdots\!39}a^{9}-\frac{69\!\cdots\!70}{14\!\cdots\!39}a^{8}+\frac{32\!\cdots\!78}{14\!\cdots\!39}a^{7}-\frac{13\!\cdots\!06}{14\!\cdots\!39}a^{6}-\frac{68\!\cdots\!53}{14\!\cdots\!39}a^{5}+\frac{19\!\cdots\!55}{14\!\cdots\!39}a^{4}+\frac{47\!\cdots\!46}{14\!\cdots\!39}a^{3}-\frac{67\!\cdots\!76}{14\!\cdots\!39}a^{2}+\frac{10\!\cdots\!46}{14\!\cdots\!39}a+\frac{16\!\cdots\!02}{14\!\cdots\!39}$, $\frac{10\!\cdots\!78}{14\!\cdots\!39}a^{21}-\frac{14\!\cdots\!91}{14\!\cdots\!39}a^{20}-\frac{49\!\cdots\!76}{14\!\cdots\!39}a^{19}-\frac{41\!\cdots\!70}{14\!\cdots\!39}a^{18}+\frac{11\!\cdots\!23}{14\!\cdots\!39}a^{17}+\frac{36\!\cdots\!56}{14\!\cdots\!39}a^{16}-\frac{13\!\cdots\!42}{14\!\cdots\!39}a^{15}-\frac{72\!\cdots\!99}{14\!\cdots\!39}a^{14}-\frac{58\!\cdots\!40}{14\!\cdots\!39}a^{13}+\frac{11\!\cdots\!62}{14\!\cdots\!39}a^{12}+\frac{38\!\cdots\!61}{14\!\cdots\!39}a^{11}+\frac{26\!\cdots\!00}{14\!\cdots\!39}a^{10}-\frac{40\!\cdots\!72}{14\!\cdots\!39}a^{9}+\frac{49\!\cdots\!86}{14\!\cdots\!39}a^{8}+\frac{55\!\cdots\!48}{14\!\cdots\!39}a^{7}-\frac{43\!\cdots\!02}{14\!\cdots\!39}a^{6}-\frac{36\!\cdots\!09}{14\!\cdots\!39}a^{5}-\frac{98\!\cdots\!45}{14\!\cdots\!39}a^{4}-\frac{65\!\cdots\!37}{14\!\cdots\!39}a^{3}-\frac{47\!\cdots\!79}{14\!\cdots\!39}a^{2}-\frac{20\!\cdots\!53}{14\!\cdots\!39}a+\frac{10\!\cdots\!88}{14\!\cdots\!39}$, $\frac{13\!\cdots\!18}{14\!\cdots\!39}a^{21}+\frac{11\!\cdots\!86}{14\!\cdots\!39}a^{20}-\frac{13\!\cdots\!36}{14\!\cdots\!39}a^{19}-\frac{10\!\cdots\!71}{14\!\cdots\!39}a^{18}+\frac{95\!\cdots\!30}{14\!\cdots\!39}a^{17}+\frac{81\!\cdots\!99}{14\!\cdots\!39}a^{16}+\frac{39\!\cdots\!10}{14\!\cdots\!39}a^{15}-\frac{19\!\cdots\!64}{14\!\cdots\!39}a^{14}-\frac{15\!\cdots\!90}{14\!\cdots\!39}a^{13}+\frac{84\!\cdots\!13}{14\!\cdots\!39}a^{12}+\frac{85\!\cdots\!76}{14\!\cdots\!39}a^{11}+\frac{94\!\cdots\!43}{14\!\cdots\!39}a^{10}+\frac{21\!\cdots\!40}{14\!\cdots\!39}a^{9}+\frac{37\!\cdots\!24}{14\!\cdots\!39}a^{8}+\frac{22\!\cdots\!40}{14\!\cdots\!39}a^{7}-\frac{80\!\cdots\!16}{14\!\cdots\!39}a^{6}-\frac{13\!\cdots\!72}{14\!\cdots\!39}a^{5}-\frac{14\!\cdots\!54}{14\!\cdots\!39}a^{4}-\frac{61\!\cdots\!42}{14\!\cdots\!39}a^{3}-\frac{39\!\cdots\!40}{14\!\cdots\!39}a^{2}+\frac{68\!\cdots\!29}{14\!\cdots\!39}a+\frac{93\!\cdots\!26}{14\!\cdots\!39}$, $\frac{59\!\cdots\!42}{14\!\cdots\!39}a^{21}-\frac{12\!\cdots\!96}{14\!\cdots\!39}a^{20}-\frac{22\!\cdots\!77}{14\!\cdots\!39}a^{19}-\frac{30\!\cdots\!89}{14\!\cdots\!39}a^{18}+\frac{83\!\cdots\!55}{14\!\cdots\!39}a^{17}+\frac{15\!\cdots\!69}{14\!\cdots\!39}a^{16}-\frac{23\!\cdots\!66}{14\!\cdots\!39}a^{15}-\frac{37\!\cdots\!94}{14\!\cdots\!39}a^{14}+\frac{17\!\cdots\!77}{14\!\cdots\!39}a^{13}+\frac{90\!\cdots\!49}{14\!\cdots\!39}a^{12}+\frac{16\!\cdots\!77}{14\!\cdots\!39}a^{11}-\frac{25\!\cdots\!54}{14\!\cdots\!39}a^{10}-\frac{13\!\cdots\!36}{14\!\cdots\!39}a^{9}+\frac{26\!\cdots\!98}{14\!\cdots\!39}a^{8}+\frac{11\!\cdots\!42}{14\!\cdots\!39}a^{7}-\frac{52\!\cdots\!78}{14\!\cdots\!39}a^{6}+\frac{14\!\cdots\!95}{14\!\cdots\!39}a^{5}+\frac{12\!\cdots\!91}{14\!\cdots\!39}a^{4}-\frac{12\!\cdots\!64}{14\!\cdots\!39}a^{3}+\frac{89\!\cdots\!24}{14\!\cdots\!39}a^{2}+\frac{11\!\cdots\!77}{14\!\cdots\!39}a-\frac{20\!\cdots\!45}{14\!\cdots\!39}$, $\frac{19\!\cdots\!48}{14\!\cdots\!39}a^{21}-\frac{29\!\cdots\!82}{14\!\cdots\!39}a^{20}-\frac{94\!\cdots\!91}{14\!\cdots\!39}a^{19}-\frac{51\!\cdots\!14}{14\!\cdots\!39}a^{18}+\frac{25\!\cdots\!74}{14\!\cdots\!39}a^{17}+\frac{67\!\cdots\!56}{14\!\cdots\!39}a^{16}-\frac{43\!\cdots\!17}{14\!\cdots\!39}a^{15}-\frac{15\!\cdots\!60}{14\!\cdots\!39}a^{14}-\frac{77\!\cdots\!66}{14\!\cdots\!39}a^{13}+\frac{26\!\cdots\!36}{14\!\cdots\!39}a^{12}+\frac{72\!\cdots\!34}{14\!\cdots\!39}a^{11}+\frac{29\!\cdots\!44}{14\!\cdots\!39}a^{10}-\frac{31\!\cdots\!36}{14\!\cdots\!39}a^{9}-\frac{10\!\cdots\!49}{14\!\cdots\!39}a^{8}+\frac{76\!\cdots\!35}{14\!\cdots\!39}a^{7}-\frac{15\!\cdots\!31}{14\!\cdots\!39}a^{6}-\frac{11\!\cdots\!44}{14\!\cdots\!39}a^{5}+\frac{98\!\cdots\!07}{14\!\cdots\!39}a^{4}+\frac{39\!\cdots\!56}{14\!\cdots\!39}a^{3}-\frac{42\!\cdots\!43}{14\!\cdots\!39}a^{2}+\frac{79\!\cdots\!39}{14\!\cdots\!39}a+\frac{48\!\cdots\!28}{14\!\cdots\!39}$, $\frac{93\!\cdots\!13}{14\!\cdots\!39}a^{21}-\frac{16\!\cdots\!91}{14\!\cdots\!39}a^{20}-\frac{43\!\cdots\!08}{14\!\cdots\!39}a^{19}-\frac{58\!\cdots\!46}{14\!\cdots\!39}a^{18}+\frac{12\!\cdots\!30}{14\!\cdots\!39}a^{17}+\frac{28\!\cdots\!42}{14\!\cdots\!39}a^{16}-\frac{31\!\cdots\!56}{14\!\cdots\!39}a^{15}-\frac{69\!\cdots\!99}{14\!\cdots\!39}a^{14}-\frac{56\!\cdots\!46}{14\!\cdots\!39}a^{13}+\frac{13\!\cdots\!53}{14\!\cdots\!39}a^{12}+\frac{30\!\cdots\!61}{14\!\cdots\!39}a^{11}+\frac{26\!\cdots\!92}{14\!\cdots\!39}a^{10}-\frac{21\!\cdots\!45}{14\!\cdots\!39}a^{9}+\frac{38\!\cdots\!74}{14\!\cdots\!39}a^{8}-\frac{56\!\cdots\!78}{14\!\cdots\!39}a^{7}-\frac{96\!\cdots\!16}{14\!\cdots\!39}a^{6}-\frac{19\!\cdots\!99}{14\!\cdots\!39}a^{5}+\frac{13\!\cdots\!70}{14\!\cdots\!39}a^{4}-\frac{34\!\cdots\!50}{14\!\cdots\!39}a^{3}-\frac{98\!\cdots\!75}{14\!\cdots\!39}a^{2}+\frac{66\!\cdots\!97}{14\!\cdots\!39}a-\frac{15\!\cdots\!72}{14\!\cdots\!39}$, $\frac{12\!\cdots\!61}{14\!\cdots\!39}a^{21}-\frac{24\!\cdots\!04}{14\!\cdots\!39}a^{20}-\frac{46\!\cdots\!07}{14\!\cdots\!39}a^{19}-\frac{11\!\cdots\!53}{14\!\cdots\!39}a^{18}+\frac{16\!\cdots\!42}{14\!\cdots\!39}a^{17}+\frac{34\!\cdots\!73}{14\!\cdots\!39}a^{16}-\frac{43\!\cdots\!76}{14\!\cdots\!39}a^{15}-\frac{75\!\cdots\!06}{14\!\cdots\!39}a^{14}-\frac{16\!\cdots\!54}{14\!\cdots\!39}a^{13}+\frac{17\!\cdots\!70}{14\!\cdots\!39}a^{12}+\frac{37\!\cdots\!14}{14\!\cdots\!39}a^{11}+\frac{97\!\cdots\!61}{14\!\cdots\!39}a^{10}-\frac{21\!\cdots\!25}{14\!\cdots\!39}a^{9}-\frac{87\!\cdots\!41}{14\!\cdots\!39}a^{8}-\frac{13\!\cdots\!03}{14\!\cdots\!39}a^{7}-\frac{70\!\cdots\!65}{14\!\cdots\!39}a^{6}-\frac{25\!\cdots\!04}{14\!\cdots\!39}a^{5}+\frac{10\!\cdots\!99}{14\!\cdots\!39}a^{4}+\frac{76\!\cdots\!60}{14\!\cdots\!39}a^{3}+\frac{13\!\cdots\!17}{14\!\cdots\!39}a^{2}+\frac{45\!\cdots\!93}{14\!\cdots\!39}a+\frac{16\!\cdots\!06}{14\!\cdots\!39}$, $\frac{33\!\cdots\!38}{14\!\cdots\!39}a^{21}-\frac{52\!\cdots\!90}{14\!\cdots\!39}a^{20}-\frac{15\!\cdots\!39}{14\!\cdots\!39}a^{19}-\frac{73\!\cdots\!12}{14\!\cdots\!39}a^{18}+\frac{42\!\cdots\!32}{14\!\cdots\!39}a^{17}+\frac{10\!\cdots\!68}{14\!\cdots\!39}a^{16}-\frac{82\!\cdots\!28}{14\!\cdots\!39}a^{15}-\frac{25\!\cdots\!20}{14\!\cdots\!39}a^{14}-\frac{10\!\cdots\!78}{14\!\cdots\!39}a^{13}+\frac{45\!\cdots\!79}{14\!\cdots\!39}a^{12}+\frac{11\!\cdots\!40}{14\!\cdots\!39}a^{11}+\frac{41\!\cdots\!78}{14\!\cdots\!39}a^{10}-\frac{52\!\cdots\!21}{14\!\cdots\!39}a^{9}-\frac{47\!\cdots\!25}{14\!\cdots\!39}a^{8}+\frac{19\!\cdots\!83}{14\!\cdots\!39}a^{7}-\frac{31\!\cdots\!06}{14\!\cdots\!39}a^{6}-\frac{16\!\cdots\!73}{14\!\cdots\!39}a^{5}+\frac{13\!\cdots\!74}{14\!\cdots\!39}a^{4}-\frac{13\!\cdots\!00}{14\!\cdots\!39}a^{3}-\frac{11\!\cdots\!62}{14\!\cdots\!39}a^{2}+\frac{19\!\cdots\!84}{14\!\cdots\!39}a+\frac{46\!\cdots\!33}{14\!\cdots\!39}$, $\frac{37\!\cdots\!01}{14\!\cdots\!39}a^{21}-\frac{55\!\cdots\!78}{14\!\cdots\!39}a^{20}-\frac{17\!\cdots\!63}{14\!\cdots\!39}a^{19}-\frac{92\!\cdots\!33}{14\!\cdots\!39}a^{18}+\frac{47\!\cdots\!78}{14\!\cdots\!39}a^{17}+\frac{12\!\cdots\!59}{14\!\cdots\!39}a^{16}-\frac{84\!\cdots\!11}{14\!\cdots\!39}a^{15}-\frac{28\!\cdots\!87}{14\!\cdots\!39}a^{14}-\frac{13\!\cdots\!55}{14\!\cdots\!39}a^{13}+\frac{50\!\cdots\!22}{14\!\cdots\!39}a^{12}+\frac{13\!\cdots\!72}{14\!\cdots\!39}a^{11}+\frac{53\!\cdots\!67}{14\!\cdots\!39}a^{10}-\frac{59\!\cdots\!72}{14\!\cdots\!39}a^{9}-\frac{15\!\cdots\!30}{14\!\cdots\!39}a^{8}-\frac{51\!\cdots\!02}{14\!\cdots\!39}a^{7}-\frac{33\!\cdots\!97}{14\!\cdots\!39}a^{6}-\frac{19\!\cdots\!50}{14\!\cdots\!39}a^{5}+\frac{17\!\cdots\!34}{14\!\cdots\!39}a^{4}-\frac{15\!\cdots\!85}{14\!\cdots\!39}a^{3}-\frac{10\!\cdots\!40}{14\!\cdots\!39}a^{2}+\frac{20\!\cdots\!79}{14\!\cdots\!39}a+\frac{74\!\cdots\!36}{14\!\cdots\!39}$
| 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: | \( 5205736.25321 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 5205736.25321 \cdot 1}{2\cdot\sqrt{854993079356720164347705078125}}\cr\approx \mathstrut & 0.177357693560 \end{aligned}\] (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 - 4*x^20 + 14*x^18 + 27*x^17 - 40*x^16 - 66*x^15 + 4*x^14 + 154*x^13 + 292*x^12 - 44*x^11 - 237*x^10 + 44*x^9 + 22*x^8 - 93*x^7 - 3*x^6 + 33*x^5 - 4*x^4 - 2*x^3 + 7*x^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 - 4*x^20 + 14*x^18 + 27*x^17 - 40*x^16 - 66*x^15 + 4*x^14 + 154*x^13 + 292*x^12 - 44*x^11 - 237*x^10 + 44*x^9 + 22*x^8 - 93*x^7 - 3*x^6 + 33*x^5 - 4*x^4 - 2*x^3 + 7*x^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 - 4*x^20 + 14*x^18 + 27*x^17 - 40*x^16 - 66*x^15 + 4*x^14 + 154*x^13 + 292*x^12 - 44*x^11 - 237*x^10 + 44*x^9 + 22*x^8 - 93*x^7 - 3*x^6 + 33*x^5 - 4*x^4 - 2*x^3 + 7*x^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 - 4*x^20 + 14*x^18 + 27*x^17 - 40*x^16 - 66*x^15 + 4*x^14 + 154*x^13 + 292*x^12 - 44*x^11 - 237*x^10 + 44*x^9 + 22*x^8 - 93*x^7 - 3*x^6 + 33*x^5 - 4*x^4 - 2*x^3 + 7*x^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
$C_2\times S_{11}$ (as 22T47):
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 79833600 |
| The 112 conjugacy class representatives for $C_2\times S_{11}$ |
| Character table for $C_2\times S_{11}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 11.5.132326332471.1 |
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 22 sibling: | data not computed |
| Degree 44 sibling: | data not computed |
| 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 | $22$ | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.6.0.1}{6} }$ | $22$ | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | $18{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.22.11.1 | $x^{22} + 220 x^{21} + 22055 x^{20} + 1331000 x^{19} + 53791375 x^{18} + 1531447500 x^{17} + 31435820625 x^{16} + 467679300000 x^{15} + 4991151206250 x^{14} + 37171668875000 x^{13} + 183624733943756 x^{12} + 553513923250726 x^{11} + 918123669784090 x^{10} + 929291725767350 x^{9} + 623894056087500 x^{8} + 292303912609500 x^{7} + 98324330218125 x^{6} + 25190924781000 x^{5} + 17099014728125 x^{4} + 90189081743750 x^{3} + 391939091809384 x^{2} + 906877245981448 x + 669277565422109$ | $2$ | $11$ | $11$ | 22T1 | $[\ ]_{2}^{11}$ |
|
\(211441\)
| $\Q_{211441}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{211441}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(625831\)
| $\Q_{625831}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{625831}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |