Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 - 7*x^20 + 14*x^19 + 18*x^18 - 67*x^17 - 101*x^16 + 234*x^15 + 497*x^14 - 273*x^13 - 893*x^12 + 23*x^11 + 753*x^10 + 147*x^9 - 418*x^8 - 88*x^7 + 226*x^6 + 6*x^5 - 90*x^4 + 7*x^3 + 16*x^2 - x - 1)
gp: K = bnfinit(y^22 - 2*y^21 - 7*y^20 + 14*y^19 + 18*y^18 - 67*y^17 - 101*y^16 + 234*y^15 + 497*y^14 - 273*y^13 - 893*y^12 + 23*y^11 + 753*y^10 + 147*y^9 - 418*y^8 - 88*y^7 + 226*y^6 + 6*y^5 - 90*y^4 + 7*y^3 + 16*y^2 - y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^21 - 7*x^20 + 14*x^19 + 18*x^18 - 67*x^17 - 101*x^16 + 234*x^15 + 497*x^14 - 273*x^13 - 893*x^12 + 23*x^11 + 753*x^10 + 147*x^9 - 418*x^8 - 88*x^7 + 226*x^6 + 6*x^5 - 90*x^4 + 7*x^3 + 16*x^2 - x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 2*x^21 - 7*x^20 + 14*x^19 + 18*x^18 - 67*x^17 - 101*x^16 + 234*x^15 + 497*x^14 - 273*x^13 - 893*x^12 + 23*x^11 + 753*x^10 + 147*x^9 - 418*x^8 - 88*x^7 + 226*x^6 + 6*x^5 - 90*x^4 + 7*x^3 + 16*x^2 - x - 1)
\( x^{22} - 2 x^{21} - 7 x^{20} + 14 x^{19} + 18 x^{18} - 67 x^{17} - 101 x^{16} + 234 x^{15} + 497 x^{14} + \cdots - 1 \)
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: | $[14, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(31919563459480622441144580078125\)
\(\medspace = 5^{11}\cdot 43^{2}\cdot 547^{2}\cdot 34374601^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.04\) | 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: | $5^{1/2}43^{1/2}547^{1/2}34374601^{1/2}\approx 2010627.9990602438$ | ||
| Ramified primes: |
\(5\), \(43\), \(547\), \(34374601\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{13\!\cdots\!21}a^{21}+\frac{44\!\cdots\!35}{13\!\cdots\!21}a^{20}+\frac{60\!\cdots\!87}{13\!\cdots\!21}a^{19}-\frac{46\!\cdots\!86}{13\!\cdots\!21}a^{18}+\frac{19\!\cdots\!09}{13\!\cdots\!21}a^{17}-\frac{24\!\cdots\!86}{13\!\cdots\!21}a^{16}+\frac{37\!\cdots\!58}{13\!\cdots\!21}a^{15}-\frac{62\!\cdots\!98}{13\!\cdots\!21}a^{14}-\frac{51\!\cdots\!72}{13\!\cdots\!21}a^{13}-\frac{35\!\cdots\!82}{13\!\cdots\!21}a^{12}-\frac{25\!\cdots\!57}{13\!\cdots\!21}a^{11}+\frac{62\!\cdots\!28}{13\!\cdots\!21}a^{10}+\frac{53\!\cdots\!13}{13\!\cdots\!21}a^{9}-\frac{54\!\cdots\!27}{13\!\cdots\!21}a^{8}+\frac{12\!\cdots\!85}{13\!\cdots\!21}a^{7}-\frac{19\!\cdots\!19}{13\!\cdots\!21}a^{6}+\frac{35\!\cdots\!02}{13\!\cdots\!21}a^{5}-\frac{42\!\cdots\!40}{13\!\cdots\!21}a^{4}+\frac{13\!\cdots\!62}{13\!\cdots\!21}a^{3}-\frac{49\!\cdots\!13}{13\!\cdots\!21}a^{2}-\frac{38\!\cdots\!93}{13\!\cdots\!21}a-\frac{38\!\cdots\!02}{13\!\cdots\!21}$
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: | $17$ | 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: |
$\frac{66\!\cdots\!55}{13\!\cdots\!21}a^{21}-\frac{27\!\cdots\!63}{13\!\cdots\!21}a^{20}-\frac{71\!\cdots\!58}{13\!\cdots\!21}a^{19}+\frac{38\!\cdots\!64}{13\!\cdots\!21}a^{18}+\frac{26\!\cdots\!17}{13\!\cdots\!21}a^{17}-\frac{36\!\cdots\!15}{13\!\cdots\!21}a^{16}-\frac{12\!\cdots\!25}{13\!\cdots\!21}a^{15}+\frac{82\!\cdots\!74}{13\!\cdots\!21}a^{14}+\frac{55\!\cdots\!98}{13\!\cdots\!21}a^{13}+\frac{17\!\cdots\!36}{13\!\cdots\!21}a^{12}-\frac{84\!\cdots\!03}{13\!\cdots\!21}a^{11}-\frac{47\!\cdots\!19}{13\!\cdots\!21}a^{10}+\frac{55\!\cdots\!62}{13\!\cdots\!21}a^{9}+\frac{40\!\cdots\!15}{13\!\cdots\!21}a^{8}-\frac{26\!\cdots\!50}{13\!\cdots\!21}a^{7}-\frac{23\!\cdots\!13}{13\!\cdots\!21}a^{6}+\frac{17\!\cdots\!98}{13\!\cdots\!21}a^{5}+\frac{83\!\cdots\!74}{13\!\cdots\!21}a^{4}-\frac{81\!\cdots\!16}{13\!\cdots\!21}a^{3}-\frac{10\!\cdots\!75}{13\!\cdots\!21}a^{2}+\frac{14\!\cdots\!27}{13\!\cdots\!21}a+\frac{25\!\cdots\!59}{13\!\cdots\!21}$, $\frac{10\!\cdots\!47}{13\!\cdots\!21}a^{21}-\frac{25\!\cdots\!73}{13\!\cdots\!21}a^{20}-\frac{53\!\cdots\!06}{13\!\cdots\!21}a^{19}+\frac{14\!\cdots\!27}{13\!\cdots\!21}a^{18}+\frac{76\!\cdots\!70}{13\!\cdots\!21}a^{17}-\frac{61\!\cdots\!70}{13\!\cdots\!21}a^{16}-\frac{71\!\cdots\!96}{13\!\cdots\!21}a^{15}+\frac{22\!\cdots\!63}{13\!\cdots\!21}a^{14}+\frac{35\!\cdots\!51}{13\!\cdots\!21}a^{13}-\frac{25\!\cdots\!88}{13\!\cdots\!21}a^{12}-\frac{49\!\cdots\!84}{13\!\cdots\!21}a^{11}+\frac{58\!\cdots\!47}{13\!\cdots\!21}a^{10}+\frac{30\!\cdots\!30}{13\!\cdots\!21}a^{9}+\frac{11\!\cdots\!40}{13\!\cdots\!21}a^{8}-\frac{17\!\cdots\!73}{13\!\cdots\!21}a^{7}+\frac{28\!\cdots\!68}{13\!\cdots\!21}a^{6}+\frac{79\!\cdots\!44}{13\!\cdots\!21}a^{5}-\frac{22\!\cdots\!66}{13\!\cdots\!21}a^{4}-\frac{14\!\cdots\!60}{13\!\cdots\!21}a^{3}+\frac{37\!\cdots\!47}{13\!\cdots\!21}a^{2}+\frac{91\!\cdots\!14}{13\!\cdots\!21}a+\frac{66\!\cdots\!55}{13\!\cdots\!21}$, $\frac{11\!\cdots\!34}{13\!\cdots\!21}a^{21}-\frac{22\!\cdots\!86}{13\!\cdots\!21}a^{20}-\frac{73\!\cdots\!27}{13\!\cdots\!21}a^{19}+\frac{15\!\cdots\!40}{13\!\cdots\!21}a^{18}+\frac{17\!\cdots\!38}{13\!\cdots\!21}a^{17}-\frac{70\!\cdots\!66}{13\!\cdots\!21}a^{16}-\frac{10\!\cdots\!90}{13\!\cdots\!21}a^{15}+\frac{24\!\cdots\!82}{13\!\cdots\!21}a^{14}+\frac{51\!\cdots\!55}{13\!\cdots\!21}a^{13}-\frac{26\!\cdots\!51}{13\!\cdots\!21}a^{12}-\frac{87\!\cdots\!69}{13\!\cdots\!21}a^{11}+\frac{27\!\cdots\!42}{13\!\cdots\!21}a^{10}+\frac{69\!\cdots\!31}{13\!\cdots\!21}a^{9}+\frac{14\!\cdots\!42}{13\!\cdots\!21}a^{8}-\frac{38\!\cdots\!54}{13\!\cdots\!21}a^{7}-\frac{74\!\cdots\!35}{13\!\cdots\!21}a^{6}+\frac{20\!\cdots\!46}{13\!\cdots\!21}a^{5}+\frac{36\!\cdots\!04}{13\!\cdots\!21}a^{4}-\frac{78\!\cdots\!34}{13\!\cdots\!21}a^{3}+\frac{57\!\cdots\!63}{13\!\cdots\!21}a^{2}+\frac{11\!\cdots\!04}{13\!\cdots\!21}a-\frac{79\!\cdots\!88}{13\!\cdots\!21}$, $\frac{28\!\cdots\!31}{13\!\cdots\!21}a^{21}-\frac{42\!\cdots\!60}{13\!\cdots\!21}a^{20}-\frac{21\!\cdots\!90}{13\!\cdots\!21}a^{19}+\frac{26\!\cdots\!94}{13\!\cdots\!21}a^{18}+\frac{64\!\cdots\!12}{13\!\cdots\!21}a^{17}-\frac{14\!\cdots\!90}{13\!\cdots\!21}a^{16}-\frac{37\!\cdots\!57}{13\!\cdots\!21}a^{15}+\frac{44\!\cdots\!90}{13\!\cdots\!21}a^{14}+\frac{16\!\cdots\!30}{13\!\cdots\!21}a^{13}+\frac{23\!\cdots\!82}{13\!\cdots\!21}a^{12}-\frac{24\!\cdots\!82}{13\!\cdots\!21}a^{11}-\frac{16\!\cdots\!96}{13\!\cdots\!21}a^{10}+\frac{14\!\cdots\!99}{13\!\cdots\!21}a^{9}+\frac{16\!\cdots\!10}{13\!\cdots\!21}a^{8}-\frac{36\!\cdots\!62}{13\!\cdots\!21}a^{7}-\frac{76\!\cdots\!93}{13\!\cdots\!21}a^{6}+\frac{21\!\cdots\!04}{13\!\cdots\!21}a^{5}+\frac{33\!\cdots\!30}{13\!\cdots\!21}a^{4}-\frac{11\!\cdots\!06}{13\!\cdots\!21}a^{3}-\frac{13\!\cdots\!42}{13\!\cdots\!21}a^{2}+\frac{16\!\cdots\!02}{13\!\cdots\!21}a+\frac{16\!\cdots\!96}{13\!\cdots\!21}$, $\frac{19\!\cdots\!12}{13\!\cdots\!21}a^{21}-\frac{32\!\cdots\!13}{13\!\cdots\!21}a^{20}-\frac{14\!\cdots\!51}{13\!\cdots\!21}a^{19}+\frac{22\!\cdots\!60}{13\!\cdots\!21}a^{18}+\frac{40\!\cdots\!83}{13\!\cdots\!21}a^{17}-\frac{11\!\cdots\!46}{13\!\cdots\!21}a^{16}-\frac{22\!\cdots\!26}{13\!\cdots\!21}a^{15}+\frac{37\!\cdots\!28}{13\!\cdots\!21}a^{14}+\frac{10\!\cdots\!72}{13\!\cdots\!21}a^{13}-\frac{17\!\cdots\!64}{13\!\cdots\!21}a^{12}-\frac{16\!\cdots\!44}{13\!\cdots\!21}a^{11}-\frac{47\!\cdots\!01}{13\!\cdots\!21}a^{10}+\frac{11\!\cdots\!08}{13\!\cdots\!21}a^{9}+\frac{59\!\cdots\!44}{13\!\cdots\!21}a^{8}-\frac{51\!\cdots\!38}{13\!\cdots\!21}a^{7}-\frac{27\!\cdots\!44}{13\!\cdots\!21}a^{6}+\frac{29\!\cdots\!07}{13\!\cdots\!21}a^{5}+\frac{77\!\cdots\!50}{13\!\cdots\!21}a^{4}-\frac{11\!\cdots\!87}{13\!\cdots\!21}a^{3}-\frac{17\!\cdots\!71}{13\!\cdots\!21}a^{2}+\frac{12\!\cdots\!35}{13\!\cdots\!21}a+\frac{22\!\cdots\!21}{13\!\cdots\!21}$, $\frac{85\!\cdots\!58}{13\!\cdots\!21}a^{21}-\frac{14\!\cdots\!31}{13\!\cdots\!21}a^{20}-\frac{65\!\cdots\!19}{13\!\cdots\!21}a^{19}+\frac{10\!\cdots\!28}{13\!\cdots\!21}a^{18}+\frac{18\!\cdots\!07}{13\!\cdots\!21}a^{17}-\frac{52\!\cdots\!21}{13\!\cdots\!21}a^{16}-\frac{10\!\cdots\!79}{13\!\cdots\!21}a^{15}+\frac{16\!\cdots\!82}{13\!\cdots\!21}a^{14}+\frac{47\!\cdots\!79}{13\!\cdots\!21}a^{13}-\frac{89\!\cdots\!15}{13\!\cdots\!21}a^{12}-\frac{78\!\cdots\!56}{13\!\cdots\!21}a^{11}-\frac{20\!\cdots\!77}{13\!\cdots\!21}a^{10}+\frac{55\!\cdots\!34}{13\!\cdots\!21}a^{9}+\frac{26\!\cdots\!11}{13\!\cdots\!21}a^{8}-\frac{25\!\cdots\!64}{13\!\cdots\!21}a^{7}-\frac{12\!\cdots\!69}{13\!\cdots\!21}a^{6}+\frac{14\!\cdots\!70}{13\!\cdots\!21}a^{5}+\frac{33\!\cdots\!72}{13\!\cdots\!21}a^{4}-\frac{57\!\cdots\!20}{13\!\cdots\!21}a^{3}-\frac{62\!\cdots\!64}{13\!\cdots\!21}a^{2}+\frac{62\!\cdots\!97}{13\!\cdots\!21}a+\frac{10\!\cdots\!11}{13\!\cdots\!21}$, $\frac{11\!\cdots\!06}{13\!\cdots\!21}a^{21}-\frac{59\!\cdots\!40}{13\!\cdots\!21}a^{20}-\frac{71\!\cdots\!65}{13\!\cdots\!21}a^{19}+\frac{41\!\cdots\!44}{13\!\cdots\!21}a^{18}-\frac{28\!\cdots\!76}{13\!\cdots\!21}a^{17}-\frac{14\!\cdots\!53}{13\!\cdots\!21}a^{16}+\frac{12\!\cdots\!66}{13\!\cdots\!21}a^{15}+\frac{64\!\cdots\!38}{13\!\cdots\!21}a^{14}-\frac{24\!\cdots\!68}{13\!\cdots\!21}a^{13}-\frac{21\!\cdots\!74}{13\!\cdots\!21}a^{12}-\frac{24\!\cdots\!19}{13\!\cdots\!21}a^{11}+\frac{30\!\cdots\!56}{13\!\cdots\!21}a^{10}+\frac{10\!\cdots\!56}{13\!\cdots\!21}a^{9}-\frac{20\!\cdots\!29}{13\!\cdots\!21}a^{8}-\frac{10\!\cdots\!96}{13\!\cdots\!21}a^{7}+\frac{10\!\cdots\!18}{13\!\cdots\!21}a^{6}+\frac{51\!\cdots\!13}{13\!\cdots\!21}a^{5}-\frac{63\!\cdots\!54}{13\!\cdots\!21}a^{4}-\frac{10\!\cdots\!05}{13\!\cdots\!21}a^{3}+\frac{23\!\cdots\!38}{13\!\cdots\!21}a^{2}-\frac{18\!\cdots\!22}{13\!\cdots\!21}a-\frac{22\!\cdots\!82}{13\!\cdots\!21}$, $\frac{33\!\cdots\!49}{13\!\cdots\!21}a^{21}-\frac{41\!\cdots\!03}{13\!\cdots\!21}a^{20}-\frac{35\!\cdots\!00}{13\!\cdots\!21}a^{19}+\frac{39\!\cdots\!61}{13\!\cdots\!21}a^{18}+\frac{14\!\cdots\!85}{13\!\cdots\!21}a^{17}-\frac{11\!\cdots\!29}{13\!\cdots\!21}a^{16}-\frac{73\!\cdots\!99}{13\!\cdots\!21}a^{15}+\frac{14\!\cdots\!06}{13\!\cdots\!21}a^{14}+\frac{30\!\cdots\!22}{13\!\cdots\!21}a^{13}+\frac{21\!\cdots\!94}{13\!\cdots\!21}a^{12}-\frac{42\!\cdots\!77}{13\!\cdots\!21}a^{11}-\frac{51\!\cdots\!27}{13\!\cdots\!21}a^{10}+\frac{19\!\cdots\!26}{13\!\cdots\!21}a^{9}+\frac{44\!\cdots\!77}{13\!\cdots\!21}a^{8}-\frac{78\!\cdots\!02}{13\!\cdots\!21}a^{7}-\frac{21\!\cdots\!90}{13\!\cdots\!21}a^{6}+\frac{15\!\cdots\!36}{13\!\cdots\!21}a^{5}+\frac{10\!\cdots\!06}{13\!\cdots\!21}a^{4}-\frac{20\!\cdots\!58}{13\!\cdots\!21}a^{3}-\frac{37\!\cdots\!07}{13\!\cdots\!21}a^{2}+\frac{41\!\cdots\!39}{13\!\cdots\!21}a+\frac{41\!\cdots\!76}{13\!\cdots\!21}$, $\frac{14\!\cdots\!28}{13\!\cdots\!21}a^{21}-\frac{19\!\cdots\!84}{13\!\cdots\!21}a^{20}-\frac{12\!\cdots\!59}{13\!\cdots\!21}a^{19}+\frac{13\!\cdots\!87}{13\!\cdots\!21}a^{18}+\frac{38\!\cdots\!56}{13\!\cdots\!21}a^{17}-\frac{81\!\cdots\!49}{13\!\cdots\!21}a^{16}-\frac{20\!\cdots\!48}{13\!\cdots\!21}a^{15}+\frac{23\!\cdots\!76}{13\!\cdots\!21}a^{14}+\frac{92\!\cdots\!82}{13\!\cdots\!21}a^{13}+\frac{97\!\cdots\!42}{13\!\cdots\!21}a^{12}-\frac{14\!\cdots\!85}{13\!\cdots\!21}a^{11}-\frac{75\!\cdots\!76}{13\!\cdots\!21}a^{10}+\frac{89\!\cdots\!77}{13\!\cdots\!21}a^{9}+\frac{74\!\cdots\!67}{13\!\cdots\!21}a^{8}-\frac{33\!\cdots\!54}{13\!\cdots\!21}a^{7}-\frac{36\!\cdots\!56}{13\!\cdots\!21}a^{6}+\frac{20\!\cdots\!15}{13\!\cdots\!21}a^{5}+\frac{13\!\cdots\!62}{13\!\cdots\!21}a^{4}-\frac{95\!\cdots\!13}{13\!\cdots\!21}a^{3}-\frac{40\!\cdots\!42}{13\!\cdots\!21}a^{2}+\frac{12\!\cdots\!12}{13\!\cdots\!21}a+\frac{44\!\cdots\!25}{13\!\cdots\!21}$, $\frac{23\!\cdots\!64}{13\!\cdots\!21}a^{21}-\frac{39\!\cdots\!60}{13\!\cdots\!21}a^{20}-\frac{17\!\cdots\!35}{13\!\cdots\!21}a^{19}+\frac{26\!\cdots\!07}{13\!\cdots\!21}a^{18}+\frac{49\!\cdots\!30}{13\!\cdots\!21}a^{17}-\frac{13\!\cdots\!42}{13\!\cdots\!21}a^{16}-\frac{28\!\cdots\!25}{13\!\cdots\!21}a^{15}+\frac{44\!\cdots\!14}{13\!\cdots\!21}a^{14}+\frac{12\!\cdots\!27}{13\!\cdots\!21}a^{13}-\frac{14\!\cdots\!23}{13\!\cdots\!21}a^{12}-\frac{20\!\cdots\!58}{13\!\cdots\!21}a^{11}-\frac{70\!\cdots\!07}{13\!\cdots\!21}a^{10}+\frac{13\!\cdots\!17}{13\!\cdots\!21}a^{9}+\frac{81\!\cdots\!05}{13\!\cdots\!21}a^{8}-\frac{58\!\cdots\!54}{13\!\cdots\!21}a^{7}-\frac{38\!\cdots\!39}{13\!\cdots\!21}a^{6}+\frac{33\!\cdots\!66}{13\!\cdots\!21}a^{5}+\frac{12\!\cdots\!89}{13\!\cdots\!21}a^{4}-\frac{13\!\cdots\!52}{13\!\cdots\!21}a^{3}-\frac{33\!\cdots\!14}{13\!\cdots\!21}a^{2}+\frac{16\!\cdots\!30}{13\!\cdots\!21}a+\frac{37\!\cdots\!48}{13\!\cdots\!21}$, $\frac{12\!\cdots\!19}{13\!\cdots\!21}a^{21}-\frac{19\!\cdots\!24}{13\!\cdots\!21}a^{20}-\frac{96\!\cdots\!99}{13\!\cdots\!21}a^{19}+\frac{12\!\cdots\!99}{13\!\cdots\!21}a^{18}+\frac{28\!\cdots\!60}{13\!\cdots\!21}a^{17}-\frac{70\!\cdots\!94}{13\!\cdots\!21}a^{16}-\frac{15\!\cdots\!03}{13\!\cdots\!21}a^{15}+\frac{21\!\cdots\!41}{13\!\cdots\!21}a^{14}+\frac{72\!\cdots\!49}{13\!\cdots\!21}a^{13}-\frac{10\!\cdots\!21}{13\!\cdots\!21}a^{12}-\frac{11\!\cdots\!93}{13\!\cdots\!21}a^{11}-\frac{48\!\cdots\!26}{13\!\cdots\!21}a^{10}+\frac{73\!\cdots\!63}{13\!\cdots\!21}a^{9}+\frac{52\!\cdots\!89}{13\!\cdots\!21}a^{8}-\frac{29\!\cdots\!59}{13\!\cdots\!21}a^{7}-\frac{25\!\cdots\!51}{13\!\cdots\!21}a^{6}+\frac{17\!\cdots\!90}{13\!\cdots\!21}a^{5}+\frac{89\!\cdots\!44}{13\!\cdots\!21}a^{4}-\frac{75\!\cdots\!38}{13\!\cdots\!21}a^{3}-\frac{25\!\cdots\!81}{13\!\cdots\!21}a^{2}+\frac{93\!\cdots\!97}{13\!\cdots\!21}a+\frac{30\!\cdots\!31}{13\!\cdots\!21}$, $\frac{59\!\cdots\!76}{13\!\cdots\!21}a^{21}-\frac{67\!\cdots\!02}{13\!\cdots\!21}a^{20}-\frac{49\!\cdots\!87}{13\!\cdots\!21}a^{19}+\frac{43\!\cdots\!97}{13\!\cdots\!21}a^{18}+\frac{16\!\cdots\!65}{13\!\cdots\!21}a^{17}-\frac{27\!\cdots\!20}{13\!\cdots\!21}a^{16}-\frac{90\!\cdots\!83}{13\!\cdots\!21}a^{15}+\frac{74\!\cdots\!85}{13\!\cdots\!21}a^{14}+\frac{39\!\cdots\!93}{13\!\cdots\!21}a^{13}+\frac{13\!\cdots\!27}{13\!\cdots\!21}a^{12}-\frac{56\!\cdots\!32}{13\!\cdots\!21}a^{11}-\frac{46\!\cdots\!90}{13\!\cdots\!21}a^{10}+\frac{29\!\cdots\!94}{13\!\cdots\!21}a^{9}+\frac{42\!\cdots\!26}{13\!\cdots\!21}a^{8}-\frac{70\!\cdots\!87}{13\!\cdots\!21}a^{7}-\frac{20\!\cdots\!34}{13\!\cdots\!21}a^{6}+\frac{47\!\cdots\!95}{13\!\cdots\!21}a^{5}+\frac{91\!\cdots\!72}{13\!\cdots\!21}a^{4}-\frac{28\!\cdots\!70}{13\!\cdots\!21}a^{3}-\frac{31\!\cdots\!26}{13\!\cdots\!21}a^{2}+\frac{48\!\cdots\!11}{13\!\cdots\!21}a+\frac{35\!\cdots\!63}{13\!\cdots\!21}$, $\frac{10\!\cdots\!99}{13\!\cdots\!21}a^{21}-\frac{16\!\cdots\!95}{13\!\cdots\!21}a^{20}-\frac{82\!\cdots\!85}{13\!\cdots\!21}a^{19}+\frac{11\!\cdots\!37}{13\!\cdots\!21}a^{18}+\frac{24\!\cdots\!53}{13\!\cdots\!21}a^{17}-\frac{61\!\cdots\!87}{13\!\cdots\!21}a^{16}-\frac{13\!\cdots\!35}{13\!\cdots\!21}a^{15}+\frac{19\!\cdots\!94}{13\!\cdots\!21}a^{14}+\frac{61\!\cdots\!50}{13\!\cdots\!21}a^{13}-\frac{33\!\cdots\!40}{13\!\cdots\!21}a^{12}-\frac{95\!\cdots\!26}{13\!\cdots\!21}a^{11}-\frac{37\!\cdots\!68}{13\!\cdots\!21}a^{10}+\frac{63\!\cdots\!64}{13\!\cdots\!21}a^{9}+\frac{40\!\cdots\!91}{13\!\cdots\!21}a^{8}-\frac{26\!\cdots\!24}{13\!\cdots\!21}a^{7}-\frac{19\!\cdots\!07}{13\!\cdots\!21}a^{6}+\frac{15\!\cdots\!10}{13\!\cdots\!21}a^{5}+\frac{65\!\cdots\!52}{13\!\cdots\!21}a^{4}-\frac{64\!\cdots\!53}{13\!\cdots\!21}a^{3}-\frac{17\!\cdots\!42}{13\!\cdots\!21}a^{2}+\frac{73\!\cdots\!75}{13\!\cdots\!21}a+\frac{18\!\cdots\!52}{13\!\cdots\!21}$, $\frac{12\!\cdots\!85}{13\!\cdots\!21}a^{21}-\frac{24\!\cdots\!21}{13\!\cdots\!21}a^{20}-\frac{91\!\cdots\!66}{13\!\cdots\!21}a^{19}+\frac{16\!\cdots\!64}{13\!\cdots\!21}a^{18}+\frac{24\!\cdots\!36}{13\!\cdots\!21}a^{17}-\frac{80\!\cdots\!08}{13\!\cdots\!21}a^{16}-\frac{13\!\cdots\!36}{13\!\cdots\!21}a^{15}+\frac{27\!\cdots\!92}{13\!\cdots\!21}a^{14}+\frac{65\!\cdots\!58}{13\!\cdots\!21}a^{13}-\frac{22\!\cdots\!84}{13\!\cdots\!21}a^{12}-\frac{10\!\cdots\!90}{13\!\cdots\!21}a^{11}-\frac{15\!\cdots\!83}{13\!\cdots\!21}a^{10}+\frac{80\!\cdots\!50}{13\!\cdots\!21}a^{9}+\frac{28\!\cdots\!48}{13\!\cdots\!21}a^{8}-\frac{38\!\cdots\!37}{13\!\cdots\!21}a^{7}-\frac{13\!\cdots\!85}{13\!\cdots\!21}a^{6}+\frac{21\!\cdots\!86}{13\!\cdots\!21}a^{5}+\frac{23\!\cdots\!19}{13\!\cdots\!21}a^{4}-\frac{82\!\cdots\!24}{13\!\cdots\!21}a^{3}-\frac{10\!\cdots\!58}{13\!\cdots\!21}a^{2}+\frac{93\!\cdots\!29}{13\!\cdots\!21}a+\frac{48\!\cdots\!24}{13\!\cdots\!21}$, $\frac{78\!\cdots\!03}{13\!\cdots\!21}a^{21}-\frac{32\!\cdots\!32}{13\!\cdots\!21}a^{20}-\frac{30\!\cdots\!82}{13\!\cdots\!21}a^{19}+\frac{16\!\cdots\!16}{13\!\cdots\!21}a^{18}-\frac{15\!\cdots\!90}{13\!\cdots\!21}a^{17}-\frac{47\!\cdots\!64}{13\!\cdots\!21}a^{16}+\frac{24\!\cdots\!63}{13\!\cdots\!21}a^{15}+\frac{22\!\cdots\!00}{13\!\cdots\!21}a^{14}-\frac{36\!\cdots\!93}{13\!\cdots\!21}a^{13}-\frac{51\!\cdots\!05}{13\!\cdots\!21}a^{12}+\frac{72\!\cdots\!15}{13\!\cdots\!21}a^{11}+\frac{47\!\cdots\!18}{13\!\cdots\!21}a^{10}-\frac{29\!\cdots\!95}{13\!\cdots\!21}a^{9}-\frac{24\!\cdots\!39}{13\!\cdots\!21}a^{8}+\frac{23\!\cdots\!36}{13\!\cdots\!21}a^{7}+\frac{18\!\cdots\!56}{13\!\cdots\!21}a^{6}-\frac{69\!\cdots\!58}{13\!\cdots\!21}a^{5}-\frac{68\!\cdots\!72}{13\!\cdots\!21}a^{4}+\frac{53\!\cdots\!14}{13\!\cdots\!21}a^{3}-\frac{60\!\cdots\!19}{13\!\cdots\!21}a^{2}-\frac{98\!\cdots\!26}{13\!\cdots\!21}a+\frac{31\!\cdots\!85}{13\!\cdots\!21}$, $\frac{60\!\cdots\!68}{13\!\cdots\!21}a^{21}-\frac{55\!\cdots\!50}{13\!\cdots\!21}a^{20}-\frac{51\!\cdots\!37}{13\!\cdots\!21}a^{19}+\frac{36\!\cdots\!51}{13\!\cdots\!21}a^{18}+\frac{16\!\cdots\!49}{13\!\cdots\!21}a^{17}-\frac{27\!\cdots\!19}{13\!\cdots\!21}a^{16}-\frac{93\!\cdots\!31}{13\!\cdots\!21}a^{15}+\frac{59\!\cdots\!55}{13\!\cdots\!21}a^{14}+\frac{39\!\cdots\!94}{13\!\cdots\!21}a^{13}+\frac{19\!\cdots\!99}{13\!\cdots\!21}a^{12}-\frac{45\!\cdots\!18}{13\!\cdots\!21}a^{11}-\frac{42\!\cdots\!14}{13\!\cdots\!21}a^{10}+\frac{16\!\cdots\!61}{13\!\cdots\!21}a^{9}+\frac{27\!\cdots\!13}{13\!\cdots\!21}a^{8}-\frac{52\!\cdots\!69}{13\!\cdots\!21}a^{7}-\frac{12\!\cdots\!10}{13\!\cdots\!21}a^{6}+\frac{50\!\cdots\!96}{13\!\cdots\!21}a^{5}+\frac{57\!\cdots\!04}{13\!\cdots\!21}a^{4}-\frac{18\!\cdots\!42}{13\!\cdots\!21}a^{3}-\frac{12\!\cdots\!91}{13\!\cdots\!21}a^{2}+\frac{39\!\cdots\!37}{13\!\cdots\!21}a+\frac{30\!\cdots\!23}{13\!\cdots\!21}$, $\frac{18\!\cdots\!65}{13\!\cdots\!21}a^{21}-\frac{30\!\cdots\!40}{13\!\cdots\!21}a^{20}-\frac{13\!\cdots\!45}{13\!\cdots\!21}a^{19}+\frac{21\!\cdots\!33}{13\!\cdots\!21}a^{18}+\frac{39\!\cdots\!13}{13\!\cdots\!21}a^{17}-\frac{10\!\cdots\!76}{13\!\cdots\!21}a^{16}-\frac{21\!\cdots\!30}{13\!\cdots\!21}a^{15}+\frac{35\!\cdots\!65}{13\!\cdots\!21}a^{14}+\frac{10\!\cdots\!21}{13\!\cdots\!21}a^{13}-\frac{15\!\cdots\!76}{13\!\cdots\!21}a^{12}-\frac{16\!\cdots\!60}{13\!\cdots\!21}a^{11}-\frac{48\!\cdots\!48}{13\!\cdots\!21}a^{10}+\frac{11\!\cdots\!78}{13\!\cdots\!21}a^{9}+\frac{59\!\cdots\!04}{13\!\cdots\!21}a^{8}-\frac{49\!\cdots\!65}{13\!\cdots\!21}a^{7}-\frac{27\!\cdots\!12}{13\!\cdots\!21}a^{6}+\frac{28\!\cdots\!63}{13\!\cdots\!21}a^{5}+\frac{80\!\cdots\!16}{13\!\cdots\!21}a^{4}-\frac{11\!\cdots\!27}{13\!\cdots\!21}a^{3}-\frac{17\!\cdots\!18}{13\!\cdots\!21}a^{2}+\frac{12\!\cdots\!21}{13\!\cdots\!21}a+\frac{20\!\cdots\!45}{13\!\cdots\!21}$
| 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: | \( 73525321.2665 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 73525321.2665 \cdot 1}{2\cdot\sqrt{31919563459480622441144580078125}}\cr\approx \mathstrut & 0.166156706924 \end{aligned}\] (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 - 7*x^20 + 14*x^19 + 18*x^18 - 67*x^17 - 101*x^16 + 234*x^15 + 497*x^14 - 273*x^13 - 893*x^12 + 23*x^11 + 753*x^10 + 147*x^9 - 418*x^8 - 88*x^7 + 226*x^6 + 6*x^5 - 90*x^4 + 7*x^3 + 16*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 - 7*x^20 + 14*x^19 + 18*x^18 - 67*x^17 - 101*x^16 + 234*x^15 + 497*x^14 - 273*x^13 - 893*x^12 + 23*x^11 + 753*x^10 + 147*x^9 - 418*x^8 - 88*x^7 + 226*x^6 + 6*x^5 - 90*x^4 + 7*x^3 + 16*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 - 7*x^20 + 14*x^19 + 18*x^18 - 67*x^17 - 101*x^16 + 234*x^15 + 497*x^14 - 273*x^13 - 893*x^12 + 23*x^11 + 753*x^10 + 147*x^9 - 418*x^8 - 88*x^7 + 226*x^6 + 6*x^5 - 90*x^4 + 7*x^3 + 16*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 - 7*x^20 + 14*x^19 + 18*x^18 - 67*x^17 - 101*x^16 + 234*x^15 + 497*x^14 - 273*x^13 - 893*x^12 + 23*x^11 + 753*x^10 + 147*x^9 - 418*x^8 - 88*x^7 + 226*x^6 + 6*x^5 - 90*x^4 + 7*x^3 + 16*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 $C_2\times S_{11}$ are not computed |
| Character table for $C_2\times S_{11}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 11.7.808524990121.1 |
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 | $22$ | $22$ | R | $22$ | ${\href{/padicField/11.9.0.1}{9} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 | $[\ ]_{2}^{11}$ |
|
\(43\)
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 43.14.0.1 | $x^{14} + 38 x^{7} + 22 x^{6} + 24 x^{5} + 37 x^{4} + 18 x^{3} + 4 x^{2} + 19 x + 3$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(547\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
|
\(34374601\)
| $\Q_{34374601}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{34374601}$ | $x$ | $1$ | $1$ | $0$ | 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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |