Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 - 14*x^20 + 65*x^19 - 57*x^18 - 197*x^17 + 729*x^16 - 508*x^15 - 1121*x^14 + 1546*x^13 - 935*x^12 + 160*x^11 + 1889*x^10 - 975*x^9 + 579*x^8 + 80*x^7 - 700*x^6 + 99*x^5 - 159*x^4 - 75*x^3 + 41*x^2 + 15*x + 1)
gp: K = bnfinit(y^22 - 3*y^21 - 14*y^20 + 65*y^19 - 57*y^18 - 197*y^17 + 729*y^16 - 508*y^15 - 1121*y^14 + 1546*y^13 - 935*y^12 + 160*y^11 + 1889*y^10 - 975*y^9 + 579*y^8 + 80*y^7 - 700*y^6 + 99*y^5 - 159*y^4 - 75*y^3 + 41*y^2 + 15*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 3*x^21 - 14*x^20 + 65*x^19 - 57*x^18 - 197*x^17 + 729*x^16 - 508*x^15 - 1121*x^14 + 1546*x^13 - 935*x^12 + 160*x^11 + 1889*x^10 - 975*x^9 + 579*x^8 + 80*x^7 - 700*x^6 + 99*x^5 - 159*x^4 - 75*x^3 + 41*x^2 + 15*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 3*x^21 - 14*x^20 + 65*x^19 - 57*x^18 - 197*x^17 + 729*x^16 - 508*x^15 - 1121*x^14 + 1546*x^13 - 935*x^12 + 160*x^11 + 1889*x^10 - 975*x^9 + 579*x^8 + 80*x^7 - 700*x^6 + 99*x^5 - 159*x^4 - 75*x^3 + 41*x^2 + 15*x + 1)
\( x^{22} - 3 x^{21} - 14 x^{20} + 65 x^{19} - 57 x^{18} - 197 x^{17} + 729 x^{16} - 508 x^{15} - 1121 x^{14} + \cdots + 1 \)
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: |
\(4098058278162967431271914143428729\)
\(\medspace = 23^{21}\cdot 47^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.72\) | 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: | $23^{21/22}47^{1/2}\approx 136.73511025656717$ | ||
| Ramified primes: |
\(23\), \(47\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{1081}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{47}a^{19}+\frac{17}{47}a^{18}-\frac{3}{47}a^{17}-\frac{19}{47}a^{16}+\frac{1}{47}a^{15}-\frac{11}{47}a^{14}-\frac{22}{47}a^{13}+\frac{15}{47}a^{12}+\frac{7}{47}a^{11}+\frac{5}{47}a^{10}-\frac{20}{47}a^{9}+\frac{15}{47}a^{8}+\frac{1}{47}a^{7}-\frac{5}{47}a^{6}-\frac{22}{47}a^{5}-\frac{8}{47}a^{4}+\frac{12}{47}a^{3}+\frac{21}{47}a^{2}-\frac{17}{47}a+\frac{2}{47}$, $\frac{1}{47}a^{20}-\frac{10}{47}a^{18}-\frac{15}{47}a^{17}-\frac{5}{47}a^{16}+\frac{19}{47}a^{15}-\frac{23}{47}a^{14}+\frac{13}{47}a^{13}-\frac{13}{47}a^{12}-\frac{20}{47}a^{11}-\frac{11}{47}a^{10}-\frac{21}{47}a^{9}-\frac{19}{47}a^{8}-\frac{22}{47}a^{7}+\frac{16}{47}a^{6}-\frac{10}{47}a^{5}+\frac{7}{47}a^{4}+\frac{5}{47}a^{3}+\frac{2}{47}a^{2}+\frac{9}{47}a+\frac{13}{47}$, $\frac{1}{88\!\cdots\!61}a^{21}-\frac{53\!\cdots\!01}{88\!\cdots\!61}a^{20}-\frac{62\!\cdots\!77}{88\!\cdots\!61}a^{19}+\frac{59\!\cdots\!86}{88\!\cdots\!61}a^{18}+\frac{82\!\cdots\!12}{88\!\cdots\!61}a^{17}+\frac{10\!\cdots\!90}{88\!\cdots\!61}a^{16}+\frac{24\!\cdots\!20}{88\!\cdots\!61}a^{15}-\frac{26\!\cdots\!93}{88\!\cdots\!61}a^{14}+\frac{29\!\cdots\!11}{88\!\cdots\!61}a^{13}-\frac{11\!\cdots\!53}{88\!\cdots\!61}a^{12}-\frac{11\!\cdots\!18}{88\!\cdots\!61}a^{11}-\frac{40\!\cdots\!05}{88\!\cdots\!61}a^{10}-\frac{23\!\cdots\!40}{88\!\cdots\!61}a^{9}-\frac{39\!\cdots\!45}{88\!\cdots\!61}a^{8}-\frac{28\!\cdots\!85}{88\!\cdots\!61}a^{7}-\frac{13\!\cdots\!59}{88\!\cdots\!61}a^{6}-\frac{14\!\cdots\!32}{88\!\cdots\!61}a^{5}+\frac{19\!\cdots\!34}{88\!\cdots\!61}a^{4}-\frac{35\!\cdots\!83}{88\!\cdots\!61}a^{3}-\frac{43\!\cdots\!21}{88\!\cdots\!61}a^{2}+\frac{36\!\cdots\!29}{88\!\cdots\!61}a+\frac{38\!\cdots\!23}{88\!\cdots\!61}$
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: |
$\frac{28\!\cdots\!29}{88\!\cdots\!61}a^{21}-\frac{87\!\cdots\!32}{88\!\cdots\!61}a^{20}-\frac{38\!\cdots\!85}{88\!\cdots\!61}a^{19}+\frac{18\!\cdots\!27}{88\!\cdots\!61}a^{18}-\frac{18\!\cdots\!86}{88\!\cdots\!61}a^{17}-\frac{52\!\cdots\!52}{88\!\cdots\!61}a^{16}+\frac{21\!\cdots\!78}{88\!\cdots\!61}a^{15}-\frac{16\!\cdots\!98}{88\!\cdots\!61}a^{14}-\frac{28\!\cdots\!18}{88\!\cdots\!61}a^{13}+\frac{46\!\cdots\!49}{88\!\cdots\!61}a^{12}-\frac{32\!\cdots\!20}{88\!\cdots\!61}a^{11}+\frac{10\!\cdots\!73}{88\!\cdots\!61}a^{10}+\frac{50\!\cdots\!85}{88\!\cdots\!61}a^{9}-\frac{33\!\cdots\!79}{88\!\cdots\!61}a^{8}+\frac{20\!\cdots\!84}{88\!\cdots\!61}a^{7}-\frac{17\!\cdots\!75}{88\!\cdots\!61}a^{6}-\frac{19\!\cdots\!27}{88\!\cdots\!61}a^{5}+\frac{46\!\cdots\!90}{88\!\cdots\!61}a^{4}-\frac{52\!\cdots\!16}{88\!\cdots\!61}a^{3}-\frac{11\!\cdots\!02}{88\!\cdots\!61}a^{2}+\frac{12\!\cdots\!98}{88\!\cdots\!61}a+\frac{29\!\cdots\!08}{88\!\cdots\!61}$, $\frac{38\!\cdots\!15}{88\!\cdots\!61}a^{21}-\frac{12\!\cdots\!49}{88\!\cdots\!61}a^{20}-\frac{51\!\cdots\!89}{88\!\cdots\!61}a^{19}+\frac{26\!\cdots\!59}{88\!\cdots\!61}a^{18}-\frac{28\!\cdots\!69}{88\!\cdots\!61}a^{17}-\frac{69\!\cdots\!33}{88\!\cdots\!61}a^{16}+\frac{29\!\cdots\!92}{88\!\cdots\!61}a^{15}-\frac{26\!\cdots\!06}{88\!\cdots\!61}a^{14}-\frac{36\!\cdots\!73}{88\!\cdots\!61}a^{13}+\frac{67\!\cdots\!61}{88\!\cdots\!61}a^{12}-\frac{53\!\cdots\!87}{88\!\cdots\!61}a^{11}+\frac{20\!\cdots\!75}{88\!\cdots\!61}a^{10}+\frac{67\!\cdots\!86}{88\!\cdots\!61}a^{9}-\frac{53\!\cdots\!36}{88\!\cdots\!61}a^{8}+\frac{37\!\cdots\!44}{88\!\cdots\!61}a^{7}-\frac{65\!\cdots\!97}{88\!\cdots\!61}a^{6}-\frac{25\!\cdots\!50}{88\!\cdots\!61}a^{5}+\frac{98\!\cdots\!04}{88\!\cdots\!61}a^{4}-\frac{91\!\cdots\!90}{88\!\cdots\!61}a^{3}-\frac{84\!\cdots\!65}{88\!\cdots\!61}a^{2}+\frac{16\!\cdots\!24}{88\!\cdots\!61}a+\frac{93\!\cdots\!55}{88\!\cdots\!61}$, $\frac{31\!\cdots\!07}{88\!\cdots\!61}a^{21}-\frac{98\!\cdots\!85}{88\!\cdots\!61}a^{20}-\frac{43\!\cdots\!31}{88\!\cdots\!61}a^{19}+\frac{21\!\cdots\!37}{88\!\cdots\!61}a^{18}-\frac{20\!\cdots\!85}{88\!\cdots\!61}a^{17}-\frac{60\!\cdots\!45}{88\!\cdots\!61}a^{16}+\frac{23\!\cdots\!29}{88\!\cdots\!61}a^{15}-\frac{18\!\cdots\!41}{88\!\cdots\!61}a^{14}-\frac{34\!\cdots\!21}{88\!\cdots\!61}a^{13}+\frac{52\!\cdots\!24}{88\!\cdots\!61}a^{12}-\frac{32\!\cdots\!37}{88\!\cdots\!61}a^{11}+\frac{74\!\cdots\!96}{88\!\cdots\!61}a^{10}+\frac{57\!\cdots\!06}{88\!\cdots\!61}a^{9}-\frac{36\!\cdots\!49}{88\!\cdots\!61}a^{8}+\frac{21\!\cdots\!87}{88\!\cdots\!61}a^{7}+\frac{25\!\cdots\!33}{88\!\cdots\!61}a^{6}-\frac{21\!\cdots\!42}{88\!\cdots\!61}a^{5}+\frac{11\!\cdots\!57}{18\!\cdots\!63}a^{4}-\frac{52\!\cdots\!89}{88\!\cdots\!61}a^{3}-\frac{16\!\cdots\!79}{88\!\cdots\!61}a^{2}+\frac{13\!\cdots\!18}{88\!\cdots\!61}a+\frac{24\!\cdots\!47}{88\!\cdots\!61}$, $\frac{24\!\cdots\!77}{88\!\cdots\!61}a^{21}-\frac{77\!\cdots\!97}{88\!\cdots\!61}a^{20}-\frac{33\!\cdots\!34}{88\!\cdots\!61}a^{19}+\frac{16\!\cdots\!82}{88\!\cdots\!61}a^{18}-\frac{16\!\cdots\!88}{88\!\cdots\!61}a^{17}-\frac{44\!\cdots\!33}{88\!\cdots\!61}a^{16}+\frac{18\!\cdots\!91}{88\!\cdots\!61}a^{15}-\frac{15\!\cdots\!73}{88\!\cdots\!61}a^{14}-\frac{23\!\cdots\!97}{88\!\cdots\!61}a^{13}+\frac{40\!\cdots\!65}{88\!\cdots\!61}a^{12}-\frac{30\!\cdots\!82}{88\!\cdots\!61}a^{11}+\frac{12\!\cdots\!07}{88\!\cdots\!61}a^{10}+\frac{42\!\cdots\!56}{88\!\cdots\!61}a^{9}-\frac{30\!\cdots\!52}{88\!\cdots\!61}a^{8}+\frac{22\!\cdots\!88}{88\!\cdots\!61}a^{7}-\frac{43\!\cdots\!72}{88\!\cdots\!61}a^{6}-\frac{16\!\cdots\!27}{88\!\cdots\!61}a^{5}+\frac{56\!\cdots\!61}{88\!\cdots\!61}a^{4}-\frac{58\!\cdots\!90}{88\!\cdots\!61}a^{3}-\frac{84\!\cdots\!40}{88\!\cdots\!61}a^{2}+\frac{11\!\cdots\!09}{88\!\cdots\!61}a+\frac{16\!\cdots\!39}{88\!\cdots\!61}$, $\frac{28\!\cdots\!50}{88\!\cdots\!61}a^{21}-\frac{89\!\cdots\!03}{88\!\cdots\!61}a^{20}-\frac{38\!\cdots\!47}{88\!\cdots\!61}a^{19}+\frac{40\!\cdots\!20}{18\!\cdots\!63}a^{18}-\frac{19\!\cdots\!13}{88\!\cdots\!61}a^{17}-\frac{51\!\cdots\!10}{88\!\cdots\!61}a^{16}+\frac{21\!\cdots\!28}{88\!\cdots\!61}a^{15}-\frac{18\!\cdots\!38}{88\!\cdots\!61}a^{14}-\frac{27\!\cdots\!35}{88\!\cdots\!61}a^{13}+\frac{47\!\cdots\!65}{88\!\cdots\!61}a^{12}-\frac{35\!\cdots\!29}{88\!\cdots\!61}a^{11}+\frac{13\!\cdots\!49}{88\!\cdots\!61}a^{10}+\frac{48\!\cdots\!92}{88\!\cdots\!61}a^{9}-\frac{35\!\cdots\!44}{88\!\cdots\!61}a^{8}+\frac{24\!\cdots\!98}{88\!\cdots\!61}a^{7}-\frac{41\!\cdots\!19}{88\!\cdots\!61}a^{6}-\frac{18\!\cdots\!92}{88\!\cdots\!61}a^{5}+\frac{59\!\cdots\!66}{88\!\cdots\!61}a^{4}-\frac{60\!\cdots\!75}{88\!\cdots\!61}a^{3}-\frac{61\!\cdots\!55}{88\!\cdots\!61}a^{2}+\frac{11\!\cdots\!81}{88\!\cdots\!61}a+\frac{21\!\cdots\!82}{88\!\cdots\!61}$, $\frac{59\!\cdots\!02}{88\!\cdots\!61}a^{21}-\frac{18\!\cdots\!00}{88\!\cdots\!61}a^{20}-\frac{80\!\cdots\!02}{88\!\cdots\!61}a^{19}+\frac{39\!\cdots\!88}{88\!\cdots\!61}a^{18}-\frac{38\!\cdots\!27}{88\!\cdots\!61}a^{17}-\frac{11\!\cdots\!47}{88\!\cdots\!61}a^{16}+\frac{44\!\cdots\!52}{88\!\cdots\!61}a^{15}-\frac{35\!\cdots\!50}{88\!\cdots\!61}a^{14}-\frac{61\!\cdots\!53}{88\!\cdots\!61}a^{13}+\frac{99\!\cdots\!90}{88\!\cdots\!61}a^{12}-\frac{67\!\cdots\!07}{88\!\cdots\!61}a^{11}+\frac{17\!\cdots\!67}{88\!\cdots\!61}a^{10}+\frac{10\!\cdots\!47}{88\!\cdots\!61}a^{9}-\frac{71\!\cdots\!44}{88\!\cdots\!61}a^{8}+\frac{43\!\cdots\!51}{88\!\cdots\!61}a^{7}-\frac{14\!\cdots\!55}{88\!\cdots\!61}a^{6}-\frac{40\!\cdots\!09}{88\!\cdots\!61}a^{5}+\frac{11\!\cdots\!71}{88\!\cdots\!61}a^{4}-\frac{10\!\cdots\!10}{88\!\cdots\!61}a^{3}-\frac{31\!\cdots\!16}{88\!\cdots\!61}a^{2}+\frac{26\!\cdots\!86}{88\!\cdots\!61}a+\frac{41\!\cdots\!21}{88\!\cdots\!61}$, $\frac{20\!\cdots\!42}{88\!\cdots\!61}a^{21}-\frac{69\!\cdots\!80}{88\!\cdots\!61}a^{20}-\frac{25\!\cdots\!48}{88\!\cdots\!61}a^{19}+\frac{14\!\cdots\!64}{88\!\cdots\!61}a^{18}-\frac{17\!\cdots\!19}{88\!\cdots\!61}a^{17}-\frac{32\!\cdots\!13}{88\!\cdots\!61}a^{16}+\frac{16\!\cdots\!71}{88\!\cdots\!61}a^{15}-\frac{17\!\cdots\!62}{88\!\cdots\!61}a^{14}-\frac{15\!\cdots\!63}{88\!\cdots\!61}a^{13}+\frac{38\!\cdots\!39}{88\!\cdots\!61}a^{12}-\frac{36\!\cdots\!02}{88\!\cdots\!61}a^{11}+\frac{18\!\cdots\!70}{88\!\cdots\!61}a^{10}+\frac{32\!\cdots\!88}{88\!\cdots\!61}a^{9}-\frac{35\!\cdots\!85}{88\!\cdots\!61}a^{8}+\frac{26\!\cdots\!35}{88\!\cdots\!61}a^{7}-\frac{84\!\cdots\!01}{88\!\cdots\!61}a^{6}-\frac{11\!\cdots\!55}{88\!\cdots\!61}a^{5}+\frac{77\!\cdots\!29}{88\!\cdots\!61}a^{4}-\frac{64\!\cdots\!66}{88\!\cdots\!61}a^{3}+\frac{83\!\cdots\!38}{88\!\cdots\!61}a^{2}+\frac{66\!\cdots\!57}{88\!\cdots\!61}a-\frac{12\!\cdots\!13}{88\!\cdots\!61}$, $\frac{20\!\cdots\!42}{88\!\cdots\!61}a^{21}-\frac{69\!\cdots\!80}{88\!\cdots\!61}a^{20}-\frac{25\!\cdots\!48}{88\!\cdots\!61}a^{19}+\frac{14\!\cdots\!64}{88\!\cdots\!61}a^{18}-\frac{17\!\cdots\!19}{88\!\cdots\!61}a^{17}-\frac{32\!\cdots\!13}{88\!\cdots\!61}a^{16}+\frac{16\!\cdots\!71}{88\!\cdots\!61}a^{15}-\frac{17\!\cdots\!62}{88\!\cdots\!61}a^{14}-\frac{15\!\cdots\!63}{88\!\cdots\!61}a^{13}+\frac{38\!\cdots\!39}{88\!\cdots\!61}a^{12}-\frac{36\!\cdots\!02}{88\!\cdots\!61}a^{11}+\frac{18\!\cdots\!70}{88\!\cdots\!61}a^{10}+\frac{32\!\cdots\!88}{88\!\cdots\!61}a^{9}-\frac{35\!\cdots\!85}{88\!\cdots\!61}a^{8}+\frac{26\!\cdots\!35}{88\!\cdots\!61}a^{7}-\frac{84\!\cdots\!01}{88\!\cdots\!61}a^{6}-\frac{11\!\cdots\!55}{88\!\cdots\!61}a^{5}+\frac{77\!\cdots\!29}{88\!\cdots\!61}a^{4}-\frac{64\!\cdots\!66}{88\!\cdots\!61}a^{3}+\frac{83\!\cdots\!38}{88\!\cdots\!61}a^{2}+\frac{66\!\cdots\!57}{88\!\cdots\!61}a-\frac{37\!\cdots\!52}{88\!\cdots\!61}$, $\frac{30\!\cdots\!45}{88\!\cdots\!61}a^{21}-\frac{97\!\cdots\!81}{88\!\cdots\!61}a^{20}-\frac{39\!\cdots\!45}{88\!\cdots\!61}a^{19}+\frac{20\!\cdots\!85}{88\!\cdots\!61}a^{18}-\frac{22\!\cdots\!43}{88\!\cdots\!61}a^{17}-\frac{53\!\cdots\!55}{88\!\cdots\!61}a^{16}+\frac{23\!\cdots\!36}{88\!\cdots\!61}a^{15}-\frac{21\!\cdots\!12}{88\!\cdots\!61}a^{14}-\frac{27\!\cdots\!88}{88\!\cdots\!61}a^{13}+\frac{52\!\cdots\!82}{88\!\cdots\!61}a^{12}-\frac{41\!\cdots\!13}{88\!\cdots\!61}a^{11}+\frac{17\!\cdots\!81}{88\!\cdots\!61}a^{10}+\frac{51\!\cdots\!34}{88\!\cdots\!61}a^{9}-\frac{42\!\cdots\!61}{88\!\cdots\!61}a^{8}+\frac{29\!\cdots\!22}{88\!\cdots\!61}a^{7}-\frac{65\!\cdots\!03}{88\!\cdots\!61}a^{6}-\frac{18\!\cdots\!63}{88\!\cdots\!61}a^{5}+\frac{80\!\cdots\!21}{88\!\cdots\!61}a^{4}-\frac{73\!\cdots\!46}{88\!\cdots\!61}a^{3}-\frac{52\!\cdots\!03}{88\!\cdots\!61}a^{2}+\frac{11\!\cdots\!23}{88\!\cdots\!61}a+\frac{17\!\cdots\!20}{88\!\cdots\!61}$, $\frac{80\!\cdots\!78}{88\!\cdots\!61}a^{21}-\frac{30\!\cdots\!94}{88\!\cdots\!61}a^{20}-\frac{92\!\cdots\!93}{88\!\cdots\!61}a^{19}+\frac{59\!\cdots\!25}{88\!\cdots\!61}a^{18}-\frac{87\!\cdots\!73}{88\!\cdots\!61}a^{17}-\frac{23\!\cdots\!26}{18\!\cdots\!63}a^{16}+\frac{68\!\cdots\!48}{88\!\cdots\!61}a^{15}-\frac{88\!\cdots\!73}{88\!\cdots\!61}a^{14}-\frac{43\!\cdots\!29}{88\!\cdots\!61}a^{13}+\frac{17\!\cdots\!50}{88\!\cdots\!61}a^{12}-\frac{18\!\cdots\!09}{88\!\cdots\!61}a^{11}+\frac{10\!\cdots\!74}{88\!\cdots\!61}a^{10}+\frac{11\!\cdots\!61}{88\!\cdots\!61}a^{9}-\frac{17\!\cdots\!86}{88\!\cdots\!61}a^{8}+\frac{13\!\cdots\!47}{88\!\cdots\!61}a^{7}-\frac{55\!\cdots\!18}{88\!\cdots\!61}a^{6}-\frac{39\!\cdots\!14}{88\!\cdots\!61}a^{5}+\frac{42\!\cdots\!28}{88\!\cdots\!61}a^{4}-\frac{31\!\cdots\!89}{88\!\cdots\!61}a^{3}+\frac{75\!\cdots\!67}{88\!\cdots\!61}a^{2}+\frac{19\!\cdots\!29}{88\!\cdots\!61}a-\frac{12\!\cdots\!14}{88\!\cdots\!61}$, $a$, $\frac{38\!\cdots\!36}{88\!\cdots\!61}a^{21}-\frac{11\!\cdots\!33}{88\!\cdots\!61}a^{20}-\frac{52\!\cdots\!17}{88\!\cdots\!61}a^{19}+\frac{25\!\cdots\!77}{88\!\cdots\!61}a^{18}-\frac{23\!\cdots\!70}{88\!\cdots\!61}a^{17}-\frac{72\!\cdots\!14}{88\!\cdots\!61}a^{16}+\frac{28\!\cdots\!82}{88\!\cdots\!61}a^{15}-\frac{21\!\cdots\!27}{88\!\cdots\!61}a^{14}-\frac{40\!\cdots\!17}{88\!\cdots\!61}a^{13}+\frac{61\!\cdots\!50}{88\!\cdots\!61}a^{12}-\frac{42\!\cdots\!57}{88\!\cdots\!61}a^{11}+\frac{11\!\cdots\!73}{88\!\cdots\!61}a^{10}+\frac{70\!\cdots\!93}{88\!\cdots\!61}a^{9}-\frac{41\!\cdots\!88}{88\!\cdots\!61}a^{8}+\frac{28\!\cdots\!09}{88\!\cdots\!61}a^{7}-\frac{13\!\cdots\!36}{88\!\cdots\!61}a^{6}-\frac{25\!\cdots\!72}{88\!\cdots\!61}a^{5}+\frac{57\!\cdots\!86}{88\!\cdots\!61}a^{4}-\frac{76\!\cdots\!50}{88\!\cdots\!61}a^{3}-\frac{24\!\cdots\!32}{88\!\cdots\!61}a^{2}+\frac{13\!\cdots\!77}{88\!\cdots\!61}a+\frac{21\!\cdots\!20}{88\!\cdots\!61}$, $\frac{10\!\cdots\!75}{88\!\cdots\!61}a^{21}-\frac{34\!\cdots\!43}{88\!\cdots\!61}a^{20}-\frac{14\!\cdots\!95}{88\!\cdots\!61}a^{19}+\frac{72\!\cdots\!27}{88\!\cdots\!61}a^{18}-\frac{76\!\cdots\!98}{88\!\cdots\!61}a^{17}-\frac{19\!\cdots\!93}{88\!\cdots\!61}a^{16}+\frac{82\!\cdots\!10}{88\!\cdots\!61}a^{15}-\frac{71\!\cdots\!21}{88\!\cdots\!61}a^{14}-\frac{10\!\cdots\!96}{88\!\cdots\!61}a^{13}+\frac{18\!\cdots\!65}{88\!\cdots\!61}a^{12}-\frac{14\!\cdots\!18}{88\!\cdots\!61}a^{11}+\frac{47\!\cdots\!65}{88\!\cdots\!61}a^{10}+\frac{19\!\cdots\!40}{88\!\cdots\!61}a^{9}-\frac{14\!\cdots\!49}{88\!\cdots\!61}a^{8}+\frac{94\!\cdots\!29}{88\!\cdots\!61}a^{7}-\frac{10\!\cdots\!43}{88\!\cdots\!61}a^{6}-\frac{73\!\cdots\!71}{88\!\cdots\!61}a^{5}+\frac{27\!\cdots\!93}{88\!\cdots\!61}a^{4}-\frac{22\!\cdots\!42}{88\!\cdots\!61}a^{3}-\frac{33\!\cdots\!86}{88\!\cdots\!61}a^{2}+\frac{53\!\cdots\!52}{88\!\cdots\!61}a+\frac{49\!\cdots\!59}{88\!\cdots\!61}$, $\frac{10\!\cdots\!30}{88\!\cdots\!61}a^{21}-\frac{36\!\cdots\!05}{88\!\cdots\!61}a^{20}-\frac{13\!\cdots\!66}{88\!\cdots\!61}a^{19}+\frac{74\!\cdots\!31}{88\!\cdots\!61}a^{18}-\frac{93\!\cdots\!48}{88\!\cdots\!61}a^{17}-\frac{17\!\cdots\!71}{88\!\cdots\!61}a^{16}+\frac{85\!\cdots\!55}{88\!\cdots\!61}a^{15}-\frac{90\!\cdots\!80}{88\!\cdots\!61}a^{14}-\frac{84\!\cdots\!23}{88\!\cdots\!61}a^{13}+\frac{20\!\cdots\!55}{88\!\cdots\!61}a^{12}-\frac{17\!\cdots\!62}{88\!\cdots\!61}a^{11}+\frac{74\!\cdots\!69}{88\!\cdots\!61}a^{10}+\frac{16\!\cdots\!51}{88\!\cdots\!61}a^{9}-\frac{17\!\cdots\!02}{88\!\cdots\!61}a^{8}+\frac{11\!\cdots\!74}{88\!\cdots\!61}a^{7}-\frac{32\!\cdots\!18}{88\!\cdots\!61}a^{6}-\frac{52\!\cdots\!25}{88\!\cdots\!61}a^{5}+\frac{33\!\cdots\!93}{88\!\cdots\!61}a^{4}-\frac{25\!\cdots\!46}{88\!\cdots\!61}a^{3}+\frac{14\!\cdots\!01}{88\!\cdots\!61}a^{2}+\frac{15\!\cdots\!65}{88\!\cdots\!61}a-\frac{32\!\cdots\!77}{88\!\cdots\!61}$, $\frac{50\!\cdots\!78}{88\!\cdots\!61}a^{21}-\frac{15\!\cdots\!70}{88\!\cdots\!61}a^{20}-\frac{68\!\cdots\!05}{88\!\cdots\!61}a^{19}+\frac{33\!\cdots\!52}{88\!\cdots\!61}a^{18}-\frac{33\!\cdots\!96}{88\!\cdots\!61}a^{17}-\frac{94\!\cdots\!50}{88\!\cdots\!61}a^{16}+\frac{38\!\cdots\!97}{88\!\cdots\!61}a^{15}-\frac{31\!\cdots\!04}{88\!\cdots\!61}a^{14}-\frac{11\!\cdots\!03}{18\!\cdots\!63}a^{13}+\frac{84\!\cdots\!85}{88\!\cdots\!61}a^{12}-\frac{60\!\cdots\!80}{88\!\cdots\!61}a^{11}+\frac{17\!\cdots\!80}{88\!\cdots\!61}a^{10}+\frac{92\!\cdots\!12}{88\!\cdots\!61}a^{9}-\frac{59\!\cdots\!49}{88\!\cdots\!61}a^{8}+\frac{40\!\cdots\!09}{88\!\cdots\!61}a^{7}-\frac{14\!\cdots\!03}{88\!\cdots\!61}a^{6}-\frac{33\!\cdots\!67}{88\!\cdots\!61}a^{5}+\frac{90\!\cdots\!99}{88\!\cdots\!61}a^{4}-\frac{10\!\cdots\!31}{88\!\cdots\!61}a^{3}-\frac{26\!\cdots\!26}{88\!\cdots\!61}a^{2}+\frac{18\!\cdots\!41}{88\!\cdots\!61}a+\frac{37\!\cdots\!93}{88\!\cdots\!61}$, $\frac{85\!\cdots\!42}{88\!\cdots\!61}a^{21}-\frac{32\!\cdots\!69}{88\!\cdots\!61}a^{20}-\frac{73\!\cdots\!52}{88\!\cdots\!61}a^{19}+\frac{56\!\cdots\!36}{88\!\cdots\!61}a^{18}-\frac{12\!\cdots\!18}{88\!\cdots\!61}a^{17}+\frac{59\!\cdots\!14}{88\!\cdots\!61}a^{16}+\frac{53\!\cdots\!31}{88\!\cdots\!61}a^{15}-\frac{14\!\cdots\!50}{88\!\cdots\!61}a^{14}+\frac{15\!\cdots\!76}{88\!\cdots\!61}a^{13}-\frac{35\!\cdots\!30}{88\!\cdots\!61}a^{12}-\frac{45\!\cdots\!15}{88\!\cdots\!61}a^{11}+\frac{65\!\cdots\!06}{88\!\cdots\!61}a^{10}-\frac{20\!\cdots\!25}{88\!\cdots\!61}a^{9}-\frac{17\!\cdots\!25}{88\!\cdots\!61}a^{8}+\frac{55\!\cdots\!20}{88\!\cdots\!61}a^{7}-\frac{44\!\cdots\!92}{88\!\cdots\!61}a^{6}+\frac{13\!\cdots\!24}{88\!\cdots\!61}a^{5}+\frac{33\!\cdots\!07}{88\!\cdots\!61}a^{4}-\frac{13\!\cdots\!43}{88\!\cdots\!61}a^{3}+\frac{74\!\cdots\!12}{88\!\cdots\!61}a^{2}-\frac{24\!\cdots\!41}{88\!\cdots\!61}a-\frac{13\!\cdots\!09}{88\!\cdots\!61}$, $\frac{39\!\cdots\!80}{88\!\cdots\!61}a^{21}-\frac{13\!\cdots\!22}{88\!\cdots\!61}a^{20}-\frac{48\!\cdots\!03}{88\!\cdots\!61}a^{19}+\frac{27\!\cdots\!58}{88\!\cdots\!61}a^{18}-\frac{36\!\cdots\!70}{88\!\cdots\!61}a^{17}-\frac{60\!\cdots\!74}{88\!\cdots\!61}a^{16}+\frac{31\!\cdots\!90}{88\!\cdots\!61}a^{15}-\frac{35\!\cdots\!10}{88\!\cdots\!61}a^{14}-\frac{27\!\cdots\!14}{88\!\cdots\!61}a^{13}+\frac{76\!\cdots\!15}{88\!\cdots\!61}a^{12}-\frac{15\!\cdots\!22}{18\!\cdots\!63}a^{11}+\frac{40\!\cdots\!33}{88\!\cdots\!61}a^{10}+\frac{12\!\cdots\!05}{18\!\cdots\!63}a^{9}-\frac{69\!\cdots\!10}{88\!\cdots\!61}a^{8}+\frac{55\!\cdots\!27}{88\!\cdots\!61}a^{7}-\frac{21\!\cdots\!41}{88\!\cdots\!61}a^{6}-\frac{19\!\cdots\!55}{88\!\cdots\!61}a^{5}+\frac{15\!\cdots\!05}{88\!\cdots\!61}a^{4}-\frac{13\!\cdots\!50}{88\!\cdots\!61}a^{3}+\frac{29\!\cdots\!41}{88\!\cdots\!61}a^{2}+\frac{82\!\cdots\!13}{88\!\cdots\!61}a-\frac{12\!\cdots\!02}{88\!\cdots\!61}$
| 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: | \( 903801457.7 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 903801457.7 \cdot 1}{2\cdot\sqrt{4098058278162967431271914143428729}}\cr\approx \mathstrut & 0.1802575191 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 - 14*x^20 + 65*x^19 - 57*x^18 - 197*x^17 + 729*x^16 - 508*x^15 - 1121*x^14 + 1546*x^13 - 935*x^12 + 160*x^11 + 1889*x^10 - 975*x^9 + 579*x^8 + 80*x^7 - 700*x^6 + 99*x^5 - 159*x^4 - 75*x^3 + 41*x^2 + 15*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 - 3*x^21 - 14*x^20 + 65*x^19 - 57*x^18 - 197*x^17 + 729*x^16 - 508*x^15 - 1121*x^14 + 1546*x^13 - 935*x^12 + 160*x^11 + 1889*x^10 - 975*x^9 + 579*x^8 + 80*x^7 - 700*x^6 + 99*x^5 - 159*x^4 - 75*x^3 + 41*x^2 + 15*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 - 3*x^21 - 14*x^20 + 65*x^19 - 57*x^18 - 197*x^17 + 729*x^16 - 508*x^15 - 1121*x^14 + 1546*x^13 - 935*x^12 + 160*x^11 + 1889*x^10 - 975*x^9 + 579*x^8 + 80*x^7 - 700*x^6 + 99*x^5 - 159*x^4 - 75*x^3 + 41*x^2 + 15*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 - 3*x^21 - 14*x^20 + 65*x^19 - 57*x^18 - 197*x^17 + 729*x^16 - 508*x^15 - 1121*x^14 + 1546*x^13 - 935*x^12 + 160*x^11 + 1889*x^10 - 975*x^9 + 579*x^8 + 80*x^7 - 700*x^6 + 99*x^5 - 159*x^4 - 75*x^3 + 41*x^2 + 15*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
$C_{15}\times C_{420}$ (as 22T28):
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 22528 |
| The 208 conjugacy class representatives for $C_{15}\times C_{420}$ |
| Character table for $C_{15}\times C_{420}$ |
Intermediate fields
| \(\Q(\zeta_{23})^+\) |
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 siblings: | data not computed |
| Degree 44 siblings: | data not computed |
| Minimal sibling: | 22.14.1855164453672687836700730712281.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | $22$ | $22$ | ${\href{/padicField/19.11.0.1}{11} }^{2}$ | R | $22$ | $22$ | $22$ | $22$ | ${\href{/padicField/43.11.0.1}{11} }^{2}$ | R | $22$ | ${\href{/padicField/59.11.0.1}{11} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.22.21.1 | $x^{22} + 506$ | $22$ | $1$ | $21$ | 22T1 | $[\ ]_{22}$ |
|
\(47\)
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |