Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 5*x^20 - 4*x^19 + 4*x^18 - 14*x^17 + 63*x^16 - 228*x^15 + 497*x^14 - 686*x^13 + 792*x^12 - 793*x^11 + 1821*x^10 - 3315*x^9 + 78*x^8 + 1215*x^7 + 1528*x^6 + 168*x^5 - 351*x^4 - 826*x^3 - 4*x^2 + 39*x - 5)
gp: K = bnfinit(y^22 - 2*y^21 + 5*y^20 - 4*y^19 + 4*y^18 - 14*y^17 + 63*y^16 - 228*y^15 + 497*y^14 - 686*y^13 + 792*y^12 - 793*y^11 + 1821*y^10 - 3315*y^9 + 78*y^8 + 1215*y^7 + 1528*y^6 + 168*y^5 - 351*y^4 - 826*y^3 - 4*y^2 + 39*y - 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^21 + 5*x^20 - 4*x^19 + 4*x^18 - 14*x^17 + 63*x^16 - 228*x^15 + 497*x^14 - 686*x^13 + 792*x^12 - 793*x^11 + 1821*x^10 - 3315*x^9 + 78*x^8 + 1215*x^7 + 1528*x^6 + 168*x^5 - 351*x^4 - 826*x^3 - 4*x^2 + 39*x - 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 2*x^21 + 5*x^20 - 4*x^19 + 4*x^18 - 14*x^17 + 63*x^16 - 228*x^15 + 497*x^14 - 686*x^13 + 792*x^12 - 793*x^11 + 1821*x^10 - 3315*x^9 + 78*x^8 + 1215*x^7 + 1528*x^6 + 168*x^5 - 351*x^4 - 826*x^3 - 4*x^2 + 39*x - 5)
\( x^{22} - 2 x^{21} + 5 x^{20} - 4 x^{19} + 4 x^{18} - 14 x^{17} + 63 x^{16} - 228 x^{15} + 497 x^{14} + \cdots - 5 \)
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: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(499131506491608208145638755501\)
\(\medspace = 3^{11}\cdot 167^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.38\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $3^{1/2}167^{1/2}\approx 22.38302928559939$ | ||
| Ramified primes: |
\(3\), \(167\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{501}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5}a^{14}+\frac{2}{5}a^{13}+\frac{2}{5}a^{12}-\frac{1}{5}a^{11}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{2}{5}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{5}a^{15}-\frac{2}{5}a^{13}-\frac{2}{5}a^{11}-\frac{1}{5}a^{10}+\frac{1}{5}a^{9}+\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{5}a^{16}-\frac{1}{5}a^{13}+\frac{2}{5}a^{12}+\frac{2}{5}a^{11}-\frac{2}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{4}-\frac{2}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{5}a^{17}-\frac{1}{5}a^{13}-\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{1}{5}a^{10}-\frac{2}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{5}a^{18}+\frac{1}{5}a^{13}-\frac{1}{5}a^{12}-\frac{2}{5}a^{11}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}+\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a$, $\frac{1}{25}a^{19}-\frac{2}{25}a^{16}+\frac{1}{25}a^{14}+\frac{11}{25}a^{13}+\frac{4}{25}a^{12}-\frac{8}{25}a^{11}-\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{9}{25}a^{8}-\frac{12}{25}a^{7}-\frac{4}{25}a^{6}-\frac{3}{25}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{8}{25}a^{2}+\frac{8}{25}a-\frac{2}{5}$, $\frac{1}{25}a^{20}-\frac{2}{25}a^{17}+\frac{1}{25}a^{15}+\frac{1}{25}a^{14}+\frac{9}{25}a^{13}-\frac{3}{25}a^{12}+\frac{1}{5}a^{11}-\frac{1}{5}a^{10}+\frac{6}{25}a^{9}+\frac{8}{25}a^{8}+\frac{1}{25}a^{7}+\frac{2}{25}a^{6}+\frac{12}{25}a^{3}-\frac{12}{25}a^{2}+\frac{2}{5}a$, $\frac{1}{74\!\cdots\!75}a^{21}+\frac{85\!\cdots\!98}{74\!\cdots\!75}a^{20}-\frac{13\!\cdots\!14}{74\!\cdots\!75}a^{19}-\frac{64\!\cdots\!42}{74\!\cdots\!75}a^{18}+\frac{67\!\cdots\!89}{74\!\cdots\!75}a^{17}+\frac{42\!\cdots\!54}{74\!\cdots\!75}a^{16}-\frac{34\!\cdots\!01}{74\!\cdots\!75}a^{15}+\frac{63\!\cdots\!33}{74\!\cdots\!75}a^{14}-\frac{18\!\cdots\!34}{14\!\cdots\!35}a^{13}-\frac{55\!\cdots\!62}{14\!\cdots\!35}a^{12}+\frac{24\!\cdots\!07}{74\!\cdots\!75}a^{11}-\frac{24\!\cdots\!74}{74\!\cdots\!75}a^{10}-\frac{17\!\cdots\!74}{74\!\cdots\!75}a^{9}+\frac{31\!\cdots\!66}{74\!\cdots\!75}a^{8}-\frac{50\!\cdots\!07}{74\!\cdots\!75}a^{7}+\frac{28\!\cdots\!47}{74\!\cdots\!75}a^{6}-\frac{31\!\cdots\!48}{74\!\cdots\!75}a^{5}-\frac{12\!\cdots\!73}{74\!\cdots\!75}a^{4}-\frac{30\!\cdots\!61}{74\!\cdots\!75}a^{3}+\frac{31\!\cdots\!56}{74\!\cdots\!75}a^{2}+\frac{18\!\cdots\!73}{74\!\cdots\!75}a+\frac{69\!\cdots\!18}{14\!\cdots\!35}$
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: | $11$ | 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: |
$\frac{23\!\cdots\!43}{74\!\cdots\!75}a^{21}-\frac{23\!\cdots\!29}{74\!\cdots\!75}a^{20}+\frac{42\!\cdots\!56}{74\!\cdots\!75}a^{19}+\frac{49\!\cdots\!79}{74\!\cdots\!75}a^{18}-\frac{10\!\cdots\!97}{74\!\cdots\!75}a^{17}-\frac{24\!\cdots\!29}{74\!\cdots\!75}a^{16}+\frac{10\!\cdots\!74}{74\!\cdots\!75}a^{15}-\frac{36\!\cdots\!66}{74\!\cdots\!75}a^{14}+\frac{48\!\cdots\!26}{74\!\cdots\!75}a^{13}+\frac{30\!\cdots\!11}{74\!\cdots\!75}a^{12}-\frac{58\!\cdots\!83}{74\!\cdots\!75}a^{11}+\frac{84\!\cdots\!23}{74\!\cdots\!75}a^{10}+\frac{29\!\cdots\!53}{14\!\cdots\!35}a^{9}-\frac{27\!\cdots\!48}{74\!\cdots\!75}a^{8}-\frac{22\!\cdots\!19}{14\!\cdots\!35}a^{7}+\frac{78\!\cdots\!23}{74\!\cdots\!75}a^{6}+\frac{12\!\cdots\!62}{74\!\cdots\!75}a^{5}+\frac{52\!\cdots\!36}{74\!\cdots\!75}a^{4}-\frac{43\!\cdots\!74}{74\!\cdots\!75}a^{3}-\frac{12\!\cdots\!02}{14\!\cdots\!35}a^{2}-\frac{40\!\cdots\!97}{74\!\cdots\!75}a+\frac{36\!\cdots\!78}{14\!\cdots\!35}$, $\frac{10\!\cdots\!53}{74\!\cdots\!75}a^{21}-\frac{20\!\cdots\!23}{74\!\cdots\!75}a^{20}+\frac{50\!\cdots\!69}{74\!\cdots\!75}a^{19}-\frac{37\!\cdots\!46}{74\!\cdots\!75}a^{18}+\frac{37\!\cdots\!31}{74\!\cdots\!75}a^{17}-\frac{27\!\cdots\!48}{14\!\cdots\!35}a^{16}+\frac{25\!\cdots\!55}{29\!\cdots\!87}a^{15}-\frac{23\!\cdots\!22}{74\!\cdots\!75}a^{14}+\frac{49\!\cdots\!18}{74\!\cdots\!75}a^{13}-\frac{13\!\cdots\!47}{14\!\cdots\!35}a^{12}+\frac{75\!\cdots\!63}{74\!\cdots\!75}a^{11}-\frac{72\!\cdots\!92}{74\!\cdots\!75}a^{10}+\frac{17\!\cdots\!01}{74\!\cdots\!75}a^{9}-\frac{32\!\cdots\!27}{74\!\cdots\!75}a^{8}-\frac{25\!\cdots\!78}{14\!\cdots\!35}a^{7}+\frac{14\!\cdots\!58}{74\!\cdots\!75}a^{6}+\frac{13\!\cdots\!48}{74\!\cdots\!75}a^{5}-\frac{10\!\cdots\!19}{74\!\cdots\!75}a^{4}-\frac{37\!\cdots\!67}{74\!\cdots\!75}a^{3}-\frac{69\!\cdots\!66}{74\!\cdots\!75}a^{2}+\frac{10\!\cdots\!72}{74\!\cdots\!75}a+\frac{31\!\cdots\!62}{14\!\cdots\!35}$, $\frac{14\!\cdots\!03}{74\!\cdots\!75}a^{21}-\frac{47\!\cdots\!57}{14\!\cdots\!35}a^{20}+\frac{64\!\cdots\!73}{74\!\cdots\!75}a^{19}-\frac{30\!\cdots\!01}{74\!\cdots\!75}a^{18}+\frac{82\!\cdots\!97}{14\!\cdots\!35}a^{17}-\frac{17\!\cdots\!03}{74\!\cdots\!75}a^{16}+\frac{85\!\cdots\!78}{74\!\cdots\!75}a^{15}-\frac{60\!\cdots\!87}{14\!\cdots\!35}a^{14}+\frac{61\!\cdots\!34}{74\!\cdots\!75}a^{13}-\frac{75\!\cdots\!53}{74\!\cdots\!75}a^{12}+\frac{80\!\cdots\!06}{74\!\cdots\!75}a^{11}-\frac{72\!\cdots\!22}{74\!\cdots\!75}a^{10}+\frac{22\!\cdots\!39}{74\!\cdots\!75}a^{9}-\frac{38\!\cdots\!09}{74\!\cdots\!75}a^{8}-\frac{33\!\cdots\!71}{14\!\cdots\!35}a^{7}+\frac{17\!\cdots\!23}{74\!\cdots\!75}a^{6}+\frac{23\!\cdots\!66}{74\!\cdots\!75}a^{5}+\frac{10\!\cdots\!76}{74\!\cdots\!75}a^{4}-\frac{31\!\cdots\!71}{74\!\cdots\!75}a^{3}-\frac{10\!\cdots\!54}{74\!\cdots\!75}a^{2}-\frac{23\!\cdots\!01}{74\!\cdots\!75}a+\frac{20\!\cdots\!24}{14\!\cdots\!35}$, $\frac{19\!\cdots\!32}{74\!\cdots\!75}a^{21}-\frac{42\!\cdots\!43}{74\!\cdots\!75}a^{20}+\frac{10\!\cdots\!13}{74\!\cdots\!75}a^{19}-\frac{96\!\cdots\!29}{74\!\cdots\!75}a^{18}+\frac{93\!\cdots\!76}{74\!\cdots\!75}a^{17}-\frac{29\!\cdots\!69}{74\!\cdots\!75}a^{16}+\frac{12\!\cdots\!39}{74\!\cdots\!75}a^{15}-\frac{46\!\cdots\!02}{74\!\cdots\!75}a^{14}+\frac{21\!\cdots\!97}{14\!\cdots\!35}a^{13}-\frac{15\!\cdots\!54}{74\!\cdots\!75}a^{12}+\frac{18\!\cdots\!16}{74\!\cdots\!75}a^{11}-\frac{18\!\cdots\!98}{74\!\cdots\!75}a^{10}+\frac{38\!\cdots\!13}{74\!\cdots\!75}a^{9}-\frac{71\!\cdots\!24}{74\!\cdots\!75}a^{8}+\frac{27\!\cdots\!16}{14\!\cdots\!35}a^{7}+\frac{22\!\cdots\!42}{74\!\cdots\!75}a^{6}+\frac{26\!\cdots\!46}{74\!\cdots\!75}a^{5}+\frac{10\!\cdots\!44}{74\!\cdots\!75}a^{4}-\frac{29\!\cdots\!32}{29\!\cdots\!87}a^{3}-\frac{16\!\cdots\!18}{74\!\cdots\!75}a^{2}+\frac{12\!\cdots\!44}{74\!\cdots\!75}a+\frac{52\!\cdots\!44}{14\!\cdots\!35}$, $\frac{69\!\cdots\!91}{74\!\cdots\!75}a^{21}-\frac{14\!\cdots\!89}{29\!\cdots\!87}a^{20}+\frac{12\!\cdots\!91}{74\!\cdots\!75}a^{19}+\frac{27\!\cdots\!93}{74\!\cdots\!75}a^{18}-\frac{40\!\cdots\!94}{14\!\cdots\!35}a^{17}-\frac{49\!\cdots\!96}{74\!\cdots\!75}a^{16}+\frac{28\!\cdots\!16}{74\!\cdots\!75}a^{15}-\frac{36\!\cdots\!61}{29\!\cdots\!87}a^{14}+\frac{10\!\cdots\!33}{74\!\cdots\!75}a^{13}+\frac{68\!\cdots\!84}{74\!\cdots\!75}a^{12}-\frac{23\!\cdots\!98}{74\!\cdots\!75}a^{11}+\frac{37\!\cdots\!86}{74\!\cdots\!75}a^{10}+\frac{32\!\cdots\!08}{74\!\cdots\!75}a^{9}-\frac{30\!\cdots\!03}{74\!\cdots\!75}a^{8}-\frac{14\!\cdots\!77}{29\!\cdots\!87}a^{7}+\frac{13\!\cdots\!21}{74\!\cdots\!75}a^{6}+\frac{21\!\cdots\!82}{74\!\cdots\!75}a^{5}+\frac{14\!\cdots\!42}{74\!\cdots\!75}a^{4}-\frac{92\!\cdots\!07}{74\!\cdots\!75}a^{3}-\frac{89\!\cdots\!88}{74\!\cdots\!75}a^{2}-\frac{76\!\cdots\!67}{74\!\cdots\!75}a+\frac{15\!\cdots\!43}{14\!\cdots\!35}$, $\frac{19\!\cdots\!32}{74\!\cdots\!75}a^{21}-\frac{42\!\cdots\!43}{74\!\cdots\!75}a^{20}+\frac{10\!\cdots\!13}{74\!\cdots\!75}a^{19}-\frac{96\!\cdots\!29}{74\!\cdots\!75}a^{18}+\frac{93\!\cdots\!76}{74\!\cdots\!75}a^{17}-\frac{29\!\cdots\!69}{74\!\cdots\!75}a^{16}+\frac{12\!\cdots\!39}{74\!\cdots\!75}a^{15}-\frac{46\!\cdots\!02}{74\!\cdots\!75}a^{14}+\frac{21\!\cdots\!97}{14\!\cdots\!35}a^{13}-\frac{15\!\cdots\!54}{74\!\cdots\!75}a^{12}+\frac{18\!\cdots\!16}{74\!\cdots\!75}a^{11}-\frac{18\!\cdots\!98}{74\!\cdots\!75}a^{10}+\frac{38\!\cdots\!13}{74\!\cdots\!75}a^{9}-\frac{71\!\cdots\!24}{74\!\cdots\!75}a^{8}+\frac{27\!\cdots\!16}{14\!\cdots\!35}a^{7}+\frac{22\!\cdots\!42}{74\!\cdots\!75}a^{6}+\frac{26\!\cdots\!46}{74\!\cdots\!75}a^{5}+\frac{10\!\cdots\!44}{74\!\cdots\!75}a^{4}-\frac{29\!\cdots\!32}{29\!\cdots\!87}a^{3}-\frac{16\!\cdots\!18}{74\!\cdots\!75}a^{2}+\frac{20\!\cdots\!19}{74\!\cdots\!75}a+\frac{52\!\cdots\!44}{14\!\cdots\!35}$, $\frac{17\!\cdots\!76}{14\!\cdots\!35}a^{21}-\frac{12\!\cdots\!69}{74\!\cdots\!75}a^{20}+\frac{34\!\cdots\!61}{74\!\cdots\!75}a^{19}-\frac{16\!\cdots\!92}{14\!\cdots\!35}a^{18}+\frac{15\!\cdots\!88}{74\!\cdots\!75}a^{17}-\frac{88\!\cdots\!57}{74\!\cdots\!75}a^{16}+\frac{46\!\cdots\!16}{74\!\cdots\!75}a^{15}-\frac{16\!\cdots\!08}{74\!\cdots\!75}a^{14}+\frac{12\!\cdots\!02}{29\!\cdots\!87}a^{13}-\frac{35\!\cdots\!19}{74\!\cdots\!75}a^{12}+\frac{32\!\cdots\!22}{74\!\cdots\!75}a^{11}-\frac{81\!\cdots\!20}{29\!\cdots\!87}a^{10}+\frac{10\!\cdots\!71}{74\!\cdots\!75}a^{9}-\frac{16\!\cdots\!86}{74\!\cdots\!75}a^{8}-\frac{17\!\cdots\!96}{74\!\cdots\!75}a^{7}+\frac{13\!\cdots\!03}{74\!\cdots\!75}a^{6}+\frac{77\!\cdots\!87}{74\!\cdots\!75}a^{5}+\frac{31\!\cdots\!19}{14\!\cdots\!35}a^{4}-\frac{18\!\cdots\!83}{74\!\cdots\!75}a^{3}-\frac{25\!\cdots\!17}{29\!\cdots\!87}a^{2}-\frac{32\!\cdots\!57}{74\!\cdots\!75}a-\frac{69\!\cdots\!82}{14\!\cdots\!35}$, $\frac{24\!\cdots\!94}{74\!\cdots\!75}a^{21}+\frac{36\!\cdots\!04}{29\!\cdots\!87}a^{20}-\frac{22\!\cdots\!91}{74\!\cdots\!75}a^{19}+\frac{22\!\cdots\!92}{74\!\cdots\!75}a^{18}-\frac{99\!\cdots\!43}{29\!\cdots\!87}a^{17}-\frac{88\!\cdots\!84}{74\!\cdots\!75}a^{16}+\frac{60\!\cdots\!19}{74\!\cdots\!75}a^{15}-\frac{31\!\cdots\!38}{14\!\cdots\!35}a^{14}-\frac{25\!\cdots\!08}{74\!\cdots\!75}a^{13}+\frac{17\!\cdots\!16}{74\!\cdots\!75}a^{12}-\frac{29\!\cdots\!52}{74\!\cdots\!75}a^{11}+\frac{38\!\cdots\!49}{74\!\cdots\!75}a^{10}-\frac{15\!\cdots\!23}{74\!\cdots\!75}a^{9}+\frac{38\!\cdots\!93}{74\!\cdots\!75}a^{8}-\frac{46\!\cdots\!17}{14\!\cdots\!35}a^{7}+\frac{95\!\cdots\!49}{74\!\cdots\!75}a^{6}+\frac{15\!\cdots\!98}{74\!\cdots\!75}a^{5}+\frac{11\!\cdots\!68}{74\!\cdots\!75}a^{4}-\frac{41\!\cdots\!93}{74\!\cdots\!75}a^{3}-\frac{85\!\cdots\!02}{74\!\cdots\!75}a^{2}-\frac{91\!\cdots\!33}{74\!\cdots\!75}a-\frac{19\!\cdots\!23}{14\!\cdots\!35}$, $\frac{51\!\cdots\!91}{74\!\cdots\!75}a^{21}-\frac{10\!\cdots\!93}{74\!\cdots\!75}a^{20}+\frac{26\!\cdots\!92}{74\!\cdots\!75}a^{19}-\frac{21\!\cdots\!27}{74\!\cdots\!75}a^{18}+\frac{25\!\cdots\!96}{74\!\cdots\!75}a^{17}-\frac{72\!\cdots\!73}{74\!\cdots\!75}a^{16}+\frac{32\!\cdots\!68}{74\!\cdots\!75}a^{15}-\frac{11\!\cdots\!12}{74\!\cdots\!75}a^{14}+\frac{25\!\cdots\!02}{74\!\cdots\!75}a^{13}-\frac{36\!\cdots\!08}{74\!\cdots\!75}a^{12}+\frac{44\!\cdots\!09}{74\!\cdots\!75}a^{11}-\frac{44\!\cdots\!69}{74\!\cdots\!75}a^{10}+\frac{19\!\cdots\!92}{14\!\cdots\!35}a^{9}-\frac{17\!\cdots\!66}{74\!\cdots\!75}a^{8}+\frac{63\!\cdots\!40}{29\!\cdots\!87}a^{7}+\frac{35\!\cdots\!21}{74\!\cdots\!75}a^{6}+\frac{64\!\cdots\!94}{74\!\cdots\!75}a^{5}+\frac{84\!\cdots\!77}{74\!\cdots\!75}a^{4}-\frac{59\!\cdots\!43}{74\!\cdots\!75}a^{3}-\frac{12\!\cdots\!33}{29\!\cdots\!87}a^{2}+\frac{61\!\cdots\!11}{74\!\cdots\!75}a+\frac{16\!\cdots\!06}{14\!\cdots\!35}$, $\frac{15\!\cdots\!85}{29\!\cdots\!87}a^{21}-\frac{13\!\cdots\!68}{14\!\cdots\!35}a^{20}+\frac{17\!\cdots\!94}{74\!\cdots\!75}a^{19}-\frac{20\!\cdots\!84}{14\!\cdots\!35}a^{18}+\frac{23\!\cdots\!29}{14\!\cdots\!35}a^{17}-\frac{46\!\cdots\!68}{74\!\cdots\!75}a^{16}+\frac{45\!\cdots\!36}{14\!\cdots\!35}a^{15}-\frac{81\!\cdots\!46}{74\!\cdots\!75}a^{14}+\frac{17\!\cdots\!54}{74\!\cdots\!75}a^{13}-\frac{21\!\cdots\!84}{74\!\cdots\!75}a^{12}+\frac{23\!\cdots\!68}{74\!\cdots\!75}a^{11}-\frac{41\!\cdots\!48}{14\!\cdots\!35}a^{10}+\frac{11\!\cdots\!73}{14\!\cdots\!35}a^{9}-\frac{10\!\cdots\!86}{74\!\cdots\!75}a^{8}-\frac{28\!\cdots\!78}{74\!\cdots\!75}a^{7}+\frac{53\!\cdots\!14}{74\!\cdots\!75}a^{6}+\frac{45\!\cdots\!73}{74\!\cdots\!75}a^{5}+\frac{38\!\cdots\!77}{14\!\cdots\!35}a^{4}-\frac{93\!\cdots\!33}{29\!\cdots\!87}a^{3}-\frac{29\!\cdots\!12}{74\!\cdots\!75}a^{2}-\frac{97\!\cdots\!88}{74\!\cdots\!75}a+\frac{28\!\cdots\!77}{14\!\cdots\!35}$, $\frac{11\!\cdots\!47}{74\!\cdots\!75}a^{21}-\frac{70\!\cdots\!28}{74\!\cdots\!75}a^{20}+\frac{11\!\cdots\!18}{74\!\cdots\!75}a^{19}-\frac{32\!\cdots\!89}{74\!\cdots\!75}a^{18}+\frac{16\!\cdots\!06}{74\!\cdots\!75}a^{17}-\frac{31\!\cdots\!59}{74\!\cdots\!75}a^{16}+\frac{92\!\cdots\!99}{74\!\cdots\!75}a^{15}-\frac{42\!\cdots\!62}{74\!\cdots\!75}a^{14}+\frac{29\!\cdots\!74}{14\!\cdots\!35}a^{13}-\frac{29\!\cdots\!29}{74\!\cdots\!75}a^{12}+\frac{39\!\cdots\!96}{74\!\cdots\!75}a^{11}-\frac{46\!\cdots\!53}{74\!\cdots\!75}a^{10}+\frac{46\!\cdots\!48}{74\!\cdots\!75}a^{9}-\frac{12\!\cdots\!59}{74\!\cdots\!75}a^{8}+\frac{37\!\cdots\!39}{14\!\cdots\!35}a^{7}+\frac{40\!\cdots\!27}{74\!\cdots\!75}a^{6}-\frac{86\!\cdots\!29}{74\!\cdots\!75}a^{5}-\frac{79\!\cdots\!96}{74\!\cdots\!75}a^{4}-\frac{94\!\cdots\!92}{14\!\cdots\!35}a^{3}-\frac{22\!\cdots\!78}{74\!\cdots\!75}a^{2}+\frac{21\!\cdots\!29}{74\!\cdots\!75}a-\frac{65\!\cdots\!31}{14\!\cdots\!35}$
| 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: | \( 1424457.4515 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 1424457.4515 \cdot 1}{2\cdot\sqrt{499131506491608208145638755501}}\cr\approx \mathstrut & 0.38669688029 \end{aligned}\] (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 + 5*x^20 - 4*x^19 + 4*x^18 - 14*x^17 + 63*x^16 - 228*x^15 + 497*x^14 - 686*x^13 + 792*x^12 - 793*x^11 + 1821*x^10 - 3315*x^9 + 78*x^8 + 1215*x^7 + 1528*x^6 + 168*x^5 - 351*x^4 - 826*x^3 - 4*x^2 + 39*x - 5)
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 + 5*x^20 - 4*x^19 + 4*x^18 - 14*x^17 + 63*x^16 - 228*x^15 + 497*x^14 - 686*x^13 + 792*x^12 - 793*x^11 + 1821*x^10 - 3315*x^9 + 78*x^8 + 1215*x^7 + 1528*x^6 + 168*x^5 - 351*x^4 - 826*x^3 - 4*x^2 + 39*x - 5, 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 + 5*x^20 - 4*x^19 + 4*x^18 - 14*x^17 + 63*x^16 - 228*x^15 + 497*x^14 - 686*x^13 + 792*x^12 - 793*x^11 + 1821*x^10 - 3315*x^9 + 78*x^8 + 1215*x^7 + 1528*x^6 + 168*x^5 - 351*x^4 - 826*x^3 - 4*x^2 + 39*x - 5);
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 + 5*x^20 - 4*x^19 + 4*x^18 - 14*x^17 + 63*x^16 - 228*x^15 + 497*x^14 - 686*x^13 + 792*x^12 - 793*x^11 + 1821*x^10 - 3315*x^9 + 78*x^8 + 1215*x^7 + 1528*x^6 + 168*x^5 - 351*x^4 - 826*x^3 - 4*x^2 + 39*x - 5);
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
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 44 |
| The 14 conjugacy class representatives for $D_{22}$ |
| Character table for $D_{22}$ |
Intermediate fields
| \(\Q(\sqrt{501}) \), 11.1.129891985607.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Galois closure: | deg 44 |
| Degree 22 sibling: | 22.0.2988811416117414420033765003.1 |
| Minimal sibling: | 22.0.2988811416117414420033765003.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $22$ | R | ${\href{/padicField/5.2.0.1}{2} }^{10}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/13.2.0.1}{2} }^{11}$ | ${\href{/padicField/17.2.0.1}{2} }^{10}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.11.0.1}{11} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{11}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{11}$ | $22$ | ${\href{/padicField/53.2.0.1}{2} }^{10}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.22.11.2 | $x^{22} + 33 x^{20} + 495 x^{18} + 4455 x^{16} + 26730 x^{14} + 4 x^{13} + 112266 x^{12} - 406 x^{11} + 336798 x^{10} + 1650 x^{9} + 721710 x^{8} + 20196 x^{7} + 1082565 x^{6} - 35640 x^{5} + 1082569 x^{4} - 37422 x^{3} + 649567 x^{2} + 20898 x + 177172$ | $2$ | $11$ | $11$ | 22T1 | $[\ ]_{2}^{11}$ |
|
\(167\)
| 167.2.1.1 | $x^{2} + 835$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |