Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 7*x^19 + 8*x^18 + 7*x^17 + 17*x^16 - 26*x^15 - 35*x^14 - 4*x^13 + 68*x^12 + 92*x^11 + 10*x^10 - 49*x^9 - 27*x^8 + 32*x^7 + 57*x^6 + 24*x^5 - 13*x^4 - 17*x^3 - 3*x^2 + 2*x + 1)
gp: K = bnfinit(y^22 - y^21 - 7*y^19 + 8*y^18 + 7*y^17 + 17*y^16 - 26*y^15 - 35*y^14 - 4*y^13 + 68*y^12 + 92*y^11 + 10*y^10 - 49*y^9 - 27*y^8 + 32*y^7 + 57*y^6 + 24*y^5 - 13*y^4 - 17*y^3 - 3*y^2 + 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^21 - 7*x^19 + 8*x^18 + 7*x^17 + 17*x^16 - 26*x^15 - 35*x^14 - 4*x^13 + 68*x^12 + 92*x^11 + 10*x^10 - 49*x^9 - 27*x^8 + 32*x^7 + 57*x^6 + 24*x^5 - 13*x^4 - 17*x^3 - 3*x^2 + 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^21 - 7*x^19 + 8*x^18 + 7*x^17 + 17*x^16 - 26*x^15 - 35*x^14 - 4*x^13 + 68*x^12 + 92*x^11 + 10*x^10 - 49*x^9 - 27*x^8 + 32*x^7 + 57*x^6 + 24*x^5 - 13*x^4 - 17*x^3 - 3*x^2 + 2*x + 1)
\( x^{22} - x^{21} - 7 x^{19} + 8 x^{18} + 7 x^{17} + 17 x^{16} - 26 x^{15} - 35 x^{14} - 4 x^{13} + \cdots + 1 \)
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: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-3273714539904703691187619\)
\(\medspace = -\,61\cdot 1279\cdot 1609^{2}\cdot 4025911^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.01\) | 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: | $61^{1/2}1279^{1/2}1609^{1/2}4025911^{1/2}\approx 22480724.15308682$ | ||
| Ramified primes: |
\(61\), \(1279\), \(1609\), \(4025911\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-78019}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{11\!\cdots\!97}a^{21}-\frac{739608894551581}{11\!\cdots\!97}a^{20}-\frac{469895265641984}{11\!\cdots\!97}a^{19}+\frac{435265577257926}{11\!\cdots\!97}a^{18}-\frac{57\!\cdots\!37}{11\!\cdots\!97}a^{17}-\frac{168388016981139}{11\!\cdots\!97}a^{16}-\frac{11\!\cdots\!67}{11\!\cdots\!97}a^{15}+\frac{30\!\cdots\!46}{11\!\cdots\!97}a^{14}-\frac{24\!\cdots\!98}{11\!\cdots\!97}a^{13}-\frac{41\!\cdots\!29}{11\!\cdots\!97}a^{12}+\frac{215875868487319}{11\!\cdots\!97}a^{11}-\frac{34\!\cdots\!06}{11\!\cdots\!97}a^{10}+\frac{25\!\cdots\!63}{11\!\cdots\!97}a^{9}+\frac{23\!\cdots\!50}{11\!\cdots\!97}a^{8}+\frac{18\!\cdots\!17}{11\!\cdots\!97}a^{7}+\frac{848460951775526}{11\!\cdots\!97}a^{6}-\frac{23\!\cdots\!10}{11\!\cdots\!97}a^{5}-\frac{31\!\cdots\!00}{11\!\cdots\!97}a^{4}+\frac{55\!\cdots\!56}{11\!\cdots\!97}a^{3}+\frac{36\!\cdots\!74}{11\!\cdots\!97}a^{2}+\frac{15\!\cdots\!48}{11\!\cdots\!97}a-\frac{10\!\cdots\!70}{11\!\cdots\!97}$
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: | $10$ | 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: |
$\frac{46\!\cdots\!15}{11\!\cdots\!97}a^{21}-\frac{96\!\cdots\!99}{11\!\cdots\!97}a^{20}+\frac{56\!\cdots\!33}{11\!\cdots\!97}a^{19}-\frac{33\!\cdots\!98}{11\!\cdots\!97}a^{18}+\frac{73\!\cdots\!49}{11\!\cdots\!97}a^{17}-\frac{12\!\cdots\!05}{11\!\cdots\!97}a^{16}+\frac{53\!\cdots\!47}{11\!\cdots\!97}a^{15}-\frac{20\!\cdots\!42}{11\!\cdots\!97}a^{14}-\frac{27\!\cdots\!22}{11\!\cdots\!97}a^{13}+\frac{13\!\cdots\!40}{11\!\cdots\!97}a^{12}+\frac{32\!\cdots\!71}{11\!\cdots\!97}a^{11}+\frac{11\!\cdots\!31}{11\!\cdots\!97}a^{10}-\frac{39\!\cdots\!52}{11\!\cdots\!97}a^{9}-\frac{25\!\cdots\!17}{11\!\cdots\!97}a^{8}+\frac{47\!\cdots\!40}{11\!\cdots\!97}a^{7}+\frac{25\!\cdots\!94}{11\!\cdots\!97}a^{6}+\frac{96\!\cdots\!93}{11\!\cdots\!97}a^{5}-\frac{15\!\cdots\!94}{11\!\cdots\!97}a^{4}-\frac{15\!\cdots\!60}{11\!\cdots\!97}a^{3}-\frac{31\!\cdots\!39}{11\!\cdots\!97}a^{2}+\frac{49\!\cdots\!54}{11\!\cdots\!97}a+\frac{13\!\cdots\!13}{11\!\cdots\!97}$, $\frac{24\!\cdots\!97}{11\!\cdots\!97}a^{21}-\frac{14\!\cdots\!04}{11\!\cdots\!97}a^{20}+\frac{14\!\cdots\!05}{11\!\cdots\!97}a^{19}-\frac{22\!\cdots\!59}{11\!\cdots\!97}a^{18}+\frac{10\!\cdots\!93}{11\!\cdots\!97}a^{17}-\frac{96\!\cdots\!45}{11\!\cdots\!97}a^{16}-\frac{66\!\cdots\!52}{11\!\cdots\!97}a^{15}-\frac{24\!\cdots\!18}{11\!\cdots\!97}a^{14}+\frac{24\!\cdots\!28}{11\!\cdots\!97}a^{13}+\frac{30\!\cdots\!14}{11\!\cdots\!97}a^{12}+\frac{13\!\cdots\!39}{11\!\cdots\!97}a^{11}-\frac{48\!\cdots\!57}{11\!\cdots\!97}a^{10}-\frac{84\!\cdots\!63}{11\!\cdots\!97}a^{9}-\frac{98\!\cdots\!39}{11\!\cdots\!97}a^{8}+\frac{31\!\cdots\!18}{11\!\cdots\!97}a^{7}+\frac{17\!\cdots\!84}{11\!\cdots\!97}a^{6}-\frac{20\!\cdots\!29}{11\!\cdots\!97}a^{5}-\frac{47\!\cdots\!01}{11\!\cdots\!97}a^{4}-\frac{20\!\cdots\!45}{11\!\cdots\!97}a^{3}+\frac{58\!\cdots\!89}{11\!\cdots\!97}a^{2}+\frac{80\!\cdots\!74}{11\!\cdots\!97}a+\frac{54\!\cdots\!80}{11\!\cdots\!97}$, $\frac{10\!\cdots\!42}{11\!\cdots\!97}a^{21}-\frac{94\!\cdots\!84}{11\!\cdots\!97}a^{20}+\frac{10\!\cdots\!95}{11\!\cdots\!97}a^{19}-\frac{10\!\cdots\!29}{11\!\cdots\!97}a^{18}+\frac{68\!\cdots\!79}{11\!\cdots\!97}a^{17}-\frac{71\!\cdots\!50}{11\!\cdots\!97}a^{16}-\frac{15\!\cdots\!57}{11\!\cdots\!97}a^{15}-\frac{17\!\cdots\!38}{11\!\cdots\!97}a^{14}+\frac{19\!\cdots\!79}{11\!\cdots\!97}a^{13}+\frac{21\!\cdots\!41}{11\!\cdots\!97}a^{12}+\frac{91\!\cdots\!03}{11\!\cdots\!97}a^{11}-\frac{42\!\cdots\!46}{11\!\cdots\!97}a^{10}-\frac{63\!\cdots\!80}{11\!\cdots\!97}a^{9}-\frac{11\!\cdots\!64}{11\!\cdots\!97}a^{8}+\frac{23\!\cdots\!78}{11\!\cdots\!97}a^{7}+\frac{12\!\cdots\!41}{11\!\cdots\!97}a^{6}-\frac{18\!\cdots\!11}{11\!\cdots\!97}a^{5}-\frac{36\!\cdots\!10}{11\!\cdots\!97}a^{4}-\frac{18\!\cdots\!75}{11\!\cdots\!97}a^{3}+\frac{31\!\cdots\!55}{11\!\cdots\!97}a^{2}+\frac{58\!\cdots\!45}{11\!\cdots\!97}a+\frac{82\!\cdots\!31}{11\!\cdots\!97}$, $\frac{24\!\cdots\!39}{11\!\cdots\!97}a^{21}-\frac{11\!\cdots\!61}{11\!\cdots\!97}a^{20}+\frac{16\!\cdots\!59}{11\!\cdots\!97}a^{19}-\frac{29\!\cdots\!40}{11\!\cdots\!97}a^{18}+\frac{86\!\cdots\!85}{11\!\cdots\!97}a^{17}-\frac{10\!\cdots\!60}{11\!\cdots\!97}a^{16}+\frac{73\!\cdots\!09}{11\!\cdots\!97}a^{15}-\frac{19\!\cdots\!11}{11\!\cdots\!97}a^{14}+\frac{24\!\cdots\!83}{11\!\cdots\!97}a^{13}+\frac{36\!\cdots\!39}{11\!\cdots\!97}a^{12}+\frac{44\!\cdots\!94}{11\!\cdots\!97}a^{11}-\frac{25\!\cdots\!28}{11\!\cdots\!97}a^{10}-\frac{29\!\cdots\!52}{11\!\cdots\!97}a^{9}+\frac{23\!\cdots\!43}{11\!\cdots\!97}a^{8}+\frac{11\!\cdots\!35}{11\!\cdots\!97}a^{7}+\frac{12\!\cdots\!95}{11\!\cdots\!97}a^{6}-\frac{12\!\cdots\!01}{11\!\cdots\!97}a^{5}-\frac{16\!\cdots\!89}{11\!\cdots\!97}a^{4}+\frac{39\!\cdots\!75}{11\!\cdots\!97}a^{3}+\frac{61\!\cdots\!36}{11\!\cdots\!97}a^{2}+\frac{53\!\cdots\!28}{11\!\cdots\!97}a-\frac{18\!\cdots\!49}{11\!\cdots\!97}$, $\frac{11\!\cdots\!73}{11\!\cdots\!97}a^{21}-\frac{53\!\cdots\!66}{11\!\cdots\!97}a^{20}-\frac{95\!\cdots\!77}{11\!\cdots\!97}a^{19}-\frac{76\!\cdots\!56}{11\!\cdots\!97}a^{18}+\frac{46\!\cdots\!13}{11\!\cdots\!97}a^{17}+\frac{15\!\cdots\!44}{11\!\cdots\!97}a^{16}+\frac{20\!\cdots\!23}{11\!\cdots\!97}a^{15}-\frac{19\!\cdots\!16}{11\!\cdots\!97}a^{14}-\frac{60\!\cdots\!71}{11\!\cdots\!97}a^{13}-\frac{16\!\cdots\!16}{11\!\cdots\!97}a^{12}+\frac{82\!\cdots\!45}{11\!\cdots\!97}a^{11}+\frac{14\!\cdots\!04}{11\!\cdots\!97}a^{10}+\frac{53\!\cdots\!42}{11\!\cdots\!97}a^{9}-\frac{61\!\cdots\!31}{11\!\cdots\!97}a^{8}-\frac{39\!\cdots\!53}{11\!\cdots\!97}a^{7}+\frac{33\!\cdots\!60}{11\!\cdots\!97}a^{6}+\frac{80\!\cdots\!46}{11\!\cdots\!97}a^{5}+\frac{52\!\cdots\!89}{11\!\cdots\!97}a^{4}-\frac{69\!\cdots\!75}{11\!\cdots\!97}a^{3}-\frac{19\!\cdots\!40}{11\!\cdots\!97}a^{2}-\frac{52\!\cdots\!29}{11\!\cdots\!97}a+\frac{85\!\cdots\!60}{11\!\cdots\!97}$, $\frac{82\!\cdots\!66}{11\!\cdots\!97}a^{21}-\frac{12\!\cdots\!59}{11\!\cdots\!97}a^{20}+\frac{20\!\cdots\!26}{11\!\cdots\!97}a^{19}-\frac{54\!\cdots\!87}{11\!\cdots\!97}a^{18}+\frac{96\!\cdots\!29}{11\!\cdots\!97}a^{17}+\frac{39\!\cdots\!25}{11\!\cdots\!97}a^{16}+\frac{83\!\cdots\!14}{11\!\cdots\!97}a^{15}-\frac{29\!\cdots\!01}{11\!\cdots\!97}a^{14}-\frac{21\!\cdots\!68}{11\!\cdots\!97}a^{13}+\frac{19\!\cdots\!63}{11\!\cdots\!97}a^{12}+\frac{62\!\cdots\!78}{11\!\cdots\!97}a^{11}+\frac{45\!\cdots\!13}{11\!\cdots\!97}a^{10}-\frac{47\!\cdots\!56}{11\!\cdots\!97}a^{9}-\frac{61\!\cdots\!60}{11\!\cdots\!97}a^{8}-\frac{33\!\cdots\!75}{11\!\cdots\!97}a^{7}+\frac{42\!\cdots\!98}{11\!\cdots\!97}a^{6}+\frac{33\!\cdots\!60}{11\!\cdots\!97}a^{5}-\frac{12\!\cdots\!49}{11\!\cdots\!97}a^{4}-\frac{28\!\cdots\!13}{11\!\cdots\!97}a^{3}-\frac{11\!\cdots\!10}{11\!\cdots\!97}a^{2}+\frac{48\!\cdots\!49}{11\!\cdots\!97}a+\frac{39\!\cdots\!50}{11\!\cdots\!97}$, $\frac{26\!\cdots\!99}{11\!\cdots\!97}a^{21}-\frac{37\!\cdots\!95}{11\!\cdots\!97}a^{20}+\frac{13\!\cdots\!77}{11\!\cdots\!97}a^{19}-\frac{18\!\cdots\!79}{11\!\cdots\!97}a^{18}+\frac{28\!\cdots\!84}{11\!\cdots\!97}a^{17}+\frac{89\!\cdots\!11}{11\!\cdots\!97}a^{16}+\frac{38\!\cdots\!86}{11\!\cdots\!97}a^{15}-\frac{81\!\cdots\!71}{11\!\cdots\!97}a^{14}-\frac{63\!\cdots\!20}{11\!\cdots\!97}a^{13}+\frac{22\!\cdots\!29}{11\!\cdots\!97}a^{12}+\frac{16\!\cdots\!33}{11\!\cdots\!97}a^{11}+\frac{17\!\cdots\!39}{11\!\cdots\!97}a^{10}-\frac{48\!\cdots\!21}{11\!\cdots\!97}a^{9}-\frac{10\!\cdots\!25}{11\!\cdots\!97}a^{8}-\frac{14\!\cdots\!30}{11\!\cdots\!97}a^{7}+\frac{92\!\cdots\!99}{11\!\cdots\!97}a^{6}+\frac{11\!\cdots\!31}{11\!\cdots\!97}a^{5}+\frac{16\!\cdots\!77}{11\!\cdots\!97}a^{4}-\frac{38\!\cdots\!10}{11\!\cdots\!97}a^{3}-\frac{21\!\cdots\!03}{11\!\cdots\!97}a^{2}+\frac{46\!\cdots\!43}{11\!\cdots\!97}a+\frac{36\!\cdots\!83}{11\!\cdots\!97}$, $\frac{33\!\cdots\!98}{11\!\cdots\!97}a^{21}+\frac{67\!\cdots\!70}{11\!\cdots\!97}a^{20}-\frac{14\!\cdots\!46}{11\!\cdots\!97}a^{19}-\frac{17\!\cdots\!46}{11\!\cdots\!97}a^{18}-\frac{44\!\cdots\!89}{11\!\cdots\!97}a^{17}+\frac{13\!\cdots\!77}{11\!\cdots\!97}a^{16}+\frac{81\!\cdots\!14}{11\!\cdots\!97}a^{15}+\frac{57\!\cdots\!84}{11\!\cdots\!97}a^{14}-\frac{42\!\cdots\!35}{11\!\cdots\!97}a^{13}-\frac{22\!\cdots\!29}{11\!\cdots\!97}a^{12}+\frac{31\!\cdots\!38}{11\!\cdots\!97}a^{11}+\frac{89\!\cdots\!47}{11\!\cdots\!97}a^{10}+\frac{65\!\cdots\!38}{11\!\cdots\!97}a^{9}-\frac{34\!\cdots\!34}{11\!\cdots\!97}a^{8}-\frac{40\!\cdots\!79}{11\!\cdots\!97}a^{7}+\frac{10\!\cdots\!04}{11\!\cdots\!97}a^{6}+\frac{50\!\cdots\!89}{11\!\cdots\!97}a^{5}+\frac{46\!\cdots\!94}{11\!\cdots\!97}a^{4}+\frac{13\!\cdots\!46}{11\!\cdots\!97}a^{3}-\frac{15\!\cdots\!18}{11\!\cdots\!97}a^{2}-\frac{59\!\cdots\!91}{11\!\cdots\!97}a+\frac{20\!\cdots\!90}{11\!\cdots\!97}$, $\frac{59\!\cdots\!49}{11\!\cdots\!97}a^{21}-\frac{63\!\cdots\!65}{11\!\cdots\!97}a^{20}-\frac{68\!\cdots\!76}{11\!\cdots\!97}a^{19}-\frac{30\!\cdots\!45}{11\!\cdots\!97}a^{18}+\frac{44\!\cdots\!21}{11\!\cdots\!97}a^{17}+\frac{91\!\cdots\!04}{11\!\cdots\!97}a^{16}+\frac{15\!\cdots\!16}{11\!\cdots\!97}a^{15}-\frac{16\!\cdots\!94}{11\!\cdots\!97}a^{14}-\frac{31\!\cdots\!15}{11\!\cdots\!97}a^{13}+\frac{22\!\cdots\!58}{11\!\cdots\!97}a^{12}+\frac{53\!\cdots\!86}{11\!\cdots\!97}a^{11}+\frac{49\!\cdots\!23}{11\!\cdots\!97}a^{10}-\frac{41\!\cdots\!75}{11\!\cdots\!97}a^{9}-\frac{74\!\cdots\!85}{11\!\cdots\!97}a^{8}-\frac{67\!\cdots\!95}{11\!\cdots\!97}a^{7}+\frac{40\!\cdots\!73}{11\!\cdots\!97}a^{6}+\frac{35\!\cdots\!35}{11\!\cdots\!97}a^{5}-\frac{89\!\cdots\!92}{11\!\cdots\!97}a^{4}-\frac{36\!\cdots\!68}{11\!\cdots\!97}a^{3}-\frac{17\!\cdots\!94}{11\!\cdots\!97}a^{2}+\frac{39\!\cdots\!56}{11\!\cdots\!97}a+\frac{53\!\cdots\!70}{11\!\cdots\!97}$, $\frac{11\!\cdots\!38}{11\!\cdots\!97}a^{21}-\frac{11\!\cdots\!26}{11\!\cdots\!97}a^{20}+\frac{52\!\cdots\!01}{11\!\cdots\!97}a^{19}-\frac{85\!\cdots\!83}{11\!\cdots\!97}a^{18}+\frac{91\!\cdots\!96}{11\!\cdots\!97}a^{17}+\frac{42\!\cdots\!84}{11\!\cdots\!97}a^{16}+\frac{23\!\cdots\!05}{11\!\cdots\!97}a^{15}-\frac{26\!\cdots\!34}{11\!\cdots\!97}a^{14}-\frac{30\!\cdots\!33}{11\!\cdots\!97}a^{13}-\frac{19\!\cdots\!16}{11\!\cdots\!97}a^{12}+\frac{60\!\cdots\!81}{11\!\cdots\!97}a^{11}+\frac{10\!\cdots\!65}{11\!\cdots\!97}a^{10}+\frac{48\!\cdots\!69}{11\!\cdots\!97}a^{9}-\frac{87\!\cdots\!48}{11\!\cdots\!97}a^{8}-\frac{20\!\cdots\!85}{11\!\cdots\!97}a^{7}+\frac{14\!\cdots\!94}{11\!\cdots\!97}a^{6}+\frac{55\!\cdots\!09}{11\!\cdots\!97}a^{5}+\frac{43\!\cdots\!31}{11\!\cdots\!97}a^{4}+\frac{14\!\cdots\!75}{11\!\cdots\!97}a^{3}-\frac{61\!\cdots\!23}{11\!\cdots\!97}a^{2}-\frac{68\!\cdots\!42}{11\!\cdots\!97}a-\frac{24\!\cdots\!85}{11\!\cdots\!97}$
| 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: | \( 1054.33143917 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 1054.33143917 \cdot 1}{2\cdot\sqrt{3273714539904703691187619}}\cr\approx \mathstrut & 0.175551809210 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 7*x^19 + 8*x^18 + 7*x^17 + 17*x^16 - 26*x^15 - 35*x^14 - 4*x^13 + 68*x^12 + 92*x^11 + 10*x^10 - 49*x^9 - 27*x^8 + 32*x^7 + 57*x^6 + 24*x^5 - 13*x^4 - 17*x^3 - 3*x^2 + 2*x + 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^22 - x^21 - 7*x^19 + 8*x^18 + 7*x^17 + 17*x^16 - 26*x^15 - 35*x^14 - 4*x^13 + 68*x^12 + 92*x^11 + 10*x^10 - 49*x^9 - 27*x^8 + 32*x^7 + 57*x^6 + 24*x^5 - 13*x^4 - 17*x^3 - 3*x^2 + 2*x + 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^22 - x^21 - 7*x^19 + 8*x^18 + 7*x^17 + 17*x^16 - 26*x^15 - 35*x^14 - 4*x^13 + 68*x^12 + 92*x^11 + 10*x^10 - 49*x^9 - 27*x^8 + 32*x^7 + 57*x^6 + 24*x^5 - 13*x^4 - 17*x^3 - 3*x^2 + 2*x + 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^21 - 7*x^19 + 8*x^18 + 7*x^17 + 17*x^16 - 26*x^15 - 35*x^14 - 4*x^13 + 68*x^12 + 92*x^11 + 10*x^10 - 49*x^9 - 27*x^8 + 32*x^7 + 57*x^6 + 24*x^5 - 13*x^4 - 17*x^3 - 3*x^2 + 2*x + 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2^{10}.(C_2\times S_{11})$ (as 22T53):
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 81749606400 |
| The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ are not computed |
| Character table for $C_2^{10}.(C_2\times S_{11})$ is not computed |
Intermediate fields
| 11.1.6477690799.1 |
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 siblings: | 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 | $22$ | $16{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | $22$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | $22$ | $16{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.9.0.1}{9} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(61\)
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 61.6.0.1 | $x^{6} + 49 x^{3} + 3 x^{2} + 29 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 61.6.0.1 | $x^{6} + 49 x^{3} + 3 x^{2} + 29 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 61.8.0.1 | $x^{8} + 57 x^{3} + x^{2} + 56 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(1279\)
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
|
\(1609\)
| $\Q_{1609}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1609}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1609}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1609}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
|
\(4025911\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ |