Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 8*x^20 + x^19 + 33*x^18 + 20*x^17 + 97*x^16 + 111*x^15 + 195*x^14 + 258*x^13 + 328*x^12 + 369*x^11 + 392*x^10 + 323*x^9 + 362*x^8 + 155*x^7 + 199*x^6 + 67*x^5 + 63*x^4 + 12*x^3 + 11*x^2 + 2*x + 1)
gp: K = bnfinit(y^22 - y^21 + 8*y^20 + y^19 + 33*y^18 + 20*y^17 + 97*y^16 + 111*y^15 + 195*y^14 + 258*y^13 + 328*y^12 + 369*y^11 + 392*y^10 + 323*y^9 + 362*y^8 + 155*y^7 + 199*y^6 + 67*y^5 + 63*y^4 + 12*y^3 + 11*y^2 + 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^21 + 8*x^20 + x^19 + 33*x^18 + 20*x^17 + 97*x^16 + 111*x^15 + 195*x^14 + 258*x^13 + 328*x^12 + 369*x^11 + 392*x^10 + 323*x^9 + 362*x^8 + 155*x^7 + 199*x^6 + 67*x^5 + 63*x^4 + 12*x^3 + 11*x^2 + 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^21 + 8*x^20 + x^19 + 33*x^18 + 20*x^17 + 97*x^16 + 111*x^15 + 195*x^14 + 258*x^13 + 328*x^12 + 369*x^11 + 392*x^10 + 323*x^9 + 362*x^8 + 155*x^7 + 199*x^6 + 67*x^5 + 63*x^4 + 12*x^3 + 11*x^2 + 2*x + 1)
\( x^{22} - x^{21} + 8 x^{20} + x^{19} + 33 x^{18} + 20 x^{17} + 97 x^{16} + 111 x^{15} + 195 x^{14} + \cdots + 1 \)
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: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-13329203788379546293009624459083\)
\(\medspace = -\,3^{11}\cdot 8674315276967^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(25.99\) | 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: | $3^{1/2}8674315276967^{1/2}\approx 5101269.041219155$ | ||
| Ramified primes: |
\(3\), \(8674315276967\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{11\!\cdots\!95}a^{21}+\frac{23\!\cdots\!31}{11\!\cdots\!95}a^{20}-\frac{77\!\cdots\!06}{23\!\cdots\!59}a^{19}+\frac{46\!\cdots\!11}{11\!\cdots\!95}a^{18}+\frac{26\!\cdots\!01}{23\!\cdots\!59}a^{17}+\frac{92\!\cdots\!28}{23\!\cdots\!59}a^{16}+\frac{28\!\cdots\!12}{11\!\cdots\!95}a^{15}+\frac{21\!\cdots\!62}{23\!\cdots\!59}a^{14}+\frac{24\!\cdots\!34}{23\!\cdots\!59}a^{13}+\frac{32\!\cdots\!53}{11\!\cdots\!95}a^{12}+\frac{22\!\cdots\!99}{11\!\cdots\!95}a^{11}-\frac{85\!\cdots\!53}{11\!\cdots\!95}a^{10}+\frac{99\!\cdots\!11}{11\!\cdots\!95}a^{9}+\frac{23\!\cdots\!55}{23\!\cdots\!59}a^{8}-\frac{42\!\cdots\!58}{11\!\cdots\!95}a^{7}-\frac{57\!\cdots\!26}{11\!\cdots\!95}a^{6}-\frac{40\!\cdots\!28}{11\!\cdots\!95}a^{5}-\frac{27\!\cdots\!64}{11\!\cdots\!95}a^{4}-\frac{10\!\cdots\!57}{23\!\cdots\!59}a^{3}-\frac{53\!\cdots\!53}{11\!\cdots\!95}a^{2}+\frac{11\!\cdots\!63}{23\!\cdots\!59}a+\frac{44\!\cdots\!37}{11\!\cdots\!95}$
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: |
\( -\frac{43777949891663214}{115784657874210295} a^{21} + \frac{56176992000648426}{115784657874210295} a^{20} - \frac{70862937565902418}{23156931574842059} a^{19} + \frac{46304238082105751}{115784657874210295} a^{18} - \frac{273845407499128465}{23156931574842059} a^{17} - \frac{92101589895594629}{23156931574842059} a^{16} - \frac{3756014963871124433}{115784657874210295} a^{15} - \frac{704799598835654261}{23156931574842059} a^{14} - \frac{1298457807946225426}{23156931574842059} a^{13} - \frac{8109584073922414987}{115784657874210295} a^{12} - \frac{9965297578375354916}{115784657874210295} a^{11} - \frac{10502743477324340478}{115784657874210295} a^{10} - \frac{10686083960975967849}{115784657874210295} a^{9} - \frac{1451060702086977237}{23156931574842059} a^{8} - \frac{9823105793066203988}{115784657874210295} a^{7} - \frac{897754812270058486}{115784657874210295} a^{6} - \frac{4998718253209464818}{115784657874210295} a^{5} - \frac{66532980044445694}{115784657874210295} a^{4} - \frac{232323650375546467}{23156931574842059} a^{3} + \frac{563169515708121062}{115784657874210295} a^{2} - \frac{32570281122904994}{23156931574842059} a + \frac{110778693642358837}{115784657874210295} \)
(order $6$)
| 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: |
$a$, $\frac{66\!\cdots\!26}{11\!\cdots\!95}a^{21}-\frac{72\!\cdots\!04}{11\!\cdots\!95}a^{20}+\frac{10\!\cdots\!85}{23\!\cdots\!59}a^{19}+\frac{50\!\cdots\!86}{11\!\cdots\!95}a^{18}+\frac{40\!\cdots\!73}{23\!\cdots\!59}a^{17}+\frac{23\!\cdots\!48}{23\!\cdots\!59}a^{16}+\frac{56\!\cdots\!52}{11\!\cdots\!95}a^{15}+\frac{13\!\cdots\!83}{23\!\cdots\!59}a^{14}+\frac{20\!\cdots\!84}{23\!\cdots\!59}a^{13}+\frac{14\!\cdots\!08}{11\!\cdots\!95}a^{12}+\frac{16\!\cdots\!24}{11\!\cdots\!95}a^{11}+\frac{18\!\cdots\!72}{11\!\cdots\!95}a^{10}+\frac{18\!\cdots\!96}{11\!\cdots\!95}a^{9}+\frac{26\!\cdots\!92}{23\!\cdots\!59}a^{8}+\frac{16\!\cdots\!47}{11\!\cdots\!95}a^{7}+\frac{43\!\cdots\!09}{11\!\cdots\!95}a^{6}+\frac{72\!\cdots\!62}{11\!\cdots\!95}a^{5}+\frac{31\!\cdots\!91}{11\!\cdots\!95}a^{4}+\frac{25\!\cdots\!30}{23\!\cdots\!59}a^{3}+\frac{64\!\cdots\!22}{11\!\cdots\!95}a^{2}+\frac{37\!\cdots\!27}{23\!\cdots\!59}a+\frac{16\!\cdots\!37}{11\!\cdots\!95}$, $\frac{19\!\cdots\!25}{23\!\cdots\!59}a^{21}-\frac{12\!\cdots\!49}{23\!\cdots\!59}a^{20}+\frac{14\!\cdots\!61}{23\!\cdots\!59}a^{19}+\frac{77\!\cdots\!38}{23\!\cdots\!59}a^{18}+\frac{63\!\cdots\!47}{23\!\cdots\!59}a^{17}+\frac{61\!\cdots\!48}{23\!\cdots\!59}a^{16}+\frac{19\!\cdots\!77}{23\!\cdots\!59}a^{15}+\frac{28\!\cdots\!48}{23\!\cdots\!59}a^{14}+\frac{43\!\cdots\!59}{23\!\cdots\!59}a^{13}+\frac{61\!\cdots\!60}{23\!\cdots\!59}a^{12}+\frac{77\!\cdots\!07}{23\!\cdots\!59}a^{11}+\frac{88\!\cdots\!46}{23\!\cdots\!59}a^{10}+\frac{94\!\cdots\!29}{23\!\cdots\!59}a^{9}+\frac{80\!\cdots\!44}{23\!\cdots\!59}a^{8}+\frac{82\!\cdots\!99}{23\!\cdots\!59}a^{7}+\frac{46\!\cdots\!91}{23\!\cdots\!59}a^{6}+\frac{40\!\cdots\!81}{23\!\cdots\!59}a^{5}+\frac{21\!\cdots\!50}{23\!\cdots\!59}a^{4}+\frac{11\!\cdots\!59}{23\!\cdots\!59}a^{3}+\frac{43\!\cdots\!99}{23\!\cdots\!59}a^{2}+\frac{11\!\cdots\!35}{23\!\cdots\!59}a+\frac{68\!\cdots\!86}{23\!\cdots\!59}$, $\frac{18\!\cdots\!81}{11\!\cdots\!95}a^{21}+\frac{72\!\cdots\!16}{11\!\cdots\!95}a^{20}+\frac{20\!\cdots\!99}{23\!\cdots\!59}a^{19}+\frac{78\!\cdots\!36}{11\!\cdots\!95}a^{18}+\frac{68\!\cdots\!90}{23\!\cdots\!59}a^{17}+\frac{65\!\cdots\!98}{23\!\cdots\!59}a^{16}+\frac{21\!\cdots\!97}{11\!\cdots\!95}a^{15}+\frac{19\!\cdots\!19}{23\!\cdots\!59}a^{14}+\frac{18\!\cdots\!47}{23\!\cdots\!59}a^{13}+\frac{16\!\cdots\!03}{11\!\cdots\!95}a^{12}+\frac{20\!\cdots\!54}{11\!\cdots\!95}a^{11}+\frac{24\!\cdots\!77}{11\!\cdots\!95}a^{10}+\frac{25\!\cdots\!46}{11\!\cdots\!95}a^{9}+\frac{49\!\cdots\!58}{23\!\cdots\!59}a^{8}+\frac{18\!\cdots\!62}{11\!\cdots\!95}a^{7}+\frac{21\!\cdots\!09}{11\!\cdots\!95}a^{6}+\frac{22\!\cdots\!57}{11\!\cdots\!95}a^{5}+\frac{12\!\cdots\!61}{11\!\cdots\!95}a^{4}-\frac{418149321821687}{23\!\cdots\!59}a^{3}+\frac{22\!\cdots\!12}{11\!\cdots\!95}a^{2}-\frac{43\!\cdots\!21}{23\!\cdots\!59}a+\frac{30\!\cdots\!82}{11\!\cdots\!95}$, $\frac{16\!\cdots\!96}{23\!\cdots\!59}a^{21}-\frac{20\!\cdots\!69}{23\!\cdots\!59}a^{20}+\frac{13\!\cdots\!47}{23\!\cdots\!59}a^{19}-\frac{566704711301760}{23\!\cdots\!59}a^{18}+\frac{50\!\cdots\!45}{23\!\cdots\!59}a^{17}+\frac{25\!\cdots\!30}{23\!\cdots\!59}a^{16}+\frac{14\!\cdots\!52}{23\!\cdots\!59}a^{15}+\frac{15\!\cdots\!04}{23\!\cdots\!59}a^{14}+\frac{24\!\cdots\!65}{23\!\cdots\!59}a^{13}+\frac{34\!\cdots\!36}{23\!\cdots\!59}a^{12}+\frac{40\!\cdots\!87}{23\!\cdots\!59}a^{11}+\frac{44\!\cdots\!56}{23\!\cdots\!59}a^{10}+\frac{45\!\cdots\!63}{23\!\cdots\!59}a^{9}+\frac{32\!\cdots\!64}{23\!\cdots\!59}a^{8}+\frac{41\!\cdots\!59}{23\!\cdots\!59}a^{7}+\frac{95\!\cdots\!00}{23\!\cdots\!59}a^{6}+\frac{18\!\cdots\!61}{23\!\cdots\!59}a^{5}+\frac{80\!\cdots\!84}{23\!\cdots\!59}a^{4}+\frac{11\!\cdots\!73}{23\!\cdots\!59}a^{3}+\frac{16\!\cdots\!79}{23\!\cdots\!59}a^{2}+\frac{48\!\cdots\!07}{23\!\cdots\!59}a+\frac{12\!\cdots\!02}{23\!\cdots\!59}$, $\frac{62\!\cdots\!48}{11\!\cdots\!95}a^{21}-\frac{77\!\cdots\!57}{11\!\cdots\!95}a^{20}+\frac{98\!\cdots\!97}{23\!\cdots\!59}a^{19}-\frac{18\!\cdots\!12}{11\!\cdots\!95}a^{18}+\frac{36\!\cdots\!63}{23\!\cdots\!59}a^{17}+\frac{16\!\cdots\!60}{23\!\cdots\!59}a^{16}+\frac{49\!\cdots\!71}{11\!\cdots\!95}a^{15}+\frac{10\!\cdots\!40}{23\!\cdots\!59}a^{14}+\frac{16\!\cdots\!55}{23\!\cdots\!59}a^{13}+\frac{11\!\cdots\!84}{11\!\cdots\!95}a^{12}+\frac{13\!\cdots\!47}{11\!\cdots\!95}a^{11}+\frac{13\!\cdots\!66}{11\!\cdots\!95}a^{10}+\frac{13\!\cdots\!63}{11\!\cdots\!95}a^{9}+\frac{17\!\cdots\!84}{23\!\cdots\!59}a^{8}+\frac{12\!\cdots\!71}{11\!\cdots\!95}a^{7}+\frac{10\!\cdots\!97}{11\!\cdots\!95}a^{6}+\frac{51\!\cdots\!61}{11\!\cdots\!95}a^{5}+\frac{18\!\cdots\!18}{11\!\cdots\!95}a^{4}+\frac{70\!\cdots\!72}{23\!\cdots\!59}a^{3}+\frac{38\!\cdots\!71}{11\!\cdots\!95}a^{2}+\frac{24\!\cdots\!91}{23\!\cdots\!59}a+\frac{11\!\cdots\!11}{11\!\cdots\!95}$, $\frac{11\!\cdots\!37}{11\!\cdots\!95}a^{21}-\frac{67\!\cdots\!23}{11\!\cdots\!95}a^{20}+\frac{16\!\cdots\!54}{23\!\cdots\!59}a^{19}+\frac{46\!\cdots\!27}{11\!\cdots\!95}a^{18}+\frac{72\!\cdots\!74}{23\!\cdots\!59}a^{17}+\frac{71\!\cdots\!13}{23\!\cdots\!59}a^{16}+\frac{11\!\cdots\!34}{11\!\cdots\!95}a^{15}+\frac{32\!\cdots\!68}{23\!\cdots\!59}a^{14}+\frac{50\!\cdots\!04}{23\!\cdots\!59}a^{13}+\frac{35\!\cdots\!76}{11\!\cdots\!95}a^{12}+\frac{44\!\cdots\!23}{11\!\cdots\!95}a^{11}+\frac{50\!\cdots\!69}{11\!\cdots\!95}a^{10}+\frac{53\!\cdots\!82}{11\!\cdots\!95}a^{9}+\frac{92\!\cdots\!40}{23\!\cdots\!59}a^{8}+\frac{47\!\cdots\!79}{11\!\cdots\!95}a^{7}+\frac{26\!\cdots\!23}{11\!\cdots\!95}a^{6}+\frac{22\!\cdots\!49}{11\!\cdots\!95}a^{5}+\frac{12\!\cdots\!97}{11\!\cdots\!95}a^{4}+\frac{14\!\cdots\!85}{23\!\cdots\!59}a^{3}+\frac{24\!\cdots\!79}{11\!\cdots\!95}a^{2}+\frac{13\!\cdots\!29}{23\!\cdots\!59}a+\frac{26\!\cdots\!49}{11\!\cdots\!95}$, $\frac{10\!\cdots\!84}{11\!\cdots\!95}a^{21}-\frac{90\!\cdots\!91}{11\!\cdots\!95}a^{20}+\frac{15\!\cdots\!76}{23\!\cdots\!59}a^{19}+\frac{20\!\cdots\!99}{11\!\cdots\!95}a^{18}+\frac{61\!\cdots\!44}{23\!\cdots\!59}a^{17}+\frac{45\!\cdots\!55}{23\!\cdots\!59}a^{16}+\frac{89\!\cdots\!18}{11\!\cdots\!95}a^{15}+\frac{22\!\cdots\!00}{23\!\cdots\!59}a^{14}+\frac{35\!\cdots\!90}{23\!\cdots\!59}a^{13}+\frac{24\!\cdots\!12}{11\!\cdots\!95}a^{12}+\frac{30\!\cdots\!11}{11\!\cdots\!95}a^{11}+\frac{33\!\cdots\!83}{11\!\cdots\!95}a^{10}+\frac{34\!\cdots\!74}{11\!\cdots\!95}a^{9}+\frac{53\!\cdots\!72}{23\!\cdots\!59}a^{8}+\frac{30\!\cdots\!23}{11\!\cdots\!95}a^{7}+\frac{12\!\cdots\!01}{11\!\cdots\!95}a^{6}+\frac{13\!\cdots\!93}{11\!\cdots\!95}a^{5}+\frac{66\!\cdots\!59}{11\!\cdots\!95}a^{4}+\frac{70\!\cdots\!91}{23\!\cdots\!59}a^{3}+\frac{13\!\cdots\!23}{11\!\cdots\!95}a^{2}+\frac{74\!\cdots\!73}{23\!\cdots\!59}a+\frac{24\!\cdots\!53}{11\!\cdots\!95}$, $\frac{65\!\cdots\!81}{11\!\cdots\!95}a^{21}-\frac{69\!\cdots\!39}{11\!\cdots\!95}a^{20}+\frac{10\!\cdots\!34}{23\!\cdots\!59}a^{19}+\frac{24\!\cdots\!71}{11\!\cdots\!95}a^{18}+\frac{43\!\cdots\!58}{23\!\cdots\!59}a^{17}+\frac{22\!\cdots\!23}{23\!\cdots\!59}a^{16}+\frac{62\!\cdots\!92}{11\!\cdots\!95}a^{15}+\frac{13\!\cdots\!63}{23\!\cdots\!59}a^{14}+\frac{24\!\cdots\!13}{23\!\cdots\!59}a^{13}+\frac{15\!\cdots\!53}{11\!\cdots\!95}a^{12}+\frac{19\!\cdots\!04}{11\!\cdots\!95}a^{11}+\frac{20\!\cdots\!52}{11\!\cdots\!95}a^{10}+\frac{21\!\cdots\!96}{11\!\cdots\!95}a^{9}+\frac{32\!\cdots\!13}{23\!\cdots\!59}a^{8}+\frac{18\!\cdots\!37}{11\!\cdots\!95}a^{7}+\frac{47\!\cdots\!59}{11\!\cdots\!95}a^{6}+\frac{91\!\cdots\!12}{11\!\cdots\!95}a^{5}+\frac{56\!\cdots\!51}{11\!\cdots\!95}a^{4}+\frac{41\!\cdots\!73}{23\!\cdots\!59}a^{3}-\frac{66\!\cdots\!23}{11\!\cdots\!95}a^{2}+\frac{31\!\cdots\!14}{23\!\cdots\!59}a-\frac{15\!\cdots\!68}{11\!\cdots\!95}$, $\frac{30\!\cdots\!44}{11\!\cdots\!95}a^{21}+\frac{43\!\cdots\!79}{11\!\cdots\!95}a^{20}-\frac{93\!\cdots\!94}{23\!\cdots\!59}a^{19}+\frac{30\!\cdots\!54}{11\!\cdots\!95}a^{18}+\frac{57\!\cdots\!16}{23\!\cdots\!59}a^{17}+\frac{25\!\cdots\!09}{23\!\cdots\!59}a^{16}+\frac{14\!\cdots\!73}{11\!\cdots\!95}a^{15}+\frac{74\!\cdots\!80}{23\!\cdots\!59}a^{14}+\frac{10\!\cdots\!82}{23\!\cdots\!59}a^{13}+\frac{73\!\cdots\!62}{11\!\cdots\!95}a^{12}+\frac{89\!\cdots\!96}{11\!\cdots\!95}a^{11}+\frac{10\!\cdots\!88}{11\!\cdots\!95}a^{10}+\frac{11\!\cdots\!44}{11\!\cdots\!95}a^{9}+\frac{20\!\cdots\!34}{23\!\cdots\!59}a^{8}+\frac{60\!\cdots\!23}{11\!\cdots\!95}a^{7}+\frac{65\!\cdots\!86}{11\!\cdots\!95}a^{6}+\frac{28\!\cdots\!13}{11\!\cdots\!95}a^{5}+\frac{21\!\cdots\!59}{11\!\cdots\!95}a^{4}+\frac{10\!\cdots\!58}{23\!\cdots\!59}a^{3}+\frac{39\!\cdots\!13}{11\!\cdots\!95}a^{2}+\frac{17\!\cdots\!51}{23\!\cdots\!59}a+\frac{35\!\cdots\!83}{11\!\cdots\!95}$
| 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: | \( 6808139.27799 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 6808139.27799 \cdot 1}{6\cdot\sqrt{13329203788379546293009624459083}}\cr\approx \mathstrut & 0.187263712663 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 8*x^20 + x^19 + 33*x^18 + 20*x^17 + 97*x^16 + 111*x^15 + 195*x^14 + 258*x^13 + 328*x^12 + 369*x^11 + 392*x^10 + 323*x^9 + 362*x^8 + 155*x^7 + 199*x^6 + 67*x^5 + 63*x^4 + 12*x^3 + 11*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 + 8*x^20 + x^19 + 33*x^18 + 20*x^17 + 97*x^16 + 111*x^15 + 195*x^14 + 258*x^13 + 328*x^12 + 369*x^11 + 392*x^10 + 323*x^9 + 362*x^8 + 155*x^7 + 199*x^6 + 67*x^5 + 63*x^4 + 12*x^3 + 11*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 + 8*x^20 + x^19 + 33*x^18 + 20*x^17 + 97*x^16 + 111*x^15 + 195*x^14 + 258*x^13 + 328*x^12 + 369*x^11 + 392*x^10 + 323*x^9 + 362*x^8 + 155*x^7 + 199*x^6 + 67*x^5 + 63*x^4 + 12*x^3 + 11*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 + 8*x^20 + x^19 + 33*x^18 + 20*x^17 + 97*x^16 + 111*x^15 + 195*x^14 + 258*x^13 + 328*x^12 + 369*x^11 + 392*x^10 + 323*x^9 + 362*x^8 + 155*x^7 + 199*x^6 + 67*x^5 + 63*x^4 + 12*x^3 + 11*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
$C_2\times S_{11}$ (as 22T47):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A non-solvable group of order 79833600 |
| The 112 conjugacy class representatives for $C_2\times S_{11}$ |
| Character table for $C_2\times S_{11}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 11.9.8674315276967.1 |
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 | $22$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.9.0.1}{9} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | $22$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | $18{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 | $[\ ]_{2}^{11}$ |
|
\(8674315276967\)
| Deg $4$ | $2$ | $2$ | $2$ | |||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |