Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 9*x^16 + 264*x^15 - 288*x^14 - 2916*x^13 + 5598*x^12 + 14526*x^11 - 39159*x^10 - 27383*x^9 + 128403*x^8 - 17730*x^7 - 188268*x^6 + 110070*x^5 + 90900*x^4 - 76710*x^3 - 9855*x^2 + 13041*x - 623)
gp: K = bnfinit(y^18 - 9*y^17 - 9*y^16 + 264*y^15 - 288*y^14 - 2916*y^13 + 5598*y^12 + 14526*y^11 - 39159*y^10 - 27383*y^9 + 128403*y^8 - 17730*y^7 - 188268*y^6 + 110070*y^5 + 90900*y^4 - 76710*y^3 - 9855*y^2 + 13041*y - 623, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 - 9*x^16 + 264*x^15 - 288*x^14 - 2916*x^13 + 5598*x^12 + 14526*x^11 - 39159*x^10 - 27383*x^9 + 128403*x^8 - 17730*x^7 - 188268*x^6 + 110070*x^5 + 90900*x^4 - 76710*x^3 - 9855*x^2 + 13041*x - 623);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 - 9*x^16 + 264*x^15 - 288*x^14 - 2916*x^13 + 5598*x^12 + 14526*x^11 - 39159*x^10 - 27383*x^9 + 128403*x^8 - 17730*x^7 - 188268*x^6 + 110070*x^5 + 90900*x^4 - 76710*x^3 - 9855*x^2 + 13041*x - 623)
\( x^{18} - 9 x^{17} - 9 x^{16} + 264 x^{15} - 288 x^{14} - 2916 x^{13} + 5598 x^{12} + 14526 x^{11} + \cdots - 623 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(178698494132131643847952616411136\)
\(\medspace = 2^{12}\cdot 3^{37}\cdot 7^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(61.91\) | 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: | $2^{2/3}3^{121/54}7^{5/6}\approx 94.19764909571116$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{21}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{56}a^{16}+\frac{1}{14}a^{14}+\frac{3}{28}a^{13}+\frac{1}{14}a^{11}+\frac{3}{28}a^{10}-\frac{3}{14}a^{9}-\frac{3}{56}a^{8}-\frac{1}{28}a^{6}+\frac{1}{14}a^{5}+\frac{3}{14}a^{3}+\frac{11}{28}a^{2}-\frac{1}{4}a-\frac{1}{8}$, $\frac{1}{85\!\cdots\!72}a^{17}+\frac{17\!\cdots\!71}{21\!\cdots\!18}a^{16}-\frac{30\!\cdots\!63}{42\!\cdots\!36}a^{15}+\frac{46\!\cdots\!99}{61\!\cdots\!48}a^{14}+\frac{94\!\cdots\!69}{10\!\cdots\!09}a^{13}-\frac{41\!\cdots\!09}{10\!\cdots\!09}a^{12}+\frac{13\!\cdots\!81}{15\!\cdots\!87}a^{11}-\frac{15\!\cdots\!17}{61\!\cdots\!48}a^{10}-\frac{15\!\cdots\!67}{85\!\cdots\!72}a^{9}+\frac{85\!\cdots\!65}{42\!\cdots\!36}a^{8}+\frac{19\!\cdots\!16}{10\!\cdots\!09}a^{7}-\frac{779286224224827}{61\!\cdots\!48}a^{6}+\frac{47\!\cdots\!27}{21\!\cdots\!18}a^{5}+\frac{12\!\cdots\!27}{42\!\cdots\!36}a^{4}+\frac{48\!\cdots\!23}{42\!\cdots\!36}a^{3}+\frac{50\!\cdots\!69}{10\!\cdots\!09}a^{2}+\frac{17\!\cdots\!91}{12\!\cdots\!96}a-\frac{26\!\cdots\!27}{61\!\cdots\!48}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
$\frac{41\!\cdots\!67}{42\!\cdots\!36}a^{17}-\frac{59\!\cdots\!79}{85\!\cdots\!72}a^{16}-\frac{45\!\cdots\!25}{21\!\cdots\!18}a^{15}+\frac{23\!\cdots\!98}{10\!\cdots\!09}a^{14}+\frac{52\!\cdots\!91}{42\!\cdots\!36}a^{13}-\frac{11\!\cdots\!73}{42\!\cdots\!36}a^{12}+\frac{22\!\cdots\!05}{42\!\cdots\!36}a^{11}+\frac{66\!\cdots\!67}{42\!\cdots\!36}a^{10}-\frac{97\!\cdots\!29}{10\!\cdots\!09}a^{9}-\frac{39\!\cdots\!79}{85\!\cdots\!72}a^{8}+\frac{16\!\cdots\!39}{42\!\cdots\!36}a^{7}+\frac{27\!\cdots\!43}{42\!\cdots\!36}a^{6}-\frac{27\!\cdots\!49}{42\!\cdots\!36}a^{5}-\frac{31\!\cdots\!64}{10\!\cdots\!09}a^{4}+\frac{15\!\cdots\!93}{42\!\cdots\!36}a^{3}+\frac{11\!\cdots\!13}{42\!\cdots\!36}a^{2}-\frac{35\!\cdots\!05}{61\!\cdots\!48}a+\frac{33\!\cdots\!97}{12\!\cdots\!96}$, $\frac{85\!\cdots\!04}{10\!\cdots\!09}a^{17}-\frac{52\!\cdots\!57}{85\!\cdots\!72}a^{16}-\frac{64\!\cdots\!69}{42\!\cdots\!36}a^{15}+\frac{40\!\cdots\!69}{21\!\cdots\!18}a^{14}+\frac{40\!\cdots\!57}{21\!\cdots\!18}a^{13}-\frac{49\!\cdots\!63}{21\!\cdots\!18}a^{12}+\frac{62\!\cdots\!03}{42\!\cdots\!36}a^{11}+\frac{14\!\cdots\!46}{10\!\cdots\!09}a^{10}-\frac{20\!\cdots\!53}{15\!\cdots\!87}a^{9}-\frac{33\!\cdots\!41}{85\!\cdots\!72}a^{8}+\frac{21\!\cdots\!57}{42\!\cdots\!36}a^{7}+\frac{55\!\cdots\!89}{10\!\cdots\!09}a^{6}-\frac{17\!\cdots\!35}{21\!\cdots\!18}a^{5}-\frac{20\!\cdots\!54}{10\!\cdots\!09}a^{4}+\frac{10\!\cdots\!17}{21\!\cdots\!18}a^{3}+\frac{15\!\cdots\!15}{10\!\cdots\!09}a^{2}-\frac{23\!\cdots\!75}{30\!\cdots\!74}a+\frac{46\!\cdots\!29}{12\!\cdots\!96}$, $\frac{90\!\cdots\!23}{85\!\cdots\!72}a^{17}-\frac{67\!\cdots\!15}{85\!\cdots\!72}a^{16}-\frac{92\!\cdots\!13}{42\!\cdots\!36}a^{15}+\frac{10\!\cdots\!23}{42\!\cdots\!36}a^{14}+\frac{30\!\cdots\!99}{42\!\cdots\!36}a^{13}-\frac{63\!\cdots\!55}{21\!\cdots\!18}a^{12}+\frac{59\!\cdots\!57}{42\!\cdots\!36}a^{11}+\frac{18\!\cdots\!07}{10\!\cdots\!09}a^{10}-\frac{18\!\cdots\!83}{12\!\cdots\!96}a^{9}-\frac{44\!\cdots\!43}{85\!\cdots\!72}a^{8}+\frac{24\!\cdots\!29}{42\!\cdots\!36}a^{7}+\frac{73\!\cdots\!88}{10\!\cdots\!09}a^{6}-\frac{10\!\cdots\!03}{10\!\cdots\!09}a^{5}-\frac{59\!\cdots\!39}{21\!\cdots\!18}a^{4}+\frac{57\!\cdots\!90}{10\!\cdots\!09}a^{3}+\frac{49\!\cdots\!71}{42\!\cdots\!36}a^{2}-\frac{10\!\cdots\!05}{12\!\cdots\!96}a+\frac{52\!\cdots\!59}{12\!\cdots\!96}$, $\frac{31\!\cdots\!59}{42\!\cdots\!36}a^{17}-\frac{48\!\cdots\!69}{85\!\cdots\!72}a^{16}-\frac{14\!\cdots\!14}{10\!\cdots\!09}a^{15}+\frac{10\!\cdots\!13}{61\!\cdots\!48}a^{14}+\frac{31\!\cdots\!43}{21\!\cdots\!18}a^{13}-\frac{91\!\cdots\!51}{42\!\cdots\!36}a^{12}+\frac{42\!\cdots\!97}{30\!\cdots\!74}a^{11}+\frac{76\!\cdots\!99}{61\!\cdots\!48}a^{10}-\frac{54\!\cdots\!65}{42\!\cdots\!36}a^{9}-\frac{31\!\cdots\!21}{85\!\cdots\!72}a^{8}+\frac{50\!\cdots\!96}{10\!\cdots\!09}a^{7}+\frac{29\!\cdots\!61}{61\!\cdots\!48}a^{6}-\frac{82\!\cdots\!52}{10\!\cdots\!09}a^{5}-\frac{38\!\cdots\!63}{21\!\cdots\!18}a^{4}+\frac{93\!\cdots\!39}{21\!\cdots\!18}a^{3}+\frac{14\!\cdots\!25}{21\!\cdots\!18}a^{2}-\frac{43\!\cdots\!19}{61\!\cdots\!48}a+\frac{42\!\cdots\!75}{12\!\cdots\!96}$, $\frac{43\!\cdots\!23}{12\!\cdots\!96}a^{17}-\frac{23\!\cdots\!45}{85\!\cdots\!72}a^{16}-\frac{10\!\cdots\!83}{15\!\cdots\!87}a^{15}+\frac{90\!\cdots\!08}{10\!\cdots\!09}a^{14}+\frac{19\!\cdots\!93}{21\!\cdots\!18}a^{13}-\frac{31\!\cdots\!95}{30\!\cdots\!74}a^{12}+\frac{27\!\cdots\!23}{42\!\cdots\!36}a^{11}+\frac{25\!\cdots\!31}{42\!\cdots\!36}a^{10}-\frac{51\!\cdots\!05}{85\!\cdots\!72}a^{9}-\frac{15\!\cdots\!59}{85\!\cdots\!72}a^{8}+\frac{13\!\cdots\!11}{61\!\cdots\!48}a^{7}+\frac{98\!\cdots\!97}{42\!\cdots\!36}a^{6}-\frac{15\!\cdots\!53}{42\!\cdots\!36}a^{5}-\frac{52\!\cdots\!03}{61\!\cdots\!48}a^{4}+\frac{88\!\cdots\!73}{42\!\cdots\!36}a^{3}+\frac{21\!\cdots\!11}{42\!\cdots\!36}a^{2}-\frac{41\!\cdots\!83}{12\!\cdots\!96}a+\frac{21\!\cdots\!79}{12\!\cdots\!96}$, $\frac{34\!\cdots\!85}{85\!\cdots\!72}a^{17}-\frac{26\!\cdots\!91}{85\!\cdots\!72}a^{16}-\frac{80\!\cdots\!12}{10\!\cdots\!09}a^{15}+\frac{40\!\cdots\!01}{42\!\cdots\!36}a^{14}+\frac{15\!\cdots\!55}{21\!\cdots\!18}a^{13}-\frac{49\!\cdots\!25}{42\!\cdots\!36}a^{12}+\frac{32\!\cdots\!91}{42\!\cdots\!36}a^{11}+\frac{28\!\cdots\!49}{42\!\cdots\!36}a^{10}-\frac{59\!\cdots\!65}{85\!\cdots\!72}a^{9}-\frac{17\!\cdots\!83}{85\!\cdots\!72}a^{8}+\frac{11\!\cdots\!99}{42\!\cdots\!36}a^{7}+\frac{11\!\cdots\!61}{42\!\cdots\!36}a^{6}-\frac{17\!\cdots\!37}{42\!\cdots\!36}a^{5}-\frac{10\!\cdots\!23}{10\!\cdots\!09}a^{4}+\frac{14\!\cdots\!67}{61\!\cdots\!48}a^{3}+\frac{45\!\cdots\!47}{21\!\cdots\!18}a^{2}-\frac{48\!\cdots\!31}{12\!\cdots\!96}a+\frac{23\!\cdots\!17}{12\!\cdots\!96}$, $\frac{17\!\cdots\!87}{85\!\cdots\!72}a^{17}-\frac{18\!\cdots\!53}{12\!\cdots\!96}a^{16}-\frac{16\!\cdots\!27}{42\!\cdots\!36}a^{15}+\frac{20\!\cdots\!05}{42\!\cdots\!36}a^{14}+\frac{43\!\cdots\!17}{61\!\cdots\!48}a^{13}-\frac{24\!\cdots\!23}{42\!\cdots\!36}a^{12}+\frac{36\!\cdots\!63}{10\!\cdots\!09}a^{11}+\frac{36\!\cdots\!34}{10\!\cdots\!09}a^{10}-\frac{27\!\cdots\!67}{85\!\cdots\!72}a^{9}-\frac{12\!\cdots\!25}{12\!\cdots\!96}a^{8}+\frac{13\!\cdots\!13}{10\!\cdots\!09}a^{7}+\frac{14\!\cdots\!90}{10\!\cdots\!09}a^{6}-\frac{12\!\cdots\!21}{61\!\cdots\!48}a^{5}-\frac{54\!\cdots\!92}{10\!\cdots\!09}a^{4}+\frac{48\!\cdots\!95}{42\!\cdots\!36}a^{3}+\frac{60\!\cdots\!39}{61\!\cdots\!48}a^{2}-\frac{22\!\cdots\!95}{12\!\cdots\!96}a+\frac{11\!\cdots\!85}{12\!\cdots\!96}$, $\frac{14\!\cdots\!03}{42\!\cdots\!36}a^{17}-\frac{10\!\cdots\!93}{42\!\cdots\!36}a^{16}-\frac{13\!\cdots\!41}{21\!\cdots\!18}a^{15}+\frac{83\!\cdots\!42}{10\!\cdots\!09}a^{14}+\frac{62\!\cdots\!05}{42\!\cdots\!36}a^{13}-\frac{40\!\cdots\!99}{42\!\cdots\!36}a^{12}+\frac{11\!\cdots\!41}{21\!\cdots\!18}a^{11}+\frac{59\!\cdots\!98}{10\!\cdots\!09}a^{10}-\frac{11\!\cdots\!73}{21\!\cdots\!18}a^{9}-\frac{69\!\cdots\!95}{42\!\cdots\!36}a^{8}+\frac{42\!\cdots\!75}{21\!\cdots\!18}a^{7}+\frac{45\!\cdots\!59}{21\!\cdots\!18}a^{6}-\frac{13\!\cdots\!45}{42\!\cdots\!36}a^{5}-\frac{17\!\cdots\!55}{21\!\cdots\!18}a^{4}+\frac{19\!\cdots\!36}{10\!\cdots\!09}a^{3}+\frac{16\!\cdots\!59}{10\!\cdots\!09}a^{2}-\frac{18\!\cdots\!73}{61\!\cdots\!48}a+\frac{45\!\cdots\!63}{30\!\cdots\!74}$, $\frac{89\!\cdots\!31}{42\!\cdots\!36}a^{17}-\frac{19\!\cdots\!89}{12\!\cdots\!96}a^{16}-\frac{42\!\cdots\!56}{10\!\cdots\!09}a^{15}+\frac{21\!\cdots\!87}{42\!\cdots\!36}a^{14}+\frac{33\!\cdots\!41}{61\!\cdots\!48}a^{13}-\frac{25\!\cdots\!99}{42\!\cdots\!36}a^{12}+\frac{16\!\cdots\!09}{42\!\cdots\!36}a^{11}+\frac{75\!\cdots\!93}{21\!\cdots\!18}a^{10}-\frac{37\!\cdots\!19}{10\!\cdots\!09}a^{9}-\frac{12\!\cdots\!51}{12\!\cdots\!96}a^{8}+\frac{56\!\cdots\!71}{42\!\cdots\!36}a^{7}+\frac{29\!\cdots\!17}{21\!\cdots\!18}a^{6}-\frac{13\!\cdots\!07}{61\!\cdots\!48}a^{5}-\frac{21\!\cdots\!33}{42\!\cdots\!36}a^{4}+\frac{52\!\cdots\!79}{42\!\cdots\!36}a^{3}+\frac{28\!\cdots\!19}{61\!\cdots\!48}a^{2}-\frac{12\!\cdots\!85}{61\!\cdots\!48}a+\frac{12\!\cdots\!47}{12\!\cdots\!96}$, $\frac{17\!\cdots\!72}{15\!\cdots\!87}a^{17}-\frac{74\!\cdots\!25}{85\!\cdots\!72}a^{16}-\frac{13\!\cdots\!79}{61\!\cdots\!48}a^{15}+\frac{28\!\cdots\!50}{10\!\cdots\!09}a^{14}+\frac{99\!\cdots\!81}{21\!\cdots\!18}a^{13}-\frac{49\!\cdots\!26}{15\!\cdots\!87}a^{12}+\frac{39\!\cdots\!51}{21\!\cdots\!18}a^{11}+\frac{20\!\cdots\!95}{10\!\cdots\!09}a^{10}-\frac{38\!\cdots\!17}{21\!\cdots\!18}a^{9}-\frac{48\!\cdots\!15}{85\!\cdots\!72}a^{8}+\frac{10\!\cdots\!76}{15\!\cdots\!87}a^{7}+\frac{15\!\cdots\!97}{21\!\cdots\!18}a^{6}-\frac{11\!\cdots\!38}{10\!\cdots\!09}a^{5}-\frac{17\!\cdots\!19}{61\!\cdots\!48}a^{4}+\frac{27\!\cdots\!53}{42\!\cdots\!36}a^{3}+\frac{76\!\cdots\!18}{10\!\cdots\!09}a^{2}-\frac{15\!\cdots\!85}{15\!\cdots\!87}a+\frac{62\!\cdots\!47}{12\!\cdots\!96}$, $\frac{65\!\cdots\!79}{42\!\cdots\!36}a^{17}-\frac{50\!\cdots\!65}{42\!\cdots\!36}a^{16}-\frac{12\!\cdots\!17}{42\!\cdots\!36}a^{15}+\frac{39\!\cdots\!60}{10\!\cdots\!09}a^{14}+\frac{15\!\cdots\!05}{42\!\cdots\!36}a^{13}-\frac{94\!\cdots\!17}{21\!\cdots\!18}a^{12}+\frac{30\!\cdots\!62}{10\!\cdots\!09}a^{11}+\frac{55\!\cdots\!85}{21\!\cdots\!18}a^{10}-\frac{28\!\cdots\!95}{10\!\cdots\!09}a^{9}-\frac{32\!\cdots\!91}{42\!\cdots\!36}a^{8}+\frac{20\!\cdots\!03}{21\!\cdots\!18}a^{7}+\frac{21\!\cdots\!19}{21\!\cdots\!18}a^{6}-\frac{68\!\cdots\!77}{42\!\cdots\!36}a^{5}-\frac{39\!\cdots\!76}{10\!\cdots\!09}a^{4}+\frac{38\!\cdots\!07}{42\!\cdots\!36}a^{3}+\frac{19\!\cdots\!71}{10\!\cdots\!09}a^{2}-\frac{90\!\cdots\!23}{61\!\cdots\!48}a+\frac{45\!\cdots\!35}{61\!\cdots\!48}$, $\frac{32\!\cdots\!65}{42\!\cdots\!36}a^{17}-\frac{50\!\cdots\!17}{85\!\cdots\!72}a^{16}-\frac{62\!\cdots\!23}{42\!\cdots\!36}a^{15}+\frac{78\!\cdots\!89}{42\!\cdots\!36}a^{14}+\frac{59\!\cdots\!11}{42\!\cdots\!36}a^{13}-\frac{95\!\cdots\!71}{42\!\cdots\!36}a^{12}+\frac{62\!\cdots\!95}{42\!\cdots\!36}a^{11}+\frac{27\!\cdots\!67}{21\!\cdots\!18}a^{10}-\frac{57\!\cdots\!09}{42\!\cdots\!36}a^{9}-\frac{32\!\cdots\!93}{85\!\cdots\!72}a^{8}+\frac{21\!\cdots\!31}{42\!\cdots\!36}a^{7}+\frac{53\!\cdots\!87}{10\!\cdots\!09}a^{6}-\frac{17\!\cdots\!83}{21\!\cdots\!18}a^{5}-\frac{19\!\cdots\!88}{10\!\cdots\!09}a^{4}+\frac{14\!\cdots\!81}{30\!\cdots\!74}a^{3}+\frac{19\!\cdots\!67}{42\!\cdots\!36}a^{2}-\frac{11\!\cdots\!06}{15\!\cdots\!87}a+\frac{45\!\cdots\!11}{12\!\cdots\!96}$, $\frac{20\!\cdots\!91}{85\!\cdots\!72}a^{17}-\frac{16\!\cdots\!57}{85\!\cdots\!72}a^{16}-\frac{48\!\cdots\!15}{10\!\cdots\!09}a^{15}+\frac{25\!\cdots\!41}{42\!\cdots\!36}a^{14}+\frac{34\!\cdots\!17}{21\!\cdots\!18}a^{13}-\frac{30\!\cdots\!79}{42\!\cdots\!36}a^{12}+\frac{53\!\cdots\!31}{10\!\cdots\!09}a^{11}+\frac{17\!\cdots\!93}{42\!\cdots\!36}a^{10}-\frac{38\!\cdots\!95}{85\!\cdots\!72}a^{9}-\frac{10\!\cdots\!23}{85\!\cdots\!72}a^{8}+\frac{17\!\cdots\!18}{10\!\cdots\!09}a^{7}+\frac{67\!\cdots\!77}{42\!\cdots\!36}a^{6}-\frac{11\!\cdots\!51}{42\!\cdots\!36}a^{5}-\frac{24\!\cdots\!69}{42\!\cdots\!36}a^{4}+\frac{16\!\cdots\!11}{10\!\cdots\!09}a^{3}-\frac{57\!\cdots\!55}{10\!\cdots\!09}a^{2}-\frac{30\!\cdots\!77}{12\!\cdots\!96}a+\frac{15\!\cdots\!37}{12\!\cdots\!96}$, $\frac{35\!\cdots\!39}{21\!\cdots\!18}a^{17}-\frac{19\!\cdots\!76}{15\!\cdots\!87}a^{16}-\frac{64\!\cdots\!25}{21\!\cdots\!18}a^{15}+\frac{16\!\cdots\!87}{42\!\cdots\!36}a^{14}+\frac{12\!\cdots\!29}{15\!\cdots\!87}a^{13}-\frac{20\!\cdots\!31}{42\!\cdots\!36}a^{12}+\frac{36\!\cdots\!20}{10\!\cdots\!09}a^{11}+\frac{29\!\cdots\!50}{10\!\cdots\!09}a^{10}-\frac{64\!\cdots\!91}{21\!\cdots\!18}a^{9}-\frac{49\!\cdots\!13}{61\!\cdots\!48}a^{8}+\frac{11\!\cdots\!50}{10\!\cdots\!09}a^{7}+\frac{11\!\cdots\!51}{10\!\cdots\!09}a^{6}-\frac{27\!\cdots\!55}{15\!\cdots\!87}a^{5}-\frac{14\!\cdots\!51}{42\!\cdots\!36}a^{4}+\frac{21\!\cdots\!87}{21\!\cdots\!18}a^{3}-\frac{87\!\cdots\!09}{61\!\cdots\!48}a^{2}-\frac{49\!\cdots\!13}{30\!\cdots\!74}a+\frac{13\!\cdots\!06}{15\!\cdots\!87}$, $\frac{84\!\cdots\!57}{85\!\cdots\!72}a^{17}-\frac{64\!\cdots\!19}{85\!\cdots\!72}a^{16}-\frac{80\!\cdots\!87}{42\!\cdots\!36}a^{15}+\frac{10\!\cdots\!35}{42\!\cdots\!36}a^{14}+\frac{26\!\cdots\!81}{10\!\cdots\!09}a^{13}-\frac{30\!\cdots\!61}{10\!\cdots\!09}a^{12}+\frac{38\!\cdots\!15}{21\!\cdots\!18}a^{11}+\frac{71\!\cdots\!69}{42\!\cdots\!36}a^{10}-\frac{14\!\cdots\!57}{85\!\cdots\!72}a^{9}-\frac{41\!\cdots\!93}{85\!\cdots\!72}a^{8}+\frac{13\!\cdots\!97}{21\!\cdots\!18}a^{7}+\frac{27\!\cdots\!03}{42\!\cdots\!36}a^{6}-\frac{43\!\cdots\!55}{42\!\cdots\!36}a^{5}-\frac{10\!\cdots\!35}{42\!\cdots\!36}a^{4}+\frac{24\!\cdots\!61}{42\!\cdots\!36}a^{3}+\frac{38\!\cdots\!93}{21\!\cdots\!18}a^{2}-\frac{11\!\cdots\!71}{12\!\cdots\!96}a+\frac{57\!\cdots\!69}{12\!\cdots\!96}$, $\frac{59\!\cdots\!51}{42\!\cdots\!36}a^{17}-\frac{46\!\cdots\!01}{42\!\cdots\!36}a^{16}-\frac{11\!\cdots\!03}{42\!\cdots\!36}a^{15}+\frac{20\!\cdots\!19}{61\!\cdots\!48}a^{14}+\frac{35\!\cdots\!71}{10\!\cdots\!09}a^{13}-\frac{17\!\cdots\!81}{42\!\cdots\!36}a^{12}+\frac{15\!\cdots\!89}{61\!\cdots\!48}a^{11}+\frac{14\!\cdots\!29}{61\!\cdots\!48}a^{10}-\frac{10\!\cdots\!31}{42\!\cdots\!36}a^{9}-\frac{29\!\cdots\!87}{42\!\cdots\!36}a^{8}+\frac{38\!\cdots\!39}{42\!\cdots\!36}a^{7}+\frac{55\!\cdots\!33}{61\!\cdots\!48}a^{6}-\frac{30\!\cdots\!07}{21\!\cdots\!18}a^{5}-\frac{36\!\cdots\!16}{10\!\cdots\!09}a^{4}+\frac{17\!\cdots\!97}{21\!\cdots\!18}a^{3}+\frac{34\!\cdots\!88}{10\!\cdots\!09}a^{2}-\frac{82\!\cdots\!77}{61\!\cdots\!48}a+\frac{10\!\cdots\!24}{15\!\cdots\!87}$, $\frac{16\!\cdots\!89}{42\!\cdots\!36}a^{17}-\frac{64\!\cdots\!63}{21\!\cdots\!18}a^{16}-\frac{82\!\cdots\!30}{10\!\cdots\!09}a^{15}+\frac{10\!\cdots\!03}{10\!\cdots\!09}a^{14}+\frac{15\!\cdots\!65}{10\!\cdots\!09}a^{13}-\frac{12\!\cdots\!28}{10\!\cdots\!09}a^{12}+\frac{14\!\cdots\!57}{21\!\cdots\!18}a^{11}+\frac{28\!\cdots\!99}{42\!\cdots\!36}a^{10}-\frac{68\!\cdots\!84}{10\!\cdots\!09}a^{9}-\frac{83\!\cdots\!55}{42\!\cdots\!36}a^{8}+\frac{51\!\cdots\!01}{21\!\cdots\!18}a^{7}+\frac{11\!\cdots\!35}{42\!\cdots\!36}a^{6}-\frac{16\!\cdots\!21}{42\!\cdots\!36}a^{5}-\frac{42\!\cdots\!07}{42\!\cdots\!36}a^{4}+\frac{48\!\cdots\!55}{21\!\cdots\!18}a^{3}+\frac{66\!\cdots\!77}{42\!\cdots\!36}a^{2}-\frac{56\!\cdots\!01}{15\!\cdots\!87}a+\frac{11\!\cdots\!73}{61\!\cdots\!48}$
| 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: | \( 102184067250 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 102184067250 \cdot 1}{2\cdot\sqrt{178698494132131643847952616411136}}\cr\approx \mathstrut & 1.00191912091289 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 9*x^16 + 264*x^15 - 288*x^14 - 2916*x^13 + 5598*x^12 + 14526*x^11 - 39159*x^10 - 27383*x^9 + 128403*x^8 - 17730*x^7 - 188268*x^6 + 110070*x^5 + 90900*x^4 - 76710*x^3 - 9855*x^2 + 13041*x - 623)
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^18 - 9*x^17 - 9*x^16 + 264*x^15 - 288*x^14 - 2916*x^13 + 5598*x^12 + 14526*x^11 - 39159*x^10 - 27383*x^9 + 128403*x^8 - 17730*x^7 - 188268*x^6 + 110070*x^5 + 90900*x^4 - 76710*x^3 - 9855*x^2 + 13041*x - 623, 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^18 - 9*x^17 - 9*x^16 + 264*x^15 - 288*x^14 - 2916*x^13 + 5598*x^12 + 14526*x^11 - 39159*x^10 - 27383*x^9 + 128403*x^8 - 17730*x^7 - 188268*x^6 + 110070*x^5 + 90900*x^4 - 76710*x^3 - 9855*x^2 + 13041*x - 623);
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^18 - 9*x^17 - 9*x^16 + 264*x^15 - 288*x^14 - 2916*x^13 + 5598*x^12 + 14526*x^11 - 39159*x^10 - 27383*x^9 + 128403*x^8 - 17730*x^7 - 188268*x^6 + 110070*x^5 + 90900*x^4 - 76710*x^3 - 9855*x^2 + 13041*x - 623);
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
$C_3^2:C_6$ (as 18T21):
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 54 |
| The 10 conjugacy class representatives for $C_3^2:C_6$ |
| Character table for $C_3^2:C_6$ |
Intermediate fields
| \(\Q(\sqrt{21}) \), 3.3.756.1 x3, 6.6.12002256.1, 9.9.2917096519063104.1 x3 |
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 9 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Minimal sibling: | 9.9.2917096519063104.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{9}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(3\)
| Deg $18$ | $18$ | $1$ | $37$ | |||
|
\(7\)
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.6.5.6 | $x^{6} + 28$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 7.6.5.6 | $x^{6} + 28$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |