Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 75*x^16 + 768*x^15 + 1986*x^14 - 25806*x^13 - 19822*x^12 + 438816*x^11 - 14037*x^10 - 4078547*x^9 + 1648491*x^8 + 21086784*x^7 - 11912402*x^6 - 58701558*x^5 + 36211158*x^4 + 78699796*x^3 - 48626967*x^2 - 36169521*x + 22115221)
gp: K = bnfinit(y^18 - 9*y^17 - 75*y^16 + 768*y^15 + 1986*y^14 - 25806*y^13 - 19822*y^12 + 438816*y^11 - 14037*y^10 - 4078547*y^9 + 1648491*y^8 + 21086784*y^7 - 11912402*y^6 - 58701558*y^5 + 36211158*y^4 + 78699796*y^3 - 48626967*y^2 - 36169521*y + 22115221, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 - 75*x^16 + 768*x^15 + 1986*x^14 - 25806*x^13 - 19822*x^12 + 438816*x^11 - 14037*x^10 - 4078547*x^9 + 1648491*x^8 + 21086784*x^7 - 11912402*x^6 - 58701558*x^5 + 36211158*x^4 + 78699796*x^3 - 48626967*x^2 - 36169521*x + 22115221);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 - 75*x^16 + 768*x^15 + 1986*x^14 - 25806*x^13 - 19822*x^12 + 438816*x^11 - 14037*x^10 - 4078547*x^9 + 1648491*x^8 + 21086784*x^7 - 11912402*x^6 - 58701558*x^5 + 36211158*x^4 + 78699796*x^3 - 48626967*x^2 - 36169521*x + 22115221)
\( x^{18} - 9 x^{17} - 75 x^{16} + 768 x^{15} + 1986 x^{14} - 25806 x^{13} - 19822 x^{12} + 438816 x^{11} + \cdots + 22115221 \)
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: |
\(56290847976736302998762533947799425024\)
\(\medspace = 2^{12}\cdot 3^{33}\cdot 47^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(125.10\) | 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: | $2^{2/3}3^{11/6}47^{3/4}\approx 213.54153367766565$ | ||
| Ramified primes: |
\(2\), \(3\), \(47\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{141}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{3}{8}a^{4}-\frac{1}{8}a^{3}+\frac{1}{4}a^{2}+\frac{1}{8}a+\frac{3}{8}$, $\frac{1}{24}a^{15}-\frac{1}{8}a^{13}-\frac{1}{24}a^{12}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}+\frac{1}{24}a^{6}-\frac{1}{8}a^{5}-\frac{3}{8}a^{3}-\frac{1}{8}a^{2}-\frac{11}{24}$, $\frac{1}{528}a^{16}-\frac{1}{264}a^{15}+\frac{1}{88}a^{14}-\frac{5}{66}a^{13}+\frac{2}{33}a^{12}+\frac{9}{88}a^{11}-\frac{3}{44}a^{10}+\frac{7}{88}a^{9}-\frac{1}{176}a^{8}-\frac{1}{264}a^{7}-\frac{23}{132}a^{6}-\frac{21}{88}a^{5}+\frac{9}{22}a^{4}-\frac{1}{2}a^{3}-\frac{13}{88}a^{2}-\frac{103}{264}a-\frac{23}{528}$, $\frac{1}{38\!\cdots\!72}a^{17}+\frac{50\!\cdots\!37}{35\!\cdots\!52}a^{16}-\frac{11\!\cdots\!63}{19\!\cdots\!36}a^{15}+\frac{21\!\cdots\!91}{96\!\cdots\!68}a^{14}-\frac{24\!\cdots\!59}{48\!\cdots\!34}a^{13}+\frac{16\!\cdots\!59}{24\!\cdots\!67}a^{12}+\frac{36\!\cdots\!33}{32\!\cdots\!56}a^{11}-\frac{18\!\cdots\!11}{21\!\cdots\!04}a^{10}-\frac{22\!\cdots\!57}{12\!\cdots\!24}a^{9}-\frac{18\!\cdots\!59}{38\!\cdots\!72}a^{8}-\frac{11\!\cdots\!75}{96\!\cdots\!68}a^{7}-\frac{11\!\cdots\!51}{19\!\cdots\!36}a^{6}-\frac{44\!\cdots\!03}{10\!\cdots\!52}a^{5}-\frac{24\!\cdots\!39}{64\!\cdots\!12}a^{4}+\frac{23\!\cdots\!09}{64\!\cdots\!12}a^{3}+\frac{73\!\cdots\!93}{19\!\cdots\!36}a^{2}-\frac{17\!\cdots\!11}{38\!\cdots\!72}a+\frac{16\!\cdots\!15}{20\!\cdots\!88}$
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
$C_{3}$, which has order $3$ (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: |
$\frac{35\!\cdots\!25}{12\!\cdots\!24}a^{17}-\frac{12\!\cdots\!89}{53\!\cdots\!72}a^{16}-\frac{32\!\cdots\!75}{16\!\cdots\!78}a^{15}+\frac{32\!\cdots\!49}{16\!\cdots\!78}a^{14}+\frac{34\!\cdots\!19}{64\!\cdots\!12}a^{13}-\frac{10\!\cdots\!65}{16\!\cdots\!78}a^{12}-\frac{57\!\cdots\!33}{10\!\cdots\!52}a^{11}+\frac{10\!\cdots\!69}{10\!\cdots\!52}a^{10}+\frac{26\!\cdots\!25}{42\!\cdots\!08}a^{9}-\frac{51\!\cdots\!01}{64\!\cdots\!12}a^{8}+\frac{71\!\cdots\!69}{32\!\cdots\!56}a^{7}+\frac{10\!\cdots\!31}{32\!\cdots\!56}a^{6}-\frac{27\!\cdots\!43}{21\!\cdots\!04}a^{5}-\frac{32\!\cdots\!95}{53\!\cdots\!26}a^{4}+\frac{12\!\cdots\!59}{53\!\cdots\!26}a^{3}+\frac{10\!\cdots\!31}{32\!\cdots\!56}a^{2}-\frac{16\!\cdots\!27}{12\!\cdots\!24}a-\frac{39\!\cdots\!95}{33\!\cdots\!48}$, $\frac{24\!\cdots\!37}{12\!\cdots\!24}a^{17}-\frac{13\!\cdots\!55}{73\!\cdots\!99}a^{16}-\frac{82\!\cdots\!53}{64\!\cdots\!12}a^{15}+\frac{12\!\cdots\!87}{80\!\cdots\!89}a^{14}+\frac{83\!\cdots\!21}{32\!\cdots\!56}a^{13}-\frac{31\!\cdots\!35}{64\!\cdots\!12}a^{12}-\frac{35\!\cdots\!77}{10\!\cdots\!52}a^{11}+\frac{16\!\cdots\!11}{21\!\cdots\!04}a^{10}-\frac{20\!\cdots\!53}{42\!\cdots\!08}a^{9}-\frac{20\!\cdots\!47}{32\!\cdots\!56}a^{8}+\frac{18\!\cdots\!19}{32\!\cdots\!56}a^{7}+\frac{17\!\cdots\!81}{64\!\cdots\!12}a^{6}-\frac{27\!\cdots\!43}{10\!\cdots\!52}a^{5}-\frac{61\!\cdots\!95}{10\!\cdots\!52}a^{4}+\frac{95\!\cdots\!91}{21\!\cdots\!04}a^{3}+\frac{31\!\cdots\!01}{64\!\cdots\!12}a^{2}-\frac{21\!\cdots\!23}{12\!\cdots\!24}a-\frac{29\!\cdots\!81}{33\!\cdots\!48}$, $\frac{14\!\cdots\!13}{64\!\cdots\!12}a^{17}-\frac{10\!\cdots\!71}{29\!\cdots\!96}a^{16}-\frac{61\!\cdots\!23}{64\!\cdots\!12}a^{15}+\frac{17\!\cdots\!83}{64\!\cdots\!12}a^{14}-\frac{67\!\cdots\!19}{80\!\cdots\!89}a^{13}-\frac{46\!\cdots\!53}{64\!\cdots\!12}a^{12}+\frac{76\!\cdots\!47}{21\!\cdots\!04}a^{11}+\frac{89\!\cdots\!83}{10\!\cdots\!52}a^{10}-\frac{64\!\cdots\!31}{10\!\cdots\!52}a^{9}-\frac{19\!\cdots\!61}{64\!\cdots\!12}a^{8}+\frac{31\!\cdots\!73}{64\!\cdots\!12}a^{7}-\frac{38\!\cdots\!43}{32\!\cdots\!56}a^{6}-\frac{42\!\cdots\!13}{21\!\cdots\!04}a^{5}+\frac{23\!\cdots\!63}{21\!\cdots\!04}a^{4}+\frac{18\!\cdots\!17}{53\!\cdots\!26}a^{3}-\frac{15\!\cdots\!33}{64\!\cdots\!12}a^{2}-\frac{61\!\cdots\!91}{32\!\cdots\!56}a+\frac{55\!\cdots\!10}{42\!\cdots\!31}$, $\frac{83\!\cdots\!39}{32\!\cdots\!56}a^{17}-\frac{28\!\cdots\!43}{11\!\cdots\!84}a^{16}-\frac{28\!\cdots\!53}{16\!\cdots\!78}a^{15}+\frac{64\!\cdots\!89}{32\!\cdots\!56}a^{14}+\frac{11\!\cdots\!95}{32\!\cdots\!56}a^{13}-\frac{40\!\cdots\!17}{64\!\cdots\!12}a^{12}-\frac{71\!\cdots\!89}{10\!\cdots\!52}a^{11}+\frac{51\!\cdots\!27}{53\!\cdots\!26}a^{10}-\frac{66\!\cdots\!57}{10\!\cdots\!52}a^{9}-\frac{95\!\cdots\!43}{12\!\cdots\!24}a^{8}+\frac{24\!\cdots\!75}{32\!\cdots\!56}a^{7}+\frac{45\!\cdots\!23}{16\!\cdots\!78}a^{6}-\frac{18\!\cdots\!91}{53\!\cdots\!26}a^{5}-\frac{96\!\cdots\!47}{21\!\cdots\!04}a^{4}+\frac{66\!\cdots\!07}{10\!\cdots\!52}a^{3}+\frac{98\!\cdots\!29}{80\!\cdots\!89}a^{2}-\frac{25\!\cdots\!22}{80\!\cdots\!89}a+\frac{57\!\cdots\!73}{67\!\cdots\!96}$, $\frac{39\!\cdots\!41}{42\!\cdots\!08}a^{17}-\frac{18\!\cdots\!43}{38\!\cdots\!28}a^{16}-\frac{10\!\cdots\!29}{10\!\cdots\!52}a^{15}+\frac{90\!\cdots\!03}{21\!\cdots\!04}a^{14}+\frac{45\!\cdots\!21}{10\!\cdots\!52}a^{13}-\frac{32\!\cdots\!99}{21\!\cdots\!04}a^{12}-\frac{20\!\cdots\!37}{21\!\cdots\!04}a^{11}+\frac{60\!\cdots\!59}{21\!\cdots\!04}a^{10}+\frac{52\!\cdots\!35}{42\!\cdots\!08}a^{9}-\frac{12\!\cdots\!69}{42\!\cdots\!08}a^{8}-\frac{18\!\cdots\!93}{21\!\cdots\!04}a^{7}+\frac{34\!\cdots\!47}{21\!\cdots\!04}a^{6}+\frac{72\!\cdots\!63}{21\!\cdots\!04}a^{5}-\frac{24\!\cdots\!83}{53\!\cdots\!26}a^{4}-\frac{12\!\cdots\!89}{21\!\cdots\!04}a^{3}+\frac{65\!\cdots\!75}{10\!\cdots\!52}a^{2}+\frac{14\!\cdots\!39}{42\!\cdots\!08}a-\frac{60\!\cdots\!41}{22\!\cdots\!32}$, $\frac{47\!\cdots\!83}{12\!\cdots\!24}a^{17}-\frac{12\!\cdots\!25}{29\!\cdots\!96}a^{16}-\frac{16\!\cdots\!03}{64\!\cdots\!12}a^{15}+\frac{13\!\cdots\!39}{32\!\cdots\!56}a^{14}+\frac{40\!\cdots\!76}{80\!\cdots\!89}a^{13}-\frac{10\!\cdots\!45}{64\!\cdots\!12}a^{12}+\frac{50\!\cdots\!09}{26\!\cdots\!63}a^{11}+\frac{69\!\cdots\!69}{21\!\cdots\!04}a^{10}-\frac{81\!\cdots\!43}{42\!\cdots\!08}a^{9}-\frac{11\!\cdots\!95}{32\!\cdots\!56}a^{8}+\frac{39\!\cdots\!63}{16\!\cdots\!78}a^{7}+\frac{15\!\cdots\!11}{64\!\cdots\!12}a^{6}-\frac{33\!\cdots\!62}{26\!\cdots\!63}a^{5}-\frac{45\!\cdots\!99}{53\!\cdots\!26}a^{4}+\frac{50\!\cdots\!39}{21\!\cdots\!04}a^{3}+\frac{89\!\cdots\!73}{64\!\cdots\!12}a^{2}-\frac{65\!\cdots\!53}{12\!\cdots\!24}a-\frac{21\!\cdots\!31}{33\!\cdots\!48}$, $\frac{54\!\cdots\!89}{64\!\cdots\!12}a^{17}-\frac{62\!\cdots\!79}{58\!\cdots\!92}a^{16}-\frac{23\!\cdots\!51}{64\!\cdots\!12}a^{15}+\frac{55\!\cdots\!13}{64\!\cdots\!12}a^{14}-\frac{19\!\cdots\!33}{32\!\cdots\!56}a^{13}-\frac{17\!\cdots\!51}{64\!\cdots\!12}a^{12}+\frac{11\!\cdots\!41}{21\!\cdots\!04}a^{11}+\frac{42\!\cdots\!71}{10\!\cdots\!52}a^{10}-\frac{60\!\cdots\!89}{53\!\cdots\!26}a^{9}-\frac{23\!\cdots\!35}{80\!\cdots\!89}a^{8}+\frac{65\!\cdots\!39}{64\!\cdots\!12}a^{7}+\frac{35\!\cdots\!41}{32\!\cdots\!56}a^{6}-\frac{92\!\cdots\!43}{21\!\cdots\!04}a^{5}-\frac{37\!\cdots\!81}{21\!\cdots\!04}a^{4}+\frac{86\!\cdots\!63}{10\!\cdots\!52}a^{3}+\frac{34\!\cdots\!93}{64\!\cdots\!12}a^{2}-\frac{14\!\cdots\!67}{32\!\cdots\!56}a+\frac{34\!\cdots\!09}{33\!\cdots\!48}$, $\frac{88\!\cdots\!45}{32\!\cdots\!56}a^{17}-\frac{15\!\cdots\!05}{58\!\cdots\!92}a^{16}-\frac{59\!\cdots\!59}{32\!\cdots\!56}a^{15}+\frac{13\!\cdots\!15}{64\!\cdots\!12}a^{14}+\frac{23\!\cdots\!69}{64\!\cdots\!12}a^{13}-\frac{21\!\cdots\!53}{32\!\cdots\!56}a^{12}-\frac{23\!\cdots\!15}{21\!\cdots\!04}a^{11}+\frac{22\!\cdots\!81}{21\!\cdots\!04}a^{10}-\frac{84\!\cdots\!31}{10\!\cdots\!52}a^{9}-\frac{25\!\cdots\!21}{32\!\cdots\!56}a^{8}+\frac{60\!\cdots\!07}{64\!\cdots\!12}a^{7}+\frac{19\!\cdots\!39}{64\!\cdots\!12}a^{6}-\frac{11\!\cdots\!35}{26\!\cdots\!63}a^{5}-\frac{10\!\cdots\!25}{21\!\cdots\!04}a^{4}+\frac{17\!\cdots\!15}{21\!\cdots\!04}a^{3}+\frac{19\!\cdots\!59}{16\!\cdots\!78}a^{2}-\frac{28\!\cdots\!67}{64\!\cdots\!12}a+\frac{41\!\cdots\!73}{42\!\cdots\!31}$, $\frac{26\!\cdots\!19}{17\!\cdots\!76}a^{17}-\frac{29\!\cdots\!09}{17\!\cdots\!76}a^{16}-\frac{13\!\cdots\!83}{17\!\cdots\!76}a^{15}+\frac{58\!\cdots\!25}{43\!\cdots\!94}a^{14}+\frac{35\!\cdots\!57}{17\!\cdots\!76}a^{13}-\frac{64\!\cdots\!17}{15\!\cdots\!16}a^{12}+\frac{16\!\cdots\!93}{29\!\cdots\!96}a^{11}+\frac{11\!\cdots\!37}{19\!\cdots\!64}a^{10}-\frac{19\!\cdots\!97}{14\!\cdots\!98}a^{9}-\frac{72\!\cdots\!33}{17\!\cdots\!76}a^{8}+\frac{11\!\cdots\!73}{87\!\cdots\!88}a^{7}+\frac{21\!\cdots\!01}{17\!\cdots\!76}a^{6}-\frac{10\!\cdots\!67}{19\!\cdots\!64}a^{5}-\frac{34\!\cdots\!88}{73\!\cdots\!99}a^{4}+\frac{60\!\cdots\!83}{58\!\cdots\!92}a^{3}-\frac{58\!\cdots\!71}{17\!\cdots\!76}a^{2}-\frac{10\!\cdots\!57}{17\!\cdots\!76}a+\frac{13\!\cdots\!33}{46\!\cdots\!52}$, $\frac{10\!\cdots\!91}{19\!\cdots\!64}a^{17}-\frac{43\!\cdots\!27}{58\!\cdots\!92}a^{16}-\frac{46\!\cdots\!11}{29\!\cdots\!96}a^{15}+\frac{28\!\cdots\!97}{48\!\cdots\!66}a^{14}-\frac{24\!\cdots\!19}{29\!\cdots\!96}a^{13}-\frac{25\!\cdots\!07}{14\!\cdots\!98}a^{12}+\frac{44\!\cdots\!19}{97\!\cdots\!32}a^{11}+\frac{11\!\cdots\!93}{48\!\cdots\!66}a^{10}-\frac{15\!\cdots\!79}{19\!\cdots\!64}a^{9}-\frac{28\!\cdots\!93}{19\!\cdots\!64}a^{8}+\frac{15\!\cdots\!23}{26\!\cdots\!36}a^{7}+\frac{20\!\cdots\!13}{73\!\cdots\!99}a^{6}-\frac{18\!\cdots\!89}{97\!\cdots\!32}a^{5}+\frac{26\!\cdots\!15}{48\!\cdots\!66}a^{4}+\frac{36\!\cdots\!77}{24\!\cdots\!33}a^{3}-\frac{29\!\cdots\!39}{24\!\cdots\!33}a^{2}+\frac{12\!\cdots\!37}{58\!\cdots\!92}a-\frac{50\!\cdots\!21}{30\!\cdots\!68}$, $\frac{10\!\cdots\!57}{38\!\cdots\!72}a^{17}-\frac{53\!\cdots\!45}{17\!\cdots\!76}a^{16}-\frac{61\!\cdots\!53}{48\!\cdots\!34}a^{15}+\frac{46\!\cdots\!57}{19\!\cdots\!36}a^{14}-\frac{49\!\cdots\!63}{96\!\cdots\!68}a^{13}-\frac{69\!\cdots\!83}{96\!\cdots\!68}a^{12}+\frac{81\!\cdots\!11}{64\!\cdots\!12}a^{11}+\frac{21\!\cdots\!07}{21\!\cdots\!04}a^{10}-\frac{34\!\cdots\!31}{12\!\cdots\!24}a^{9}-\frac{30\!\cdots\!97}{48\!\cdots\!34}a^{8}+\frac{47\!\cdots\!37}{19\!\cdots\!36}a^{7}+\frac{25\!\cdots\!27}{19\!\cdots\!36}a^{6}-\frac{22\!\cdots\!13}{21\!\cdots\!04}a^{5}+\frac{17\!\cdots\!93}{64\!\cdots\!12}a^{4}+\frac{12\!\cdots\!13}{64\!\cdots\!12}a^{3}-\frac{11\!\cdots\!27}{96\!\cdots\!68}a^{2}-\frac{42\!\cdots\!57}{38\!\cdots\!72}a+\frac{35\!\cdots\!41}{50\!\cdots\!72}$, $\frac{81\!\cdots\!77}{21\!\cdots\!04}a^{17}-\frac{29\!\cdots\!21}{73\!\cdots\!99}a^{16}-\frac{14\!\cdots\!53}{64\!\cdots\!12}a^{15}+\frac{69\!\cdots\!11}{21\!\cdots\!04}a^{14}+\frac{89\!\cdots\!23}{32\!\cdots\!56}a^{13}-\frac{64\!\cdots\!49}{64\!\cdots\!12}a^{12}+\frac{14\!\cdots\!83}{21\!\cdots\!04}a^{11}+\frac{40\!\cdots\!04}{26\!\cdots\!63}a^{10}-\frac{23\!\cdots\!15}{10\!\cdots\!52}a^{9}-\frac{24\!\cdots\!49}{21\!\cdots\!04}a^{8}+\frac{13\!\cdots\!29}{64\!\cdots\!12}a^{7}+\frac{32\!\cdots\!67}{80\!\cdots\!89}a^{6}-\frac{20\!\cdots\!99}{21\!\cdots\!04}a^{5}-\frac{10\!\cdots\!77}{21\!\cdots\!04}a^{4}+\frac{18\!\cdots\!45}{10\!\cdots\!52}a^{3}-\frac{34\!\cdots\!51}{21\!\cdots\!04}a^{2}-\frac{14\!\cdots\!99}{16\!\cdots\!78}a+\frac{58\!\cdots\!07}{16\!\cdots\!24}$, $\frac{25\!\cdots\!87}{12\!\cdots\!24}a^{17}+\frac{27\!\cdots\!25}{38\!\cdots\!28}a^{16}-\frac{12\!\cdots\!94}{26\!\cdots\!63}a^{15}+\frac{33\!\cdots\!43}{64\!\cdots\!12}a^{14}+\frac{31\!\cdots\!87}{10\!\cdots\!52}a^{13}-\frac{15\!\cdots\!61}{21\!\cdots\!04}a^{12}-\frac{17\!\cdots\!47}{21\!\cdots\!04}a^{11}+\frac{55\!\cdots\!59}{21\!\cdots\!04}a^{10}+\frac{50\!\cdots\!95}{42\!\cdots\!08}a^{9}-\frac{54\!\cdots\!51}{12\!\cdots\!24}a^{8}-\frac{18\!\cdots\!59}{21\!\cdots\!04}a^{7}+\frac{73\!\cdots\!47}{21\!\cdots\!04}a^{6}+\frac{65\!\cdots\!83}{21\!\cdots\!04}a^{5}-\frac{74\!\cdots\!01}{53\!\cdots\!26}a^{4}-\frac{99\!\cdots\!49}{21\!\cdots\!04}a^{3}+\frac{19\!\cdots\!95}{80\!\cdots\!89}a^{2}+\frac{85\!\cdots\!67}{42\!\cdots\!08}a-\frac{28\!\cdots\!77}{22\!\cdots\!32}$, $\frac{63\!\cdots\!41}{38\!\cdots\!72}a^{17}-\frac{55\!\cdots\!79}{31\!\cdots\!32}a^{16}-\frac{17\!\cdots\!69}{19\!\cdots\!36}a^{15}+\frac{13\!\cdots\!83}{96\!\cdots\!68}a^{14}+\frac{67\!\cdots\!49}{96\!\cdots\!68}a^{13}-\frac{38\!\cdots\!35}{96\!\cdots\!68}a^{12}+\frac{13\!\cdots\!53}{32\!\cdots\!56}a^{11}+\frac{11\!\cdots\!51}{21\!\cdots\!04}a^{10}-\frac{12\!\cdots\!09}{12\!\cdots\!24}a^{9}-\frac{14\!\cdots\!69}{38\!\cdots\!72}a^{8}+\frac{79\!\cdots\!99}{96\!\cdots\!68}a^{7}+\frac{20\!\cdots\!23}{19\!\cdots\!36}a^{6}-\frac{16\!\cdots\!45}{53\!\cdots\!26}a^{5}-\frac{63\!\cdots\!83}{64\!\cdots\!12}a^{4}+\frac{30\!\cdots\!95}{64\!\cdots\!12}a^{3}-\frac{16\!\cdots\!97}{19\!\cdots\!36}a^{2}-\frac{89\!\cdots\!59}{38\!\cdots\!72}a+\frac{20\!\cdots\!37}{20\!\cdots\!88}$, $\frac{99\!\cdots\!39}{19\!\cdots\!36}a^{17}-\frac{10\!\cdots\!99}{35\!\cdots\!52}a^{16}-\frac{10\!\cdots\!73}{19\!\cdots\!36}a^{15}+\frac{47\!\cdots\!17}{19\!\cdots\!36}a^{14}+\frac{21\!\cdots\!63}{96\!\cdots\!68}a^{13}-\frac{20\!\cdots\!01}{24\!\cdots\!67}a^{12}-\frac{30\!\cdots\!37}{64\!\cdots\!12}a^{11}+\frac{15\!\cdots\!95}{10\!\cdots\!52}a^{10}+\frac{95\!\cdots\!19}{16\!\cdots\!78}a^{9}-\frac{54\!\cdots\!49}{38\!\cdots\!72}a^{8}-\frac{82\!\cdots\!03}{19\!\cdots\!36}a^{7}+\frac{69\!\cdots\!79}{96\!\cdots\!68}a^{6}+\frac{35\!\cdots\!61}{21\!\cdots\!04}a^{5}-\frac{30\!\cdots\!47}{16\!\cdots\!78}a^{4}-\frac{92\!\cdots\!57}{32\!\cdots\!56}a^{3}+\frac{44\!\cdots\!31}{19\!\cdots\!36}a^{2}+\frac{15\!\cdots\!59}{96\!\cdots\!68}a-\frac{21\!\cdots\!43}{20\!\cdots\!88}$, $\frac{33\!\cdots\!23}{67\!\cdots\!96}a^{17}-\frac{58\!\cdots\!91}{27\!\cdots\!88}a^{16}+\frac{19\!\cdots\!45}{16\!\cdots\!24}a^{15}+\frac{13\!\cdots\!75}{84\!\cdots\!62}a^{14}-\frac{37\!\cdots\!37}{33\!\cdots\!48}a^{13}-\frac{67\!\cdots\!23}{16\!\cdots\!24}a^{12}+\frac{20\!\cdots\!01}{56\!\cdots\!08}a^{11}+\frac{55\!\cdots\!02}{14\!\cdots\!77}a^{10}-\frac{12\!\cdots\!81}{22\!\cdots\!32}a^{9}-\frac{12\!\cdots\!27}{33\!\cdots\!48}a^{8}+\frac{73\!\cdots\!23}{16\!\cdots\!24}a^{7}-\frac{14\!\cdots\!85}{84\!\cdots\!62}a^{6}-\frac{18\!\cdots\!07}{11\!\cdots\!16}a^{5}+\frac{13\!\cdots\!91}{14\!\cdots\!77}a^{4}+\frac{14\!\cdots\!43}{56\!\cdots\!08}a^{3}-\frac{71\!\cdots\!12}{42\!\cdots\!31}a^{2}-\frac{76\!\cdots\!61}{67\!\cdots\!96}a+\frac{24\!\cdots\!63}{33\!\cdots\!48}$, $\frac{15\!\cdots\!09}{19\!\cdots\!36}a^{17}-\frac{20\!\cdots\!47}{35\!\cdots\!52}a^{16}-\frac{13\!\cdots\!63}{19\!\cdots\!36}a^{15}+\frac{99\!\cdots\!39}{19\!\cdots\!36}a^{14}+\frac{11\!\cdots\!39}{48\!\cdots\!34}a^{13}-\frac{16\!\cdots\!21}{96\!\cdots\!68}a^{12}-\frac{28\!\cdots\!05}{64\!\cdots\!12}a^{11}+\frac{16\!\cdots\!13}{53\!\cdots\!26}a^{10}+\frac{15\!\cdots\!07}{32\!\cdots\!56}a^{9}-\frac{11\!\cdots\!97}{38\!\cdots\!72}a^{8}-\frac{62\!\cdots\!23}{19\!\cdots\!36}a^{7}+\frac{76\!\cdots\!63}{48\!\cdots\!34}a^{6}+\frac{29\!\cdots\!13}{21\!\cdots\!04}a^{5}-\frac{14\!\cdots\!59}{32\!\cdots\!56}a^{4}-\frac{49\!\cdots\!97}{16\!\cdots\!78}a^{3}+\frac{10\!\cdots\!19}{19\!\cdots\!36}a^{2}+\frac{27\!\cdots\!51}{96\!\cdots\!68}a-\frac{29\!\cdots\!03}{20\!\cdots\!88}$
| 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: | \( 15891448025500 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 15891448025500 \cdot 3}{2\cdot\sqrt{56290847976736302998762533947799425024}}\cr\approx \mathstrut & 0.832867196766136 \end{aligned}\] (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 - 75*x^16 + 768*x^15 + 1986*x^14 - 25806*x^13 - 19822*x^12 + 438816*x^11 - 14037*x^10 - 4078547*x^9 + 1648491*x^8 + 21086784*x^7 - 11912402*x^6 - 58701558*x^5 + 36211158*x^4 + 78699796*x^3 - 48626967*x^2 - 36169521*x + 22115221)
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 - 75*x^16 + 768*x^15 + 1986*x^14 - 25806*x^13 - 19822*x^12 + 438816*x^11 - 14037*x^10 - 4078547*x^9 + 1648491*x^8 + 21086784*x^7 - 11912402*x^6 - 58701558*x^5 + 36211158*x^4 + 78699796*x^3 - 48626967*x^2 - 36169521*x + 22115221, 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 - 75*x^16 + 768*x^15 + 1986*x^14 - 25806*x^13 - 19822*x^12 + 438816*x^11 - 14037*x^10 - 4078547*x^9 + 1648491*x^8 + 21086784*x^7 - 11912402*x^6 - 58701558*x^5 + 36211158*x^4 + 78699796*x^3 - 48626967*x^2 - 36169521*x + 22115221);
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 - 75*x^16 + 768*x^15 + 1986*x^14 - 25806*x^13 - 19822*x^12 + 438816*x^11 - 14037*x^10 - 4078547*x^9 + 1648491*x^8 + 21086784*x^7 - 11912402*x^6 - 58701558*x^5 + 36211158*x^4 + 78699796*x^3 - 48626967*x^2 - 36169521*x + 22115221);
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 72 |
| The 9 conjugacy class representatives for $C_3:S_4$ |
| Character table for $C_3:S_4$ |
Intermediate fields
| 3.3.11421.1, 3.3.45684.2, 3.3.45684.1, 3.3.564.1, 6.6.44851536.1, 9.9.13443473060864064.2 |
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 12 siblings: | data not computed |
| Degree 18 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.12.749216001849307784743212864.1 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{6}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{9}$ | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(3\)
| 3.6.11.9 | $x^{6} + 3$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ |
| 3.12.22.67 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 816 x^{8} + 1920 x^{7} + 3512 x^{6} + 4560 x^{5} + 3444 x^{4} + 272 x^{3} - 720 x^{2} + 960 x + 928$ | $6$ | $2$ | $22$ | $D_6$ | $[5/2]_{2}^{2}$ | |
|
\(47\)
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 47.4.3.2 | $x^{4} + 235$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.3.2 | $x^{4} + 235$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 47.4.3.2 | $x^{4} + 235$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |