Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9097*x^16 + 21918784*x^14 - 8722330342*x^12 - 13442398479722*x^10 + 8439306339453536*x^8 + 1882836967318101*x^6 - 314874705862769661*x^4 + 759018943953897696*x^2 - 232463222451360000)
gp: K = bnfinit(y^18 - 9097*y^16 + 21918784*y^14 - 8722330342*y^12 - 13442398479722*y^10 + 8439306339453536*y^8 + 1882836967318101*y^6 - 314874705862769661*y^4 + 759018943953897696*y^2 - 232463222451360000, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9097*x^16 + 21918784*x^14 - 8722330342*x^12 - 13442398479722*x^10 + 8439306339453536*x^8 + 1882836967318101*x^6 - 314874705862769661*x^4 + 759018943953897696*x^2 - 232463222451360000);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9097*x^16 + 21918784*x^14 - 8722330342*x^12 - 13442398479722*x^10 + 8439306339453536*x^8 + 1882836967318101*x^6 - 314874705862769661*x^4 + 759018943953897696*x^2 - 232463222451360000)
\( x^{18} - 9097 x^{16} + 21918784 x^{14} - 8722330342 x^{12} - 13442398479722 x^{10} + \cdots - 23\!\cdots\!00 \)
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: | $[14, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2402705811545396560791327146734278062629093830398539967179801088896102378329\)
\(\medspace = 3^{10}\cdot 37^{12}\cdot 2977717^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15\,410.50\) | 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: | not computed | ||
| Ramified primes: |
\(3\), \(37\), \(2977717\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{4}a$, $\frac{1}{12}a^{8}-\frac{1}{6}a^{6}+\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{12}a^{2}-\frac{1}{2}a$, $\frac{1}{180}a^{9}-\frac{11}{180}a^{7}+\frac{11}{45}a^{5}+\frac{23}{180}a^{3}-\frac{9}{20}a$, $\frac{1}{360}a^{10}+\frac{1}{90}a^{8}-\frac{1}{8}a^{7}-\frac{19}{90}a^{6}-\frac{37}{360}a^{4}+\frac{7}{30}a^{2}-\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{360}a^{11}-\frac{1}{24}a^{8}-\frac{4}{45}a^{7}-\frac{1}{6}a^{6}-\frac{11}{120}a^{5}+\frac{1}{6}a^{4}+\frac{43}{90}a^{3}+\frac{1}{24}a^{2}-\frac{1}{10}a$, $\frac{1}{1080}a^{12}-\frac{1}{360}a^{9}-\frac{4}{135}a^{8}-\frac{17}{180}a^{7}+\frac{49}{360}a^{6}-\frac{11}{90}a^{5}-\frac{47}{270}a^{4}+\frac{157}{360}a^{3}+\frac{3}{10}a^{2}+\frac{7}{20}a$, $\frac{1}{3240}a^{13}+\frac{1}{810}a^{9}-\frac{19}{540}a^{7}-\frac{28}{405}a^{5}-\frac{14}{45}a^{3}-\frac{53}{120}a-\frac{1}{2}$, $\frac{1}{19440}a^{14}+\frac{11}{9720}a^{10}+\frac{83}{3240}a^{8}-\frac{1}{8}a^{7}+\frac{71}{2430}a^{6}+\frac{79}{360}a^{4}+\frac{103}{720}a^{2}+\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{38880}a^{15}-\frac{1}{38880}a^{14}-\frac{1}{1215}a^{11}+\frac{1}{1215}a^{10}+\frac{11}{6480}a^{9}-\frac{47}{6480}a^{8}-\frac{167}{2430}a^{7}-\frac{146}{1215}a^{6}+\frac{1}{6}a^{5}+\frac{4}{45}a^{4}-\frac{83}{480}a^{3}+\frac{13}{288}a^{2}-\frac{3}{10}a-\frac{1}{2}$, $\frac{1}{10\!\cdots\!60}a^{16}+\frac{20\!\cdots\!07}{10\!\cdots\!60}a^{14}+\frac{11\!\cdots\!43}{43\!\cdots\!08}a^{12}+\frac{11\!\cdots\!81}{50\!\cdots\!80}a^{10}-\frac{29\!\cdots\!81}{50\!\cdots\!80}a^{8}-\frac{1}{8}a^{7}-\frac{15\!\cdots\!28}{15\!\cdots\!15}a^{6}-\frac{15\!\cdots\!99}{11\!\cdots\!40}a^{4}-\frac{1}{2}a^{3}+\frac{18\!\cdots\!17}{37\!\cdots\!80}a^{2}+\frac{1}{8}a+\frac{19\!\cdots\!77}{52\!\cdots\!24}$, $\frac{1}{18\!\cdots\!00}a^{17}-\frac{1}{20\!\cdots\!20}a^{16}-\frac{53\!\cdots\!37}{18\!\cdots\!00}a^{15}-\frac{20\!\cdots\!07}{20\!\cdots\!20}a^{14}+\frac{54\!\cdots\!91}{38\!\cdots\!00}a^{13}-\frac{11\!\cdots\!43}{87\!\cdots\!16}a^{12}-\frac{24\!\cdots\!57}{39\!\cdots\!00}a^{11}+\frac{12\!\cdots\!67}{10\!\cdots\!60}a^{10}+\frac{84\!\cdots\!53}{39\!\cdots\!00}a^{9}-\frac{33\!\cdots\!67}{10\!\cdots\!60}a^{8}-\frac{41\!\cdots\!69}{70\!\cdots\!50}a^{7}+\frac{43\!\cdots\!27}{15\!\cdots\!15}a^{6}-\frac{20\!\cdots\!91}{20\!\cdots\!00}a^{5}-\frac{15\!\cdots\!69}{22\!\cdots\!80}a^{4}-\frac{13\!\cdots\!43}{67\!\cdots\!00}a^{3}+\frac{61\!\cdots\!47}{15\!\cdots\!12}a^{2}-\frac{14\!\cdots\!53}{46\!\cdots\!00}a-\frac{46\!\cdots\!39}{10\!\cdots\!48}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
$C_{21}$, which has order $21$ (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: | $15$ | 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: |
$\frac{51\!\cdots\!37}{10\!\cdots\!60}a^{16}-\frac{51\!\cdots\!97}{11\!\cdots\!40}a^{14}+\frac{48\!\cdots\!31}{43\!\cdots\!08}a^{12}-\frac{73\!\cdots\!61}{16\!\cdots\!60}a^{10}-\frac{34\!\cdots\!77}{50\!\cdots\!80}a^{8}+\frac{74\!\cdots\!71}{17\!\cdots\!35}a^{6}+\frac{46\!\cdots\!39}{37\!\cdots\!80}a^{4}-\frac{15\!\cdots\!53}{12\!\cdots\!60}a^{2}+\frac{22\!\cdots\!79}{52\!\cdots\!24}$, $\frac{64\!\cdots\!25}{20\!\cdots\!12}a^{16}-\frac{29\!\cdots\!69}{10\!\cdots\!60}a^{14}+\frac{15\!\cdots\!53}{21\!\cdots\!54}a^{12}-\frac{14\!\cdots\!79}{50\!\cdots\!80}a^{10}-\frac{21\!\cdots\!01}{50\!\cdots\!80}a^{8}+\frac{17\!\cdots\!73}{63\!\cdots\!60}a^{6}-\frac{70\!\cdots\!97}{37\!\cdots\!80}a^{4}+\frac{13\!\cdots\!73}{37\!\cdots\!80}a^{2}-\frac{53\!\cdots\!89}{52\!\cdots\!24}$, $\frac{15\!\cdots\!03}{28\!\cdots\!60}a^{16}-\frac{27\!\cdots\!43}{56\!\cdots\!92}a^{14}+\frac{28\!\cdots\!81}{24\!\cdots\!60}a^{12}-\frac{13\!\cdots\!43}{28\!\cdots\!96}a^{10}-\frac{10\!\cdots\!11}{14\!\cdots\!80}a^{8}+\frac{32\!\cdots\!63}{70\!\cdots\!40}a^{6}+\frac{12\!\cdots\!77}{93\!\cdots\!20}a^{4}-\frac{42\!\cdots\!79}{31\!\cdots\!40}a^{2}+\frac{60\!\cdots\!42}{13\!\cdots\!81}$, $\frac{14\!\cdots\!09}{10\!\cdots\!60}a^{16}-\frac{10\!\cdots\!41}{20\!\cdots\!12}a^{14}+\frac{62\!\cdots\!91}{21\!\cdots\!40}a^{12}+\frac{15\!\cdots\!73}{50\!\cdots\!80}a^{10}-\frac{11\!\cdots\!97}{50\!\cdots\!80}a^{8}+\frac{20\!\cdots\!08}{15\!\cdots\!15}a^{6}+\frac{37\!\cdots\!11}{41\!\cdots\!20}a^{4}-\frac{52\!\cdots\!71}{75\!\cdots\!56}a^{2}+\frac{57\!\cdots\!03}{52\!\cdots\!24}$, $\frac{89\!\cdots\!33}{12\!\cdots\!80}a^{17}-\frac{92\!\cdots\!83}{20\!\cdots\!20}a^{16}-\frac{88\!\cdots\!17}{12\!\cdots\!80}a^{15}+\frac{91\!\cdots\!79}{20\!\cdots\!20}a^{14}+\frac{57\!\cdots\!19}{25\!\cdots\!20}a^{13}-\frac{58\!\cdots\!83}{43\!\cdots\!80}a^{12}-\frac{61\!\cdots\!97}{26\!\cdots\!80}a^{11}+\frac{14\!\cdots\!73}{10\!\cdots\!60}a^{10}+\frac{42\!\cdots\!85}{52\!\cdots\!56}a^{9}-\frac{50\!\cdots\!21}{10\!\cdots\!60}a^{8}+\frac{52\!\cdots\!47}{23\!\cdots\!15}a^{7}-\frac{17\!\cdots\!07}{12\!\cdots\!32}a^{6}-\frac{81\!\cdots\!39}{26\!\cdots\!64}a^{5}+\frac{28\!\cdots\!79}{15\!\cdots\!12}a^{4}+\frac{65\!\cdots\!37}{89\!\cdots\!88}a^{3}-\frac{33\!\cdots\!51}{75\!\cdots\!60}a^{2}-\frac{13\!\cdots\!25}{62\!\cdots\!52}a+\frac{14\!\cdots\!63}{10\!\cdots\!48}$, $\frac{13\!\cdots\!19}{33\!\cdots\!20}a^{16}-\frac{12\!\cdots\!07}{33\!\cdots\!20}a^{14}+\frac{12\!\cdots\!43}{14\!\cdots\!36}a^{12}-\frac{60\!\cdots\!29}{16\!\cdots\!60}a^{10}-\frac{91\!\cdots\!51}{16\!\cdots\!60}a^{8}+\frac{36\!\cdots\!29}{10\!\cdots\!10}a^{6}-\frac{27\!\cdots\!19}{11\!\cdots\!40}a^{4}+\frac{18\!\cdots\!97}{41\!\cdots\!20}a^{2}-\frac{68\!\cdots\!13}{52\!\cdots\!24}$, $\frac{89\!\cdots\!33}{12\!\cdots\!80}a^{17}+\frac{92\!\cdots\!83}{20\!\cdots\!20}a^{16}-\frac{88\!\cdots\!17}{12\!\cdots\!80}a^{15}-\frac{91\!\cdots\!79}{20\!\cdots\!20}a^{14}+\frac{57\!\cdots\!19}{25\!\cdots\!20}a^{13}+\frac{58\!\cdots\!83}{43\!\cdots\!80}a^{12}-\frac{61\!\cdots\!97}{26\!\cdots\!80}a^{11}-\frac{14\!\cdots\!73}{10\!\cdots\!60}a^{10}+\frac{42\!\cdots\!85}{52\!\cdots\!56}a^{9}+\frac{50\!\cdots\!21}{10\!\cdots\!60}a^{8}+\frac{52\!\cdots\!47}{23\!\cdots\!15}a^{7}+\frac{17\!\cdots\!07}{12\!\cdots\!32}a^{6}-\frac{81\!\cdots\!39}{26\!\cdots\!64}a^{5}-\frac{28\!\cdots\!79}{15\!\cdots\!12}a^{4}+\frac{65\!\cdots\!37}{89\!\cdots\!88}a^{3}+\frac{33\!\cdots\!51}{75\!\cdots\!60}a^{2}-\frac{13\!\cdots\!25}{62\!\cdots\!52}a-\frac{14\!\cdots\!63}{10\!\cdots\!48}$, $\frac{10\!\cdots\!83}{60\!\cdots\!00}a^{17}+\frac{72\!\cdots\!57}{67\!\cdots\!40}a^{16}-\frac{97\!\cdots\!31}{60\!\cdots\!00}a^{15}-\frac{65\!\cdots\!77}{67\!\cdots\!40}a^{14}+\frac{50\!\cdots\!93}{12\!\cdots\!00}a^{13}+\frac{42\!\cdots\!79}{18\!\cdots\!45}a^{12}-\frac{20\!\cdots\!11}{13\!\cdots\!00}a^{11}-\frac{31\!\cdots\!43}{33\!\cdots\!20}a^{10}-\frac{31\!\cdots\!21}{13\!\cdots\!00}a^{9}-\frac{96\!\cdots\!49}{67\!\cdots\!04}a^{8}+\frac{14\!\cdots\!87}{94\!\cdots\!00}a^{7}+\frac{37\!\cdots\!07}{42\!\cdots\!40}a^{6}+\frac{58\!\cdots\!87}{67\!\cdots\!00}a^{5}+\frac{39\!\cdots\!01}{75\!\cdots\!60}a^{4}-\frac{41\!\cdots\!23}{74\!\cdots\!00}a^{3}-\frac{83\!\cdots\!59}{25\!\cdots\!20}a^{2}+\frac{17\!\cdots\!61}{15\!\cdots\!00}a+\frac{71\!\cdots\!65}{10\!\cdots\!48}$, $\frac{10\!\cdots\!83}{60\!\cdots\!00}a^{17}-\frac{72\!\cdots\!57}{67\!\cdots\!40}a^{16}-\frac{97\!\cdots\!31}{60\!\cdots\!00}a^{15}+\frac{65\!\cdots\!77}{67\!\cdots\!40}a^{14}+\frac{50\!\cdots\!93}{12\!\cdots\!00}a^{13}-\frac{42\!\cdots\!79}{18\!\cdots\!45}a^{12}-\frac{20\!\cdots\!11}{13\!\cdots\!00}a^{11}+\frac{31\!\cdots\!43}{33\!\cdots\!20}a^{10}-\frac{31\!\cdots\!21}{13\!\cdots\!00}a^{9}+\frac{96\!\cdots\!49}{67\!\cdots\!04}a^{8}+\frac{14\!\cdots\!87}{94\!\cdots\!00}a^{7}-\frac{37\!\cdots\!07}{42\!\cdots\!40}a^{6}+\frac{58\!\cdots\!87}{67\!\cdots\!00}a^{5}-\frac{39\!\cdots\!01}{75\!\cdots\!60}a^{4}-\frac{41\!\cdots\!23}{74\!\cdots\!00}a^{3}+\frac{83\!\cdots\!59}{25\!\cdots\!20}a^{2}+\frac{17\!\cdots\!61}{15\!\cdots\!00}a-\frac{71\!\cdots\!65}{10\!\cdots\!48}$, $\frac{34\!\cdots\!23}{45\!\cdots\!00}a^{17}+\frac{15\!\cdots\!39}{33\!\cdots\!52}a^{16}-\frac{31\!\cdots\!71}{45\!\cdots\!00}a^{15}-\frac{10\!\cdots\!87}{25\!\cdots\!40}a^{14}+\frac{16\!\cdots\!13}{97\!\cdots\!00}a^{13}+\frac{14\!\cdots\!27}{14\!\cdots\!60}a^{12}-\frac{65\!\cdots\!31}{98\!\cdots\!00}a^{11}-\frac{10\!\cdots\!99}{25\!\cdots\!40}a^{10}-\frac{10\!\cdots\!61}{98\!\cdots\!00}a^{9}-\frac{12\!\cdots\!51}{21\!\cdots\!20}a^{8}+\frac{45\!\cdots\!07}{70\!\cdots\!50}a^{7}+\frac{97\!\cdots\!27}{25\!\cdots\!64}a^{6}+\frac{18\!\cdots\!67}{50\!\cdots\!00}a^{5}+\frac{12\!\cdots\!77}{56\!\cdots\!20}a^{4}-\frac{39\!\cdots\!09}{16\!\cdots\!00}a^{3}-\frac{26\!\cdots\!03}{18\!\cdots\!64}a^{2}+\frac{57\!\cdots\!61}{11\!\cdots\!00}a+\frac{38\!\cdots\!90}{13\!\cdots\!81}$, $\frac{34\!\cdots\!23}{45\!\cdots\!00}a^{17}-\frac{15\!\cdots\!39}{33\!\cdots\!52}a^{16}-\frac{31\!\cdots\!71}{45\!\cdots\!00}a^{15}+\frac{10\!\cdots\!87}{25\!\cdots\!40}a^{14}+\frac{16\!\cdots\!13}{97\!\cdots\!00}a^{13}-\frac{14\!\cdots\!27}{14\!\cdots\!60}a^{12}-\frac{65\!\cdots\!31}{98\!\cdots\!00}a^{11}+\frac{10\!\cdots\!99}{25\!\cdots\!40}a^{10}-\frac{10\!\cdots\!61}{98\!\cdots\!00}a^{9}+\frac{12\!\cdots\!51}{21\!\cdots\!20}a^{8}+\frac{45\!\cdots\!07}{70\!\cdots\!50}a^{7}-\frac{97\!\cdots\!27}{25\!\cdots\!64}a^{6}+\frac{18\!\cdots\!67}{50\!\cdots\!00}a^{5}-\frac{12\!\cdots\!77}{56\!\cdots\!20}a^{4}-\frac{39\!\cdots\!09}{16\!\cdots\!00}a^{3}+\frac{26\!\cdots\!03}{18\!\cdots\!64}a^{2}+\frac{57\!\cdots\!61}{11\!\cdots\!00}a-\frac{38\!\cdots\!90}{13\!\cdots\!81}$, $\frac{57\!\cdots\!49}{18\!\cdots\!00}a^{17}+\frac{11\!\cdots\!49}{67\!\cdots\!40}a^{16}-\frac{37\!\cdots\!73}{18\!\cdots\!00}a^{15}-\frac{14\!\cdots\!57}{13\!\cdots\!08}a^{14}+\frac{57\!\cdots\!79}{38\!\cdots\!00}a^{13}+\frac{56\!\cdots\!97}{72\!\cdots\!80}a^{12}+\frac{48\!\cdots\!47}{39\!\cdots\!00}a^{11}+\frac{21\!\cdots\!57}{33\!\cdots\!20}a^{10}-\frac{39\!\cdots\!83}{39\!\cdots\!00}a^{9}-\frac{17\!\cdots\!13}{33\!\cdots\!20}a^{8}-\frac{32\!\cdots\!87}{14\!\cdots\!00}a^{7}-\frac{63\!\cdots\!27}{42\!\cdots\!40}a^{6}+\frac{57\!\cdots\!81}{20\!\cdots\!00}a^{5}+\frac{14\!\cdots\!29}{75\!\cdots\!60}a^{4}-\frac{30\!\cdots\!27}{67\!\cdots\!00}a^{3}-\frac{22\!\cdots\!07}{50\!\cdots\!04}a^{2}+\frac{59\!\cdots\!63}{46\!\cdots\!00}a+\frac{14\!\cdots\!53}{10\!\cdots\!48}$, $\frac{94\!\cdots\!43}{60\!\cdots\!00}a^{17}-\frac{10\!\cdots\!21}{40\!\cdots\!24}a^{16}-\frac{86\!\cdots\!31}{60\!\cdots\!00}a^{15}+\frac{47\!\cdots\!29}{20\!\cdots\!20}a^{14}+\frac{44\!\cdots\!13}{12\!\cdots\!00}a^{13}-\frac{12\!\cdots\!21}{21\!\cdots\!40}a^{12}-\frac{17\!\cdots\!11}{13\!\cdots\!00}a^{11}+\frac{22\!\cdots\!91}{10\!\cdots\!60}a^{10}-\frac{27\!\cdots\!81}{13\!\cdots\!00}a^{9}+\frac{35\!\cdots\!97}{10\!\cdots\!60}a^{8}+\frac{12\!\cdots\!77}{94\!\cdots\!00}a^{7}-\frac{27\!\cdots\!53}{12\!\cdots\!20}a^{6}+\frac{86\!\cdots\!49}{22\!\cdots\!00}a^{5}-\frac{28\!\cdots\!49}{45\!\cdots\!36}a^{4}-\frac{96\!\cdots\!01}{24\!\cdots\!00}a^{3}+\frac{48\!\cdots\!59}{75\!\cdots\!60}a^{2}+\frac{20\!\cdots\!61}{15\!\cdots\!00}a-\frac{23\!\cdots\!55}{10\!\cdots\!48}$, $\frac{67\!\cdots\!03}{36\!\cdots\!64}a^{17}+\frac{38\!\cdots\!63}{10\!\cdots\!60}a^{16}-\frac{30\!\cdots\!67}{18\!\cdots\!20}a^{15}-\frac{35\!\cdots\!97}{10\!\cdots\!60}a^{14}+\frac{15\!\cdots\!09}{38\!\cdots\!80}a^{13}+\frac{36\!\cdots\!39}{43\!\cdots\!80}a^{12}-\frac{63\!\cdots\!19}{39\!\cdots\!20}a^{11}-\frac{16\!\cdots\!09}{50\!\cdots\!80}a^{10}-\frac{99\!\cdots\!37}{39\!\cdots\!20}a^{9}-\frac{26\!\cdots\!49}{50\!\cdots\!80}a^{8}+\frac{17\!\cdots\!73}{11\!\cdots\!52}a^{7}+\frac{40\!\cdots\!83}{12\!\cdots\!20}a^{6}+\frac{13\!\cdots\!91}{20\!\cdots\!80}a^{5}+\frac{15\!\cdots\!39}{11\!\cdots\!40}a^{4}-\frac{40\!\cdots\!57}{13\!\cdots\!32}a^{3}-\frac{23\!\cdots\!23}{37\!\cdots\!80}a^{2}+\frac{46\!\cdots\!63}{46\!\cdots\!40}a+\frac{10\!\cdots\!77}{52\!\cdots\!24}$, $\frac{62\!\cdots\!97}{18\!\cdots\!00}a^{17}-\frac{13\!\cdots\!27}{20\!\cdots\!20}a^{16}-\frac{57\!\cdots\!89}{18\!\cdots\!00}a^{15}+\frac{84\!\cdots\!09}{13\!\cdots\!08}a^{14}+\frac{29\!\cdots\!27}{38\!\cdots\!00}a^{13}-\frac{65\!\cdots\!03}{43\!\cdots\!80}a^{12}-\frac{11\!\cdots\!29}{39\!\cdots\!00}a^{11}+\frac{20\!\cdots\!83}{33\!\cdots\!20}a^{10}-\frac{18\!\cdots\!19}{39\!\cdots\!00}a^{9}+\frac{94\!\cdots\!63}{10\!\cdots\!60}a^{8}+\frac{82\!\cdots\!93}{28\!\cdots\!00}a^{7}-\frac{12\!\cdots\!17}{21\!\cdots\!20}a^{6}+\frac{24\!\cdots\!13}{20\!\cdots\!00}a^{5}-\frac{17\!\cdots\!93}{75\!\cdots\!60}a^{4}-\frac{41\!\cdots\!11}{67\!\cdots\!00}a^{3}+\frac{10\!\cdots\!53}{83\!\cdots\!40}a^{2}+\frac{95\!\cdots\!79}{46\!\cdots\!00}a-\frac{42\!\cdots\!21}{10\!\cdots\!48}$
| 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: | \( 235897989389000000000000000000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{2}\cdot 235897989389000000000000000000000 \cdot 21}{2\cdot\sqrt{2402705811545396560791327146734278062629093830398539967179801088896102378329}}\cr\approx \mathstrut & 32.6845804791017 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 9097*x^16 + 21918784*x^14 - 8722330342*x^12 - 13442398479722*x^10 + 8439306339453536*x^8 + 1882836967318101*x^6 - 314874705862769661*x^4 + 759018943953897696*x^2 - 232463222451360000)
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 - 9097*x^16 + 21918784*x^14 - 8722330342*x^12 - 13442398479722*x^10 + 8439306339453536*x^8 + 1882836967318101*x^6 - 314874705862769661*x^4 + 759018943953897696*x^2 - 232463222451360000, 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 - 9097*x^16 + 21918784*x^14 - 8722330342*x^12 - 13442398479722*x^10 + 8439306339453536*x^8 + 1882836967318101*x^6 - 314874705862769661*x^4 + 759018943953897696*x^2 - 232463222451360000);
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 - 9097*x^16 + 21918784*x^14 - 8722330342*x^12 - 13442398479722*x^10 + 8439306339453536*x^8 + 1882836967318101*x^6 - 314874705862769661*x^4 + 759018943953897696*x^2 - 232463222451360000);
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
$C_2^8:\SL(2,8)$ (as 18T802):
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 129024 |
| The 29 conjugacy class representatives for $C_2^8:\SL(2,8)$ |
| Character table for $C_2^8:\SL(2,8)$ |
Intermediate fields
| 9.9.16339134383250936197651486708561524809.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | ${\href{/padicField/19.7.0.1}{7} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/41.7.0.1}{7} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(37\)
| 37.9.6.3 | $x^{9} - 444 x^{6} + 49284 x^{3} + 62049925$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ |
| 37.9.6.3 | $x^{9} - 444 x^{6} + 49284 x^{3} + 62049925$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ | |
|
\(2977717\)
| $\Q_{2977717}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2977717}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $8$ | $2$ | $4$ | $4$ |