Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 120*x^16 + 20*x^15 + 6198*x^14 + 20970*x^13 - 178442*x^12 - 665868*x^11 - 647649*x^10 + 9439297*x^9 + 106641522*x^8 - 165400536*x^7 - 2012219134*x^6 + 2401149960*x^5 + 14014572888*x^4 - 21798805792*x^3 - 7493040096*x^2 + 52913142528*x - 91792099456)
gp: K = bnfinit(y^18 - 3*y^17 - 120*y^16 + 20*y^15 + 6198*y^14 + 20970*y^13 - 178442*y^12 - 665868*y^11 - 647649*y^10 + 9439297*y^9 + 106641522*y^8 - 165400536*y^7 - 2012219134*y^6 + 2401149960*y^5 + 14014572888*y^4 - 21798805792*y^3 - 7493040096*y^2 + 52913142528*y - 91792099456, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 120*x^16 + 20*x^15 + 6198*x^14 + 20970*x^13 - 178442*x^12 - 665868*x^11 - 647649*x^10 + 9439297*x^9 + 106641522*x^8 - 165400536*x^7 - 2012219134*x^6 + 2401149960*x^5 + 14014572888*x^4 - 21798805792*x^3 - 7493040096*x^2 + 52913142528*x - 91792099456);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 3*x^17 - 120*x^16 + 20*x^15 + 6198*x^14 + 20970*x^13 - 178442*x^12 - 665868*x^11 - 647649*x^10 + 9439297*x^9 + 106641522*x^8 - 165400536*x^7 - 2012219134*x^6 + 2401149960*x^5 + 14014572888*x^4 - 21798805792*x^3 - 7493040096*x^2 + 52913142528*x - 91792099456)
\( x^{18} - 3 x^{17} - 120 x^{16} + 20 x^{15} + 6198 x^{14} + 20970 x^{13} - 178442 x^{12} + \cdots - 91792099456 \)
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: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(115613898980631103446799946287500356340584448\)
\(\medspace = 2^{12}\cdot 3^{30}\cdot 7^{12}\cdot 17^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(280.51\) | 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: | $2^{2/3}3^{11/6}7^{2/3}17^{5/6}\approx 461.510683112854$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{4}a^{7}+\frac{1}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{8}a^{7}+\frac{3}{8}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{11424}a^{12}-\frac{1}{136}a^{11}+\frac{5}{3808}a^{10}+\frac{19}{816}a^{9}+\frac{349}{3808}a^{8}+\frac{5}{136}a^{7}+\frac{29}{3808}a^{6}-\frac{31}{68}a^{5}-\frac{827}{1904}a^{4}-\frac{335}{816}a^{3}+\frac{145}{952}a^{2}+\frac{1}{4}a-\frac{115}{714}$, $\frac{1}{11424}a^{13}+\frac{33}{3808}a^{11}+\frac{7}{816}a^{10}+\frac{181}{3808}a^{9}-\frac{1}{68}a^{8}+\frac{841}{3808}a^{7}+\frac{1}{17}a^{6}+\frac{517}{1904}a^{5}+\frac{5}{48}a^{4}+\frac{397}{952}a^{3}-\frac{31}{68}a^{2}-\frac{115}{714}a+\frac{8}{17}$, $\frac{1}{4889472}a^{14}+\frac{31}{4889472}a^{13}+\frac{27}{1629824}a^{12}+\frac{81791}{4889472}a^{11}-\frac{39919}{698496}a^{10}-\frac{11273}{1629824}a^{9}+\frac{99447}{1629824}a^{8}+\frac{348407}{1629824}a^{7}-\frac{27985}{203728}a^{6}-\frac{223157}{611184}a^{5}+\frac{105235}{349248}a^{4}-\frac{4555}{101864}a^{3}+\frac{39245}{305592}a^{2}-\frac{16787}{152796}a+\frac{14457}{50932}$, $\frac{1}{4889472}a^{15}-\frac{1}{203728}a^{13}+\frac{25}{1222368}a^{12}+\frac{6043}{116416}a^{11}+\frac{7941}{203728}a^{10}+\frac{153773}{2444736}a^{9}+\frac{71497}{814912}a^{8}+\frac{205367}{1629824}a^{7}+\frac{124619}{1222368}a^{6}+\frac{165}{116416}a^{5}-\frac{59285}{814912}a^{4}-\frac{41067}{203728}a^{3}+\frac{16285}{50932}a^{2}-\frac{19479}{50932}a-\frac{9355}{152796}$, $\frac{1}{87893148672}a^{16}+\frac{359}{14648858112}a^{15}-\frac{1111}{21973287168}a^{14}-\frac{459853}{21973287168}a^{13}+\frac{112813}{2585092608}a^{12}+\frac{2379715}{3139041024}a^{11}-\frac{678341627}{14648858112}a^{10}+\frac{29912291}{410715648}a^{9}-\frac{648987555}{9765905408}a^{8}-\frac{10367182325}{43946574336}a^{7}+\frac{3545896883}{14648858112}a^{6}+\frac{499661441}{6278082048}a^{5}-\frac{10428276013}{21973287168}a^{4}+\frac{684395405}{5493321792}a^{3}-\frac{2833381}{2746660896}a^{2}+\frac{457240493}{915553632}a-\frac{402092899}{1373330448}$, $\frac{1}{29\!\cdots\!52}a^{17}+\frac{22\!\cdots\!05}{73\!\cdots\!88}a^{16}-\frac{12\!\cdots\!85}{21\!\cdots\!32}a^{15}-\frac{76\!\cdots\!11}{81\!\cdots\!32}a^{14}-\frac{36\!\cdots\!01}{20\!\cdots\!68}a^{13}+\frac{23\!\cdots\!61}{12\!\cdots\!48}a^{12}+\frac{35\!\cdots\!15}{14\!\cdots\!76}a^{11}+\frac{98\!\cdots\!21}{20\!\cdots\!68}a^{10}-\frac{75\!\cdots\!91}{29\!\cdots\!52}a^{9}+\frac{69\!\cdots\!59}{31\!\cdots\!56}a^{8}-\frac{33\!\cdots\!97}{14\!\cdots\!76}a^{7}+\frac{26\!\cdots\!57}{14\!\cdots\!76}a^{6}+\frac{15\!\cdots\!11}{36\!\cdots\!44}a^{5}+\frac{16\!\cdots\!99}{12\!\cdots\!48}a^{4}-\frac{61\!\cdots\!35}{57\!\cdots\!46}a^{3}+\frac{71\!\cdots\!73}{18\!\cdots\!64}a^{2}-\frac{21\!\cdots\!71}{57\!\cdots\!46}a+\frac{55\!\cdots\!21}{22\!\cdots\!84}$
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
$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (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: | $11$ | 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: |
$\frac{65\!\cdots\!53}{56\!\cdots\!47}a^{17}-\frac{91\!\cdots\!85}{56\!\cdots\!47}a^{16}-\frac{77\!\cdots\!59}{56\!\cdots\!47}a^{15}-\frac{11\!\cdots\!89}{56\!\cdots\!47}a^{14}+\frac{35\!\cdots\!50}{56\!\cdots\!47}a^{13}+\frac{38\!\cdots\!33}{11\!\cdots\!94}a^{12}-\frac{71\!\cdots\!09}{56\!\cdots\!47}a^{11}-\frac{97\!\cdots\!37}{11\!\cdots\!94}a^{10}-\frac{15\!\cdots\!91}{56\!\cdots\!47}a^{9}+\frac{40\!\cdots\!41}{11\!\cdots\!94}a^{8}+\frac{67\!\cdots\!70}{56\!\cdots\!47}a^{7}+\frac{42\!\cdots\!35}{11\!\cdots\!94}a^{6}-\frac{10\!\cdots\!06}{56\!\cdots\!47}a^{5}-\frac{23\!\cdots\!54}{56\!\cdots\!47}a^{4}+\frac{47\!\cdots\!36}{56\!\cdots\!47}a^{3}-\frac{63\!\cdots\!84}{56\!\cdots\!47}a^{2}+\frac{67\!\cdots\!76}{56\!\cdots\!47}a+\frac{51\!\cdots\!09}{56\!\cdots\!47}$, $\frac{26\!\cdots\!79}{10\!\cdots\!84}a^{17}-\frac{19\!\cdots\!17}{41\!\cdots\!36}a^{16}-\frac{63\!\cdots\!41}{20\!\cdots\!68}a^{15}-\frac{31\!\cdots\!81}{10\!\cdots\!84}a^{14}+\frac{17\!\cdots\!95}{11\!\cdots\!76}a^{13}+\frac{14\!\cdots\!59}{20\!\cdots\!68}a^{12}-\frac{38\!\cdots\!23}{10\!\cdots\!84}a^{11}-\frac{43\!\cdots\!27}{20\!\cdots\!68}a^{10}-\frac{11\!\cdots\!57}{29\!\cdots\!24}a^{9}+\frac{80\!\cdots\!39}{41\!\cdots\!36}a^{8}+\frac{59\!\cdots\!97}{20\!\cdots\!68}a^{7}-\frac{21\!\cdots\!45}{20\!\cdots\!68}a^{6}-\frac{11\!\cdots\!59}{23\!\cdots\!52}a^{5}+\frac{78\!\cdots\!53}{10\!\cdots\!84}a^{4}+\frac{32\!\cdots\!81}{87\!\cdots\!32}a^{3}-\frac{12\!\cdots\!43}{13\!\cdots\!48}a^{2}-\frac{89\!\cdots\!59}{13\!\cdots\!48}a+\frac{42\!\cdots\!03}{65\!\cdots\!24}$, $\frac{34\!\cdots\!73}{23\!\cdots\!92}a^{17}-\frac{65\!\cdots\!07}{18\!\cdots\!36}a^{16}-\frac{17\!\cdots\!35}{92\!\cdots\!68}a^{15}-\frac{30\!\cdots\!39}{58\!\cdots\!48}a^{14}+\frac{16\!\cdots\!33}{16\!\cdots\!14}a^{13}+\frac{32\!\cdots\!51}{92\!\cdots\!68}a^{12}-\frac{43\!\cdots\!67}{16\!\cdots\!44}a^{11}-\frac{10\!\cdots\!99}{92\!\cdots\!68}a^{10}-\frac{38\!\cdots\!03}{64\!\cdots\!76}a^{9}+\frac{23\!\cdots\!69}{18\!\cdots\!36}a^{8}+\frac{10\!\cdots\!83}{64\!\cdots\!76}a^{7}-\frac{20\!\cdots\!43}{92\!\cdots\!68}a^{6}-\frac{73\!\cdots\!73}{21\!\cdots\!92}a^{5}+\frac{19\!\cdots\!85}{46\!\cdots\!84}a^{4}+\frac{76\!\cdots\!77}{27\!\cdots\!24}a^{3}-\frac{23\!\cdots\!31}{58\!\cdots\!48}a^{2}-\frac{27\!\cdots\!79}{40\!\cdots\!36}a+\frac{28\!\cdots\!79}{29\!\cdots\!24}$, $\frac{87\!\cdots\!01}{15\!\cdots\!56}a^{17}+\frac{85\!\cdots\!23}{20\!\cdots\!08}a^{16}+\frac{29\!\cdots\!85}{30\!\cdots\!12}a^{15}-\frac{36\!\cdots\!33}{76\!\cdots\!28}a^{14}-\frac{11\!\cdots\!83}{38\!\cdots\!64}a^{13}+\frac{52\!\cdots\!31}{43\!\cdots\!16}a^{12}+\frac{15\!\cdots\!07}{76\!\cdots\!28}a^{11}+\frac{15\!\cdots\!19}{30\!\cdots\!12}a^{10}-\frac{24\!\cdots\!37}{10\!\cdots\!04}a^{9}-\frac{16\!\cdots\!63}{60\!\cdots\!24}a^{8}-\frac{10\!\cdots\!25}{10\!\cdots\!04}a^{7}+\frac{44\!\cdots\!97}{43\!\cdots\!16}a^{6}+\frac{15\!\cdots\!53}{10\!\cdots\!04}a^{5}+\frac{16\!\cdots\!15}{15\!\cdots\!56}a^{4}-\frac{41\!\cdots\!15}{76\!\cdots\!28}a^{3}+\frac{12\!\cdots\!49}{19\!\cdots\!82}a^{2}+\frac{23\!\cdots\!69}{95\!\cdots\!41}a-\frac{10\!\cdots\!96}{45\!\cdots\!21}$, $\frac{33\!\cdots\!19}{60\!\cdots\!24}a^{17}-\frac{18\!\cdots\!77}{48\!\cdots\!92}a^{16}-\frac{15\!\cdots\!69}{24\!\cdots\!96}a^{15}+\frac{18\!\cdots\!83}{60\!\cdots\!24}a^{14}+\frac{79\!\cdots\!07}{20\!\cdots\!08}a^{13}-\frac{16\!\cdots\!03}{49\!\cdots\!04}a^{12}-\frac{52\!\cdots\!45}{30\!\cdots\!12}a^{11}-\frac{98\!\cdots\!05}{24\!\cdots\!96}a^{10}+\frac{52\!\cdots\!97}{24\!\cdots\!96}a^{9}+\frac{40\!\cdots\!19}{48\!\cdots\!92}a^{8}+\frac{10\!\cdots\!55}{24\!\cdots\!96}a^{7}-\frac{19\!\cdots\!29}{49\!\cdots\!04}a^{6}-\frac{96\!\cdots\!05}{81\!\cdots\!32}a^{5}+\frac{89\!\cdots\!27}{12\!\cdots\!48}a^{4}+\frac{93\!\cdots\!21}{10\!\cdots\!04}a^{3}-\frac{83\!\cdots\!73}{15\!\cdots\!56}a^{2}-\frac{28\!\cdots\!91}{15\!\cdots\!56}a+\frac{13\!\cdots\!75}{10\!\cdots\!04}$, $\frac{15\!\cdots\!61}{91\!\cdots\!36}a^{17}+\frac{85\!\cdots\!37}{73\!\cdots\!88}a^{16}-\frac{79\!\cdots\!55}{36\!\cdots\!44}a^{15}-\frac{66\!\cdots\!71}{91\!\cdots\!36}a^{14}+\frac{98\!\cdots\!51}{10\!\cdots\!04}a^{13}+\frac{37\!\cdots\!81}{52\!\cdots\!92}a^{12}-\frac{57\!\cdots\!91}{45\!\cdots\!68}a^{11}-\frac{70\!\cdots\!15}{36\!\cdots\!44}a^{10}-\frac{22\!\cdots\!65}{36\!\cdots\!44}a^{9}+\frac{21\!\cdots\!57}{73\!\cdots\!88}a^{8}+\frac{78\!\cdots\!57}{36\!\cdots\!44}a^{7}+\frac{21\!\cdots\!39}{52\!\cdots\!92}a^{6}-\frac{42\!\cdots\!31}{12\!\cdots\!48}a^{5}-\frac{12\!\cdots\!39}{18\!\cdots\!72}a^{4}+\frac{34\!\cdots\!51}{15\!\cdots\!56}a^{3}+\frac{49\!\cdots\!69}{22\!\cdots\!84}a^{2}-\frac{58\!\cdots\!33}{13\!\cdots\!52}a+\frac{21\!\cdots\!21}{16\!\cdots\!56}$, $\frac{64\!\cdots\!01}{18\!\cdots\!72}a^{17}-\frac{28\!\cdots\!07}{14\!\cdots\!76}a^{16}-\frac{25\!\cdots\!15}{73\!\cdots\!88}a^{15}+\frac{19\!\cdots\!99}{22\!\cdots\!84}a^{14}+\frac{25\!\cdots\!29}{15\!\cdots\!56}a^{13}+\frac{35\!\cdots\!65}{10\!\cdots\!84}a^{12}-\frac{10\!\cdots\!89}{18\!\cdots\!72}a^{11}-\frac{17\!\cdots\!03}{43\!\cdots\!64}a^{10}-\frac{38\!\cdots\!25}{73\!\cdots\!88}a^{9}+\frac{55\!\cdots\!49}{14\!\cdots\!76}a^{8}+\frac{16\!\cdots\!13}{73\!\cdots\!88}a^{7}-\frac{97\!\cdots\!69}{10\!\cdots\!84}a^{6}-\frac{63\!\cdots\!31}{24\!\cdots\!96}a^{5}+\frac{38\!\cdots\!53}{36\!\cdots\!44}a^{4}-\frac{49\!\cdots\!39}{10\!\cdots\!04}a^{3}-\frac{82\!\cdots\!03}{45\!\cdots\!68}a^{2}+\frac{21\!\cdots\!27}{45\!\cdots\!68}a-\frac{12\!\cdots\!11}{32\!\cdots\!12}$, $\frac{50\!\cdots\!99}{14\!\cdots\!76}a^{17}-\frac{11\!\cdots\!45}{29\!\cdots\!52}a^{16}-\frac{60\!\cdots\!49}{14\!\cdots\!76}a^{15}-\frac{54\!\cdots\!73}{81\!\cdots\!32}a^{14}+\frac{10\!\cdots\!67}{52\!\cdots\!92}a^{13}+\frac{50\!\cdots\!13}{48\!\cdots\!92}a^{12}-\frac{85\!\cdots\!71}{21\!\cdots\!32}a^{11}-\frac{33\!\cdots\!71}{12\!\cdots\!04}a^{10}-\frac{50\!\cdots\!19}{73\!\cdots\!88}a^{9}+\frac{27\!\cdots\!43}{17\!\cdots\!56}a^{8}+\frac{53\!\cdots\!35}{14\!\cdots\!76}a^{7}-\frac{34\!\cdots\!87}{14\!\cdots\!76}a^{6}-\frac{97\!\cdots\!67}{14\!\cdots\!76}a^{5}-\frac{69\!\cdots\!61}{24\!\cdots\!96}a^{4}+\frac{72\!\cdots\!87}{18\!\cdots\!72}a^{3}-\frac{11\!\cdots\!95}{13\!\cdots\!48}a^{2}+\frac{43\!\cdots\!65}{91\!\cdots\!36}a+\frac{65\!\cdots\!63}{45\!\cdots\!68}$, $\frac{35\!\cdots\!23}{17\!\cdots\!64}a^{17}-\frac{53\!\cdots\!31}{14\!\cdots\!76}a^{16}-\frac{71\!\cdots\!47}{24\!\cdots\!96}a^{15}+\frac{21\!\cdots\!49}{57\!\cdots\!46}a^{14}+\frac{53\!\cdots\!15}{18\!\cdots\!72}a^{13}-\frac{68\!\cdots\!69}{73\!\cdots\!88}a^{12}-\frac{44\!\cdots\!27}{26\!\cdots\!96}a^{11}-\frac{31\!\cdots\!25}{24\!\cdots\!96}a^{10}+\frac{19\!\cdots\!13}{73\!\cdots\!88}a^{9}+\frac{79\!\cdots\!55}{48\!\cdots\!92}a^{8}+\frac{31\!\cdots\!83}{73\!\cdots\!88}a^{7}-\frac{81\!\cdots\!89}{24\!\cdots\!96}a^{6}-\frac{20\!\cdots\!53}{14\!\cdots\!12}a^{5}+\frac{15\!\cdots\!75}{36\!\cdots\!44}a^{4}+\frac{14\!\cdots\!05}{91\!\cdots\!36}a^{3}-\frac{50\!\cdots\!57}{45\!\cdots\!68}a^{2}-\frac{33\!\cdots\!79}{15\!\cdots\!56}a+\frac{96\!\cdots\!85}{22\!\cdots\!84}$, $\frac{83\!\cdots\!11}{48\!\cdots\!92}a^{17}-\frac{15\!\cdots\!87}{14\!\cdots\!76}a^{16}-\frac{43\!\cdots\!23}{24\!\cdots\!96}a^{15}+\frac{33\!\cdots\!95}{52\!\cdots\!92}a^{14}+\frac{65\!\cdots\!53}{73\!\cdots\!88}a^{13}+\frac{38\!\cdots\!83}{73\!\cdots\!88}a^{12}-\frac{24\!\cdots\!93}{73\!\cdots\!88}a^{11}-\frac{40\!\cdots\!17}{60\!\cdots\!24}a^{10}-\frac{48\!\cdots\!15}{20\!\cdots\!68}a^{9}+\frac{73\!\cdots\!95}{48\!\cdots\!92}a^{8}+\frac{53\!\cdots\!41}{36\!\cdots\!44}a^{7}-\frac{33\!\cdots\!05}{40\!\cdots\!16}a^{6}-\frac{41\!\cdots\!77}{43\!\cdots\!64}a^{5}+\frac{33\!\cdots\!65}{36\!\cdots\!44}a^{4}-\frac{83\!\cdots\!59}{91\!\cdots\!36}a^{3}-\frac{11\!\cdots\!19}{11\!\cdots\!92}a^{2}+\frac{51\!\cdots\!65}{15\!\cdots\!56}a-\frac{97\!\cdots\!73}{22\!\cdots\!84}$, $\frac{62\!\cdots\!25}{14\!\cdots\!76}a^{17}-\frac{24\!\cdots\!35}{14\!\cdots\!76}a^{16}-\frac{39\!\cdots\!43}{73\!\cdots\!88}a^{15}-\frac{67\!\cdots\!21}{52\!\cdots\!92}a^{14}+\frac{64\!\cdots\!63}{24\!\cdots\!96}a^{13}+\frac{11\!\cdots\!75}{73\!\cdots\!88}a^{12}-\frac{36\!\cdots\!65}{73\!\cdots\!88}a^{11}-\frac{11\!\cdots\!73}{22\!\cdots\!84}a^{10}-\frac{24\!\cdots\!83}{20\!\cdots\!68}a^{9}+\frac{48\!\cdots\!57}{14\!\cdots\!76}a^{8}+\frac{52\!\cdots\!03}{91\!\cdots\!36}a^{7}+\frac{20\!\cdots\!17}{36\!\cdots\!44}a^{6}-\frac{24\!\cdots\!27}{24\!\cdots\!96}a^{5}-\frac{55\!\cdots\!25}{36\!\cdots\!44}a^{4}+\frac{23\!\cdots\!95}{30\!\cdots\!12}a^{3}+\frac{26\!\cdots\!75}{22\!\cdots\!84}a^{2}-\frac{88\!\cdots\!73}{45\!\cdots\!68}a-\frac{64\!\cdots\!51}{22\!\cdots\!84}$
| 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: | \( 2131338973787866.8 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 2131338973787866.8 \cdot 27}{2\cdot\sqrt{115613898980631103446799946287500356340584448}}\cr\approx \mathstrut & 10.5375695811798 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 120*x^16 + 20*x^15 + 6198*x^14 + 20970*x^13 - 178442*x^12 - 665868*x^11 - 647649*x^10 + 9439297*x^9 + 106641522*x^8 - 165400536*x^7 - 2012219134*x^6 + 2401149960*x^5 + 14014572888*x^4 - 21798805792*x^3 - 7493040096*x^2 + 52913142528*x - 91792099456)
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 - 3*x^17 - 120*x^16 + 20*x^15 + 6198*x^14 + 20970*x^13 - 178442*x^12 - 665868*x^11 - 647649*x^10 + 9439297*x^9 + 106641522*x^8 - 165400536*x^7 - 2012219134*x^6 + 2401149960*x^5 + 14014572888*x^4 - 21798805792*x^3 - 7493040096*x^2 + 52913142528*x - 91792099456, 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 - 3*x^17 - 120*x^16 + 20*x^15 + 6198*x^14 + 20970*x^13 - 178442*x^12 - 665868*x^11 - 647649*x^10 + 9439297*x^9 + 106641522*x^8 - 165400536*x^7 - 2012219134*x^6 + 2401149960*x^5 + 14014572888*x^4 - 21798805792*x^3 - 7493040096*x^2 + 52913142528*x - 91792099456);
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 - 3*x^17 - 120*x^16 + 20*x^15 + 6198*x^14 + 20970*x^13 - 178442*x^12 - 665868*x^11 - 647649*x^10 + 9439297*x^9 + 106641522*x^8 - 165400536*x^7 - 2012219134*x^6 + 2401149960*x^5 + 14014572888*x^4 - 21798805792*x^3 - 7493040096*x^2 + 52913142528*x - 91792099456);
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
$C_3^2:D_6$ (as 18T52):
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 108 |
| The 20 conjugacy class representatives for $C_3^2:D_6$ |
| Character table for $C_3^2:D_6$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 3.1.972.2, 6.2.4641723792.9, 9.3.9023659939580352192.1 |
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 18 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 18.6.318286244526390688310901391528421762902272.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(3\)
| 3.6.10.3 | $x^{6} + 6 x^{5} + 36 x^{4} + 128 x^{3} + 297 x^{2} + 474 x + 482$ | $3$ | $2$ | $10$ | $D_{6}$ | $[5/2]_{2}^{2}$ |
| 3.6.10.3 | $x^{6} + 6 x^{5} + 36 x^{4} + 128 x^{3} + 297 x^{2} + 474 x + 482$ | $3$ | $2$ | $10$ | $D_{6}$ | $[5/2]_{2}^{2}$ | |
| 3.6.10.3 | $x^{6} + 6 x^{5} + 36 x^{4} + 128 x^{3} + 297 x^{2} + 474 x + 482$ | $3$ | $2$ | $10$ | $D_{6}$ | $[5/2]_{2}^{2}$ | |
|
\(7\)
| 7.6.4.2 | $x^{6} - 42 x^{3} + 147$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.6.4.1 | $x^{6} + 14 x^{3} - 245$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(17\)
| 17.6.3.1 | $x^{6} + 459 x^{5} + 70280 x^{4} + 3597823 x^{3} + 1271380 x^{2} + 4696159 x + 50437479$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 17.12.10.2 | $x^{12} - 3060 x^{6} - 197676$ | $6$ | $2$ | $10$ | $C_6\times S_3$ | $[\ ]_{6}^{6}$ |