Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 27*x^16 + 270*x^14 - 1269*x^12 + 3042*x^10 - 76*x^9 - 3888*x^8 + 261*x^7 + 2601*x^6 - 297*x^5 - 810*x^4 + 120*x^3 + 81*x^2 - 9*x - 1)
gp: K = bnfinit(y^18 - 27*y^16 + 270*y^14 - 1269*y^12 + 3042*y^10 - 76*y^9 - 3888*y^8 + 261*y^7 + 2601*y^6 - 297*y^5 - 810*y^4 + 120*y^3 + 81*y^2 - 9*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 27*x^16 + 270*x^14 - 1269*x^12 + 3042*x^10 - 76*x^9 - 3888*x^8 + 261*x^7 + 2601*x^6 - 297*x^5 - 810*x^4 + 120*x^3 + 81*x^2 - 9*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 27*x^16 + 270*x^14 - 1269*x^12 + 3042*x^10 - 76*x^9 - 3888*x^8 + 261*x^7 + 2601*x^6 - 297*x^5 - 810*x^4 + 120*x^3 + 81*x^2 - 9*x - 1)
\( x^{18} - 27 x^{16} + 270 x^{14} - 1269 x^{12} + 3042 x^{10} - 76 x^{9} - 3888 x^{8} + 261 x^{7} + \cdots - 1 \)
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: |
\(1923380668327365689220703125\)
\(\medspace = 3^{44}\cdot 5^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.79\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $3^{22/9}5^{1/2}\approx 32.793019146867586$ | ||
| Ramified primes: |
\(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(135=3^{3}\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{135}(64,·)$, $\chi_{135}(1,·)$, $\chi_{135}(4,·)$, $\chi_{135}(76,·)$, $\chi_{135}(79,·)$, $\chi_{135}(16,·)$, $\chi_{135}(19,·)$, $\chi_{135}(91,·)$, $\chi_{135}(94,·)$, $\chi_{135}(31,·)$, $\chi_{135}(34,·)$, $\chi_{135}(106,·)$, $\chi_{135}(109,·)$, $\chi_{135}(46,·)$, $\chi_{135}(49,·)$, $\chi_{135}(121,·)$, $\chi_{135}(124,·)$, $\chi_{135}(61,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{252470006105599}a^{17}-\frac{125005862637351}{252470006105599}a^{16}-\frac{55802239894318}{252470006105599}a^{15}+\frac{20683531428349}{252470006105599}a^{14}+\frac{75527407604533}{252470006105599}a^{13}+\frac{3308342371889}{252470006105599}a^{12}+\frac{59644436246245}{252470006105599}a^{11}+\frac{3762488467568}{252470006105599}a^{10}-\frac{125164617995790}{252470006105599}a^{9}-\frac{30792598734353}{252470006105599}a^{8}+\frac{99294107283645}{252470006105599}a^{7}-\frac{103580502572143}{252470006105599}a^{6}-\frac{79925047077669}{252470006105599}a^{5}-\frac{114505757343394}{252470006105599}a^{4}-\frac{39216739194526}{252470006105599}a^{3}-\frac{87257331108262}{252470006105599}a^{2}+\frac{15847820505903}{252470006105599}a+\frac{97497629787280}{252470006105599}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $17$ | 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: |
$\frac{11712989862153}{252470006105599}a^{17}-\frac{40866475659723}{252470006105599}a^{16}-\frac{330839044962651}{252470006105599}a^{15}+\frac{10\!\cdots\!89}{252470006105599}a^{14}+\frac{35\!\cdots\!55}{252470006105599}a^{13}-\frac{97\!\cdots\!22}{252470006105599}a^{12}-\frac{18\!\cdots\!37}{252470006105599}a^{11}+\frac{39\!\cdots\!74}{252470006105599}a^{10}+\frac{53\!\cdots\!44}{252470006105599}a^{9}-\frac{74\!\cdots\!48}{252470006105599}a^{8}-\frac{81\!\cdots\!28}{252470006105599}a^{7}+\frac{65\!\cdots\!07}{252470006105599}a^{6}+\frac{63\!\cdots\!26}{252470006105599}a^{5}-\frac{24\!\cdots\!18}{252470006105599}a^{4}-\frac{22\!\cdots\!85}{252470006105599}a^{3}+\frac{25\!\cdots\!28}{252470006105599}a^{2}+\frac{25\!\cdots\!44}{252470006105599}a+\frac{5021109822822}{252470006105599}$, $\frac{72939326445}{938550208571}a^{17}+\frac{157487640930}{938550208571}a^{16}-\frac{1849357980339}{938550208571}a^{15}-\frac{4091003678655}{938550208571}a^{14}+\frac{16545732643260}{938550208571}a^{13}+\frac{38329381837864}{938550208571}a^{12}-\frac{62584036301070}{938550208571}a^{11}-\frac{160467859962042}{938550208571}a^{10}+\frac{92794646069280}{938550208571}a^{9}+\frac{307678238244396}{938550208571}a^{8}-\frac{31318016672610}{938550208571}a^{7}-\frac{273940128450879}{938550208571}a^{6}-\frac{37535610928617}{938550208571}a^{5}+\frac{102894590558223}{938550208571}a^{4}+\frac{23072401941867}{938550208571}a^{3}-\frac{11466641398509}{938550208571}a^{2}+\frac{303989869626}{938550208571}a+\frac{699458732900}{938550208571}$, $\frac{172187583030}{938550208571}a^{17}+\frac{8860548420}{938550208571}a^{16}-\frac{4496113816979}{938550208571}a^{15}-\frac{256608048870}{938550208571}a^{14}+\frac{42547679998920}{938550208571}a^{13}+\frac{2864412960831}{938550208571}a^{12}-\frac{182019526733250}{938550208571}a^{11}-\frac{15989712215418}{938550208571}a^{10}+\frac{374154905406120}{938550208571}a^{9}+\frac{36069074513334}{938550208571}a^{8}-\frac{386623258584840}{938550208571}a^{7}-\frac{51997695863381}{938550208571}a^{6}+\frac{198274357195488}{938550208571}a^{5}+\frac{52789153964352}{938550208571}a^{4}-\frac{46300304260737}{938550208571}a^{3}-\frac{24554594891646}{938550208571}a^{2}+\frac{3698001189054}{938550208571}a+\frac{2138297772511}{938550208571}$, $\frac{70409548104990}{252470006105599}a^{17}+\frac{98084054951325}{252470006105599}a^{16}-\frac{18\!\cdots\!20}{252470006105599}a^{15}-\frac{25\!\cdots\!25}{252470006105599}a^{14}+\frac{18\!\cdots\!85}{252470006105599}a^{13}+\frac{24\!\cdots\!75}{252470006105599}a^{12}-\frac{80\!\cdots\!70}{252470006105599}a^{11}-\frac{10\!\cdots\!21}{252470006105599}a^{10}+\frac{17\!\cdots\!50}{252470006105599}a^{9}+\frac{22\!\cdots\!90}{252470006105599}a^{8}-\frac{19\!\cdots\!20}{252470006105599}a^{7}-\frac{23\!\cdots\!10}{252470006105599}a^{6}+\frac{10\!\cdots\!85}{252470006105599}a^{5}+\frac{10\!\cdots\!00}{252470006105599}a^{4}-\frac{28\!\cdots\!90}{252470006105599}a^{3}-\frac{19\!\cdots\!75}{252470006105599}a^{2}+\frac{24\!\cdots\!30}{252470006105599}a+\frac{682715852263958}{252470006105599}$, $\frac{40866475659723}{252470006105599}a^{17}+\frac{14588318684520}{252470006105599}a^{16}-\frac{10\!\cdots\!89}{252470006105599}a^{15}-\frac{393184214160345}{252470006105599}a^{14}+\frac{97\!\cdots\!22}{252470006105599}a^{13}+\frac{38\!\cdots\!80}{252470006105599}a^{12}-\frac{39\!\cdots\!74}{252470006105599}a^{11}-\frac{17\!\cdots\!18}{252470006105599}a^{10}+\frac{73\!\cdots\!20}{252470006105599}a^{9}+\frac{36\!\cdots\!64}{252470006105599}a^{8}-\frac{62\!\cdots\!74}{252470006105599}a^{7}-\frac{33\!\cdots\!73}{252470006105599}a^{6}+\frac{20\!\cdots\!77}{252470006105599}a^{5}+\frac{12\!\cdots\!55}{252470006105599}a^{4}-\frac{11\!\cdots\!68}{252470006105599}a^{3}-\frac{15\!\cdots\!51}{252470006105599}a^{2}+\frac{142031987523400}{252470006105599}a-\frac{11712989862153}{252470006105599}$, $\frac{56252103772995}{252470006105599}a^{17}+\frac{22676859846900}{252470006105599}a^{16}-\frac{14\!\cdots\!20}{252470006105599}a^{15}-\frac{605612020625571}{252470006105599}a^{14}+\frac{13\!\cdots\!34}{252470006105599}a^{13}+\frac{59\!\cdots\!66}{252470006105599}a^{12}-\frac{55\!\cdots\!88}{252470006105599}a^{11}-\frac{26\!\cdots\!62}{252470006105599}a^{10}+\frac{10\!\cdots\!00}{252470006105599}a^{9}+\frac{53\!\cdots\!50}{252470006105599}a^{8}-\frac{96\!\cdots\!46}{252470006105599}a^{7}-\frac{48\!\cdots\!94}{252470006105599}a^{6}+\frac{37\!\cdots\!65}{252470006105599}a^{5}+\frac{17\!\cdots\!79}{252470006105599}a^{4}-\frac{49\!\cdots\!26}{252470006105599}a^{3}+\frac{332703731408411}{252470006105599}a^{2}-\frac{9636891985908}{252470006105599}a-\frac{495942231820502}{252470006105599}$, $\frac{47443994916000}{252470006105599}a^{17}-\frac{41010903545704}{252470006105599}a^{16}-\frac{12\!\cdots\!50}{252470006105599}a^{15}+\frac{10\!\cdots\!36}{252470006105599}a^{14}+\frac{12\!\cdots\!50}{252470006105599}a^{13}-\frac{94\!\cdots\!66}{252470006105599}a^{12}-\frac{57\!\cdots\!66}{252470006105599}a^{11}+\frac{37\!\cdots\!12}{252470006105599}a^{10}+\frac{13\!\cdots\!49}{252470006105599}a^{9}-\frac{68\!\cdots\!00}{252470006105599}a^{8}-\frac{15\!\cdots\!00}{252470006105599}a^{7}+\frac{59\!\cdots\!52}{252470006105599}a^{6}+\frac{81\!\cdots\!86}{252470006105599}a^{5}-\frac{21\!\cdots\!74}{252470006105599}a^{4}-\frac{14\!\cdots\!84}{252470006105599}a^{3}+\frac{20\!\cdots\!96}{252470006105599}a^{2}+\frac{184541591757092}{252470006105599}a+\frac{197814949858791}{252470006105599}$, $\frac{14588318684520}{252470006105599}a^{17}+\frac{51489374082732}{252470006105599}a^{16}-\frac{393184214160345}{252470006105599}a^{15}-\frac{13\!\cdots\!88}{252470006105599}a^{14}+\frac{38\!\cdots\!80}{252470006105599}a^{13}+\frac{12\!\cdots\!13}{252470006105599}a^{12}-\frac{17\!\cdots\!18}{252470006105599}a^{11}-\frac{50\!\cdots\!46}{252470006105599}a^{10}+\frac{39\!\cdots\!12}{252470006105599}a^{9}+\frac{96\!\cdots\!50}{252470006105599}a^{8}-\frac{43\!\cdots\!76}{252470006105599}a^{7}-\frac{85\!\cdots\!46}{252470006105599}a^{6}+\frac{24\!\cdots\!86}{252470006105599}a^{5}+\frac{31\!\cdots\!62}{252470006105599}a^{4}-\frac{64\!\cdots\!11}{252470006105599}a^{3}-\frac{29\!\cdots\!64}{252470006105599}a^{2}+\frac{356085291075354}{252470006105599}a-\frac{464073536551475}{252470006105599}$, $\frac{62909581934493}{252470006105599}a^{17}-\frac{56588370961482}{252470006105599}a^{16}-\frac{16\!\cdots\!73}{252470006105599}a^{15}+\frac{14\!\cdots\!67}{252470006105599}a^{14}+\frac{14\!\cdots\!36}{252470006105599}a^{13}-\frac{12\!\cdots\!83}{252470006105599}a^{12}-\frac{60\!\cdots\!14}{252470006105599}a^{11}+\frac{49\!\cdots\!64}{252470006105599}a^{10}+\frac{11\!\cdots\!88}{252470006105599}a^{9}-\frac{89\!\cdots\!60}{252470006105599}a^{8}-\frac{10\!\cdots\!54}{252470006105599}a^{7}+\frac{76\!\cdots\!30}{252470006105599}a^{6}+\frac{43\!\cdots\!29}{252470006105599}a^{5}-\frac{29\!\cdots\!15}{252470006105599}a^{4}-\frac{57\!\cdots\!06}{252470006105599}a^{3}+\frac{47\!\cdots\!45}{252470006105599}a^{2}-\frac{305463470467256}{252470006105599}a-\frac{231736084281280}{252470006105599}$, $\frac{31333668675858}{252470006105599}a^{17}+\frac{1497699750447}{252470006105599}a^{16}-\frac{828316341673842}{252470006105599}a^{15}-\frac{48574520828406}{252470006105599}a^{14}+\frac{80\!\cdots\!95}{252470006105599}a^{13}+\frac{605774840999294}{252470006105599}a^{12}-\frac{35\!\cdots\!67}{252470006105599}a^{11}-\frac{36\!\cdots\!24}{252470006105599}a^{10}+\frac{78\!\cdots\!64}{252470006105599}a^{9}+\frac{84\!\cdots\!76}{252470006105599}a^{8}-\frac{90\!\cdots\!18}{252470006105599}a^{7}-\frac{83\!\cdots\!44}{252470006105599}a^{6}+\frac{53\!\cdots\!53}{252470006105599}a^{5}+\frac{33\!\cdots\!69}{252470006105599}a^{4}-\frac{16\!\cdots\!62}{252470006105599}a^{3}-\frac{552782606858793}{252470006105599}a^{2}+\frac{26\!\cdots\!38}{252470006105599}a-\frac{311764503238276}{252470006105599}$, $\frac{77652128510928}{252470006105599}a^{17}+\frac{3881187275427}{252470006105599}a^{16}-\frac{20\!\cdots\!93}{252470006105599}a^{15}-\frac{117602085974436}{252470006105599}a^{14}+\frac{19\!\cdots\!75}{252470006105599}a^{13}+\frac{13\!\cdots\!33}{252470006105599}a^{12}-\frac{84\!\cdots\!17}{252470006105599}a^{11}-\frac{79\!\cdots\!66}{252470006105599}a^{10}+\frac{17\!\cdots\!44}{252470006105599}a^{9}+\frac{18\!\cdots\!22}{252470006105599}a^{8}-\frac{19\!\cdots\!78}{252470006105599}a^{7}-\frac{22\!\cdots\!33}{252470006105599}a^{6}+\frac{10\!\cdots\!25}{252470006105599}a^{5}+\frac{17\!\cdots\!57}{252470006105599}a^{4}-\frac{28\!\cdots\!15}{252470006105599}a^{3}-\frac{71\!\cdots\!67}{252470006105599}a^{2}+\frac{36\!\cdots\!64}{252470006105599}a+\frac{515907603672782}{252470006105599}$, $\frac{58696558242837}{252470006105599}a^{17}+\frac{138950530611048}{252470006105599}a^{16}-\frac{15\!\cdots\!69}{252470006105599}a^{15}-\frac{36\!\cdots\!14}{252470006105599}a^{14}+\frac{14\!\cdots\!30}{252470006105599}a^{13}+\frac{34\!\cdots\!97}{252470006105599}a^{12}-\frac{61\!\cdots\!33}{252470006105599}a^{11}-\frac{14\!\cdots\!95}{252470006105599}a^{10}+\frac{12\!\cdots\!06}{252470006105599}a^{9}+\frac{30\!\cdots\!38}{252470006105599}a^{8}-\frac{11\!\cdots\!92}{252470006105599}a^{7}-\frac{29\!\cdots\!17}{252470006105599}a^{6}+\frac{44\!\cdots\!59}{252470006105599}a^{5}+\frac{13\!\cdots\!18}{252470006105599}a^{4}-\frac{61\!\cdots\!05}{252470006105599}a^{3}-\frac{21\!\cdots\!03}{252470006105599}a^{2}-\frac{95134814068514}{252470006105599}a+\frac{677694742441136}{252470006105599}$, $\frac{5262221664741}{252470006105599}a^{17}-\frac{11647813611741}{252470006105599}a^{16}-\frac{145998030031259}{252470006105599}a^{15}+\frac{293291584598626}{252470006105599}a^{14}+\frac{15\!\cdots\!22}{252470006105599}a^{13}-\frac{26\!\cdots\!35}{252470006105599}a^{12}-\frac{76\!\cdots\!59}{252470006105599}a^{11}+\frac{10\!\cdots\!28}{252470006105599}a^{10}+\frac{19\!\cdots\!56}{252470006105599}a^{9}-\frac{18\!\cdots\!43}{252470006105599}a^{8}-\frac{28\!\cdots\!33}{252470006105599}a^{7}+\frac{18\!\cdots\!51}{252470006105599}a^{6}+\frac{23\!\cdots\!64}{252470006105599}a^{5}-\frac{11\!\cdots\!83}{252470006105599}a^{4}-\frac{11\!\cdots\!67}{252470006105599}a^{3}+\frac{30\!\cdots\!35}{252470006105599}a^{2}+\frac{17\!\cdots\!95}{252470006105599}a-\frac{107549617026924}{252470006105599}$, $\frac{73058728690473}{252470006105599}a^{17}+\frac{19001632859802}{252470006105599}a^{16}-\frac{19\!\cdots\!71}{252470006105599}a^{15}-\frac{505935413148817}{252470006105599}a^{14}+\frac{19\!\cdots\!99}{252470006105599}a^{13}+\frac{49\!\cdots\!29}{252470006105599}a^{12}-\frac{90\!\cdots\!45}{252470006105599}a^{11}-\frac{22\!\cdots\!08}{252470006105599}a^{10}+\frac{21\!\cdots\!94}{252470006105599}a^{9}+\frac{45\!\cdots\!52}{252470006105599}a^{8}-\frac{26\!\cdots\!39}{252470006105599}a^{7}-\frac{41\!\cdots\!52}{252470006105599}a^{6}+\frac{17\!\cdots\!69}{252470006105599}a^{5}+\frac{14\!\cdots\!56}{252470006105599}a^{4}-\frac{49\!\cdots\!61}{252470006105599}a^{3}+\frac{395229725082569}{252470006105599}a^{2}+\frac{32\!\cdots\!31}{252470006105599}a-\frac{289974937450190}{252470006105599}$, $\frac{82122537967143}{252470006105599}a^{17}+\frac{57217579291602}{252470006105599}a^{16}-\frac{21\!\cdots\!71}{252470006105599}a^{15}-\frac{15\!\cdots\!36}{252470006105599}a^{14}+\frac{21\!\cdots\!40}{252470006105599}a^{13}+\frac{15\!\cdots\!53}{252470006105599}a^{12}-\frac{99\!\cdots\!07}{252470006105599}a^{11}-\frac{70\!\cdots\!47}{252470006105599}a^{10}+\frac{22\!\cdots\!94}{252470006105599}a^{9}+\frac{15\!\cdots\!42}{252470006105599}a^{8}-\frac{27\!\cdots\!48}{252470006105599}a^{7}-\frac{16\!\cdots\!03}{252470006105599}a^{6}+\frac{17\!\cdots\!11}{252470006105599}a^{5}+\frac{85\!\cdots\!82}{252470006105599}a^{4}-\frac{50\!\cdots\!75}{252470006105599}a^{3}-\frac{16\!\cdots\!47}{252470006105599}a^{2}+\frac{49\!\cdots\!74}{252470006105599}a+\frac{687736962086780}{252470006105599}$, $\frac{26301308546673}{252470006105599}a^{17}+\frac{10622898423009}{252470006105599}a^{16}-\frac{724023259122996}{252470006105599}a^{15}-\frac{277214086009299}{252470006105599}a^{14}+\frac{74\!\cdots\!35}{252470006105599}a^{13}+\frac{26\!\cdots\!91}{252470006105599}a^{12}-\frac{36\!\cdots\!55}{252470006105599}a^{11}-\frac{11\!\cdots\!72}{252470006105599}a^{10}+\frac{92\!\cdots\!56}{252470006105599}a^{9}+\frac{22\!\cdots\!02}{252470006105599}a^{8}-\frac{12\!\cdots\!04}{252470006105599}a^{7}-\frac{20\!\cdots\!39}{252470006105599}a^{6}+\frac{88\!\cdots\!12}{252470006105599}a^{5}+\frac{76\!\cdots\!44}{252470006105599}a^{4}-\frac{28\!\cdots\!96}{252470006105599}a^{3}-\frac{383938605468436}{252470006105599}a^{2}+\frac{28\!\cdots\!98}{252470006105599}a-\frac{711522432834252}{252470006105599}$, $\frac{29153485797570}{252470006105599}a^{17}+\frac{55454794344243}{252470006105599}a^{16}-\frac{721066423767138}{252470006105599}a^{15}-\frac{14\!\cdots\!34}{252470006105599}a^{14}+\frac{61\!\cdots\!67}{252470006105599}a^{13}+\frac{13\!\cdots\!02}{252470006105599}a^{12}-\frac{20\!\cdots\!37}{252470006105599}a^{11}-\frac{57\!\cdots\!92}{252470006105599}a^{10}+\frac{19\!\cdots\!76}{252470006105599}a^{9}+\frac{11\!\cdots\!12}{252470006105599}a^{8}+\frac{19\!\cdots\!54}{252470006105599}a^{7}-\frac{98\!\cdots\!80}{252470006105599}a^{6}-\frac{42\!\cdots\!49}{252470006105599}a^{5}+\frac{37\!\cdots\!73}{252470006105599}a^{4}+\frac{21\!\cdots\!17}{252470006105599}a^{3}-\frac{41\!\cdots\!79}{252470006105599}a^{2}-\frac{23\!\cdots\!44}{252470006105599}a+\frac{235735906420624}{252470006105599}$
| 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: | \( 49974435.7673 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 49974435.7673 \cdot 1}{2\cdot\sqrt{1923380668327365689220703125}}\cr\approx \mathstrut & 0.149356869537 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 27*x^16 + 270*x^14 - 1269*x^12 + 3042*x^10 - 76*x^9 - 3888*x^8 + 261*x^7 + 2601*x^6 - 297*x^5 - 810*x^4 + 120*x^3 + 81*x^2 - 9*x - 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^18 - 27*x^16 + 270*x^14 - 1269*x^12 + 3042*x^10 - 76*x^9 - 3888*x^8 + 261*x^7 + 2601*x^6 - 297*x^5 - 810*x^4 + 120*x^3 + 81*x^2 - 9*x - 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^18 - 27*x^16 + 270*x^14 - 1269*x^12 + 3042*x^10 - 76*x^9 - 3888*x^8 + 261*x^7 + 2601*x^6 - 297*x^5 - 810*x^4 + 120*x^3 + 81*x^2 - 9*x - 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 27*x^16 + 270*x^14 - 1269*x^12 + 3042*x^10 - 76*x^9 - 3888*x^8 + 261*x^7 + 2601*x^6 - 297*x^5 - 810*x^4 + 120*x^3 + 81*x^2 - 9*x - 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
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 cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 6.6.820125.1, \(\Q(\zeta_{27})^+\) |
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 | $18$ | R | R | $18$ | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | $18$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $9$ | $2$ | $44$ | |||
|
\(5\)
| 5.18.9.1 | $x^{18} + 180 x^{17} + 14445 x^{16} + 679200 x^{15} + 20664900 x^{14} + 423486000 x^{13} + 5887570504 x^{12} + 54397260480 x^{11} + 316143109712 x^{10} + 1034969211206 x^{9} + 1580754753720 x^{8} + 1360718216520 x^{7} + 746510415004 x^{6} + 357128191140 x^{5} + 552895560364 x^{4} + 1314509471572 x^{3} + 1121303668936 x^{2} + 1315877100296 x + 1500010785049$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.27.9t1.a.a | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.135.18t1.a.a | $1$ | $ 3^{3} \cdot 5 $ | 18.18.1923380668327365689220703125.1 | $C_{18}$ (as 18T1) | $0$ | $1$ |
| * | 1.27.9t1.a.b | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.135.18t1.a.b | $1$ | $ 3^{3} \cdot 5 $ | 18.18.1923380668327365689220703125.1 | $C_{18}$ (as 18T1) | $0$ | $1$ |
| * | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.45.6t1.a.a | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ |
| * | 1.27.9t1.a.c | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.135.18t1.a.c | $1$ | $ 3^{3} \cdot 5 $ | 18.18.1923380668327365689220703125.1 | $C_{18}$ (as 18T1) | $0$ | $1$ |
| * | 1.27.9t1.a.d | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.135.18t1.a.d | $1$ | $ 3^{3} \cdot 5 $ | 18.18.1923380668327365689220703125.1 | $C_{18}$ (as 18T1) | $0$ | $1$ |
| * | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.45.6t1.a.b | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ |
| * | 1.27.9t1.a.e | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.135.18t1.a.e | $1$ | $ 3^{3} \cdot 5 $ | 18.18.1923380668327365689220703125.1 | $C_{18}$ (as 18T1) | $0$ | $1$ |
| * | 1.27.9t1.a.f | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.135.18t1.a.f | $1$ | $ 3^{3} \cdot 5 $ | 18.18.1923380668327365689220703125.1 | $C_{18}$ (as 18T1) | $0$ | $1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.