Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 42*x^16 - 21*x^15 + 639*x^14 + 477*x^13 - 4557*x^12 - 3465*x^11 + 17562*x^10 + 10481*x^9 - 40194*x^8 - 12870*x^7 + 55341*x^6 - 41745*x^4 + 13068*x^3 + 11979*x^2 - 7986*x + 1331)
gp: K = bnfinit(y^18 - 42*y^16 - 21*y^15 + 639*y^14 + 477*y^13 - 4557*y^12 - 3465*y^11 + 17562*y^10 + 10481*y^9 - 40194*y^8 - 12870*y^7 + 55341*y^6 - 41745*y^4 + 13068*y^3 + 11979*y^2 - 7986*y + 1331, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 42*x^16 - 21*x^15 + 639*x^14 + 477*x^13 - 4557*x^12 - 3465*x^11 + 17562*x^10 + 10481*x^9 - 40194*x^8 - 12870*x^7 + 55341*x^6 - 41745*x^4 + 13068*x^3 + 11979*x^2 - 7986*x + 1331);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 42*x^16 - 21*x^15 + 639*x^14 + 477*x^13 - 4557*x^12 - 3465*x^11 + 17562*x^10 + 10481*x^9 - 40194*x^8 - 12870*x^7 + 55341*x^6 - 41745*x^4 + 13068*x^3 + 11979*x^2 - 7986*x + 1331)
\( x^{18} - 42 x^{16} - 21 x^{15} + 639 x^{14} + 477 x^{13} - 4557 x^{12} - 3465 x^{11} + 17562 x^{10} + \cdots + 1331 \)
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: |
\(89049975552912225851806640625\)
\(\medspace = 3^{30}\cdot 5^{12}\cdot 11^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(40.58\) | 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: | $3^{97/54}5^{3/4}11^{2/3}\approx 118.99821203205568$ | ||
| Ramified primes: |
\(3\), \(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11}a^{12}+\frac{2}{11}a^{10}+\frac{1}{11}a^{9}+\frac{1}{11}a^{8}+\frac{4}{11}a^{7}-\frac{3}{11}a^{6}-\frac{5}{11}a^{4}-\frac{2}{11}a^{3}$, $\frac{1}{11}a^{13}+\frac{2}{11}a^{11}+\frac{1}{11}a^{10}+\frac{1}{11}a^{9}+\frac{4}{11}a^{8}-\frac{3}{11}a^{7}-\frac{5}{11}a^{5}-\frac{2}{11}a^{4}$, $\frac{1}{11}a^{14}+\frac{1}{11}a^{11}-\frac{3}{11}a^{10}+\frac{2}{11}a^{9}-\frac{5}{11}a^{8}+\frac{3}{11}a^{7}+\frac{1}{11}a^{6}-\frac{2}{11}a^{5}-\frac{1}{11}a^{4}+\frac{4}{11}a^{3}$, $\frac{1}{121}a^{15}+\frac{2}{121}a^{13}+\frac{1}{121}a^{12}+\frac{1}{121}a^{11}-\frac{40}{121}a^{10}-\frac{14}{121}a^{9}-\frac{27}{121}a^{7}+\frac{9}{121}a^{6}$, $\frac{1}{121}a^{16}+\frac{2}{121}a^{14}+\frac{1}{121}a^{13}+\frac{1}{121}a^{12}-\frac{40}{121}a^{11}-\frac{14}{121}a^{10}-\frac{27}{121}a^{8}+\frac{9}{121}a^{7}$, $\frac{1}{12\!\cdots\!91}a^{17}+\frac{756892607111}{12\!\cdots\!91}a^{16}+\frac{2808228545985}{12\!\cdots\!91}a^{15}-\frac{6223420009448}{12\!\cdots\!91}a^{14}-\frac{7879504343151}{12\!\cdots\!91}a^{13}-\frac{11026180341144}{12\!\cdots\!91}a^{12}+\frac{488638947002702}{12\!\cdots\!91}a^{11}-\frac{353641250793333}{12\!\cdots\!91}a^{10}+\frac{560341597393311}{12\!\cdots\!91}a^{9}-\frac{387647791431654}{12\!\cdots\!91}a^{8}-\frac{212895146722202}{12\!\cdots\!91}a^{7}-\frac{257887976904036}{12\!\cdots\!91}a^{6}+\frac{53746452185243}{114908978923981}a^{5}+\frac{42793611677669}{114908978923981}a^{4}-\frac{9746563271805}{114908978923981}a^{3}+\frac{2534497677756}{10446270811271}a^{2}-\frac{251989881344}{549803726909}a+\frac{2186566119375}{10446270811271}$
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: |
$\frac{678037366449}{10446270811271}a^{17}+\frac{1075480148715}{10446270811271}a^{16}-\frac{27978869036058}{10446270811271}a^{15}-\frac{58274849837871}{10446270811271}a^{14}+\frac{389889093424209}{10446270811271}a^{13}+\frac{953960149209090}{10446270811271}a^{12}-\frac{22\!\cdots\!03}{10446270811271}a^{11}-\frac{63\!\cdots\!62}{10446270811271}a^{10}+\frac{65\!\cdots\!12}{10446270811271}a^{9}+\frac{19\!\cdots\!34}{10446270811271}a^{8}-\frac{11\!\cdots\!05}{10446270811271}a^{7}-\frac{32\!\cdots\!19}{10446270811271}a^{6}+\frac{15\!\cdots\!05}{10446270811271}a^{5}+\frac{26\!\cdots\!03}{10446270811271}a^{4}-\frac{14\!\cdots\!57}{10446270811271}a^{3}-\frac{79\!\cdots\!81}{10446270811271}a^{2}+\frac{346243065079824}{549803726909}a-\frac{10\!\cdots\!44}{10446270811271}$, $\frac{77591089467019}{12\!\cdots\!91}a^{17}+\frac{40311882927880}{12\!\cdots\!91}a^{16}-\frac{32\!\cdots\!80}{12\!\cdots\!91}a^{15}-\frac{33\!\cdots\!04}{12\!\cdots\!91}a^{14}+\frac{47\!\cdots\!94}{12\!\cdots\!91}a^{13}+\frac{61\!\cdots\!13}{12\!\cdots\!91}a^{12}-\frac{28\!\cdots\!34}{114908978923981}a^{11}-\frac{43\!\cdots\!40}{12\!\cdots\!91}a^{10}+\frac{10\!\cdots\!12}{12\!\cdots\!91}a^{9}+\frac{13\!\cdots\!28}{12\!\cdots\!91}a^{8}-\frac{21\!\cdots\!30}{12\!\cdots\!91}a^{7}-\frac{21\!\cdots\!68}{12\!\cdots\!91}a^{6}+\frac{23\!\cdots\!63}{114908978923981}a^{5}+\frac{13\!\cdots\!14}{114908978923981}a^{4}-\frac{16\!\cdots\!90}{114908978923981}a^{3}-\frac{20\!\cdots\!01}{10446270811271}a^{2}+\frac{247001560124472}{549803726909}a-\frac{10\!\cdots\!89}{10446270811271}$, $\frac{19384536728581}{12\!\cdots\!91}a^{17}+\frac{45731440598970}{12\!\cdots\!91}a^{16}-\frac{821260957438535}{12\!\cdots\!91}a^{15}-\frac{23\!\cdots\!98}{12\!\cdots\!91}a^{14}+\frac{11\!\cdots\!68}{12\!\cdots\!91}a^{13}+\frac{34\!\cdots\!55}{114908978923981}a^{12}-\frac{70\!\cdots\!48}{12\!\cdots\!91}a^{11}-\frac{26\!\cdots\!76}{12\!\cdots\!91}a^{10}+\frac{21\!\cdots\!12}{12\!\cdots\!91}a^{9}+\frac{90\!\cdots\!30}{12\!\cdots\!91}a^{8}-\frac{39\!\cdots\!02}{114908978923981}a^{7}-\frac{16\!\cdots\!46}{12\!\cdots\!91}a^{6}+\frac{67\!\cdots\!61}{114908978923981}a^{5}+\frac{14\!\cdots\!40}{114908978923981}a^{4}-\frac{82\!\cdots\!86}{114908978923981}a^{3}-\frac{42\!\cdots\!07}{10446270811271}a^{2}+\frac{207364400972256}{549803726909}a-\frac{768643030595798}{10446270811271}$, $\frac{14933104855271}{12\!\cdots\!91}a^{17}-\frac{44089774467665}{12\!\cdots\!91}a^{16}-\frac{659983789464997}{12\!\cdots\!91}a^{15}+\frac{14\!\cdots\!89}{12\!\cdots\!91}a^{14}+\frac{11\!\cdots\!73}{12\!\cdots\!91}a^{13}-\frac{16\!\cdots\!72}{12\!\cdots\!91}a^{12}-\frac{10\!\cdots\!59}{12\!\cdots\!91}a^{11}+\frac{73\!\cdots\!86}{12\!\cdots\!91}a^{10}+\frac{44\!\cdots\!52}{114908978923981}a^{9}-\frac{14\!\cdots\!56}{12\!\cdots\!91}a^{8}-\frac{11\!\cdots\!47}{12\!\cdots\!91}a^{7}+\frac{15\!\cdots\!85}{12\!\cdots\!91}a^{6}+\frac{13\!\cdots\!69}{114908978923981}a^{5}-\frac{18\!\cdots\!79}{114908978923981}a^{4}-\frac{87\!\cdots\!49}{114908978923981}a^{3}+\frac{17\!\cdots\!73}{10446270811271}a^{2}+\frac{108122896016904}{549803726909}a-\frac{714316390522643}{10446270811271}$, $\frac{101427058068910}{12\!\cdots\!91}a^{17}+\frac{175864538593485}{12\!\cdots\!91}a^{16}-\frac{42\!\cdots\!53}{12\!\cdots\!91}a^{15}-\frac{93\!\cdots\!89}{12\!\cdots\!91}a^{14}+\frac{58\!\cdots\!57}{12\!\cdots\!91}a^{13}+\frac{13\!\cdots\!45}{114908978923981}a^{12}-\frac{34\!\cdots\!11}{12\!\cdots\!91}a^{11}-\frac{10\!\cdots\!78}{12\!\cdots\!91}a^{10}+\frac{10\!\cdots\!64}{12\!\cdots\!91}a^{9}+\frac{33\!\cdots\!44}{12\!\cdots\!91}a^{8}-\frac{16\!\cdots\!57}{114908978923981}a^{7}-\frac{56\!\cdots\!45}{12\!\cdots\!91}a^{6}+\frac{23\!\cdots\!16}{114908978923981}a^{5}+\frac{43\!\cdots\!73}{114908978923981}a^{4}-\frac{24\!\cdots\!13}{114908978923981}a^{3}-\frac{12\!\cdots\!88}{10446270811271}a^{2}+\frac{553607466052080}{549803726909}a-\frac{18\!\cdots\!84}{10446270811271}$, $\frac{93229590320763}{12\!\cdots\!91}a^{17}-\frac{8078450861057}{12\!\cdots\!91}a^{16}-\frac{39\!\cdots\!63}{12\!\cdots\!91}a^{15}-\frac{16\!\cdots\!24}{12\!\cdots\!91}a^{14}+\frac{59\!\cdots\!02}{12\!\cdots\!91}a^{13}+\frac{41\!\cdots\!85}{12\!\cdots\!91}a^{12}-\frac{43\!\cdots\!87}{12\!\cdots\!91}a^{11}-\frac{32\!\cdots\!12}{12\!\cdots\!91}a^{10}+\frac{15\!\cdots\!19}{114908978923981}a^{9}+\frac{10\!\cdots\!31}{12\!\cdots\!91}a^{8}-\frac{37\!\cdots\!18}{12\!\cdots\!91}a^{7}-\frac{16\!\cdots\!88}{12\!\cdots\!91}a^{6}+\frac{46\!\cdots\!37}{114908978923981}a^{5}+\frac{84\!\cdots\!59}{114908978923981}a^{4}-\frac{35\!\cdots\!89}{114908978923981}a^{3}+\frac{26\!\cdots\!44}{10446270811271}a^{2}+\frac{546326575764799}{549803726909}a-\frac{32\!\cdots\!78}{10446270811271}$, $\frac{30656619270803}{114908978923981}a^{17}+\frac{201677847197764}{12\!\cdots\!91}a^{16}-\frac{13\!\cdots\!32}{12\!\cdots\!91}a^{15}-\frac{15\!\cdots\!57}{12\!\cdots\!91}a^{14}+\frac{20\!\cdots\!99}{12\!\cdots\!91}a^{13}+\frac{28\!\cdots\!88}{12\!\cdots\!91}a^{12}-\frac{13\!\cdots\!45}{12\!\cdots\!91}a^{11}-\frac{19\!\cdots\!27}{12\!\cdots\!91}a^{10}+\frac{43\!\cdots\!60}{12\!\cdots\!91}a^{9}+\frac{61\!\cdots\!65}{12\!\cdots\!91}a^{8}-\frac{86\!\cdots\!52}{12\!\cdots\!91}a^{7}-\frac{95\!\cdots\!62}{12\!\cdots\!91}a^{6}+\frac{95\!\cdots\!53}{114908978923981}a^{5}+\frac{62\!\cdots\!82}{114908978923981}a^{4}-\frac{68\!\cdots\!21}{114908978923981}a^{3}-\frac{11\!\cdots\!12}{10446270811271}a^{2}+\frac{10\!\cdots\!62}{549803726909}a-\frac{41\!\cdots\!28}{10446270811271}$, $\frac{181241810116201}{12\!\cdots\!91}a^{17}+\frac{108108506679660}{12\!\cdots\!91}a^{16}-\frac{75\!\cdots\!65}{12\!\cdots\!91}a^{15}-\frac{83\!\cdots\!66}{12\!\cdots\!91}a^{14}+\frac{11\!\cdots\!85}{12\!\cdots\!91}a^{13}+\frac{15\!\cdots\!90}{12\!\cdots\!91}a^{12}-\frac{73\!\cdots\!14}{12\!\cdots\!91}a^{11}-\frac{10\!\cdots\!30}{12\!\cdots\!91}a^{10}+\frac{25\!\cdots\!94}{12\!\cdots\!91}a^{9}+\frac{34\!\cdots\!63}{12\!\cdots\!91}a^{8}-\frac{51\!\cdots\!16}{12\!\cdots\!91}a^{7}-\frac{49\!\cdots\!83}{114908978923981}a^{6}+\frac{60\!\cdots\!18}{114908978923981}a^{5}+\frac{36\!\cdots\!07}{114908978923981}a^{4}-\frac{45\!\cdots\!61}{114908978923981}a^{3}-\frac{59\!\cdots\!96}{10446270811271}a^{2}+\frac{747075177343236}{549803726909}a-\frac{32\!\cdots\!10}{10446270811271}$, $\frac{32521776240123}{114908978923981}a^{17}+\frac{156813681588610}{12\!\cdots\!91}a^{16}-\frac{14\!\cdots\!81}{12\!\cdots\!91}a^{15}-\frac{14\!\cdots\!67}{12\!\cdots\!91}a^{14}+\frac{21\!\cdots\!41}{12\!\cdots\!91}a^{13}+\frac{26\!\cdots\!41}{12\!\cdots\!91}a^{12}-\frac{13\!\cdots\!81}{12\!\cdots\!91}a^{11}-\frac{17\!\cdots\!93}{12\!\cdots\!91}a^{10}+\frac{43\!\cdots\!06}{12\!\cdots\!91}a^{9}+\frac{51\!\cdots\!98}{12\!\cdots\!91}a^{8}-\frac{72\!\cdots\!25}{114908978923981}a^{7}-\frac{71\!\cdots\!63}{12\!\cdots\!91}a^{6}+\frac{80\!\cdots\!23}{114908978923981}a^{5}+\frac{37\!\cdots\!94}{114908978923981}a^{4}-\frac{49\!\cdots\!18}{114908978923981}a^{3}-\frac{29\!\cdots\!46}{10446270811271}a^{2}+\frac{568382137304574}{549803726909}a-\frac{22\!\cdots\!45}{10446270811271}$, $\frac{11578661153955}{12\!\cdots\!91}a^{17}+\frac{25996086702882}{12\!\cdots\!91}a^{16}-\frac{486484959204006}{12\!\cdots\!91}a^{15}-\frac{13\!\cdots\!15}{12\!\cdots\!91}a^{14}+\frac{68\!\cdots\!15}{12\!\cdots\!91}a^{13}+\frac{20\!\cdots\!03}{12\!\cdots\!91}a^{12}-\frac{37\!\cdots\!83}{114908978923981}a^{11}-\frac{13\!\cdots\!23}{12\!\cdots\!91}a^{10}+\frac{12\!\cdots\!32}{12\!\cdots\!91}a^{9}+\frac{44\!\cdots\!83}{12\!\cdots\!91}a^{8}-\frac{27\!\cdots\!95}{12\!\cdots\!91}a^{7}-\frac{75\!\cdots\!75}{12\!\cdots\!91}a^{6}+\frac{45\!\cdots\!67}{114908978923981}a^{5}+\frac{58\!\cdots\!12}{114908978923981}a^{4}-\frac{51\!\cdots\!94}{114908978923981}a^{3}-\frac{13\!\cdots\!40}{10446270811271}a^{2}+\frac{119525631024583}{549803726909}a-\frac{563024243361875}{10446270811271}$, $\frac{39512473430766}{12\!\cdots\!91}a^{17}+\frac{17119376605508}{12\!\cdots\!91}a^{16}-\frac{16\!\cdots\!29}{12\!\cdots\!91}a^{15}-\frac{15\!\cdots\!11}{12\!\cdots\!91}a^{14}+\frac{24\!\cdots\!19}{12\!\cdots\!91}a^{13}+\frac{29\!\cdots\!29}{12\!\cdots\!91}a^{12}-\frac{17\!\cdots\!69}{12\!\cdots\!91}a^{11}-\frac{21\!\cdots\!89}{12\!\cdots\!91}a^{10}+\frac{64\!\cdots\!27}{12\!\cdots\!91}a^{9}+\frac{73\!\cdots\!23}{12\!\cdots\!91}a^{8}-\frac{14\!\cdots\!41}{12\!\cdots\!91}a^{7}-\frac{12\!\cdots\!71}{12\!\cdots\!91}a^{6}+\frac{18\!\cdots\!68}{114908978923981}a^{5}+\frac{81\!\cdots\!21}{10446270811271}a^{4}-\frac{15\!\cdots\!00}{114908978923981}a^{3}-\frac{11\!\cdots\!40}{10446270811271}a^{2}+\frac{281033465314550}{549803726909}a-\frac{13\!\cdots\!39}{10446270811271}$, $\frac{39147584476228}{12\!\cdots\!91}a^{17}-\frac{92892314291011}{12\!\cdots\!91}a^{16}-\frac{15\!\cdots\!70}{12\!\cdots\!91}a^{15}+\frac{263997157075275}{114908978923981}a^{14}+\frac{23\!\cdots\!07}{12\!\cdots\!91}a^{13}-\frac{35\!\cdots\!60}{12\!\cdots\!91}a^{12}-\frac{16\!\cdots\!42}{12\!\cdots\!91}a^{11}+\frac{22\!\cdots\!88}{12\!\cdots\!91}a^{10}+\frac{63\!\cdots\!03}{12\!\cdots\!91}a^{9}-\frac{84\!\cdots\!87}{12\!\cdots\!91}a^{8}-\frac{11\!\cdots\!74}{12\!\cdots\!91}a^{7}+\frac{17\!\cdots\!36}{12\!\cdots\!91}a^{6}+\frac{71\!\cdots\!10}{10446270811271}a^{5}-\frac{18\!\cdots\!16}{114908978923981}a^{4}+\frac{20\!\cdots\!96}{114908978923981}a^{3}+\frac{68\!\cdots\!69}{10446270811271}a^{2}-\frac{193263147336454}{549803726909}a+\frac{559405626047189}{10446270811271}$, $\frac{219611685549741}{12\!\cdots\!91}a^{17}+\frac{252627763737540}{12\!\cdots\!91}a^{16}-\frac{89\!\cdots\!20}{12\!\cdots\!91}a^{15}-\frac{14\!\cdots\!44}{12\!\cdots\!91}a^{14}+\frac{12\!\cdots\!95}{12\!\cdots\!91}a^{13}+\frac{24\!\cdots\!09}{12\!\cdots\!91}a^{12}-\frac{74\!\cdots\!35}{12\!\cdots\!91}a^{11}-\frac{16\!\cdots\!54}{12\!\cdots\!91}a^{10}+\frac{21\!\cdots\!90}{12\!\cdots\!91}a^{9}+\frac{44\!\cdots\!31}{114908978923981}a^{8}-\frac{35\!\cdots\!99}{114908978923981}a^{7}-\frac{74\!\cdots\!01}{12\!\cdots\!91}a^{6}+\frac{43\!\cdots\!71}{114908978923981}a^{5}+\frac{50\!\cdots\!68}{114908978923981}a^{4}-\frac{36\!\cdots\!29}{114908978923981}a^{3}-\frac{10\!\cdots\!33}{10446270811271}a^{2}+\frac{658772560492917}{549803726909}a-\frac{24\!\cdots\!53}{10446270811271}$, $\frac{196056193011738}{12\!\cdots\!91}a^{17}+\frac{125315626683900}{12\!\cdots\!91}a^{16}-\frac{80\!\cdots\!63}{12\!\cdots\!91}a^{15}-\frac{93\!\cdots\!62}{12\!\cdots\!91}a^{14}+\frac{11\!\cdots\!53}{12\!\cdots\!91}a^{13}+\frac{16\!\cdots\!81}{12\!\cdots\!91}a^{12}-\frac{74\!\cdots\!40}{12\!\cdots\!91}a^{11}-\frac{11\!\cdots\!01}{12\!\cdots\!91}a^{10}+\frac{24\!\cdots\!91}{12\!\cdots\!91}a^{9}+\frac{34\!\cdots\!17}{12\!\cdots\!91}a^{8}-\frac{42\!\cdots\!18}{114908978923981}a^{7}-\frac{52\!\cdots\!73}{12\!\cdots\!91}a^{6}+\frac{52\!\cdots\!09}{114908978923981}a^{5}+\frac{33\!\cdots\!45}{114908978923981}a^{4}-\frac{37\!\cdots\!53}{114908978923981}a^{3}-\frac{54\!\cdots\!14}{10446270811271}a^{2}+\frac{578402388544607}{549803726909}a-\frac{24\!\cdots\!53}{10446270811271}$, $\frac{227905127447532}{12\!\cdots\!91}a^{17}+\frac{169552444829829}{12\!\cdots\!91}a^{16}-\frac{93\!\cdots\!92}{12\!\cdots\!91}a^{15}-\frac{11\!\cdots\!50}{12\!\cdots\!91}a^{14}+\frac{13\!\cdots\!92}{12\!\cdots\!91}a^{13}+\frac{20\!\cdots\!00}{12\!\cdots\!91}a^{12}-\frac{85\!\cdots\!45}{12\!\cdots\!91}a^{11}-\frac{14\!\cdots\!23}{12\!\cdots\!91}a^{10}+\frac{27\!\cdots\!13}{12\!\cdots\!91}a^{9}+\frac{39\!\cdots\!85}{114908978923981}a^{8}-\frac{48\!\cdots\!47}{114908978923981}a^{7}-\frac{66\!\cdots\!45}{12\!\cdots\!91}a^{6}+\frac{53\!\cdots\!63}{10446270811271}a^{5}+\frac{43\!\cdots\!54}{114908978923981}a^{4}-\frac{43\!\cdots\!56}{114908978923981}a^{3}-\frac{77\!\cdots\!57}{10446270811271}a^{2}+\frac{689659869203195}{549803726909}a-\frac{27\!\cdots\!92}{10446270811271}$, $\frac{681016008424}{10446270811271}a^{17}-\frac{88019702592130}{12\!\cdots\!91}a^{16}-\frac{307681933801024}{114908978923981}a^{15}+\frac{18\!\cdots\!57}{12\!\cdots\!91}a^{14}+\frac{51\!\cdots\!82}{12\!\cdots\!91}a^{13}-\frac{12\!\cdots\!05}{12\!\cdots\!91}a^{12}-\frac{36\!\cdots\!11}{12\!\cdots\!91}a^{11}+\frac{58\!\cdots\!78}{12\!\cdots\!91}a^{10}+\frac{12\!\cdots\!64}{114908978923981}a^{9}-\frac{32\!\cdots\!70}{12\!\cdots\!91}a^{8}-\frac{27\!\cdots\!27}{12\!\cdots\!91}a^{7}+\frac{10\!\cdots\!72}{114908978923981}a^{6}+\frac{25\!\cdots\!15}{114908978923981}a^{5}-\frac{16\!\cdots\!27}{114908978923981}a^{4}-\frac{77\!\cdots\!79}{114908978923981}a^{3}+\frac{88\!\cdots\!96}{10446270811271}a^{2}-\frac{128301058687718}{549803726909}a+\frac{162555666593032}{10446270811271}$, $\frac{109936891858617}{12\!\cdots\!91}a^{17}+\frac{249883988695199}{12\!\cdots\!91}a^{16}-\frac{44\!\cdots\!37}{12\!\cdots\!91}a^{15}-\frac{12\!\cdots\!63}{12\!\cdots\!91}a^{14}+\frac{57\!\cdots\!58}{12\!\cdots\!91}a^{13}+\frac{19\!\cdots\!66}{12\!\cdots\!91}a^{12}-\frac{28\!\cdots\!58}{12\!\cdots\!91}a^{11}-\frac{12\!\cdots\!85}{12\!\cdots\!91}a^{10}+\frac{46\!\cdots\!08}{12\!\cdots\!91}a^{9}+\frac{38\!\cdots\!64}{12\!\cdots\!91}a^{8}-\frac{38\!\cdots\!30}{12\!\cdots\!91}a^{7}-\frac{60\!\cdots\!17}{12\!\cdots\!91}a^{6}-\frac{32\!\cdots\!32}{10446270811271}a^{5}+\frac{44\!\cdots\!71}{114908978923981}a^{4}-\frac{45\!\cdots\!58}{114908978923981}a^{3}-\frac{14\!\cdots\!41}{10446270811271}a^{2}+\frac{112897974542772}{549803726909}a+\frac{759130831346848}{10446270811271}$
| 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: | \( 473091347.54 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 473091347.54 \cdot 1}{2\cdot\sqrt{89049975552912225851806640625}}\cr\approx \mathstrut & 0.20779640476 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 42*x^16 - 21*x^15 + 639*x^14 + 477*x^13 - 4557*x^12 - 3465*x^11 + 17562*x^10 + 10481*x^9 - 40194*x^8 - 12870*x^7 + 55341*x^6 - 41745*x^4 + 13068*x^3 + 11979*x^2 - 7986*x + 1331)
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 - 42*x^16 - 21*x^15 + 639*x^14 + 477*x^13 - 4557*x^12 - 3465*x^11 + 17562*x^10 + 10481*x^9 - 40194*x^8 - 12870*x^7 + 55341*x^6 - 41745*x^4 + 13068*x^3 + 11979*x^2 - 7986*x + 1331, 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 - 42*x^16 - 21*x^15 + 639*x^14 + 477*x^13 - 4557*x^12 - 3465*x^11 + 17562*x^10 + 10481*x^9 - 40194*x^8 - 12870*x^7 + 55341*x^6 - 41745*x^4 + 13068*x^3 + 11979*x^2 - 7986*x + 1331);
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 - 42*x^16 - 21*x^15 + 639*x^14 + 477*x^13 - 4557*x^12 - 3465*x^11 + 17562*x^10 + 10481*x^9 - 40194*x^8 - 12870*x^7 + 55341*x^6 - 41745*x^4 + 13068*x^3 + 11979*x^2 - 7986*x + 1331);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_3^2:C_{12}$ (as 18T44):
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 18 conjugacy class representatives for $C_3^2:C_{12}$ |
| Character table for $C_3^2:C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 6.6.820125.1, 6.6.55130625.1 |
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 12 siblings: | data not computed |
| Degree 18 sibling: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.12.11078561287986328125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | R | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | R | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $9$ | $2$ | $30$ | |||
|
\(5\)
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 5.12.9.1 | $x^{12} - 30 x^{8} + 225 x^{4} + 1125$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
|
\(11\)
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 11.6.0.1 | $x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 11.9.6.1 | $x^{9} + 6 x^{7} + 60 x^{6} + 12 x^{5} + 42 x^{4} - 1465 x^{3} + 240 x^{2} - 1560 x + 8088$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |