Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 45*x^16 - x^15 + 756*x^14 + 57*x^13 - 5953*x^12 - 1089*x^11 + 23253*x^10 + 8945*x^9 - 46602*x^8 - 30060*x^7 + 42927*x^6 + 43173*x^5 - 7479*x^4 - 21483*x^3 - 8262*x^2 - 972*x - 27)
gp: K = bnfinit(y^18 - 45*y^16 - y^15 + 756*y^14 + 57*y^13 - 5953*y^12 - 1089*y^11 + 23253*y^10 + 8945*y^9 - 46602*y^8 - 30060*y^7 + 42927*y^6 + 43173*y^5 - 7479*y^4 - 21483*y^3 - 8262*y^2 - 972*y - 27, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 45*x^16 - x^15 + 756*x^14 + 57*x^13 - 5953*x^12 - 1089*x^11 + 23253*x^10 + 8945*x^9 - 46602*x^8 - 30060*x^7 + 42927*x^6 + 43173*x^5 - 7479*x^4 - 21483*x^3 - 8262*x^2 - 972*x - 27);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 45*x^16 - x^15 + 756*x^14 + 57*x^13 - 5953*x^12 - 1089*x^11 + 23253*x^10 + 8945*x^9 - 46602*x^8 - 30060*x^7 + 42927*x^6 + 43173*x^5 - 7479*x^4 - 21483*x^3 - 8262*x^2 - 972*x - 27)
\( x^{18} - 45 x^{16} - x^{15} + 756 x^{14} + 57 x^{13} - 5953 x^{12} - 1089 x^{11} + 23253 x^{10} + \cdots - 27 \)
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: |
\(187933304498364210515293279521\)
\(\medspace = 3^{24}\cdot 13^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(42.30\) | 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: | $3^{37/18}13^{11/12}\approx 100.43012088175487$ | ||
| Ramified primes: |
\(3\), \(13\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{9}-\frac{1}{3}a^{6}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{10}-\frac{1}{3}a^{7}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{11}-\frac{1}{3}a^{8}-\frac{1}{3}a^{5}$, $\frac{1}{9}a^{15}-\frac{1}{9}a^{12}+\frac{1}{3}a^{10}-\frac{4}{9}a^{9}-\frac{1}{3}a^{7}-\frac{1}{9}a^{6}-\frac{1}{3}a^{3}$, $\frac{1}{9}a^{16}-\frac{1}{9}a^{13}+\frac{1}{3}a^{11}-\frac{4}{9}a^{10}-\frac{1}{3}a^{8}-\frac{1}{9}a^{7}-\frac{1}{3}a^{4}$, $\frac{1}{21\!\cdots\!87}a^{17}-\frac{11\!\cdots\!62}{21\!\cdots\!87}a^{16}-\frac{85\!\cdots\!30}{21\!\cdots\!87}a^{15}-\frac{33\!\cdots\!23}{21\!\cdots\!87}a^{14}+\frac{18\!\cdots\!98}{21\!\cdots\!87}a^{13}+\frac{38\!\cdots\!01}{21\!\cdots\!87}a^{12}-\frac{65\!\cdots\!44}{21\!\cdots\!87}a^{11}+\frac{94\!\cdots\!50}{21\!\cdots\!87}a^{10}-\frac{55\!\cdots\!81}{21\!\cdots\!87}a^{9}+\frac{73\!\cdots\!00}{21\!\cdots\!87}a^{8}-\frac{10\!\cdots\!50}{21\!\cdots\!87}a^{7}+\frac{33\!\cdots\!25}{21\!\cdots\!87}a^{6}+\frac{82\!\cdots\!56}{72\!\cdots\!29}a^{5}-\frac{16\!\cdots\!34}{72\!\cdots\!29}a^{4}-\frac{64\!\cdots\!35}{24\!\cdots\!43}a^{3}+\frac{18\!\cdots\!44}{24\!\cdots\!43}a^{2}-\frac{97\!\cdots\!18}{24\!\cdots\!43}a-\frac{85\!\cdots\!46}{24\!\cdots\!43}$
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: |
$\frac{23\!\cdots\!91}{24\!\cdots\!43}a^{17}-\frac{60\!\cdots\!11}{72\!\cdots\!29}a^{16}-\frac{94\!\cdots\!56}{21\!\cdots\!87}a^{15}+\frac{86\!\cdots\!34}{24\!\cdots\!43}a^{14}+\frac{51\!\cdots\!32}{72\!\cdots\!29}a^{13}-\frac{11\!\cdots\!37}{21\!\cdots\!87}a^{12}-\frac{12\!\cdots\!26}{24\!\cdots\!43}a^{11}+\frac{25\!\cdots\!12}{72\!\cdots\!29}a^{10}+\frac{43\!\cdots\!78}{21\!\cdots\!87}a^{9}-\frac{19\!\cdots\!78}{24\!\cdots\!43}a^{8}-\frac{93\!\cdots\!91}{24\!\cdots\!43}a^{7}+\frac{76\!\cdots\!39}{21\!\cdots\!87}a^{6}+\frac{94\!\cdots\!32}{24\!\cdots\!43}a^{5}+\frac{22\!\cdots\!74}{24\!\cdots\!43}a^{4}-\frac{10\!\cdots\!73}{72\!\cdots\!29}a^{3}-\frac{19\!\cdots\!15}{24\!\cdots\!43}a^{2}-\frac{26\!\cdots\!57}{24\!\cdots\!43}a-\frac{76\!\cdots\!73}{24\!\cdots\!43}$, $\frac{42\!\cdots\!74}{24\!\cdots\!43}a^{17}-\frac{10\!\cdots\!39}{72\!\cdots\!29}a^{16}-\frac{16\!\cdots\!38}{21\!\cdots\!87}a^{15}+\frac{15\!\cdots\!93}{24\!\cdots\!43}a^{14}+\frac{92\!\cdots\!86}{72\!\cdots\!29}a^{13}-\frac{21\!\cdots\!62}{21\!\cdots\!87}a^{12}-\frac{23\!\cdots\!89}{24\!\cdots\!43}a^{11}+\frac{44\!\cdots\!46}{72\!\cdots\!29}a^{10}+\frac{77\!\cdots\!09}{21\!\cdots\!87}a^{9}-\frac{34\!\cdots\!17}{24\!\cdots\!43}a^{8}-\frac{16\!\cdots\!02}{24\!\cdots\!43}a^{7}+\frac{12\!\cdots\!52}{21\!\cdots\!87}a^{6}+\frac{16\!\cdots\!94}{24\!\cdots\!43}a^{5}+\frac{41\!\cdots\!96}{24\!\cdots\!43}a^{4}-\frac{19\!\cdots\!34}{72\!\cdots\!29}a^{3}-\frac{36\!\cdots\!26}{24\!\cdots\!43}a^{2}-\frac{49\!\cdots\!47}{24\!\cdots\!43}a-\frac{14\!\cdots\!24}{24\!\cdots\!43}$, $\frac{94\!\cdots\!82}{24\!\cdots\!43}a^{17}-\frac{80\!\cdots\!14}{24\!\cdots\!43}a^{16}-\frac{12\!\cdots\!27}{72\!\cdots\!29}a^{15}+\frac{34\!\cdots\!42}{24\!\cdots\!43}a^{14}+\frac{68\!\cdots\!41}{24\!\cdots\!43}a^{13}-\frac{15\!\cdots\!50}{72\!\cdots\!29}a^{12}-\frac{51\!\cdots\!17}{24\!\cdots\!43}a^{11}+\frac{33\!\cdots\!01}{24\!\cdots\!43}a^{10}+\frac{19\!\cdots\!52}{24\!\cdots\!43}a^{9}-\frac{78\!\cdots\!21}{24\!\cdots\!43}a^{8}-\frac{37\!\cdots\!35}{24\!\cdots\!43}a^{7}+\frac{34\!\cdots\!58}{24\!\cdots\!43}a^{6}+\frac{37\!\cdots\!44}{24\!\cdots\!43}a^{5}+\frac{87\!\cdots\!62}{24\!\cdots\!43}a^{4}-\frac{43\!\cdots\!21}{72\!\cdots\!29}a^{3}-\frac{79\!\cdots\!28}{24\!\cdots\!43}a^{2}-\frac{10\!\cdots\!35}{24\!\cdots\!43}a-\frac{29\!\cdots\!46}{24\!\cdots\!43}$, $\frac{17\!\cdots\!26}{24\!\cdots\!43}a^{17}-\frac{45\!\cdots\!41}{72\!\cdots\!29}a^{16}-\frac{70\!\cdots\!57}{21\!\cdots\!87}a^{15}+\frac{65\!\cdots\!10}{24\!\cdots\!43}a^{14}+\frac{38\!\cdots\!76}{72\!\cdots\!29}a^{13}-\frac{89\!\cdots\!61}{21\!\cdots\!87}a^{12}-\frac{97\!\cdots\!14}{24\!\cdots\!43}a^{11}+\frac{19\!\cdots\!85}{72\!\cdots\!29}a^{10}+\frac{32\!\cdots\!35}{21\!\cdots\!87}a^{9}-\frac{14\!\cdots\!10}{24\!\cdots\!43}a^{8}-\frac{70\!\cdots\!77}{24\!\cdots\!43}a^{7}+\frac{59\!\cdots\!58}{21\!\cdots\!87}a^{6}+\frac{70\!\cdots\!24}{24\!\cdots\!43}a^{5}+\frac{16\!\cdots\!44}{24\!\cdots\!43}a^{4}-\frac{27\!\cdots\!18}{24\!\cdots\!43}a^{3}-\frac{14\!\cdots\!34}{24\!\cdots\!43}a^{2}-\frac{19\!\cdots\!61}{24\!\cdots\!43}a-\frac{55\!\cdots\!49}{24\!\cdots\!43}$, $\frac{99\!\cdots\!35}{24\!\cdots\!43}a^{17}-\frac{25\!\cdots\!30}{72\!\cdots\!29}a^{16}-\frac{39\!\cdots\!37}{21\!\cdots\!87}a^{15}+\frac{36\!\cdots\!99}{24\!\cdots\!43}a^{14}+\frac{21\!\cdots\!23}{72\!\cdots\!29}a^{13}-\frac{50\!\cdots\!25}{21\!\cdots\!87}a^{12}-\frac{54\!\cdots\!20}{24\!\cdots\!43}a^{11}+\frac{10\!\cdots\!63}{72\!\cdots\!29}a^{10}+\frac{18\!\cdots\!01}{21\!\cdots\!87}a^{9}-\frac{82\!\cdots\!40}{24\!\cdots\!43}a^{8}-\frac{39\!\cdots\!54}{24\!\cdots\!43}a^{7}+\frac{33\!\cdots\!65}{21\!\cdots\!87}a^{6}+\frac{39\!\cdots\!94}{24\!\cdots\!43}a^{5}+\frac{92\!\cdots\!16}{24\!\cdots\!43}a^{4}-\frac{15\!\cdots\!06}{24\!\cdots\!43}a^{3}-\frac{83\!\cdots\!61}{24\!\cdots\!43}a^{2}-\frac{10\!\cdots\!75}{24\!\cdots\!43}a-\frac{32\!\cdots\!03}{24\!\cdots\!43}$, $\frac{33\!\cdots\!29}{21\!\cdots\!87}a^{17}-\frac{28\!\cdots\!61}{21\!\cdots\!87}a^{16}-\frac{14\!\cdots\!17}{21\!\cdots\!87}a^{15}+\frac{12\!\cdots\!85}{21\!\cdots\!87}a^{14}+\frac{24\!\cdots\!22}{21\!\cdots\!87}a^{13}-\frac{19\!\cdots\!00}{21\!\cdots\!87}a^{12}-\frac{18\!\cdots\!51}{21\!\cdots\!87}a^{11}+\frac{12\!\cdots\!70}{21\!\cdots\!87}a^{10}+\frac{67\!\cdots\!02}{21\!\cdots\!87}a^{9}-\frac{28\!\cdots\!81}{21\!\cdots\!87}a^{8}-\frac{13\!\cdots\!44}{21\!\cdots\!87}a^{7}+\frac{13\!\cdots\!12}{21\!\cdots\!87}a^{6}+\frac{14\!\cdots\!64}{24\!\cdots\!43}a^{5}+\frac{10\!\cdots\!20}{72\!\cdots\!29}a^{4}-\frac{17\!\cdots\!18}{72\!\cdots\!29}a^{3}-\frac{30\!\cdots\!41}{24\!\cdots\!43}a^{2}-\frac{40\!\cdots\!09}{24\!\cdots\!43}a-\frac{11\!\cdots\!93}{24\!\cdots\!43}$, $\frac{35\!\cdots\!73}{72\!\cdots\!29}a^{17}-\frac{92\!\cdots\!93}{21\!\cdots\!87}a^{16}-\frac{47\!\cdots\!59}{21\!\cdots\!87}a^{15}+\frac{44\!\cdots\!38}{24\!\cdots\!43}a^{14}+\frac{77\!\cdots\!71}{21\!\cdots\!87}a^{13}-\frac{60\!\cdots\!06}{21\!\cdots\!87}a^{12}-\frac{65\!\cdots\!52}{24\!\cdots\!43}a^{11}+\frac{38\!\cdots\!59}{21\!\cdots\!87}a^{10}+\frac{21\!\cdots\!80}{21\!\cdots\!87}a^{9}-\frac{29\!\cdots\!51}{72\!\cdots\!29}a^{8}-\frac{42\!\cdots\!02}{21\!\cdots\!87}a^{7}+\frac{40\!\cdots\!53}{21\!\cdots\!87}a^{6}+\frac{14\!\cdots\!55}{72\!\cdots\!29}a^{5}+\frac{32\!\cdots\!18}{72\!\cdots\!29}a^{4}-\frac{54\!\cdots\!20}{72\!\cdots\!29}a^{3}-\frac{99\!\cdots\!68}{24\!\cdots\!43}a^{2}-\frac{13\!\cdots\!69}{24\!\cdots\!43}a-\frac{37\!\cdots\!64}{24\!\cdots\!43}$, $\frac{37\!\cdots\!87}{21\!\cdots\!87}a^{17}-\frac{33\!\cdots\!89}{21\!\cdots\!87}a^{16}-\frac{16\!\cdots\!74}{21\!\cdots\!87}a^{15}+\frac{14\!\cdots\!05}{21\!\cdots\!87}a^{14}+\frac{27\!\cdots\!37}{21\!\cdots\!87}a^{13}-\frac{22\!\cdots\!35}{21\!\cdots\!87}a^{12}-\frac{20\!\cdots\!42}{21\!\cdots\!87}a^{11}+\frac{14\!\cdots\!85}{21\!\cdots\!87}a^{10}+\frac{74\!\cdots\!38}{21\!\cdots\!87}a^{9}-\frac{33\!\cdots\!29}{21\!\cdots\!87}a^{8}-\frac{14\!\cdots\!96}{21\!\cdots\!87}a^{7}+\frac{17\!\cdots\!58}{21\!\cdots\!87}a^{6}+\frac{48\!\cdots\!08}{72\!\cdots\!29}a^{5}+\frac{34\!\cdots\!52}{24\!\cdots\!43}a^{4}-\frac{19\!\cdots\!09}{72\!\cdots\!29}a^{3}-\frac{33\!\cdots\!18}{24\!\cdots\!43}a^{2}-\frac{41\!\cdots\!10}{24\!\cdots\!43}a-\frac{94\!\cdots\!36}{24\!\cdots\!43}$, $\frac{50\!\cdots\!42}{72\!\cdots\!29}a^{17}-\frac{14\!\cdots\!12}{24\!\cdots\!43}a^{16}-\frac{67\!\cdots\!38}{21\!\cdots\!87}a^{15}+\frac{62\!\cdots\!61}{24\!\cdots\!43}a^{14}+\frac{36\!\cdots\!65}{72\!\cdots\!29}a^{13}-\frac{86\!\cdots\!88}{21\!\cdots\!87}a^{12}-\frac{27\!\cdots\!16}{72\!\cdots\!29}a^{11}+\frac{61\!\cdots\!00}{24\!\cdots\!43}a^{10}+\frac{30\!\cdots\!64}{21\!\cdots\!87}a^{9}-\frac{42\!\cdots\!53}{72\!\cdots\!29}a^{8}-\frac{19\!\cdots\!97}{72\!\cdots\!29}a^{7}+\frac{57\!\cdots\!52}{21\!\cdots\!87}a^{6}+\frac{20\!\cdots\!00}{72\!\cdots\!29}a^{5}+\frac{46\!\cdots\!39}{72\!\cdots\!29}a^{4}-\frac{25\!\cdots\!21}{24\!\cdots\!43}a^{3}-\frac{14\!\cdots\!51}{24\!\cdots\!43}a^{2}-\frac{18\!\cdots\!01}{24\!\cdots\!43}a-\frac{53\!\cdots\!23}{24\!\cdots\!43}$, $\frac{25\!\cdots\!71}{21\!\cdots\!87}a^{17}-\frac{21\!\cdots\!49}{21\!\cdots\!87}a^{16}-\frac{11\!\cdots\!42}{21\!\cdots\!87}a^{15}+\frac{91\!\cdots\!13}{21\!\cdots\!87}a^{14}+\frac{18\!\cdots\!18}{21\!\cdots\!87}a^{13}-\frac{13\!\cdots\!50}{21\!\cdots\!87}a^{12}-\frac{14\!\cdots\!89}{21\!\cdots\!87}a^{11}+\frac{88\!\cdots\!67}{21\!\cdots\!87}a^{10}+\frac{52\!\cdots\!62}{21\!\cdots\!87}a^{9}-\frac{20\!\cdots\!74}{21\!\cdots\!87}a^{8}-\frac{10\!\cdots\!15}{21\!\cdots\!87}a^{7}+\frac{76\!\cdots\!92}{21\!\cdots\!87}a^{6}+\frac{11\!\cdots\!55}{24\!\cdots\!43}a^{5}+\frac{84\!\cdots\!43}{72\!\cdots\!29}a^{4}-\frac{44\!\cdots\!41}{24\!\cdots\!43}a^{3}-\frac{24\!\cdots\!24}{24\!\cdots\!43}a^{2}-\frac{32\!\cdots\!22}{24\!\cdots\!43}a-\frac{94\!\cdots\!46}{24\!\cdots\!43}$, $\frac{22\!\cdots\!16}{21\!\cdots\!87}a^{17}-\frac{18\!\cdots\!40}{21\!\cdots\!87}a^{16}-\frac{97\!\cdots\!27}{21\!\cdots\!87}a^{15}+\frac{78\!\cdots\!33}{21\!\cdots\!87}a^{14}+\frac{16\!\cdots\!50}{21\!\cdots\!87}a^{13}-\frac{11\!\cdots\!58}{21\!\cdots\!87}a^{12}-\frac{12\!\cdots\!73}{21\!\cdots\!87}a^{11}+\frac{75\!\cdots\!61}{21\!\cdots\!87}a^{10}+\frac{45\!\cdots\!63}{21\!\cdots\!87}a^{9}-\frac{17\!\cdots\!57}{21\!\cdots\!87}a^{8}-\frac{88\!\cdots\!35}{21\!\cdots\!87}a^{7}+\frac{63\!\cdots\!97}{21\!\cdots\!87}a^{6}+\frac{29\!\cdots\!18}{72\!\cdots\!29}a^{5}+\frac{24\!\cdots\!30}{24\!\cdots\!43}a^{4}-\frac{11\!\cdots\!83}{72\!\cdots\!29}a^{3}-\frac{21\!\cdots\!61}{24\!\cdots\!43}a^{2}-\frac{28\!\cdots\!49}{24\!\cdots\!43}a-\frac{81\!\cdots\!07}{24\!\cdots\!43}$, $\frac{72\!\cdots\!08}{21\!\cdots\!87}a^{17}-\frac{60\!\cdots\!53}{21\!\cdots\!87}a^{16}-\frac{10\!\cdots\!43}{72\!\cdots\!29}a^{15}+\frac{26\!\cdots\!76}{21\!\cdots\!87}a^{14}+\frac{52\!\cdots\!09}{21\!\cdots\!87}a^{13}-\frac{44\!\cdots\!55}{24\!\cdots\!43}a^{12}-\frac{39\!\cdots\!83}{21\!\cdots\!87}a^{11}+\frac{25\!\cdots\!29}{21\!\cdots\!87}a^{10}+\frac{49\!\cdots\!41}{72\!\cdots\!29}a^{9}-\frac{57\!\cdots\!55}{21\!\cdots\!87}a^{8}-\frac{29\!\cdots\!38}{21\!\cdots\!87}a^{7}+\frac{24\!\cdots\!72}{24\!\cdots\!43}a^{6}+\frac{97\!\cdots\!38}{72\!\cdots\!29}a^{5}+\frac{79\!\cdots\!98}{24\!\cdots\!43}a^{4}-\frac{37\!\cdots\!33}{72\!\cdots\!29}a^{3}-\frac{69\!\cdots\!31}{24\!\cdots\!43}a^{2}-\frac{90\!\cdots\!16}{24\!\cdots\!43}a-\frac{24\!\cdots\!01}{24\!\cdots\!43}$, $\frac{27\!\cdots\!41}{72\!\cdots\!29}a^{17}-\frac{75\!\cdots\!66}{21\!\cdots\!87}a^{16}-\frac{36\!\cdots\!54}{21\!\cdots\!87}a^{15}+\frac{10\!\cdots\!46}{72\!\cdots\!29}a^{14}+\frac{60\!\cdots\!39}{21\!\cdots\!87}a^{13}-\frac{49\!\cdots\!43}{21\!\cdots\!87}a^{12}-\frac{15\!\cdots\!89}{72\!\cdots\!29}a^{11}+\frac{31\!\cdots\!28}{21\!\cdots\!87}a^{10}+\frac{17\!\cdots\!97}{21\!\cdots\!87}a^{9}-\frac{83\!\cdots\!77}{24\!\cdots\!43}a^{8}-\frac{34\!\cdots\!01}{21\!\cdots\!87}a^{7}+\frac{37\!\cdots\!02}{21\!\cdots\!87}a^{6}+\frac{38\!\cdots\!20}{24\!\cdots\!43}a^{5}+\frac{25\!\cdots\!36}{72\!\cdots\!29}a^{4}-\frac{46\!\cdots\!50}{72\!\cdots\!29}a^{3}-\frac{82\!\cdots\!31}{24\!\cdots\!43}a^{2}-\frac{10\!\cdots\!03}{24\!\cdots\!43}a-\frac{23\!\cdots\!95}{24\!\cdots\!43}$, $\frac{34\!\cdots\!55}{72\!\cdots\!29}a^{17}-\frac{92\!\cdots\!91}{21\!\cdots\!87}a^{16}-\frac{45\!\cdots\!92}{21\!\cdots\!87}a^{15}+\frac{44\!\cdots\!78}{24\!\cdots\!43}a^{14}+\frac{75\!\cdots\!09}{21\!\cdots\!87}a^{13}-\frac{61\!\cdots\!53}{21\!\cdots\!87}a^{12}-\frac{18\!\cdots\!46}{72\!\cdots\!29}a^{11}+\frac{39\!\cdots\!85}{21\!\cdots\!87}a^{10}+\frac{20\!\cdots\!15}{21\!\cdots\!87}a^{9}-\frac{10\!\cdots\!15}{24\!\cdots\!43}a^{8}-\frac{40\!\cdots\!45}{21\!\cdots\!87}a^{7}+\frac{47\!\cdots\!83}{21\!\cdots\!87}a^{6}+\frac{13\!\cdots\!01}{72\!\cdots\!29}a^{5}+\frac{29\!\cdots\!38}{72\!\cdots\!29}a^{4}-\frac{52\!\cdots\!47}{72\!\cdots\!29}a^{3}-\frac{93\!\cdots\!03}{24\!\cdots\!43}a^{2}-\frac{12\!\cdots\!04}{24\!\cdots\!43}a-\frac{33\!\cdots\!19}{24\!\cdots\!43}$, $\frac{59\!\cdots\!91}{72\!\cdots\!29}a^{17}-\frac{14\!\cdots\!75}{21\!\cdots\!87}a^{16}-\frac{78\!\cdots\!04}{21\!\cdots\!87}a^{15}+\frac{71\!\cdots\!29}{24\!\cdots\!43}a^{14}+\frac{12\!\cdots\!51}{21\!\cdots\!87}a^{13}-\frac{97\!\cdots\!06}{21\!\cdots\!87}a^{12}-\frac{10\!\cdots\!54}{24\!\cdots\!43}a^{11}+\frac{62\!\cdots\!99}{21\!\cdots\!87}a^{10}+\frac{36\!\cdots\!57}{21\!\cdots\!87}a^{9}-\frac{47\!\cdots\!22}{72\!\cdots\!29}a^{8}-\frac{70\!\cdots\!02}{21\!\cdots\!87}a^{7}+\frac{58\!\cdots\!35}{21\!\cdots\!87}a^{6}+\frac{23\!\cdots\!79}{72\!\cdots\!29}a^{5}+\frac{56\!\cdots\!30}{72\!\cdots\!29}a^{4}-\frac{91\!\cdots\!98}{72\!\cdots\!29}a^{3}-\frac{16\!\cdots\!69}{24\!\cdots\!43}a^{2}-\frac{22\!\cdots\!21}{24\!\cdots\!43}a-\frac{64\!\cdots\!54}{24\!\cdots\!43}$, $\frac{15\!\cdots\!42}{72\!\cdots\!29}a^{17}-\frac{39\!\cdots\!17}{21\!\cdots\!87}a^{16}-\frac{20\!\cdots\!23}{21\!\cdots\!87}a^{15}+\frac{57\!\cdots\!37}{72\!\cdots\!29}a^{14}+\frac{33\!\cdots\!89}{21\!\cdots\!87}a^{13}-\frac{26\!\cdots\!87}{21\!\cdots\!87}a^{12}-\frac{28\!\cdots\!97}{24\!\cdots\!43}a^{11}+\frac{16\!\cdots\!02}{21\!\cdots\!87}a^{10}+\frac{93\!\cdots\!03}{21\!\cdots\!87}a^{9}-\frac{12\!\cdots\!03}{72\!\cdots\!29}a^{8}-\frac{18\!\cdots\!26}{21\!\cdots\!87}a^{7}+\frac{17\!\cdots\!97}{21\!\cdots\!87}a^{6}+\frac{20\!\cdots\!83}{24\!\cdots\!43}a^{5}+\frac{14\!\cdots\!58}{72\!\cdots\!29}a^{4}-\frac{79\!\cdots\!56}{24\!\cdots\!43}a^{3}-\frac{43\!\cdots\!78}{24\!\cdots\!43}a^{2}-\frac{56\!\cdots\!27}{24\!\cdots\!43}a-\frac{16\!\cdots\!80}{24\!\cdots\!43}$, $\frac{11\!\cdots\!45}{21\!\cdots\!87}a^{17}-\frac{37\!\cdots\!86}{72\!\cdots\!29}a^{16}-\frac{50\!\cdots\!43}{21\!\cdots\!87}a^{15}+\frac{47\!\cdots\!72}{21\!\cdots\!87}a^{14}+\frac{91\!\cdots\!50}{24\!\cdots\!43}a^{13}-\frac{73\!\cdots\!68}{21\!\cdots\!87}a^{12}-\frac{61\!\cdots\!63}{21\!\cdots\!87}a^{11}+\frac{15\!\cdots\!68}{72\!\cdots\!29}a^{10}+\frac{22\!\cdots\!65}{21\!\cdots\!87}a^{9}-\frac{11\!\cdots\!57}{21\!\cdots\!87}a^{8}-\frac{47\!\cdots\!96}{24\!\cdots\!43}a^{7}+\frac{62\!\cdots\!97}{21\!\cdots\!87}a^{6}+\frac{14\!\cdots\!67}{72\!\cdots\!29}a^{5}+\frac{30\!\cdots\!17}{72\!\cdots\!29}a^{4}-\frac{17\!\cdots\!64}{24\!\cdots\!43}a^{3}-\frac{10\!\cdots\!96}{24\!\cdots\!43}a^{2}-\frac{15\!\cdots\!26}{24\!\cdots\!43}a-\frac{52\!\cdots\!86}{24\!\cdots\!43}$
| 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: | \( 637998345.319 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 637998345.319 \cdot 1}{2\cdot\sqrt{187933304498364210515293279521}}\cr\approx \mathstrut & 0.192897946136 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 45*x^16 - x^15 + 756*x^14 + 57*x^13 - 5953*x^12 - 1089*x^11 + 23253*x^10 + 8945*x^9 - 46602*x^8 - 30060*x^7 + 42927*x^6 + 43173*x^5 - 7479*x^4 - 21483*x^3 - 8262*x^2 - 972*x - 27)
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 - 45*x^16 - x^15 + 756*x^14 + 57*x^13 - 5953*x^12 - 1089*x^11 + 23253*x^10 + 8945*x^9 - 46602*x^8 - 30060*x^7 + 42927*x^6 + 43173*x^5 - 7479*x^4 - 21483*x^3 - 8262*x^2 - 972*x - 27, 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 - 45*x^16 - x^15 + 756*x^14 + 57*x^13 - 5953*x^12 - 1089*x^11 + 23253*x^10 + 8945*x^9 - 46602*x^8 - 30060*x^7 + 42927*x^6 + 43173*x^5 - 7479*x^4 - 21483*x^3 - 8262*x^2 - 972*x - 27);
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 - 45*x^16 - x^15 + 756*x^14 + 57*x^13 - 5953*x^12 - 1089*x^11 + 23253*x^10 + 8945*x^9 - 46602*x^8 - 30060*x^7 + 42927*x^6 + 43173*x^5 - 7479*x^4 - 21483*x^3 - 8262*x^2 - 972*x - 27);
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
| \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\zeta_{13})^+\), 6.6.187388721.1 |
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: | 12.12.694319656224247224093.3 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | R | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.9.15.3 | $x^{9} - 36 x^{7} - 36 x^{6} + 324 x^{5} + 918 x^{4} + 2349 x^{3} + 5346 x^{2} + 13284 x + 14688$ | $3$ | $3$ | $15$ | $S_3\times C_3$ | $[5/2]_{2}^{3}$ |
| 3.9.9.6 | $x^{9} - 6 x^{8} + 45 x^{7} + 594 x^{6} + 99 x^{5} + 108 x^{4} - 54 x^{3} + 27 x^{2} + 81 x + 27$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $[3/2]_{2}^{3}$ | |
|
\(13\)
| 13.6.5.2 | $x^{6} + 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 13.12.11.4 | $x^{12} + 13$ | $12$ | $1$ | $11$ | $C_{12}$ | $[\ ]_{12}$ |