Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 42*x^16 - 6*x^15 + 621*x^14 + 174*x^13 - 4243*x^12 - 1224*x^11 + 14907*x^10 + 2716*x^9 - 27504*x^8 - 90*x^7 + 24358*x^6 - 4302*x^5 - 7317*x^4 + 1174*x^3 + 657*x^2 + 42*x - 1)
gp: K = bnfinit(y^18 - 42*y^16 - 6*y^15 + 621*y^14 + 174*y^13 - 4243*y^12 - 1224*y^11 + 14907*y^10 + 2716*y^9 - 27504*y^8 - 90*y^7 + 24358*y^6 - 4302*y^5 - 7317*y^4 + 1174*y^3 + 657*y^2 + 42*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 42*x^16 - 6*x^15 + 621*x^14 + 174*x^13 - 4243*x^12 - 1224*x^11 + 14907*x^10 + 2716*x^9 - 27504*x^8 - 90*x^7 + 24358*x^6 - 4302*x^5 - 7317*x^4 + 1174*x^3 + 657*x^2 + 42*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 42*x^16 - 6*x^15 + 621*x^14 + 174*x^13 - 4243*x^12 - 1224*x^11 + 14907*x^10 + 2716*x^9 - 27504*x^8 - 90*x^7 + 24358*x^6 - 4302*x^5 - 7317*x^4 + 1174*x^3 + 657*x^2 + 42*x - 1)
\( x^{18} - 42 x^{16} - 6 x^{15} + 621 x^{14} + 174 x^{13} - 4243 x^{12} - 1224 x^{11} + 14907 x^{10} + \cdots - 1 \)
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: |
\(4459726608577599249580032000000\)
\(\medspace = 2^{33}\cdot 3^{24}\cdot 5^{6}\cdot 7^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(50.44\) | 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: | $2^{11/6}3^{4/3}5^{1/2}7^{2/3}\approx 126.16348990930013$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{72\!\cdots\!14}a^{17}-\frac{37\!\cdots\!45}{72\!\cdots\!14}a^{16}+\frac{11\!\cdots\!02}{36\!\cdots\!57}a^{15}+\frac{27\!\cdots\!00}{36\!\cdots\!57}a^{14}+\frac{17\!\cdots\!90}{36\!\cdots\!57}a^{13}-\frac{76\!\cdots\!28}{36\!\cdots\!57}a^{12}+\frac{15\!\cdots\!13}{72\!\cdots\!14}a^{11}+\frac{85\!\cdots\!51}{36\!\cdots\!57}a^{10}+\frac{10\!\cdots\!27}{36\!\cdots\!57}a^{9}-\frac{10\!\cdots\!26}{36\!\cdots\!57}a^{8}+\frac{18\!\cdots\!73}{72\!\cdots\!14}a^{7}+\frac{26\!\cdots\!96}{36\!\cdots\!57}a^{6}-\frac{15\!\cdots\!13}{72\!\cdots\!14}a^{5}+\frac{65\!\cdots\!54}{36\!\cdots\!57}a^{4}+\frac{26\!\cdots\!43}{72\!\cdots\!14}a^{3}-\frac{31\!\cdots\!91}{16\!\cdots\!98}a^{2}-\frac{55\!\cdots\!30}{36\!\cdots\!57}a-\frac{83\!\cdots\!79}{36\!\cdots\!57}$
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: |
$\frac{5273018701398}{218033592643117}a^{17}-\frac{2729057963040}{218033592643117}a^{16}-\frac{225009549434064}{218033592643117}a^{15}+\frac{81867652678386}{218033592643117}a^{14}+\frac{34\!\cdots\!79}{218033592643117}a^{13}-\frac{14\!\cdots\!09}{436067185286234}a^{12}-\frac{24\!\cdots\!94}{218033592643117}a^{11}+\frac{37\!\cdots\!49}{218033592643117}a^{10}+\frac{94\!\cdots\!47}{218033592643117}a^{9}-\frac{16\!\cdots\!55}{218033592643117}a^{8}-\frac{18\!\cdots\!00}{218033592643117}a^{7}+\frac{47\!\cdots\!67}{218033592643117}a^{6}+\frac{18\!\cdots\!67}{218033592643117}a^{5}-\frac{11\!\cdots\!33}{436067185286234}a^{4}-\frac{59\!\cdots\!40}{218033592643117}a^{3}+\frac{291028788838926}{5070548666119}a^{2}+\frac{63\!\cdots\!68}{218033592643117}a-\frac{61586298440265}{436067185286234}$, $\frac{43\!\cdots\!20}{36\!\cdots\!57}a^{17}-\frac{38\!\cdots\!93}{72\!\cdots\!14}a^{16}-\frac{18\!\cdots\!57}{36\!\cdots\!57}a^{15}+\frac{10\!\cdots\!53}{72\!\cdots\!14}a^{14}+\frac{27\!\cdots\!50}{36\!\cdots\!57}a^{13}-\frac{41\!\cdots\!17}{36\!\cdots\!57}a^{12}-\frac{38\!\cdots\!69}{72\!\cdots\!14}a^{11}+\frac{49\!\cdots\!55}{72\!\cdots\!14}a^{10}+\frac{13\!\cdots\!65}{72\!\cdots\!14}a^{9}-\frac{29\!\cdots\!81}{72\!\cdots\!14}a^{8}-\frac{12\!\cdots\!03}{36\!\cdots\!57}a^{7}+\frac{45\!\cdots\!16}{36\!\cdots\!57}a^{6}+\frac{11\!\cdots\!35}{36\!\cdots\!57}a^{5}-\frac{55\!\cdots\!38}{36\!\cdots\!57}a^{4}-\frac{27\!\cdots\!34}{36\!\cdots\!57}a^{3}+\frac{61\!\cdots\!83}{16\!\cdots\!98}a^{2}+\frac{32\!\cdots\!61}{72\!\cdots\!14}a-\frac{48\!\cdots\!94}{36\!\cdots\!57}$, $\frac{47\!\cdots\!89}{36\!\cdots\!57}a^{17}+\frac{66\!\cdots\!49}{72\!\cdots\!14}a^{16}-\frac{39\!\cdots\!83}{72\!\cdots\!14}a^{15}-\frac{33\!\cdots\!11}{72\!\cdots\!14}a^{14}+\frac{55\!\cdots\!13}{72\!\cdots\!14}a^{13}+\frac{57\!\cdots\!21}{72\!\cdots\!14}a^{12}-\frac{34\!\cdots\!71}{72\!\cdots\!14}a^{11}-\frac{18\!\cdots\!78}{36\!\cdots\!57}a^{10}+\frac{10\!\cdots\!29}{72\!\cdots\!14}a^{9}+\frac{54\!\cdots\!28}{36\!\cdots\!57}a^{8}-\frac{78\!\cdots\!11}{36\!\cdots\!57}a^{7}-\frac{13\!\cdots\!27}{72\!\cdots\!14}a^{6}+\frac{10\!\cdots\!63}{72\!\cdots\!14}a^{5}+\frac{62\!\cdots\!19}{72\!\cdots\!14}a^{4}-\frac{66\!\cdots\!06}{36\!\cdots\!57}a^{3}-\frac{11\!\cdots\!80}{83\!\cdots\!99}a^{2}-\frac{28\!\cdots\!03}{72\!\cdots\!14}a-\frac{25\!\cdots\!97}{72\!\cdots\!14}$, $\frac{10\!\cdots\!61}{72\!\cdots\!14}a^{17}+\frac{63\!\cdots\!07}{36\!\cdots\!57}a^{16}-\frac{43\!\cdots\!55}{72\!\cdots\!14}a^{15}-\frac{59\!\cdots\!19}{72\!\cdots\!14}a^{14}+\frac{61\!\cdots\!97}{72\!\cdots\!14}a^{13}+\frac{46\!\cdots\!97}{36\!\cdots\!57}a^{12}-\frac{38\!\cdots\!75}{72\!\cdots\!14}a^{11}-\frac{30\!\cdots\!31}{36\!\cdots\!57}a^{10}+\frac{11\!\cdots\!11}{72\!\cdots\!14}a^{9}+\frac{18\!\cdots\!55}{72\!\cdots\!14}a^{8}-\frac{16\!\cdots\!75}{72\!\cdots\!14}a^{7}-\frac{12\!\cdots\!63}{36\!\cdots\!57}a^{6}+\frac{11\!\cdots\!41}{72\!\cdots\!14}a^{5}+\frac{14\!\cdots\!39}{72\!\cdots\!14}a^{4}-\frac{10\!\cdots\!26}{36\!\cdots\!57}a^{3}-\frac{69\!\cdots\!09}{16\!\cdots\!98}a^{2}-\frac{12\!\cdots\!87}{36\!\cdots\!57}a+\frac{46\!\cdots\!17}{72\!\cdots\!14}$, $\frac{96\!\cdots\!58}{36\!\cdots\!57}a^{17}+\frac{33\!\cdots\!01}{72\!\cdots\!14}a^{16}-\frac{40\!\cdots\!10}{36\!\cdots\!57}a^{15}-\frac{14\!\cdots\!63}{72\!\cdots\!14}a^{14}+\frac{11\!\cdots\!89}{72\!\cdots\!14}a^{13}+\frac{21\!\cdots\!23}{72\!\cdots\!14}a^{12}-\frac{69\!\cdots\!89}{72\!\cdots\!14}a^{11}-\frac{66\!\cdots\!25}{36\!\cdots\!57}a^{10}+\frac{90\!\cdots\!33}{36\!\cdots\!57}a^{9}+\frac{18\!\cdots\!25}{36\!\cdots\!57}a^{8}-\frac{63\!\cdots\!28}{36\!\cdots\!57}a^{7}-\frac{25\!\cdots\!27}{36\!\cdots\!57}a^{6}-\frac{14\!\cdots\!98}{36\!\cdots\!57}a^{5}+\frac{41\!\cdots\!89}{72\!\cdots\!14}a^{4}+\frac{48\!\cdots\!89}{72\!\cdots\!14}a^{3}-\frac{73\!\cdots\!47}{16\!\cdots\!98}a^{2}-\frac{21\!\cdots\!03}{36\!\cdots\!57}a+\frac{27\!\cdots\!97}{36\!\cdots\!57}$, $\frac{15\!\cdots\!57}{72\!\cdots\!14}a^{17}+\frac{44\!\cdots\!75}{36\!\cdots\!57}a^{16}-\frac{66\!\cdots\!13}{72\!\cdots\!14}a^{15}-\frac{23\!\cdots\!86}{36\!\cdots\!57}a^{14}+\frac{47\!\cdots\!32}{36\!\cdots\!57}a^{13}+\frac{40\!\cdots\!84}{36\!\cdots\!57}a^{12}-\frac{30\!\cdots\!02}{36\!\cdots\!57}a^{11}-\frac{26\!\cdots\!91}{36\!\cdots\!57}a^{10}+\frac{97\!\cdots\!18}{36\!\cdots\!57}a^{9}+\frac{72\!\cdots\!85}{36\!\cdots\!57}a^{8}-\frac{15\!\cdots\!61}{36\!\cdots\!57}a^{7}-\frac{15\!\cdots\!45}{72\!\cdots\!14}a^{6}+\frac{12\!\cdots\!68}{36\!\cdots\!57}a^{5}+\frac{21\!\cdots\!49}{36\!\cdots\!57}a^{4}-\frac{52\!\cdots\!43}{72\!\cdots\!14}a^{3}-\frac{10\!\cdots\!67}{16\!\cdots\!98}a^{2}+\frac{35\!\cdots\!75}{72\!\cdots\!14}a-\frac{91\!\cdots\!91}{72\!\cdots\!14}$, $\frac{26\!\cdots\!57}{36\!\cdots\!57}a^{17}-\frac{10\!\cdots\!03}{72\!\cdots\!14}a^{16}-\frac{23\!\cdots\!41}{72\!\cdots\!14}a^{15}+\frac{19\!\cdots\!86}{36\!\cdots\!57}a^{14}+\frac{18\!\cdots\!92}{36\!\cdots\!57}a^{13}-\frac{25\!\cdots\!26}{36\!\cdots\!57}a^{12}-\frac{13\!\cdots\!79}{36\!\cdots\!57}a^{11}+\frac{16\!\cdots\!64}{36\!\cdots\!57}a^{10}+\frac{11\!\cdots\!57}{72\!\cdots\!14}a^{9}-\frac{11\!\cdots\!43}{72\!\cdots\!14}a^{8}-\frac{11\!\cdots\!08}{36\!\cdots\!57}a^{7}+\frac{22\!\cdots\!83}{72\!\cdots\!14}a^{6}+\frac{23\!\cdots\!11}{72\!\cdots\!14}a^{5}-\frac{19\!\cdots\!39}{72\!\cdots\!14}a^{4}-\frac{71\!\cdots\!65}{72\!\cdots\!14}a^{3}+\frac{11\!\cdots\!09}{16\!\cdots\!98}a^{2}+\frac{77\!\cdots\!69}{72\!\cdots\!14}a-\frac{25\!\cdots\!53}{72\!\cdots\!14}$, $\frac{22\!\cdots\!31}{72\!\cdots\!14}a^{17}+\frac{99\!\cdots\!50}{36\!\cdots\!57}a^{16}-\frac{91\!\cdots\!61}{72\!\cdots\!14}a^{15}-\frac{96\!\cdots\!53}{72\!\cdots\!14}a^{14}+\frac{63\!\cdots\!14}{36\!\cdots\!57}a^{13}+\frac{78\!\cdots\!54}{36\!\cdots\!57}a^{12}-\frac{37\!\cdots\!15}{36\!\cdots\!57}a^{11}-\frac{50\!\cdots\!41}{36\!\cdots\!57}a^{10}+\frac{10\!\cdots\!38}{36\!\cdots\!57}a^{9}+\frac{14\!\cdots\!36}{36\!\cdots\!57}a^{8}-\frac{13\!\cdots\!03}{36\!\cdots\!57}a^{7}-\frac{17\!\cdots\!37}{36\!\cdots\!57}a^{6}+\frac{70\!\cdots\!29}{36\!\cdots\!57}a^{5}+\frac{76\!\cdots\!03}{36\!\cdots\!57}a^{4}-\frac{57\!\cdots\!03}{72\!\cdots\!14}a^{3}-\frac{31\!\cdots\!42}{83\!\cdots\!99}a^{2}-\frac{39\!\cdots\!49}{72\!\cdots\!14}a+\frac{49\!\cdots\!01}{72\!\cdots\!14}$, $\frac{18\!\cdots\!21}{72\!\cdots\!14}a^{17}-\frac{51\!\cdots\!90}{36\!\cdots\!57}a^{16}-\frac{39\!\cdots\!87}{36\!\cdots\!57}a^{15}+\frac{14\!\cdots\!62}{36\!\cdots\!57}a^{14}+\frac{11\!\cdots\!79}{72\!\cdots\!14}a^{13}-\frac{10\!\cdots\!39}{36\!\cdots\!57}a^{12}-\frac{40\!\cdots\!62}{36\!\cdots\!57}a^{11}+\frac{37\!\cdots\!17}{36\!\cdots\!57}a^{10}+\frac{28\!\cdots\!37}{72\!\cdots\!14}a^{9}-\frac{12\!\cdots\!70}{36\!\cdots\!57}a^{8}-\frac{26\!\cdots\!35}{36\!\cdots\!57}a^{7}+\frac{35\!\cdots\!22}{36\!\cdots\!57}a^{6}+\frac{22\!\cdots\!78}{36\!\cdots\!57}a^{5}-\frac{41\!\cdots\!81}{36\!\cdots\!57}a^{4}-\frac{58\!\cdots\!46}{36\!\cdots\!57}a^{3}+\frac{18\!\cdots\!04}{83\!\cdots\!99}a^{2}+\frac{43\!\cdots\!73}{72\!\cdots\!14}a+\frac{13\!\cdots\!45}{36\!\cdots\!57}$, $\frac{28\!\cdots\!41}{72\!\cdots\!14}a^{17}+\frac{18\!\cdots\!55}{72\!\cdots\!14}a^{16}-\frac{11\!\cdots\!89}{72\!\cdots\!14}a^{15}-\frac{47\!\cdots\!18}{36\!\cdots\!57}a^{14}+\frac{16\!\cdots\!93}{72\!\cdots\!14}a^{13}+\frac{16\!\cdots\!03}{72\!\cdots\!14}a^{12}-\frac{52\!\cdots\!92}{36\!\cdots\!57}a^{11}-\frac{11\!\cdots\!95}{72\!\cdots\!14}a^{10}+\frac{16\!\cdots\!93}{36\!\cdots\!57}a^{9}+\frac{17\!\cdots\!53}{36\!\cdots\!57}a^{8}-\frac{49\!\cdots\!43}{72\!\cdots\!14}a^{7}-\frac{23\!\cdots\!59}{36\!\cdots\!57}a^{6}+\frac{36\!\cdots\!59}{72\!\cdots\!14}a^{5}+\frac{12\!\cdots\!64}{36\!\cdots\!57}a^{4}-\frac{51\!\cdots\!41}{36\!\cdots\!57}a^{3}-\frac{63\!\cdots\!66}{83\!\cdots\!99}a^{2}+\frac{68\!\cdots\!03}{72\!\cdots\!14}a+\frac{11\!\cdots\!39}{36\!\cdots\!57}$, $\frac{64\!\cdots\!89}{36\!\cdots\!57}a^{17}+\frac{25\!\cdots\!03}{72\!\cdots\!14}a^{16}-\frac{52\!\cdots\!53}{72\!\cdots\!14}a^{15}-\frac{57\!\cdots\!68}{36\!\cdots\!57}a^{14}+\frac{36\!\cdots\!87}{36\!\cdots\!57}a^{13}+\frac{86\!\cdots\!36}{36\!\cdots\!57}a^{12}-\frac{20\!\cdots\!13}{36\!\cdots\!57}a^{11}-\frac{11\!\cdots\!31}{72\!\cdots\!14}a^{10}+\frac{46\!\cdots\!48}{36\!\cdots\!57}a^{9}+\frac{16\!\cdots\!21}{36\!\cdots\!57}a^{8}-\frac{28\!\cdots\!70}{36\!\cdots\!57}a^{7}-\frac{44\!\cdots\!53}{72\!\cdots\!14}a^{6}-\frac{40\!\cdots\!39}{36\!\cdots\!57}a^{5}+\frac{26\!\cdots\!05}{72\!\cdots\!14}a^{4}+\frac{51\!\cdots\!83}{36\!\cdots\!57}a^{3}-\frac{11\!\cdots\!53}{16\!\cdots\!98}a^{2}-\frac{28\!\cdots\!41}{72\!\cdots\!14}a-\frac{12\!\cdots\!05}{36\!\cdots\!57}$, $\frac{43\!\cdots\!40}{36\!\cdots\!57}a^{17}+\frac{22\!\cdots\!97}{36\!\cdots\!57}a^{16}-\frac{18\!\cdots\!99}{36\!\cdots\!57}a^{15}-\frac{11\!\cdots\!35}{36\!\cdots\!57}a^{14}+\frac{53\!\cdots\!87}{72\!\cdots\!14}a^{13}+\frac{40\!\cdots\!97}{72\!\cdots\!14}a^{12}-\frac{35\!\cdots\!13}{72\!\cdots\!14}a^{11}-\frac{26\!\cdots\!87}{72\!\cdots\!14}a^{10}+\frac{58\!\cdots\!12}{36\!\cdots\!57}a^{9}+\frac{36\!\cdots\!37}{36\!\cdots\!57}a^{8}-\frac{20\!\cdots\!39}{72\!\cdots\!14}a^{7}-\frac{38\!\cdots\!01}{36\!\cdots\!57}a^{6}+\frac{81\!\cdots\!80}{36\!\cdots\!57}a^{5}+\frac{68\!\cdots\!82}{36\!\cdots\!57}a^{4}-\frac{19\!\cdots\!86}{36\!\cdots\!57}a^{3}-\frac{22\!\cdots\!05}{16\!\cdots\!98}a^{2}+\frac{89\!\cdots\!33}{72\!\cdots\!14}a-\frac{54\!\cdots\!02}{36\!\cdots\!57}$, $\frac{43\!\cdots\!40}{36\!\cdots\!57}a^{17}+\frac{22\!\cdots\!97}{36\!\cdots\!57}a^{16}-\frac{18\!\cdots\!99}{36\!\cdots\!57}a^{15}-\frac{11\!\cdots\!35}{36\!\cdots\!57}a^{14}+\frac{53\!\cdots\!87}{72\!\cdots\!14}a^{13}+\frac{40\!\cdots\!97}{72\!\cdots\!14}a^{12}-\frac{35\!\cdots\!13}{72\!\cdots\!14}a^{11}-\frac{26\!\cdots\!87}{72\!\cdots\!14}a^{10}+\frac{58\!\cdots\!12}{36\!\cdots\!57}a^{9}+\frac{36\!\cdots\!37}{36\!\cdots\!57}a^{8}-\frac{20\!\cdots\!39}{72\!\cdots\!14}a^{7}-\frac{38\!\cdots\!01}{36\!\cdots\!57}a^{6}+\frac{81\!\cdots\!80}{36\!\cdots\!57}a^{5}+\frac{68\!\cdots\!82}{36\!\cdots\!57}a^{4}-\frac{19\!\cdots\!86}{36\!\cdots\!57}a^{3}-\frac{22\!\cdots\!05}{16\!\cdots\!98}a^{2}+\frac{89\!\cdots\!33}{72\!\cdots\!14}a+\frac{30\!\cdots\!55}{36\!\cdots\!57}$, $\frac{10\!\cdots\!67}{72\!\cdots\!14}a^{17}-\frac{11\!\cdots\!07}{72\!\cdots\!14}a^{16}-\frac{45\!\cdots\!43}{36\!\cdots\!57}a^{15}+\frac{53\!\cdots\!93}{72\!\cdots\!14}a^{14}+\frac{12\!\cdots\!05}{36\!\cdots\!57}a^{13}-\frac{86\!\cdots\!71}{72\!\cdots\!14}a^{12}-\frac{30\!\cdots\!71}{72\!\cdots\!14}a^{11}+\frac{61\!\cdots\!39}{72\!\cdots\!14}a^{10}+\frac{84\!\cdots\!05}{36\!\cdots\!57}a^{9}-\frac{22\!\cdots\!43}{72\!\cdots\!14}a^{8}-\frac{22\!\cdots\!26}{36\!\cdots\!57}a^{7}+\frac{21\!\cdots\!37}{36\!\cdots\!57}a^{6}+\frac{50\!\cdots\!55}{72\!\cdots\!14}a^{5}-\frac{40\!\cdots\!15}{72\!\cdots\!14}a^{4}-\frac{82\!\cdots\!50}{36\!\cdots\!57}a^{3}+\frac{22\!\cdots\!35}{16\!\cdots\!98}a^{2}+\frac{18\!\cdots\!33}{72\!\cdots\!14}a-\frac{13\!\cdots\!29}{72\!\cdots\!14}$, $\frac{47\!\cdots\!31}{72\!\cdots\!14}a^{17}+\frac{17\!\cdots\!98}{36\!\cdots\!57}a^{16}-\frac{19\!\cdots\!27}{72\!\cdots\!14}a^{15}-\frac{17\!\cdots\!75}{72\!\cdots\!14}a^{14}+\frac{14\!\cdots\!15}{36\!\cdots\!57}a^{13}+\frac{29\!\cdots\!75}{72\!\cdots\!14}a^{12}-\frac{17\!\cdots\!15}{72\!\cdots\!14}a^{11}-\frac{19\!\cdots\!99}{72\!\cdots\!14}a^{10}+\frac{55\!\cdots\!47}{72\!\cdots\!14}a^{9}+\frac{27\!\cdots\!72}{36\!\cdots\!57}a^{8}-\frac{42\!\cdots\!39}{36\!\cdots\!57}a^{7}-\frac{65\!\cdots\!75}{72\!\cdots\!14}a^{6}+\frac{28\!\cdots\!03}{36\!\cdots\!57}a^{5}+\frac{13\!\cdots\!87}{36\!\cdots\!57}a^{4}-\frac{93\!\cdots\!17}{72\!\cdots\!14}a^{3}-\frac{39\!\cdots\!04}{83\!\cdots\!99}a^{2}-\frac{63\!\cdots\!55}{36\!\cdots\!57}a-\frac{94\!\cdots\!96}{36\!\cdots\!57}$, $\frac{14\!\cdots\!75}{36\!\cdots\!57}a^{17}-\frac{24\!\cdots\!55}{72\!\cdots\!14}a^{16}-\frac{60\!\cdots\!53}{36\!\cdots\!57}a^{15}-\frac{33\!\cdots\!80}{36\!\cdots\!57}a^{14}+\frac{17\!\cdots\!83}{72\!\cdots\!14}a^{13}+\frac{17\!\cdots\!68}{36\!\cdots\!57}a^{12}-\frac{12\!\cdots\!27}{72\!\cdots\!14}a^{11}-\frac{12\!\cdots\!34}{36\!\cdots\!57}a^{10}+\frac{43\!\cdots\!97}{72\!\cdots\!14}a^{9}+\frac{18\!\cdots\!90}{36\!\cdots\!57}a^{8}-\frac{79\!\cdots\!31}{72\!\cdots\!14}a^{7}+\frac{38\!\cdots\!26}{36\!\cdots\!57}a^{6}+\frac{69\!\cdots\!59}{72\!\cdots\!14}a^{5}-\frac{97\!\cdots\!31}{36\!\cdots\!57}a^{4}-\frac{18\!\cdots\!91}{72\!\cdots\!14}a^{3}+\frac{12\!\cdots\!67}{16\!\cdots\!98}a^{2}+\frac{12\!\cdots\!09}{72\!\cdots\!14}a-\frac{14\!\cdots\!50}{36\!\cdots\!57}$, $\frac{27\!\cdots\!07}{72\!\cdots\!14}a^{17}+\frac{23\!\cdots\!83}{72\!\cdots\!14}a^{16}-\frac{11\!\cdots\!93}{72\!\cdots\!14}a^{15}-\frac{11\!\cdots\!53}{72\!\cdots\!14}a^{14}+\frac{83\!\cdots\!67}{36\!\cdots\!57}a^{13}+\frac{19\!\cdots\!99}{72\!\cdots\!14}a^{12}-\frac{53\!\cdots\!49}{36\!\cdots\!57}a^{11}-\frac{64\!\cdots\!05}{36\!\cdots\!57}a^{10}+\frac{34\!\cdots\!73}{72\!\cdots\!14}a^{9}+\frac{38\!\cdots\!75}{72\!\cdots\!14}a^{8}-\frac{58\!\cdots\!65}{72\!\cdots\!14}a^{7}-\frac{53\!\cdots\!75}{72\!\cdots\!14}a^{6}+\frac{50\!\cdots\!69}{72\!\cdots\!14}a^{5}+\frac{14\!\cdots\!84}{36\!\cdots\!57}a^{4}-\frac{19\!\cdots\!71}{72\!\cdots\!14}a^{3}-\frac{10\!\cdots\!97}{16\!\cdots\!98}a^{2}+\frac{85\!\cdots\!97}{36\!\cdots\!57}a+\frac{24\!\cdots\!21}{72\!\cdots\!14}$
| 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: | \( 3927677460.06 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 3927677460.06 \cdot 1}{2\cdot\sqrt{4459726608577599249580032000000}}\cr\approx \mathstrut & 0.243776380527 \end{aligned}\] (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 - 6*x^15 + 621*x^14 + 174*x^13 - 4243*x^12 - 1224*x^11 + 14907*x^10 + 2716*x^9 - 27504*x^8 - 90*x^7 + 24358*x^6 - 4302*x^5 - 7317*x^4 + 1174*x^3 + 657*x^2 + 42*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 - 42*x^16 - 6*x^15 + 621*x^14 + 174*x^13 - 4243*x^12 - 1224*x^11 + 14907*x^10 + 2716*x^9 - 27504*x^8 - 90*x^7 + 24358*x^6 - 4302*x^5 - 7317*x^4 + 1174*x^3 + 657*x^2 + 42*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 - 42*x^16 - 6*x^15 + 621*x^14 + 174*x^13 - 4243*x^12 - 1224*x^11 + 14907*x^10 + 2716*x^9 - 27504*x^8 - 90*x^7 + 24358*x^6 - 4302*x^5 - 7317*x^4 + 1174*x^3 + 657*x^2 + 42*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 - 42*x^16 - 6*x^15 + 621*x^14 + 174*x^13 - 4243*x^12 - 1224*x^11 + 14907*x^10 + 2716*x^9 - 27504*x^8 - 90*x^7 + 24358*x^6 - 4302*x^5 - 7317*x^4 + 1174*x^3 + 657*x^2 + 42*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
$C_3\times S_3^2$ (as 18T46):
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 27 conjugacy class representatives for $C_3\times S_3^2$ |
| Character table for $C_3\times S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 3.3.1620.1, 6.6.335923200.1, 6.6.164602368.3 |
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 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.12.1032386052096000000.1 |
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 | R | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{9}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{9}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.33.308 | $x^{18} + 8 x^{16} - 8 x^{15} + 28 x^{14} - 176 x^{13} - 182 x^{12} - 128 x^{11} - 684 x^{10} - 536 x^{9} - 24 x^{8} + 544 x^{7} + 404 x^{6} + 1904 x^{5} + 3768 x^{4} + 3152 x^{3} + 1072 x^{2} + 3424 x + 6392$ | $6$ | $3$ | $33$ | $S_3 \times C_6$ | $[3]_{3}^{6}$ |
|
\(3\)
| Deg $18$ | $3$ | $6$ | $24$ | |||
|
\(5\)
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(7\)
| 7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 7.6.4.1 | $x^{6} + 14 x^{3} - 245$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |