Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 72*x^16 - 48*x^15 + 1917*x^14 + 1944*x^13 - 24780*x^12 - 30708*x^11 + 164547*x^10 + 240556*x^9 - 512883*x^8 - 957456*x^7 + 410361*x^6 + 1650123*x^5 + 829629*x^4 - 421407*x^3 - 566379*x^2 - 206577*x - 26241)
gp: K = bnfinit(y^18 - 72*y^16 - 48*y^15 + 1917*y^14 + 1944*y^13 - 24780*y^12 - 30708*y^11 + 164547*y^10 + 240556*y^9 - 512883*y^8 - 957456*y^7 + 410361*y^6 + 1650123*y^5 + 829629*y^4 - 421407*y^3 - 566379*y^2 - 206577*y - 26241, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 72*x^16 - 48*x^15 + 1917*x^14 + 1944*x^13 - 24780*x^12 - 30708*x^11 + 164547*x^10 + 240556*x^9 - 512883*x^8 - 957456*x^7 + 410361*x^6 + 1650123*x^5 + 829629*x^4 - 421407*x^3 - 566379*x^2 - 206577*x - 26241);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 72*x^16 - 48*x^15 + 1917*x^14 + 1944*x^13 - 24780*x^12 - 30708*x^11 + 164547*x^10 + 240556*x^9 - 512883*x^8 - 957456*x^7 + 410361*x^6 + 1650123*x^5 + 829629*x^4 - 421407*x^3 - 566379*x^2 - 206577*x - 26241)
\( x^{18} - 72 x^{16} - 48 x^{15} + 1917 x^{14} + 1944 x^{13} - 24780 x^{12} - 30708 x^{11} + 164547 x^{10} + \cdots - 26241 \)
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: |
\(126149432166859917805140950806401\)
\(\medspace = 3^{44}\cdot 71^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(60.73\) | 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: | $3^{22/9}71^{1/2}\approx 123.5735646798655$ | ||
| Ramified primes: |
\(3\), \(71\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{11\!\cdots\!31}a^{17}-\frac{17\!\cdots\!94}{11\!\cdots\!31}a^{16}+\frac{25\!\cdots\!45}{11\!\cdots\!31}a^{15}+\frac{31\!\cdots\!73}{11\!\cdots\!31}a^{14}-\frac{48\!\cdots\!62}{11\!\cdots\!31}a^{13}+\frac{39\!\cdots\!44}{11\!\cdots\!31}a^{12}-\frac{35\!\cdots\!54}{11\!\cdots\!31}a^{11}+\frac{21\!\cdots\!85}{11\!\cdots\!31}a^{10}-\frac{39\!\cdots\!79}{11\!\cdots\!31}a^{9}-\frac{31\!\cdots\!92}{11\!\cdots\!31}a^{8}+\frac{51\!\cdots\!09}{11\!\cdots\!31}a^{7}+\frac{39\!\cdots\!58}{11\!\cdots\!31}a^{6}+\frac{18\!\cdots\!43}{11\!\cdots\!31}a^{5}-\frac{19\!\cdots\!67}{11\!\cdots\!31}a^{4}-\frac{14\!\cdots\!75}{11\!\cdots\!31}a^{3}+\frac{39\!\cdots\!75}{11\!\cdots\!31}a^{2}-\frac{41\!\cdots\!00}{11\!\cdots\!31}a-\frac{53\!\cdots\!47}{11\!\cdots\!31}$
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: |
$\frac{33\!\cdots\!87}{98\!\cdots\!41}a^{17}-\frac{23\!\cdots\!40}{98\!\cdots\!41}a^{16}-\frac{23\!\cdots\!30}{98\!\cdots\!41}a^{15}+\frac{10\!\cdots\!19}{98\!\cdots\!41}a^{14}+\frac{63\!\cdots\!24}{98\!\cdots\!41}a^{13}+\frac{19\!\cdots\!96}{98\!\cdots\!41}a^{12}-\frac{83\!\cdots\!89}{98\!\cdots\!41}a^{11}-\frac{41\!\cdots\!83}{98\!\cdots\!41}a^{10}+\frac{57\!\cdots\!17}{98\!\cdots\!41}a^{9}+\frac{38\!\cdots\!53}{98\!\cdots\!41}a^{8}-\frac{19\!\cdots\!02}{98\!\cdots\!41}a^{7}-\frac{17\!\cdots\!96}{98\!\cdots\!41}a^{6}+\frac{26\!\cdots\!27}{98\!\cdots\!41}a^{5}+\frac{35\!\cdots\!37}{98\!\cdots\!41}a^{4}+\frac{11\!\cdots\!18}{98\!\cdots\!41}a^{3}-\frac{15\!\cdots\!39}{98\!\cdots\!41}a^{2}-\frac{75\!\cdots\!57}{98\!\cdots\!41}a-\frac{11\!\cdots\!38}{98\!\cdots\!41}$, $\frac{34\!\cdots\!56}{98\!\cdots\!41}a^{17}-\frac{20\!\cdots\!52}{98\!\cdots\!41}a^{16}-\frac{24\!\cdots\!85}{98\!\cdots\!41}a^{15}-\frac{19\!\cdots\!77}{98\!\cdots\!41}a^{14}+\frac{65\!\cdots\!57}{98\!\cdots\!41}a^{13}+\frac{27\!\cdots\!91}{98\!\cdots\!41}a^{12}-\frac{86\!\cdots\!74}{98\!\cdots\!41}a^{11}-\frac{54\!\cdots\!95}{98\!\cdots\!41}a^{10}+\frac{59\!\cdots\!58}{98\!\cdots\!41}a^{9}+\frac{47\!\cdots\!03}{98\!\cdots\!41}a^{8}-\frac{20\!\cdots\!91}{98\!\cdots\!41}a^{7}-\frac{20\!\cdots\!06}{98\!\cdots\!41}a^{6}+\frac{26\!\cdots\!81}{98\!\cdots\!41}a^{5}+\frac{41\!\cdots\!32}{98\!\cdots\!41}a^{4}+\frac{43\!\cdots\!69}{98\!\cdots\!41}a^{3}-\frac{16\!\cdots\!20}{98\!\cdots\!41}a^{2}-\frac{94\!\cdots\!71}{98\!\cdots\!41}a-\frac{15\!\cdots\!60}{98\!\cdots\!41}$, $\frac{12\!\cdots\!42}{11\!\cdots\!31}a^{17}-\frac{74\!\cdots\!85}{11\!\cdots\!31}a^{16}-\frac{86\!\cdots\!15}{11\!\cdots\!31}a^{15}-\frac{43\!\cdots\!37}{11\!\cdots\!31}a^{14}+\frac{23\!\cdots\!65}{11\!\cdots\!31}a^{13}+\frac{90\!\cdots\!18}{11\!\cdots\!31}a^{12}-\frac{30\!\cdots\!65}{11\!\cdots\!31}a^{11}-\frac{18\!\cdots\!72}{11\!\cdots\!31}a^{10}+\frac{20\!\cdots\!89}{11\!\cdots\!31}a^{9}+\frac{15\!\cdots\!54}{11\!\cdots\!31}a^{8}-\frac{71\!\cdots\!26}{11\!\cdots\!31}a^{7}-\frac{70\!\cdots\!03}{11\!\cdots\!31}a^{6}+\frac{92\!\cdots\!04}{11\!\cdots\!31}a^{5}+\frac{14\!\cdots\!26}{11\!\cdots\!31}a^{4}+\frac{13\!\cdots\!66}{11\!\cdots\!31}a^{3}-\frac{58\!\cdots\!87}{11\!\cdots\!31}a^{2}-\frac{32\!\cdots\!19}{11\!\cdots\!31}a-\frac{51\!\cdots\!58}{11\!\cdots\!31}$, $\frac{16\!\cdots\!25}{11\!\cdots\!31}a^{17}-\frac{10\!\cdots\!46}{11\!\cdots\!31}a^{16}-\frac{11\!\cdots\!19}{11\!\cdots\!31}a^{15}-\frac{77\!\cdots\!68}{11\!\cdots\!31}a^{14}+\frac{32\!\cdots\!19}{11\!\cdots\!31}a^{13}+\frac{13\!\cdots\!00}{11\!\cdots\!31}a^{12}-\frac{42\!\cdots\!02}{11\!\cdots\!31}a^{11}-\frac{25\!\cdots\!49}{11\!\cdots\!31}a^{10}+\frac{28\!\cdots\!00}{11\!\cdots\!31}a^{9}+\frac{22\!\cdots\!24}{11\!\cdots\!31}a^{8}-\frac{99\!\cdots\!90}{11\!\cdots\!31}a^{7}-\frac{99\!\cdots\!77}{11\!\cdots\!31}a^{6}+\frac{12\!\cdots\!87}{11\!\cdots\!31}a^{5}+\frac{19\!\cdots\!36}{11\!\cdots\!31}a^{4}+\frac{19\!\cdots\!52}{11\!\cdots\!31}a^{3}-\frac{81\!\cdots\!78}{11\!\cdots\!31}a^{2}-\frac{45\!\cdots\!40}{11\!\cdots\!31}a-\frac{73\!\cdots\!65}{11\!\cdots\!31}$, $\frac{90\!\cdots\!35}{11\!\cdots\!31}a^{17}-\frac{55\!\cdots\!96}{11\!\cdots\!31}a^{16}-\frac{64\!\cdots\!03}{11\!\cdots\!31}a^{15}-\frac{37\!\cdots\!32}{11\!\cdots\!31}a^{14}+\frac{17\!\cdots\!37}{11\!\cdots\!31}a^{13}+\frac{69\!\cdots\!39}{11\!\cdots\!31}a^{12}-\frac{22\!\cdots\!93}{11\!\cdots\!31}a^{11}-\frac{13\!\cdots\!82}{11\!\cdots\!31}a^{10}+\frac{15\!\cdots\!69}{11\!\cdots\!31}a^{9}+\frac{12\!\cdots\!61}{11\!\cdots\!31}a^{8}-\frac{53\!\cdots\!69}{11\!\cdots\!31}a^{7}-\frac{53\!\cdots\!90}{11\!\cdots\!31}a^{6}+\frac{69\!\cdots\!22}{11\!\cdots\!31}a^{5}+\frac{10\!\cdots\!24}{11\!\cdots\!31}a^{4}+\frac{10\!\cdots\!55}{11\!\cdots\!31}a^{3}-\frac{44\!\cdots\!28}{11\!\cdots\!31}a^{2}-\frac{24\!\cdots\!28}{11\!\cdots\!31}a-\frac{38\!\cdots\!92}{11\!\cdots\!31}$, $\frac{94\!\cdots\!50}{11\!\cdots\!31}a^{17}-\frac{66\!\cdots\!52}{11\!\cdots\!31}a^{16}-\frac{67\!\cdots\!49}{11\!\cdots\!31}a^{15}+\frac{24\!\cdots\!50}{11\!\cdots\!31}a^{14}+\frac{18\!\cdots\!37}{11\!\cdots\!31}a^{13}+\frac{55\!\cdots\!73}{11\!\cdots\!31}a^{12}-\frac{23\!\cdots\!58}{11\!\cdots\!31}a^{11}-\frac{12\!\cdots\!93}{11\!\cdots\!31}a^{10}+\frac{16\!\cdots\!21}{11\!\cdots\!31}a^{9}+\frac{11\!\cdots\!25}{11\!\cdots\!31}a^{8}-\frac{56\!\cdots\!92}{11\!\cdots\!31}a^{7}-\frac{50\!\cdots\!52}{11\!\cdots\!31}a^{6}+\frac{73\!\cdots\!89}{11\!\cdots\!31}a^{5}+\frac{10\!\cdots\!64}{11\!\cdots\!31}a^{4}+\frac{63\!\cdots\!04}{11\!\cdots\!31}a^{3}-\frac{43\!\cdots\!87}{11\!\cdots\!31}a^{2}-\frac{22\!\cdots\!11}{11\!\cdots\!31}a-\frac{36\!\cdots\!18}{11\!\cdots\!31}$, $\frac{88\!\cdots\!20}{11\!\cdots\!31}a^{17}-\frac{48\!\cdots\!53}{11\!\cdots\!31}a^{16}-\frac{63\!\cdots\!07}{11\!\cdots\!31}a^{15}-\frac{74\!\cdots\!69}{11\!\cdots\!31}a^{14}+\frac{17\!\cdots\!39}{11\!\cdots\!31}a^{13}+\frac{78\!\cdots\!33}{11\!\cdots\!31}a^{12}-\frac{22\!\cdots\!58}{11\!\cdots\!31}a^{11}-\frac{14\!\cdots\!93}{11\!\cdots\!31}a^{10}+\frac{15\!\cdots\!74}{11\!\cdots\!31}a^{9}+\frac{12\!\cdots\!26}{11\!\cdots\!31}a^{8}-\frac{52\!\cdots\!15}{11\!\cdots\!31}a^{7}-\frac{55\!\cdots\!56}{11\!\cdots\!31}a^{6}+\frac{67\!\cdots\!05}{11\!\cdots\!31}a^{5}+\frac{10\!\cdots\!53}{11\!\cdots\!31}a^{4}+\frac{13\!\cdots\!23}{11\!\cdots\!31}a^{3}-\frac{44\!\cdots\!74}{11\!\cdots\!31}a^{2}-\frac{25\!\cdots\!37}{11\!\cdots\!31}a-\frac{42\!\cdots\!34}{11\!\cdots\!31}$, $\frac{72\!\cdots\!60}{11\!\cdots\!31}a^{17}-\frac{30\!\cdots\!13}{11\!\cdots\!31}a^{16}-\frac{51\!\cdots\!48}{11\!\cdots\!31}a^{15}-\frac{12\!\cdots\!00}{11\!\cdots\!31}a^{14}+\frac{13\!\cdots\!08}{11\!\cdots\!31}a^{13}+\frac{80\!\cdots\!12}{11\!\cdots\!31}a^{12}-\frac{18\!\cdots\!28}{11\!\cdots\!31}a^{11}-\frac{14\!\cdots\!48}{11\!\cdots\!31}a^{10}+\frac{12\!\cdots\!58}{11\!\cdots\!31}a^{9}+\frac{12\!\cdots\!04}{11\!\cdots\!31}a^{8}-\frac{42\!\cdots\!00}{11\!\cdots\!31}a^{7}-\frac{51\!\cdots\!44}{11\!\cdots\!31}a^{6}+\frac{53\!\cdots\!16}{11\!\cdots\!31}a^{5}+\frac{97\!\cdots\!00}{11\!\cdots\!31}a^{4}+\frac{16\!\cdots\!88}{11\!\cdots\!31}a^{3}-\frac{38\!\cdots\!29}{11\!\cdots\!31}a^{2}-\frac{23\!\cdots\!60}{11\!\cdots\!31}a-\frac{40\!\cdots\!66}{11\!\cdots\!31}$, $\frac{29\!\cdots\!25}{11\!\cdots\!31}a^{17}-\frac{17\!\cdots\!31}{11\!\cdots\!31}a^{16}-\frac{21\!\cdots\!74}{11\!\cdots\!31}a^{15}-\frac{16\!\cdots\!25}{11\!\cdots\!31}a^{14}+\frac{57\!\cdots\!65}{11\!\cdots\!31}a^{13}+\frac{24\!\cdots\!63}{11\!\cdots\!31}a^{12}-\frac{75\!\cdots\!43}{11\!\cdots\!31}a^{11}-\frac{47\!\cdots\!30}{11\!\cdots\!31}a^{10}+\frac{52\!\cdots\!45}{11\!\cdots\!31}a^{9}+\frac{41\!\cdots\!19}{11\!\cdots\!31}a^{8}-\frac{17\!\cdots\!98}{11\!\cdots\!31}a^{7}-\frac{18\!\cdots\!60}{11\!\cdots\!31}a^{6}+\frac{22\!\cdots\!61}{11\!\cdots\!31}a^{5}+\frac{35\!\cdots\!48}{11\!\cdots\!31}a^{4}+\frac{38\!\cdots\!74}{11\!\cdots\!31}a^{3}-\frac{14\!\cdots\!17}{11\!\cdots\!31}a^{2}-\frac{82\!\cdots\!08}{11\!\cdots\!31}a-\frac{13\!\cdots\!25}{11\!\cdots\!31}$, $\frac{32\!\cdots\!11}{11\!\cdots\!31}a^{17}-\frac{19\!\cdots\!47}{11\!\cdots\!31}a^{16}-\frac{23\!\cdots\!04}{11\!\cdots\!31}a^{15}-\frac{20\!\cdots\!97}{11\!\cdots\!31}a^{14}+\frac{62\!\cdots\!31}{11\!\cdots\!31}a^{13}+\frac{26\!\cdots\!54}{11\!\cdots\!31}a^{12}-\frac{82\!\cdots\!71}{11\!\cdots\!31}a^{11}-\frac{52\!\cdots\!57}{11\!\cdots\!31}a^{10}+\frac{56\!\cdots\!74}{11\!\cdots\!31}a^{9}+\frac{45\!\cdots\!64}{11\!\cdots\!31}a^{8}-\frac{19\!\cdots\!97}{11\!\cdots\!31}a^{7}-\frac{20\!\cdots\!14}{11\!\cdots\!31}a^{6}+\frac{25\!\cdots\!98}{11\!\cdots\!31}a^{5}+\frac{39\!\cdots\!04}{11\!\cdots\!31}a^{4}+\frac{41\!\cdots\!82}{11\!\cdots\!31}a^{3}-\frac{16\!\cdots\!16}{11\!\cdots\!31}a^{2}-\frac{90\!\cdots\!86}{11\!\cdots\!31}a-\frac{14\!\cdots\!29}{11\!\cdots\!31}$, $\frac{14\!\cdots\!77}{11\!\cdots\!31}a^{17}-\frac{96\!\cdots\!30}{11\!\cdots\!31}a^{16}-\frac{10\!\cdots\!27}{11\!\cdots\!31}a^{15}-\frac{20\!\cdots\!65}{11\!\cdots\!31}a^{14}+\frac{28\!\cdots\!89}{11\!\cdots\!31}a^{13}+\frac{10\!\cdots\!85}{11\!\cdots\!31}a^{12}-\frac{37\!\cdots\!29}{11\!\cdots\!31}a^{11}-\frac{21\!\cdots\!08}{11\!\cdots\!31}a^{10}+\frac{25\!\cdots\!54}{11\!\cdots\!31}a^{9}+\frac{18\!\cdots\!39}{11\!\cdots\!31}a^{8}-\frac{88\!\cdots\!65}{11\!\cdots\!31}a^{7}-\frac{84\!\cdots\!48}{11\!\cdots\!31}a^{6}+\frac{11\!\cdots\!88}{11\!\cdots\!31}a^{5}+\frac{16\!\cdots\!25}{11\!\cdots\!31}a^{4}+\frac{13\!\cdots\!03}{11\!\cdots\!31}a^{3}-\frac{70\!\cdots\!19}{11\!\cdots\!31}a^{2}-\frac{37\!\cdots\!84}{11\!\cdots\!31}a-\frac{59\!\cdots\!88}{11\!\cdots\!31}$, $\frac{20\!\cdots\!78}{11\!\cdots\!31}a^{17}-\frac{12\!\cdots\!68}{11\!\cdots\!31}a^{16}-\frac{14\!\cdots\!88}{11\!\cdots\!31}a^{15}-\frac{10\!\cdots\!05}{11\!\cdots\!31}a^{14}+\frac{39\!\cdots\!91}{11\!\cdots\!31}a^{13}+\frac{16\!\cdots\!94}{11\!\cdots\!31}a^{12}-\frac{51\!\cdots\!75}{11\!\cdots\!31}a^{11}-\frac{32\!\cdots\!35}{11\!\cdots\!31}a^{10}+\frac{35\!\cdots\!44}{11\!\cdots\!31}a^{9}+\frac{28\!\cdots\!27}{11\!\cdots\!31}a^{8}-\frac{12\!\cdots\!23}{11\!\cdots\!31}a^{7}-\frac{12\!\cdots\!38}{11\!\cdots\!31}a^{6}+\frac{15\!\cdots\!82}{11\!\cdots\!31}a^{5}+\frac{24\!\cdots\!57}{11\!\cdots\!31}a^{4}+\frac{25\!\cdots\!48}{11\!\cdots\!31}a^{3}-\frac{10\!\cdots\!95}{11\!\cdots\!31}a^{2}-\frac{56\!\cdots\!90}{11\!\cdots\!31}a-\frac{91\!\cdots\!29}{11\!\cdots\!31}$, $\frac{15\!\cdots\!50}{11\!\cdots\!31}a^{17}-\frac{10\!\cdots\!45}{11\!\cdots\!31}a^{16}-\frac{10\!\cdots\!75}{11\!\cdots\!31}a^{15}+\frac{83\!\cdots\!80}{11\!\cdots\!31}a^{14}+\frac{29\!\cdots\!24}{11\!\cdots\!31}a^{13}+\frac{98\!\cdots\!84}{11\!\cdots\!31}a^{12}-\frac{38\!\cdots\!68}{11\!\cdots\!31}a^{11}-\frac{20\!\cdots\!05}{11\!\cdots\!31}a^{10}+\frac{26\!\cdots\!63}{11\!\cdots\!31}a^{9}+\frac{18\!\cdots\!23}{11\!\cdots\!31}a^{8}-\frac{90\!\cdots\!55}{11\!\cdots\!31}a^{7}-\frac{84\!\cdots\!61}{11\!\cdots\!31}a^{6}+\frac{11\!\cdots\!50}{11\!\cdots\!31}a^{5}+\frac{17\!\cdots\!33}{11\!\cdots\!31}a^{4}+\frac{14\!\cdots\!86}{11\!\cdots\!31}a^{3}-\frac{71\!\cdots\!08}{11\!\cdots\!31}a^{2}-\frac{38\!\cdots\!27}{11\!\cdots\!31}a-\frac{62\!\cdots\!13}{11\!\cdots\!31}$, $\frac{23\!\cdots\!19}{11\!\cdots\!31}a^{17}-\frac{13\!\cdots\!83}{11\!\cdots\!31}a^{16}-\frac{16\!\cdots\!16}{11\!\cdots\!31}a^{15}-\frac{17\!\cdots\!13}{11\!\cdots\!31}a^{14}+\frac{45\!\cdots\!68}{11\!\cdots\!31}a^{13}+\frac{20\!\cdots\!48}{11\!\cdots\!31}a^{12}-\frac{59\!\cdots\!26}{11\!\cdots\!31}a^{11}-\frac{38\!\cdots\!10}{11\!\cdots\!31}a^{10}+\frac{40\!\cdots\!19}{11\!\cdots\!31}a^{9}+\frac{33\!\cdots\!33}{11\!\cdots\!31}a^{8}-\frac{13\!\cdots\!94}{11\!\cdots\!31}a^{7}-\frac{14\!\cdots\!44}{11\!\cdots\!31}a^{6}+\frac{17\!\cdots\!99}{11\!\cdots\!31}a^{5}+\frac{28\!\cdots\!46}{11\!\cdots\!31}a^{4}+\frac{33\!\cdots\!17}{11\!\cdots\!31}a^{3}-\frac{11\!\cdots\!38}{11\!\cdots\!31}a^{2}-\frac{66\!\cdots\!70}{11\!\cdots\!31}a-\frac{10\!\cdots\!00}{11\!\cdots\!31}$, $\frac{20\!\cdots\!37}{11\!\cdots\!31}a^{17}-\frac{10\!\cdots\!08}{11\!\cdots\!31}a^{16}-\frac{14\!\cdots\!13}{11\!\cdots\!31}a^{15}-\frac{27\!\cdots\!03}{11\!\cdots\!31}a^{14}+\frac{40\!\cdots\!92}{11\!\cdots\!31}a^{13}+\frac{21\!\cdots\!46}{11\!\cdots\!31}a^{12}-\frac{52\!\cdots\!38}{11\!\cdots\!31}a^{11}-\frac{38\!\cdots\!82}{11\!\cdots\!31}a^{10}+\frac{36\!\cdots\!32}{11\!\cdots\!31}a^{9}+\frac{32\!\cdots\!38}{11\!\cdots\!31}a^{8}-\frac{12\!\cdots\!61}{11\!\cdots\!31}a^{7}-\frac{13\!\cdots\!92}{11\!\cdots\!31}a^{6}+\frac{15\!\cdots\!12}{11\!\cdots\!31}a^{5}+\frac{26\!\cdots\!14}{11\!\cdots\!31}a^{4}+\frac{40\!\cdots\!92}{11\!\cdots\!31}a^{3}-\frac{10\!\cdots\!38}{11\!\cdots\!31}a^{2}-\frac{64\!\cdots\!85}{11\!\cdots\!31}a-\frac{10\!\cdots\!29}{11\!\cdots\!31}$, $\frac{90\!\cdots\!27}{11\!\cdots\!31}a^{17}-\frac{71\!\cdots\!34}{11\!\cdots\!31}a^{16}-\frac{64\!\cdots\!18}{11\!\cdots\!31}a^{15}+\frac{79\!\cdots\!82}{11\!\cdots\!31}a^{14}+\frac{17\!\cdots\!24}{11\!\cdots\!31}a^{13}+\frac{38\!\cdots\!38}{11\!\cdots\!31}a^{12}-\frac{22\!\cdots\!05}{11\!\cdots\!31}a^{11}-\frac{96\!\cdots\!05}{11\!\cdots\!31}a^{10}+\frac{15\!\cdots\!27}{11\!\cdots\!31}a^{9}+\frac{92\!\cdots\!69}{11\!\cdots\!31}a^{8}-\frac{53\!\cdots\!45}{11\!\cdots\!31}a^{7}-\frac{43\!\cdots\!50}{11\!\cdots\!31}a^{6}+\frac{72\!\cdots\!94}{11\!\cdots\!31}a^{5}+\frac{91\!\cdots\!89}{11\!\cdots\!31}a^{4}+\frac{95\!\cdots\!70}{11\!\cdots\!31}a^{3}-\frac{39\!\cdots\!96}{11\!\cdots\!31}a^{2}-\frac{19\!\cdots\!46}{11\!\cdots\!31}a-\frac{28\!\cdots\!52}{11\!\cdots\!31}$, $\frac{19\!\cdots\!18}{11\!\cdots\!31}a^{17}-\frac{12\!\cdots\!17}{11\!\cdots\!31}a^{16}-\frac{14\!\cdots\!56}{11\!\cdots\!31}a^{15}-\frac{26\!\cdots\!65}{11\!\cdots\!31}a^{14}+\frac{37\!\cdots\!31}{11\!\cdots\!31}a^{13}+\frac{13\!\cdots\!69}{11\!\cdots\!31}a^{12}-\frac{49\!\cdots\!23}{11\!\cdots\!31}a^{11}-\frac{28\!\cdots\!04}{11\!\cdots\!31}a^{10}+\frac{34\!\cdots\!32}{11\!\cdots\!31}a^{9}+\frac{25\!\cdots\!55}{11\!\cdots\!31}a^{8}-\frac{11\!\cdots\!25}{11\!\cdots\!31}a^{7}-\frac{11\!\cdots\!01}{11\!\cdots\!31}a^{6}+\frac{15\!\cdots\!37}{11\!\cdots\!31}a^{5}+\frac{22\!\cdots\!85}{11\!\cdots\!31}a^{4}+\frac{17\!\cdots\!83}{11\!\cdots\!31}a^{3}-\frac{94\!\cdots\!21}{11\!\cdots\!31}a^{2}-\frac{50\!\cdots\!11}{11\!\cdots\!31}a-\frac{80\!\cdots\!17}{11\!\cdots\!31}$
| 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: | \( 25129615009.6 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 25129615009.6 \cdot 1}{2\cdot\sqrt{126149432166859917805140950806401}}\cr\approx \mathstrut & 0.293260189904 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 72*x^16 - 48*x^15 + 1917*x^14 + 1944*x^13 - 24780*x^12 - 30708*x^11 + 164547*x^10 + 240556*x^9 - 512883*x^8 - 957456*x^7 + 410361*x^6 + 1650123*x^5 + 829629*x^4 - 421407*x^3 - 566379*x^2 - 206577*x - 26241)
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 - 72*x^16 - 48*x^15 + 1917*x^14 + 1944*x^13 - 24780*x^12 - 30708*x^11 + 164547*x^10 + 240556*x^9 - 512883*x^8 - 957456*x^7 + 410361*x^6 + 1650123*x^5 + 829629*x^4 - 421407*x^3 - 566379*x^2 - 206577*x - 26241, 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 - 72*x^16 - 48*x^15 + 1917*x^14 + 1944*x^13 - 24780*x^12 - 30708*x^11 + 164547*x^10 + 240556*x^9 - 512883*x^8 - 957456*x^7 + 410361*x^6 + 1650123*x^5 + 829629*x^4 - 421407*x^3 - 566379*x^2 - 206577*x - 26241);
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 - 72*x^16 - 48*x^15 + 1917*x^14 + 1944*x^13 - 24780*x^12 - 30708*x^11 + 164547*x^10 + 240556*x^9 - 512883*x^8 - 957456*x^7 + 410361*x^6 + 1650123*x^5 + 829629*x^4 - 421407*x^3 - 566379*x^2 - 206577*x - 26241);
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
$C_2^2:C_9$ (as 18T7):
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 36 |
| The 12 conjugacy class representatives for $C_2^2 : C_9$ |
| Character table for $C_2^2 : C_9$ |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 6.6.33074001.1, \(\Q(\zeta_{27})^+\) |
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]
Sibling fields
| Galois closure: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | ${\href{/padicField/7.9.0.1}{9} }^{2}$ | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.9.22.8 | $x^{9} + 24 x^{6} + 18 x^{5} + 9 x^{3} + 3$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 3.9.22.8 | $x^{9} + 24 x^{6} + 18 x^{5} + 9 x^{3} + 3$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
|
\(71\)
| 71.6.3.2 | $x^{6} + 221 x^{4} + 128 x^{3} + 15139 x^{2} - 26752 x + 322815$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 71.6.0.1 | $x^{6} + x^{4} + 10 x^{3} + 13 x^{2} + 29 x + 7$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 71.6.3.1 | $x^{6} - 3550 x^{5} + 161634624 x^{4} + 10165888006904 x^{3} + 6625668596 x^{2} - 569794312 x - 22906304$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |