Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 18*x^25 - 9*x^24 + 135*x^23 + 63*x^22 - 642*x^21 - 135*x^20 + 2133*x^19 + 336*x^18 - 6156*x^17 - 270*x^16 + 15525*x^15 - 4347*x^14 - 26667*x^13 + 16194*x^12 + 26730*x^11 - 24912*x^10 - 13179*x^9 + 19197*x^8 + 693*x^7 - 6957*x^6 + 1917*x^5 + 648*x^4 - 507*x^3 + 135*x^2 - 18*x + 1)
gp: K = bnfinit(y^27 - 18*y^25 - 9*y^24 + 135*y^23 + 63*y^22 - 642*y^21 - 135*y^20 + 2133*y^19 + 336*y^18 - 6156*y^17 - 270*y^16 + 15525*y^15 - 4347*y^14 - 26667*y^13 + 16194*y^12 + 26730*y^11 - 24912*y^10 - 13179*y^9 + 19197*y^8 + 693*y^7 - 6957*y^6 + 1917*y^5 + 648*y^4 - 507*y^3 + 135*y^2 - 18*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 18*x^25 - 9*x^24 + 135*x^23 + 63*x^22 - 642*x^21 - 135*x^20 + 2133*x^19 + 336*x^18 - 6156*x^17 - 270*x^16 + 15525*x^15 - 4347*x^14 - 26667*x^13 + 16194*x^12 + 26730*x^11 - 24912*x^10 - 13179*x^9 + 19197*x^8 + 693*x^7 - 6957*x^6 + 1917*x^5 + 648*x^4 - 507*x^3 + 135*x^2 - 18*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 18*x^25 - 9*x^24 + 135*x^23 + 63*x^22 - 642*x^21 - 135*x^20 + 2133*x^19 + 336*x^18 - 6156*x^17 - 270*x^16 + 15525*x^15 - 4347*x^14 - 26667*x^13 + 16194*x^12 + 26730*x^11 - 24912*x^10 - 13179*x^9 + 19197*x^8 + 693*x^7 - 6957*x^6 + 1917*x^5 + 648*x^4 - 507*x^3 + 135*x^2 - 18*x + 1)
\( x^{27} - 18 x^{25} - 9 x^{24} + 135 x^{23} + 63 x^{22} - 642 x^{21} - 135 x^{20} + 2133 x^{19} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $27$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[9, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-218729065566982775445089767573496266752\)
\(\medspace = -\,2^{18}\cdot 3^{69}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(26.30\) | 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: | $2^{2/3}3^{139/54}\approx 26.84306578592094$ | ||
| Ramified primes: |
\(2\), \(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $9$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{53}a^{25}-\frac{19}{53}a^{24}+\frac{10}{53}a^{23}-\frac{20}{53}a^{22}-\frac{6}{53}a^{21}-\frac{22}{53}a^{19}+\frac{18}{53}a^{18}+\frac{1}{53}a^{17}-\frac{6}{53}a^{16}-\frac{15}{53}a^{15}-\frac{1}{53}a^{14}-\frac{25}{53}a^{13}+\frac{12}{53}a^{12}-\frac{20}{53}a^{11}+\frac{17}{53}a^{10}-\frac{5}{53}a^{9}-\frac{3}{53}a^{8}-\frac{9}{53}a^{7}+\frac{15}{53}a^{6}+\frac{13}{53}a^{5}-\frac{9}{53}a^{4}-\frac{15}{53}a^{3}+\frac{8}{53}a^{2}-\frac{10}{53}a-\frac{7}{53}$, $\frac{1}{28\!\cdots\!89}a^{26}+\frac{51\!\cdots\!83}{28\!\cdots\!89}a^{25}+\frac{10\!\cdots\!86}{28\!\cdots\!89}a^{24}+\frac{10\!\cdots\!08}{28\!\cdots\!89}a^{23}+\frac{13\!\cdots\!01}{28\!\cdots\!89}a^{22}+\frac{87\!\cdots\!28}{28\!\cdots\!89}a^{21}-\frac{20\!\cdots\!35}{28\!\cdots\!89}a^{20}-\frac{23\!\cdots\!71}{28\!\cdots\!89}a^{19}-\frac{38\!\cdots\!47}{28\!\cdots\!89}a^{18}-\frac{20\!\cdots\!67}{28\!\cdots\!89}a^{17}-\frac{58\!\cdots\!14}{28\!\cdots\!89}a^{16}-\frac{90\!\cdots\!07}{28\!\cdots\!89}a^{15}+\frac{84\!\cdots\!71}{28\!\cdots\!89}a^{14}+\frac{11\!\cdots\!51}{28\!\cdots\!89}a^{13}+\frac{13\!\cdots\!37}{28\!\cdots\!89}a^{12}+\frac{10\!\cdots\!25}{28\!\cdots\!89}a^{11}+\frac{97\!\cdots\!84}{28\!\cdots\!89}a^{10}+\frac{77\!\cdots\!51}{28\!\cdots\!89}a^{9}-\frac{68\!\cdots\!59}{28\!\cdots\!89}a^{8}+\frac{10\!\cdots\!67}{28\!\cdots\!89}a^{7}+\frac{10\!\cdots\!64}{28\!\cdots\!89}a^{6}+\frac{11\!\cdots\!66}{28\!\cdots\!89}a^{5}+\frac{11\!\cdots\!19}{28\!\cdots\!89}a^{4}+\frac{82\!\cdots\!66}{28\!\cdots\!89}a^{3}-\frac{11\!\cdots\!80}{28\!\cdots\!89}a^{2}+\frac{44\!\cdots\!18}{28\!\cdots\!89}a-\frac{29\!\cdots\!43}{28\!\cdots\!89}$
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: |
$\frac{29\!\cdots\!43}{28\!\cdots\!89}a^{26}+\frac{63\!\cdots\!35}{28\!\cdots\!89}a^{25}-\frac{52\!\cdots\!70}{28\!\cdots\!89}a^{24}-\frac{37\!\cdots\!25}{28\!\cdots\!89}a^{23}+\frac{38\!\cdots\!87}{28\!\cdots\!89}a^{22}+\frac{26\!\cdots\!56}{28\!\cdots\!89}a^{21}-\frac{18\!\cdots\!56}{28\!\cdots\!89}a^{20}-\frac{79\!\cdots\!72}{28\!\cdots\!89}a^{19}+\frac{61\!\cdots\!86}{28\!\cdots\!89}a^{18}+\frac{23\!\cdots\!39}{28\!\cdots\!89}a^{17}-\frac{17\!\cdots\!03}{28\!\cdots\!89}a^{16}-\frac{46\!\cdots\!85}{28\!\cdots\!89}a^{15}+\frac{44\!\cdots\!24}{28\!\cdots\!89}a^{14}-\frac{31\!\cdots\!31}{28\!\cdots\!89}a^{13}-\frac{79\!\cdots\!56}{28\!\cdots\!89}a^{12}+\frac{30\!\cdots\!58}{28\!\cdots\!89}a^{11}+\frac{86\!\cdots\!84}{28\!\cdots\!89}a^{10}-\frac{55\!\cdots\!04}{28\!\cdots\!89}a^{9}-\frac{51\!\cdots\!46}{28\!\cdots\!89}a^{8}+\frac{46\!\cdots\!15}{28\!\cdots\!89}a^{7}+\frac{11\!\cdots\!39}{28\!\cdots\!89}a^{6}-\frac{18\!\cdots\!07}{28\!\cdots\!89}a^{5}+\frac{18\!\cdots\!29}{28\!\cdots\!89}a^{4}+\frac{23\!\cdots\!89}{28\!\cdots\!89}a^{3}-\frac{10\!\cdots\!05}{28\!\cdots\!89}a^{2}+\frac{18\!\cdots\!53}{28\!\cdots\!89}a-\frac{14\!\cdots\!34}{28\!\cdots\!89}$, $\frac{36\!\cdots\!53}{28\!\cdots\!89}a^{26}-\frac{38\!\cdots\!09}{28\!\cdots\!89}a^{25}-\frac{65\!\cdots\!28}{28\!\cdots\!89}a^{24}-\frac{26\!\cdots\!44}{28\!\cdots\!89}a^{23}+\frac{48\!\cdots\!27}{28\!\cdots\!89}a^{22}+\frac{18\!\cdots\!43}{28\!\cdots\!89}a^{21}-\frac{23\!\cdots\!16}{28\!\cdots\!89}a^{20}-\frac{27\!\cdots\!33}{28\!\cdots\!89}a^{19}+\frac{76\!\cdots\!13}{28\!\cdots\!89}a^{18}+\frac{59\!\cdots\!49}{28\!\cdots\!89}a^{17}-\frac{21\!\cdots\!28}{28\!\cdots\!89}a^{16}+\frac{75\!\cdots\!40}{28\!\cdots\!89}a^{15}+\frac{54\!\cdots\!60}{28\!\cdots\!89}a^{14}-\frac{19\!\cdots\!07}{28\!\cdots\!89}a^{13}-\frac{91\!\cdots\!48}{28\!\cdots\!89}a^{12}+\frac{62\!\cdots\!79}{28\!\cdots\!89}a^{11}+\frac{86\!\cdots\!73}{28\!\cdots\!89}a^{10}-\frac{88\!\cdots\!07}{28\!\cdots\!89}a^{9}-\frac{39\!\cdots\!25}{28\!\cdots\!89}a^{8}+\frac{62\!\cdots\!21}{28\!\cdots\!89}a^{7}+\frac{10\!\cdots\!21}{28\!\cdots\!89}a^{6}-\frac{20\!\cdots\!27}{28\!\cdots\!89}a^{5}+\frac{51\!\cdots\!54}{28\!\cdots\!89}a^{4}+\frac{18\!\cdots\!37}{28\!\cdots\!89}a^{3}-\frac{11\!\cdots\!73}{28\!\cdots\!89}a^{2}+\frac{25\!\cdots\!07}{28\!\cdots\!89}a-\frac{24\!\cdots\!68}{28\!\cdots\!89}$, $\frac{36\!\cdots\!16}{28\!\cdots\!89}a^{26}-\frac{11\!\cdots\!96}{28\!\cdots\!89}a^{25}-\frac{49\!\cdots\!75}{28\!\cdots\!89}a^{24}+\frac{15\!\cdots\!14}{28\!\cdots\!89}a^{23}+\frac{31\!\cdots\!43}{28\!\cdots\!89}a^{22}-\frac{12\!\cdots\!11}{28\!\cdots\!89}a^{21}-\frac{99\!\cdots\!85}{28\!\cdots\!89}a^{20}+\frac{64\!\cdots\!13}{28\!\cdots\!89}a^{19}-\frac{20\!\cdots\!36}{28\!\cdots\!89}a^{18}-\frac{19\!\cdots\!61}{28\!\cdots\!89}a^{17}+\frac{31\!\cdots\!46}{28\!\cdots\!89}a^{16}+\frac{56\!\cdots\!95}{28\!\cdots\!89}a^{15}-\frac{23\!\cdots\!67}{28\!\cdots\!89}a^{14}-\frac{14\!\cdots\!56}{28\!\cdots\!89}a^{13}+\frac{15\!\cdots\!32}{28\!\cdots\!89}a^{12}+\frac{17\!\cdots\!33}{28\!\cdots\!89}a^{11}-\frac{35\!\cdots\!04}{28\!\cdots\!89}a^{10}-\frac{93\!\cdots\!84}{28\!\cdots\!89}a^{9}+\frac{36\!\cdots\!47}{28\!\cdots\!89}a^{8}-\frac{18\!\cdots\!53}{28\!\cdots\!89}a^{7}-\frac{14\!\cdots\!93}{28\!\cdots\!89}a^{6}+\frac{15\!\cdots\!80}{28\!\cdots\!89}a^{5}-\frac{93\!\cdots\!54}{28\!\cdots\!89}a^{4}-\frac{33\!\cdots\!03}{28\!\cdots\!89}a^{3}+\frac{14\!\cdots\!37}{28\!\cdots\!89}a^{2}-\frac{28\!\cdots\!74}{28\!\cdots\!89}a+\frac{24\!\cdots\!70}{28\!\cdots\!89}$, $\frac{11\!\cdots\!24}{28\!\cdots\!89}a^{26}+\frac{16\!\cdots\!40}{28\!\cdots\!89}a^{25}-\frac{27\!\cdots\!91}{28\!\cdots\!89}a^{24}-\frac{30\!\cdots\!61}{28\!\cdots\!89}a^{23}+\frac{11\!\cdots\!16}{28\!\cdots\!89}a^{22}+\frac{22\!\cdots\!59}{28\!\cdots\!89}a^{21}-\frac{50\!\cdots\!70}{28\!\cdots\!89}a^{20}-\frac{10\!\cdots\!56}{28\!\cdots\!89}a^{19}+\frac{40\!\cdots\!06}{28\!\cdots\!89}a^{18}+\frac{32\!\cdots\!55}{28\!\cdots\!89}a^{17}-\frac{12\!\cdots\!80}{28\!\cdots\!89}a^{16}-\frac{92\!\cdots\!82}{28\!\cdots\!89}a^{15}+\frac{85\!\cdots\!34}{53\!\cdots\!13}a^{14}+\frac{21\!\cdots\!63}{28\!\cdots\!89}a^{13}-\frac{17\!\cdots\!62}{28\!\cdots\!89}a^{12}-\frac{30\!\cdots\!55}{28\!\cdots\!89}a^{11}+\frac{37\!\cdots\!35}{28\!\cdots\!89}a^{10}+\frac{18\!\cdots\!32}{28\!\cdots\!89}a^{9}-\frac{39\!\cdots\!19}{28\!\cdots\!89}a^{8}+\frac{11\!\cdots\!50}{28\!\cdots\!89}a^{7}+\frac{19\!\cdots\!19}{28\!\cdots\!89}a^{6}-\frac{67\!\cdots\!86}{28\!\cdots\!89}a^{5}-\frac{24\!\cdots\!50}{28\!\cdots\!89}a^{4}+\frac{19\!\cdots\!85}{28\!\cdots\!89}a^{3}-\frac{66\!\cdots\!58}{28\!\cdots\!89}a^{2}+\frac{13\!\cdots\!69}{28\!\cdots\!89}a-\frac{11\!\cdots\!21}{28\!\cdots\!89}$, $\frac{11\!\cdots\!24}{28\!\cdots\!89}a^{26}+\frac{16\!\cdots\!40}{28\!\cdots\!89}a^{25}-\frac{27\!\cdots\!91}{28\!\cdots\!89}a^{24}-\frac{30\!\cdots\!61}{28\!\cdots\!89}a^{23}+\frac{11\!\cdots\!16}{28\!\cdots\!89}a^{22}+\frac{22\!\cdots\!59}{28\!\cdots\!89}a^{21}-\frac{50\!\cdots\!70}{28\!\cdots\!89}a^{20}-\frac{10\!\cdots\!56}{28\!\cdots\!89}a^{19}+\frac{40\!\cdots\!06}{28\!\cdots\!89}a^{18}+\frac{32\!\cdots\!55}{28\!\cdots\!89}a^{17}-\frac{12\!\cdots\!80}{28\!\cdots\!89}a^{16}-\frac{92\!\cdots\!82}{28\!\cdots\!89}a^{15}+\frac{85\!\cdots\!34}{53\!\cdots\!13}a^{14}+\frac{21\!\cdots\!63}{28\!\cdots\!89}a^{13}-\frac{17\!\cdots\!62}{28\!\cdots\!89}a^{12}-\frac{30\!\cdots\!55}{28\!\cdots\!89}a^{11}+\frac{37\!\cdots\!35}{28\!\cdots\!89}a^{10}+\frac{18\!\cdots\!32}{28\!\cdots\!89}a^{9}-\frac{39\!\cdots\!19}{28\!\cdots\!89}a^{8}+\frac{11\!\cdots\!50}{28\!\cdots\!89}a^{7}+\frac{19\!\cdots\!19}{28\!\cdots\!89}a^{6}-\frac{67\!\cdots\!86}{28\!\cdots\!89}a^{5}-\frac{24\!\cdots\!50}{28\!\cdots\!89}a^{4}+\frac{19\!\cdots\!85}{28\!\cdots\!89}a^{3}-\frac{66\!\cdots\!58}{28\!\cdots\!89}a^{2}+\frac{13\!\cdots\!69}{28\!\cdots\!89}a-\frac{91\!\cdots\!32}{28\!\cdots\!89}$, $\frac{20\!\cdots\!35}{28\!\cdots\!89}a^{26}+\frac{59\!\cdots\!74}{28\!\cdots\!89}a^{25}-\frac{37\!\cdots\!27}{28\!\cdots\!89}a^{24}-\frac{29\!\cdots\!14}{28\!\cdots\!89}a^{23}+\frac{27\!\cdots\!99}{28\!\cdots\!89}a^{22}+\frac{20\!\cdots\!71}{28\!\cdots\!89}a^{21}-\frac{12\!\cdots\!88}{28\!\cdots\!89}a^{20}-\frac{65\!\cdots\!04}{28\!\cdots\!89}a^{19}+\frac{43\!\cdots\!22}{28\!\cdots\!89}a^{18}+\frac{19\!\cdots\!76}{28\!\cdots\!89}a^{17}-\frac{12\!\cdots\!23}{28\!\cdots\!89}a^{16}-\frac{40\!\cdots\!37}{28\!\cdots\!89}a^{15}+\frac{32\!\cdots\!72}{28\!\cdots\!89}a^{14}-\frac{33\!\cdots\!36}{28\!\cdots\!89}a^{13}-\frac{58\!\cdots\!98}{28\!\cdots\!89}a^{12}+\frac{19\!\cdots\!49}{28\!\cdots\!89}a^{11}+\frac{65\!\cdots\!49}{28\!\cdots\!89}a^{10}-\frac{38\!\cdots\!90}{28\!\cdots\!89}a^{9}-\frac{40\!\cdots\!51}{28\!\cdots\!89}a^{8}+\frac{34\!\cdots\!93}{28\!\cdots\!89}a^{7}+\frac{10\!\cdots\!44}{28\!\cdots\!89}a^{6}-\frac{14\!\cdots\!35}{28\!\cdots\!89}a^{5}+\frac{11\!\cdots\!33}{28\!\cdots\!89}a^{4}+\frac{20\!\cdots\!06}{28\!\cdots\!89}a^{3}-\frac{84\!\cdots\!13}{28\!\cdots\!89}a^{2}+\frac{15\!\cdots\!55}{28\!\cdots\!89}a-\frac{11\!\cdots\!92}{28\!\cdots\!89}$, $\frac{72\!\cdots\!14}{28\!\cdots\!89}a^{26}+\frac{16\!\cdots\!23}{28\!\cdots\!89}a^{25}-\frac{12\!\cdots\!66}{28\!\cdots\!89}a^{24}-\frac{94\!\cdots\!59}{28\!\cdots\!89}a^{23}+\frac{96\!\cdots\!67}{28\!\cdots\!89}a^{22}+\frac{67\!\cdots\!94}{28\!\cdots\!89}a^{21}-\frac{45\!\cdots\!30}{28\!\cdots\!89}a^{20}-\frac{19\!\cdots\!60}{28\!\cdots\!89}a^{19}+\frac{15\!\cdots\!52}{28\!\cdots\!89}a^{18}+\frac{57\!\cdots\!51}{28\!\cdots\!89}a^{17}-\frac{44\!\cdots\!41}{28\!\cdots\!89}a^{16}-\frac{11\!\cdots\!08}{28\!\cdots\!89}a^{15}+\frac{11\!\cdots\!40}{28\!\cdots\!89}a^{14}-\frac{76\!\cdots\!78}{28\!\cdots\!89}a^{13}-\frac{20\!\cdots\!49}{28\!\cdots\!89}a^{12}+\frac{78\!\cdots\!01}{28\!\cdots\!89}a^{11}+\frac{21\!\cdots\!44}{28\!\cdots\!89}a^{10}-\frac{14\!\cdots\!99}{28\!\cdots\!89}a^{9}-\frac{13\!\cdots\!57}{28\!\cdots\!89}a^{8}+\frac{12\!\cdots\!40}{28\!\cdots\!89}a^{7}+\frac{29\!\cdots\!16}{28\!\cdots\!89}a^{6}-\frac{50\!\cdots\!51}{28\!\cdots\!89}a^{5}+\frac{67\!\cdots\!12}{28\!\cdots\!89}a^{4}+\frac{66\!\cdots\!54}{28\!\cdots\!89}a^{3}-\frac{32\!\cdots\!09}{28\!\cdots\!89}a^{2}+\frac{64\!\cdots\!90}{28\!\cdots\!89}a-\frac{51\!\cdots\!56}{28\!\cdots\!89}$, $\frac{17\!\cdots\!66}{28\!\cdots\!89}a^{26}+\frac{29\!\cdots\!34}{28\!\cdots\!89}a^{25}-\frac{30\!\cdots\!38}{28\!\cdots\!89}a^{24}-\frac{20\!\cdots\!47}{28\!\cdots\!89}a^{23}+\frac{22\!\cdots\!82}{28\!\cdots\!89}a^{22}+\frac{14\!\cdots\!81}{28\!\cdots\!89}a^{21}-\frac{10\!\cdots\!59}{28\!\cdots\!89}a^{20}-\frac{41\!\cdots\!84}{28\!\cdots\!89}a^{19}+\frac{35\!\cdots\!29}{28\!\cdots\!89}a^{18}+\frac{12\!\cdots\!05}{28\!\cdots\!89}a^{17}-\frac{10\!\cdots\!92}{28\!\cdots\!89}a^{16}-\frac{22\!\cdots\!03}{28\!\cdots\!89}a^{15}+\frac{26\!\cdots\!23}{28\!\cdots\!89}a^{14}-\frac{28\!\cdots\!48}{28\!\cdots\!89}a^{13}-\frac{45\!\cdots\!38}{28\!\cdots\!89}a^{12}+\frac{19\!\cdots\!22}{28\!\cdots\!89}a^{11}+\frac{48\!\cdots\!20}{28\!\cdots\!89}a^{10}-\frac{33\!\cdots\!52}{28\!\cdots\!89}a^{9}-\frac{28\!\cdots\!98}{28\!\cdots\!89}a^{8}+\frac{27\!\cdots\!20}{28\!\cdots\!89}a^{7}+\frac{61\!\cdots\!62}{28\!\cdots\!89}a^{6}-\frac{10\!\cdots\!04}{28\!\cdots\!89}a^{5}+\frac{12\!\cdots\!23}{28\!\cdots\!89}a^{4}+\frac{13\!\cdots\!49}{28\!\cdots\!89}a^{3}-\frac{58\!\cdots\!51}{28\!\cdots\!89}a^{2}+\frac{10\!\cdots\!30}{28\!\cdots\!89}a-\frac{79\!\cdots\!81}{28\!\cdots\!89}$, $\frac{23\!\cdots\!49}{28\!\cdots\!89}a^{26}+\frac{14\!\cdots\!34}{28\!\cdots\!89}a^{25}-\frac{42\!\cdots\!68}{28\!\cdots\!89}a^{24}-\frac{46\!\cdots\!45}{28\!\cdots\!89}a^{23}+\frac{30\!\cdots\!91}{28\!\cdots\!89}a^{22}+\frac{33\!\cdots\!57}{28\!\cdots\!89}a^{21}-\frac{14\!\cdots\!32}{28\!\cdots\!89}a^{20}-\frac{11\!\cdots\!96}{28\!\cdots\!89}a^{19}+\frac{49\!\cdots\!82}{28\!\cdots\!89}a^{18}+\frac{35\!\cdots\!14}{28\!\cdots\!89}a^{17}-\frac{14\!\cdots\!52}{28\!\cdots\!89}a^{16}-\frac{86\!\cdots\!10}{28\!\cdots\!89}a^{15}+\frac{36\!\cdots\!40}{28\!\cdots\!89}a^{14}+\frac{94\!\cdots\!31}{28\!\cdots\!89}a^{13}-\frac{69\!\cdots\!25}{28\!\cdots\!89}a^{12}+\frac{61\!\cdots\!14}{28\!\cdots\!89}a^{11}+\frac{84\!\cdots\!68}{28\!\cdots\!89}a^{10}-\frac{29\!\cdots\!82}{28\!\cdots\!89}a^{9}-\frac{59\!\cdots\!31}{28\!\cdots\!89}a^{8}+\frac{32\!\cdots\!40}{28\!\cdots\!89}a^{7}+\frac{20\!\cdots\!53}{28\!\cdots\!89}a^{6}-\frac{15\!\cdots\!49}{28\!\cdots\!89}a^{5}-\frac{88\!\cdots\!07}{28\!\cdots\!89}a^{4}+\frac{26\!\cdots\!71}{28\!\cdots\!89}a^{3}-\frac{85\!\cdots\!37}{28\!\cdots\!89}a^{2}+\frac{12\!\cdots\!14}{28\!\cdots\!89}a-\frac{64\!\cdots\!08}{28\!\cdots\!89}$, $\frac{15\!\cdots\!00}{28\!\cdots\!89}a^{26}+\frac{22\!\cdots\!31}{28\!\cdots\!89}a^{25}-\frac{27\!\cdots\!37}{28\!\cdots\!89}a^{24}-\frac{17\!\cdots\!65}{28\!\cdots\!89}a^{23}+\frac{20\!\cdots\!62}{28\!\cdots\!89}a^{22}+\frac{12\!\cdots\!72}{28\!\cdots\!89}a^{21}-\frac{97\!\cdots\!37}{28\!\cdots\!89}a^{20}-\frac{35\!\cdots\!23}{28\!\cdots\!89}a^{19}+\frac{32\!\cdots\!29}{28\!\cdots\!89}a^{18}+\frac{98\!\cdots\!77}{28\!\cdots\!89}a^{17}-\frac{94\!\cdots\!13}{28\!\cdots\!89}a^{16}-\frac{17\!\cdots\!94}{28\!\cdots\!89}a^{15}+\frac{23\!\cdots\!03}{28\!\cdots\!89}a^{14}-\frac{32\!\cdots\!87}{28\!\cdots\!89}a^{13}-\frac{42\!\cdots\!65}{28\!\cdots\!89}a^{12}+\frac{19\!\cdots\!90}{28\!\cdots\!89}a^{11}+\frac{44\!\cdots\!28}{28\!\cdots\!89}a^{10}-\frac{32\!\cdots\!10}{28\!\cdots\!89}a^{9}-\frac{25\!\cdots\!68}{28\!\cdots\!89}a^{8}+\frac{26\!\cdots\!65}{28\!\cdots\!89}a^{7}+\frac{48\!\cdots\!73}{28\!\cdots\!89}a^{6}-\frac{10\!\cdots\!11}{28\!\cdots\!89}a^{5}+\frac{15\!\cdots\!86}{28\!\cdots\!89}a^{4}+\frac{12\!\cdots\!34}{28\!\cdots\!89}a^{3}-\frac{64\!\cdots\!41}{28\!\cdots\!89}a^{2}+\frac{12\!\cdots\!87}{28\!\cdots\!89}a-\frac{10\!\cdots\!62}{28\!\cdots\!89}$, $\frac{40\!\cdots\!90}{28\!\cdots\!89}a^{26}+\frac{30\!\cdots\!63}{28\!\cdots\!89}a^{25}-\frac{72\!\cdots\!68}{28\!\cdots\!89}a^{24}-\frac{91\!\cdots\!25}{28\!\cdots\!89}a^{23}+\frac{51\!\cdots\!33}{28\!\cdots\!89}a^{22}+\frac{65\!\cdots\!28}{28\!\cdots\!89}a^{21}-\frac{23\!\cdots\!93}{28\!\cdots\!89}a^{20}-\frac{24\!\cdots\!42}{28\!\cdots\!89}a^{19}+\frac{80\!\cdots\!11}{28\!\cdots\!89}a^{18}+\frac{75\!\cdots\!08}{28\!\cdots\!89}a^{17}-\frac{23\!\cdots\!02}{28\!\cdots\!89}a^{16}-\frac{18\!\cdots\!01}{28\!\cdots\!89}a^{15}+\frac{60\!\cdots\!52}{28\!\cdots\!89}a^{14}+\frac{26\!\cdots\!37}{28\!\cdots\!89}a^{13}-\frac{11\!\cdots\!65}{28\!\cdots\!89}a^{12}-\frac{10\!\cdots\!15}{28\!\cdots\!89}a^{11}+\frac{14\!\cdots\!67}{28\!\cdots\!89}a^{10}-\frac{25\!\cdots\!90}{28\!\cdots\!89}a^{9}-\frac{11\!\cdots\!12}{28\!\cdots\!89}a^{8}+\frac{37\!\cdots\!25}{28\!\cdots\!89}a^{7}+\frac{44\!\cdots\!52}{28\!\cdots\!89}a^{6}-\frac{20\!\cdots\!29}{28\!\cdots\!89}a^{5}-\frac{62\!\cdots\!06}{28\!\cdots\!89}a^{4}+\frac{41\!\cdots\!36}{28\!\cdots\!89}a^{3}-\frac{64\!\cdots\!69}{28\!\cdots\!89}a^{2}-\frac{13\!\cdots\!42}{28\!\cdots\!89}a+\frac{13\!\cdots\!77}{28\!\cdots\!89}$, $\frac{11\!\cdots\!87}{28\!\cdots\!89}a^{26}+\frac{74\!\cdots\!89}{28\!\cdots\!89}a^{25}-\frac{20\!\cdots\!35}{28\!\cdots\!89}a^{24}-\frac{11\!\cdots\!02}{28\!\cdots\!89}a^{23}+\frac{15\!\cdots\!60}{28\!\cdots\!89}a^{22}+\frac{82\!\cdots\!81}{28\!\cdots\!89}a^{21}-\frac{70\!\cdots\!19}{28\!\cdots\!89}a^{20}-\frac{20\!\cdots\!78}{28\!\cdots\!89}a^{19}+\frac{23\!\cdots\!78}{28\!\cdots\!89}a^{18}+\frac{57\!\cdots\!39}{28\!\cdots\!89}a^{17}-\frac{66\!\cdots\!33}{28\!\cdots\!89}a^{16}-\frac{86\!\cdots\!00}{28\!\cdots\!89}a^{15}+\frac{16\!\cdots\!14}{28\!\cdots\!89}a^{14}-\frac{34\!\cdots\!98}{28\!\cdots\!89}a^{13}-\frac{28\!\cdots\!55}{28\!\cdots\!89}a^{12}+\frac{14\!\cdots\!31}{28\!\cdots\!89}a^{11}+\frac{28\!\cdots\!30}{28\!\cdots\!89}a^{10}-\frac{22\!\cdots\!27}{28\!\cdots\!89}a^{9}-\frac{14\!\cdots\!03}{28\!\cdots\!89}a^{8}+\frac{16\!\cdots\!93}{28\!\cdots\!89}a^{7}+\frac{24\!\cdots\!08}{28\!\cdots\!89}a^{6}-\frac{56\!\cdots\!17}{28\!\cdots\!89}a^{5}+\frac{83\!\cdots\!18}{28\!\cdots\!89}a^{4}+\frac{56\!\cdots\!28}{28\!\cdots\!89}a^{3}-\frac{27\!\cdots\!47}{28\!\cdots\!89}a^{2}+\frac{51\!\cdots\!63}{28\!\cdots\!89}a-\frac{40\!\cdots\!10}{28\!\cdots\!89}$, $\frac{50\!\cdots\!44}{53\!\cdots\!13}a^{26}+\frac{69\!\cdots\!30}{28\!\cdots\!89}a^{25}-\frac{48\!\cdots\!71}{28\!\cdots\!89}a^{24}-\frac{36\!\cdots\!38}{28\!\cdots\!89}a^{23}+\frac{35\!\cdots\!74}{28\!\cdots\!89}a^{22}+\frac{26\!\cdots\!29}{28\!\cdots\!89}a^{21}-\frac{31\!\cdots\!69}{53\!\cdots\!13}a^{20}-\frac{79\!\cdots\!40}{28\!\cdots\!89}a^{19}+\frac{56\!\cdots\!05}{28\!\cdots\!89}a^{18}+\frac{23\!\cdots\!46}{28\!\cdots\!89}a^{17}-\frac{16\!\cdots\!61}{28\!\cdots\!89}a^{16}-\frac{47\!\cdots\!10}{28\!\cdots\!89}a^{15}+\frac{41\!\cdots\!02}{28\!\cdots\!89}a^{14}-\frac{16\!\cdots\!23}{28\!\cdots\!89}a^{13}-\frac{74\!\cdots\!03}{28\!\cdots\!89}a^{12}+\frac{26\!\cdots\!90}{28\!\cdots\!89}a^{11}+\frac{82\!\cdots\!82}{28\!\cdots\!89}a^{10}-\frac{50\!\cdots\!03}{28\!\cdots\!89}a^{9}-\frac{50\!\cdots\!33}{28\!\cdots\!89}a^{8}+\frac{43\!\cdots\!80}{28\!\cdots\!89}a^{7}+\frac{12\!\cdots\!99}{28\!\cdots\!89}a^{6}-\frac{17\!\cdots\!12}{28\!\cdots\!89}a^{5}+\frac{17\!\cdots\!24}{28\!\cdots\!89}a^{4}+\frac{24\!\cdots\!41}{28\!\cdots\!89}a^{3}-\frac{10\!\cdots\!05}{28\!\cdots\!89}a^{2}+\frac{19\!\cdots\!39}{28\!\cdots\!89}a-\frac{14\!\cdots\!10}{28\!\cdots\!89}$, $\frac{44\!\cdots\!12}{28\!\cdots\!89}a^{26}+\frac{33\!\cdots\!98}{28\!\cdots\!89}a^{25}-\frac{76\!\cdots\!34}{28\!\cdots\!89}a^{24}-\frac{10\!\cdots\!92}{28\!\cdots\!89}a^{23}+\frac{50\!\cdots\!36}{28\!\cdots\!89}a^{22}+\frac{70\!\cdots\!86}{28\!\cdots\!89}a^{21}-\frac{21\!\cdots\!83}{28\!\cdots\!89}a^{20}-\frac{25\!\cdots\!49}{28\!\cdots\!89}a^{19}+\frac{67\!\cdots\!95}{28\!\cdots\!89}a^{18}+\frac{83\!\cdots\!98}{28\!\cdots\!89}a^{17}-\frac{18\!\cdots\!33}{28\!\cdots\!89}a^{16}-\frac{20\!\cdots\!68}{28\!\cdots\!89}a^{15}+\frac{47\!\cdots\!47}{28\!\cdots\!89}a^{14}+\frac{32\!\cdots\!23}{28\!\cdots\!89}a^{13}-\frac{80\!\cdots\!89}{28\!\cdots\!89}a^{12}-\frac{36\!\cdots\!42}{28\!\cdots\!89}a^{11}+\frac{88\!\cdots\!14}{28\!\cdots\!89}a^{10}+\frac{37\!\cdots\!95}{28\!\cdots\!89}a^{9}-\frac{67\!\cdots\!30}{28\!\cdots\!89}a^{8}-\frac{35\!\cdots\!33}{28\!\cdots\!89}a^{7}+\frac{38\!\cdots\!81}{28\!\cdots\!89}a^{6}+\frac{18\!\cdots\!42}{28\!\cdots\!89}a^{5}-\frac{29\!\cdots\!87}{53\!\cdots\!13}a^{4}-\frac{27\!\cdots\!76}{28\!\cdots\!89}a^{3}+\frac{28\!\cdots\!92}{28\!\cdots\!89}a^{2}-\frac{76\!\cdots\!64}{28\!\cdots\!89}a+\frac{78\!\cdots\!77}{28\!\cdots\!89}$, $\frac{11\!\cdots\!44}{28\!\cdots\!89}a^{26}+\frac{43\!\cdots\!19}{28\!\cdots\!89}a^{25}-\frac{20\!\cdots\!20}{28\!\cdots\!89}a^{24}-\frac{18\!\cdots\!26}{28\!\cdots\!89}a^{23}+\frac{15\!\cdots\!96}{28\!\cdots\!89}a^{22}+\frac{13\!\cdots\!10}{28\!\cdots\!89}a^{21}-\frac{70\!\cdots\!35}{28\!\cdots\!89}a^{20}-\frac{42\!\cdots\!70}{28\!\cdots\!89}a^{19}+\frac{23\!\cdots\!01}{28\!\cdots\!89}a^{18}+\frac{12\!\cdots\!47}{28\!\cdots\!89}a^{17}-\frac{67\!\cdots\!81}{28\!\cdots\!89}a^{16}-\frac{28\!\cdots\!19}{28\!\cdots\!89}a^{15}+\frac{17\!\cdots\!58}{28\!\cdots\!89}a^{14}+\frac{13\!\cdots\!65}{28\!\cdots\!89}a^{13}-\frac{31\!\cdots\!07}{28\!\cdots\!89}a^{12}+\frac{74\!\cdots\!23}{28\!\cdots\!89}a^{11}+\frac{34\!\cdots\!28}{28\!\cdots\!89}a^{10}-\frac{16\!\cdots\!40}{28\!\cdots\!89}a^{9}-\frac{22\!\cdots\!00}{28\!\cdots\!89}a^{8}+\frac{14\!\cdots\!31}{28\!\cdots\!89}a^{7}+\frac{65\!\cdots\!73}{28\!\cdots\!89}a^{6}-\frac{59\!\cdots\!85}{28\!\cdots\!89}a^{5}-\frac{48\!\cdots\!27}{28\!\cdots\!89}a^{4}+\frac{82\!\cdots\!75}{28\!\cdots\!89}a^{3}-\frac{29\!\cdots\!35}{28\!\cdots\!89}a^{2}+\frac{42\!\cdots\!52}{28\!\cdots\!89}a-\frac{20\!\cdots\!48}{28\!\cdots\!89}$, $\frac{11\!\cdots\!54}{28\!\cdots\!89}a^{26}+\frac{31\!\cdots\!96}{28\!\cdots\!89}a^{25}-\frac{20\!\cdots\!33}{28\!\cdots\!89}a^{24}-\frac{15\!\cdots\!71}{28\!\cdots\!89}a^{23}+\frac{14\!\cdots\!41}{28\!\cdots\!89}a^{22}+\frac{11\!\cdots\!26}{28\!\cdots\!89}a^{21}-\frac{70\!\cdots\!93}{28\!\cdots\!89}a^{20}-\frac{34\!\cdots\!15}{28\!\cdots\!89}a^{19}+\frac{44\!\cdots\!19}{53\!\cdots\!13}a^{18}+\frac{10\!\cdots\!49}{28\!\cdots\!89}a^{17}-\frac{68\!\cdots\!07}{28\!\cdots\!89}a^{16}-\frac{21\!\cdots\!63}{28\!\cdots\!89}a^{15}+\frac{17\!\cdots\!39}{28\!\cdots\!89}a^{14}-\frac{25\!\cdots\!89}{28\!\cdots\!89}a^{13}-\frac{31\!\cdots\!69}{28\!\cdots\!89}a^{12}+\frac{10\!\cdots\!76}{28\!\cdots\!89}a^{11}+\frac{34\!\cdots\!88}{28\!\cdots\!89}a^{10}-\frac{20\!\cdots\!52}{28\!\cdots\!89}a^{9}-\frac{39\!\cdots\!79}{53\!\cdots\!13}a^{8}+\frac{17\!\cdots\!61}{28\!\cdots\!89}a^{7}+\frac{54\!\cdots\!15}{28\!\cdots\!89}a^{6}-\frac{71\!\cdots\!78}{28\!\cdots\!89}a^{5}+\frac{57\!\cdots\!22}{28\!\cdots\!89}a^{4}+\frac{18\!\cdots\!94}{53\!\cdots\!13}a^{3}-\frac{41\!\cdots\!82}{28\!\cdots\!89}a^{2}+\frac{74\!\cdots\!96}{28\!\cdots\!89}a-\frac{54\!\cdots\!47}{28\!\cdots\!89}$, $\frac{18\!\cdots\!61}{28\!\cdots\!89}a^{26}+\frac{22\!\cdots\!10}{28\!\cdots\!89}a^{25}-\frac{33\!\cdots\!68}{28\!\cdots\!89}a^{24}-\frac{20\!\cdots\!97}{28\!\cdots\!89}a^{23}+\frac{24\!\cdots\!07}{28\!\cdots\!89}a^{22}+\frac{14\!\cdots\!04}{28\!\cdots\!89}a^{21}-\frac{11\!\cdots\!39}{28\!\cdots\!89}a^{20}-\frac{39\!\cdots\!73}{28\!\cdots\!89}a^{19}+\frac{39\!\cdots\!25}{28\!\cdots\!89}a^{18}+\frac{11\!\cdots\!83}{28\!\cdots\!89}a^{17}-\frac{11\!\cdots\!92}{28\!\cdots\!89}a^{16}-\frac{18\!\cdots\!71}{28\!\cdots\!89}a^{15}+\frac{28\!\cdots\!13}{28\!\cdots\!89}a^{14}-\frac{44\!\cdots\!61}{28\!\cdots\!89}a^{13}-\frac{50\!\cdots\!02}{28\!\cdots\!89}a^{12}+\frac{23\!\cdots\!23}{28\!\cdots\!89}a^{11}+\frac{52\!\cdots\!88}{28\!\cdots\!89}a^{10}-\frac{39\!\cdots\!65}{28\!\cdots\!89}a^{9}-\frac{29\!\cdots\!39}{28\!\cdots\!89}a^{8}+\frac{32\!\cdots\!37}{28\!\cdots\!89}a^{7}+\frac{56\!\cdots\!30}{28\!\cdots\!89}a^{6}-\frac{12\!\cdots\!03}{28\!\cdots\!89}a^{5}+\frac{18\!\cdots\!62}{28\!\cdots\!89}a^{4}+\frac{15\!\cdots\!14}{28\!\cdots\!89}a^{3}-\frac{73\!\cdots\!84}{28\!\cdots\!89}a^{2}+\frac{14\!\cdots\!01}{28\!\cdots\!89}a-\frac{11\!\cdots\!64}{28\!\cdots\!89}$
| 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: | \( 932907706.847076 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{9}\cdot 932907706.847076 \cdot 1}{2\cdot\sqrt{218729065566982775445089767573496266752}}\cr\approx \mathstrut & 0.246458733596961 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 - 18*x^25 - 9*x^24 + 135*x^23 + 63*x^22 - 642*x^21 - 135*x^20 + 2133*x^19 + 336*x^18 - 6156*x^17 - 270*x^16 + 15525*x^15 - 4347*x^14 - 26667*x^13 + 16194*x^12 + 26730*x^11 - 24912*x^10 - 13179*x^9 + 19197*x^8 + 693*x^7 - 6957*x^6 + 1917*x^5 + 648*x^4 - 507*x^3 + 135*x^2 - 18*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^27 - 18*x^25 - 9*x^24 + 135*x^23 + 63*x^22 - 642*x^21 - 135*x^20 + 2133*x^19 + 336*x^18 - 6156*x^17 - 270*x^16 + 15525*x^15 - 4347*x^14 - 26667*x^13 + 16194*x^12 + 26730*x^11 - 24912*x^10 - 13179*x^9 + 19197*x^8 + 693*x^7 - 6957*x^6 + 1917*x^5 + 648*x^4 - 507*x^3 + 135*x^2 - 18*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^27 - 18*x^25 - 9*x^24 + 135*x^23 + 63*x^22 - 642*x^21 - 135*x^20 + 2133*x^19 + 336*x^18 - 6156*x^17 - 270*x^16 + 15525*x^15 - 4347*x^14 - 26667*x^13 + 16194*x^12 + 26730*x^11 - 24912*x^10 - 13179*x^9 + 19197*x^8 + 693*x^7 - 6957*x^6 + 1917*x^5 + 648*x^4 - 507*x^3 + 135*x^2 - 18*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^27 - 18*x^25 - 9*x^24 + 135*x^23 + 63*x^22 - 642*x^21 - 135*x^20 + 2133*x^19 + 336*x^18 - 6156*x^17 - 270*x^16 + 15525*x^15 - 4347*x^14 - 26667*x^13 + 16194*x^12 + 26730*x^11 - 24912*x^10 - 13179*x^9 + 19197*x^8 + 693*x^7 - 6957*x^6 + 1917*x^5 + 648*x^4 - 507*x^3 + 135*x^2 - 18*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
$S_3\times C_9$ (as 27T12):
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 54 |
| The 27 conjugacy class representatives for $S_3\times C_9$ |
| Character table for $S_3\times C_9$ |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 3.1.108.1, \(\Q(\zeta_{27})^+\), 9.3.918330048.1 |
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 18 sibling: | data not computed |
| Minimal sibling: | 18.0.12100864846032214829641728.4 |
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 | $18{,}\,{\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }^{3}$ | $18{,}\,{\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{9}$ | $18{,}\,{\href{/padicField/23.9.0.1}{9} }$ | $18{,}\,{\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{9}$ | $18{,}\,{\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }^{3}$ | $18{,}\,{\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.2.0.1}{2} }^{9}{,}\,{\href{/padicField/53.1.0.1}{1} }^{9}$ | $18{,}\,{\href{/padicField/59.9.0.1}{9} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $27$ | $3$ | $9$ | $18$ | |||
|
\(3\)
| Deg $27$ | $27$ | $1$ | $69$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| 1.9.6t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.6t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| 1.27.18t1.a.a | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| * | 1.27.9t1.a.a | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.27.9t1.a.b | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.27.9t1.a.c | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.27.9t1.a.d | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| 1.27.18t1.a.b | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.18t1.a.c | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.18t1.a.d | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.18t1.a.e | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| * | 1.27.9t1.a.e | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.27.9t1.a.f | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| 1.27.18t1.a.f | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| * | 2.108.3t2.b.a | $2$ | $ 2^{2} \cdot 3^{3}$ | 3.1.108.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.324.6t5.c.a | $2$ | $ 2^{2} \cdot 3^{4}$ | 6.0.314928.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.324.6t5.c.b | $2$ | $ 2^{2} \cdot 3^{4}$ | 6.0.314928.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.2916.18t16.b.a | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.218729065566982775445089767573496266752.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.b.b | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.218729065566982775445089767573496266752.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.b.c | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.218729065566982775445089767573496266752.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.b.d | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.218729065566982775445089767573496266752.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.b.e | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.218729065566982775445089767573496266752.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.b.f | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.218729065566982775445089767573496266752.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.