Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 + 27*x^25 - 24*x^24 - 9*x^23 - 63*x^22 + 273*x^21 + 144*x^20 - 2376*x^19 + 4050*x^18 - 396*x^17 - 3465*x^16 - 141*x^15 + 441*x^14 + 17019*x^13 - 33225*x^12 + 12852*x^11 + 17379*x^10 - 14277*x^9 + 1188*x^8 + 1143*x^7 - 501*x^6 + 891*x^5 + 216*x^4 + 129*x^3 + 18*x^2 + 9*x - 1)
gp: K = bnfinit(y^27 - 9*y^26 + 27*y^25 - 24*y^24 - 9*y^23 - 63*y^22 + 273*y^21 + 144*y^20 - 2376*y^19 + 4050*y^18 - 396*y^17 - 3465*y^16 - 141*y^15 + 441*y^14 + 17019*y^13 - 33225*y^12 + 12852*y^11 + 17379*y^10 - 14277*y^9 + 1188*y^8 + 1143*y^7 - 501*y^6 + 891*y^5 + 216*y^4 + 129*y^3 + 18*y^2 + 9*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 9*x^26 + 27*x^25 - 24*x^24 - 9*x^23 - 63*x^22 + 273*x^21 + 144*x^20 - 2376*x^19 + 4050*x^18 - 396*x^17 - 3465*x^16 - 141*x^15 + 441*x^14 + 17019*x^13 - 33225*x^12 + 12852*x^11 + 17379*x^10 - 14277*x^9 + 1188*x^8 + 1143*x^7 - 501*x^6 + 891*x^5 + 216*x^4 + 129*x^3 + 18*x^2 + 9*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 9*x^26 + 27*x^25 - 24*x^24 - 9*x^23 - 63*x^22 + 273*x^21 + 144*x^20 - 2376*x^19 + 4050*x^18 - 396*x^17 - 3465*x^16 - 141*x^15 + 441*x^14 + 17019*x^13 - 33225*x^12 + 12852*x^11 + 17379*x^10 - 14277*x^9 + 1188*x^8 + 1143*x^7 - 501*x^6 + 891*x^5 + 216*x^4 + 129*x^3 + 18*x^2 + 9*x - 1)
\( x^{27} - 9 x^{26} + 27 x^{25} - 24 x^{24} - 9 x^{23} - 63 x^{22} + 273 x^{21} + 144 x^{20} - 2376 x^{19} + \cdots - 1 \)
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: |
\(-656187196700948326335269302720488800256\)
\(\medspace = -\,2^{18}\cdot 3^{70}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.39\) | 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: | $2\cdot 3^{70/27}\approx 34.51525023937336$ | ||
| 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: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $9$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{53}a^{21}-\frac{22}{53}a^{20}+\frac{15}{53}a^{19}-\frac{22}{53}a^{18}+\frac{18}{53}a^{17}+\frac{8}{53}a^{16}-\frac{11}{53}a^{15}+\frac{3}{53}a^{14}+\frac{26}{53}a^{13}-\frac{6}{53}a^{12}-\frac{16}{53}a^{11}+\frac{13}{53}a^{10}-\frac{8}{53}a^{9}+\frac{8}{53}a^{8}+\frac{7}{53}a^{7}-\frac{8}{53}a^{6}-\frac{13}{53}a^{5}+\frac{7}{53}a^{4}+\frac{24}{53}a^{3}+\frac{25}{53}a^{2}-\frac{26}{53}a+\frac{10}{53}$, $\frac{1}{53}a^{22}+\frac{8}{53}a^{20}-\frac{10}{53}a^{19}+\frac{11}{53}a^{18}-\frac{20}{53}a^{17}+\frac{6}{53}a^{16}+\frac{26}{53}a^{15}-\frac{14}{53}a^{14}-\frac{17}{53}a^{13}+\frac{11}{53}a^{12}-\frac{21}{53}a^{11}+\frac{13}{53}a^{10}-\frac{9}{53}a^{9}+\frac{24}{53}a^{8}-\frac{13}{53}a^{7}+\frac{23}{53}a^{6}-\frac{14}{53}a^{5}+\frac{19}{53}a^{4}+\frac{23}{53}a^{3}-\frac{6}{53}a^{2}+\frac{21}{53}a+\frac{8}{53}$, $\frac{1}{53}a^{23}+\frac{7}{53}a^{20}-\frac{3}{53}a^{19}-\frac{3}{53}a^{18}+\frac{21}{53}a^{17}+\frac{15}{53}a^{16}+\frac{21}{53}a^{15}+\frac{12}{53}a^{14}+\frac{15}{53}a^{13}-\frac{26}{53}a^{12}-\frac{18}{53}a^{11}-\frac{7}{53}a^{10}-\frac{18}{53}a^{9}-\frac{24}{53}a^{8}+\frac{20}{53}a^{7}-\frac{3}{53}a^{6}+\frac{17}{53}a^{5}+\frac{20}{53}a^{4}+\frac{14}{53}a^{3}-\frac{20}{53}a^{2}+\frac{4}{53}a+\frac{26}{53}$, $\frac{1}{53}a^{24}-\frac{8}{53}a^{20}-\frac{2}{53}a^{19}+\frac{16}{53}a^{18}-\frac{5}{53}a^{17}+\frac{18}{53}a^{16}-\frac{17}{53}a^{15}-\frac{6}{53}a^{14}+\frac{4}{53}a^{13}+\frac{24}{53}a^{12}-\frac{1}{53}a^{11}-\frac{3}{53}a^{10}-\frac{21}{53}a^{9}+\frac{17}{53}a^{8}+\frac{1}{53}a^{7}+\frac{20}{53}a^{6}+\frac{5}{53}a^{5}+\frac{18}{53}a^{4}+\frac{24}{53}a^{3}-\frac{12}{53}a^{2}-\frac{4}{53}a-\frac{17}{53}$, $\frac{1}{2809}a^{25}+\frac{15}{2809}a^{24}+\frac{22}{2809}a^{23}+\frac{11}{2809}a^{22}+\frac{16}{2809}a^{21}-\frac{408}{2809}a^{20}+\frac{1071}{2809}a^{19}+\frac{345}{2809}a^{18}+\frac{299}{2809}a^{17}+\frac{1159}{2809}a^{16}+\frac{488}{2809}a^{15}-\frac{911}{2809}a^{14}-\frac{209}{2809}a^{13}-\frac{1243}{2809}a^{12}+\frac{1356}{2809}a^{11}+\frac{1401}{2809}a^{10}+\frac{1400}{2809}a^{9}-\frac{1088}{2809}a^{8}+\frac{606}{2809}a^{7}+\frac{1201}{2809}a^{6}-\frac{264}{2809}a^{5}-\frac{638}{2809}a^{4}-\frac{635}{2809}a^{3}+\frac{334}{2809}a^{2}+\frac{254}{2809}a+\frac{645}{2809}$, $\frac{1}{65\!\cdots\!49}a^{26}+\frac{14\!\cdots\!71}{12\!\cdots\!33}a^{25}+\frac{98\!\cdots\!49}{65\!\cdots\!49}a^{24}-\frac{44\!\cdots\!12}{65\!\cdots\!49}a^{23}-\frac{37\!\cdots\!00}{65\!\cdots\!49}a^{22}-\frac{18\!\cdots\!59}{65\!\cdots\!49}a^{21}-\frac{19\!\cdots\!42}{65\!\cdots\!49}a^{20}+\frac{21\!\cdots\!56}{65\!\cdots\!49}a^{19}+\frac{47\!\cdots\!62}{12\!\cdots\!33}a^{18}-\frac{23\!\cdots\!89}{65\!\cdots\!49}a^{17}+\frac{22\!\cdots\!56}{65\!\cdots\!49}a^{16}+\frac{14\!\cdots\!90}{65\!\cdots\!49}a^{15}-\frac{54\!\cdots\!21}{65\!\cdots\!49}a^{14}+\frac{12\!\cdots\!82}{65\!\cdots\!49}a^{13}-\frac{20\!\cdots\!64}{65\!\cdots\!49}a^{12}+\frac{80\!\cdots\!75}{65\!\cdots\!49}a^{11}+\frac{30\!\cdots\!71}{65\!\cdots\!49}a^{10}+\frac{19\!\cdots\!48}{65\!\cdots\!49}a^{9}+\frac{12\!\cdots\!91}{65\!\cdots\!49}a^{8}+\frac{56\!\cdots\!91}{65\!\cdots\!49}a^{7}-\frac{16\!\cdots\!68}{65\!\cdots\!49}a^{6}+\frac{10\!\cdots\!89}{65\!\cdots\!49}a^{5}-\frac{21\!\cdots\!21}{65\!\cdots\!49}a^{4}-\frac{22\!\cdots\!26}{65\!\cdots\!49}a^{3}-\frac{17\!\cdots\!14}{65\!\cdots\!49}a^{2}+\frac{29\!\cdots\!53}{65\!\cdots\!49}a+\frac{24\!\cdots\!02}{65\!\cdots\!49}$
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: |
$\frac{47\!\cdots\!08}{65\!\cdots\!49}a^{26}-\frac{43\!\cdots\!50}{65\!\cdots\!49}a^{25}+\frac{13\!\cdots\!11}{65\!\cdots\!49}a^{24}-\frac{12\!\cdots\!11}{65\!\cdots\!49}a^{23}-\frac{14\!\cdots\!04}{65\!\cdots\!49}a^{22}-\frac{31\!\cdots\!77}{65\!\cdots\!49}a^{21}+\frac{13\!\cdots\!04}{65\!\cdots\!49}a^{20}+\frac{54\!\cdots\!70}{65\!\cdots\!49}a^{19}-\frac{11\!\cdots\!33}{65\!\cdots\!49}a^{18}+\frac{20\!\cdots\!39}{65\!\cdots\!49}a^{17}-\frac{49\!\cdots\!90}{65\!\cdots\!49}a^{16}-\frac{13\!\cdots\!47}{65\!\cdots\!49}a^{15}-\frac{13\!\cdots\!66}{65\!\cdots\!49}a^{14}-\frac{72\!\cdots\!65}{65\!\cdots\!49}a^{13}+\frac{80\!\cdots\!91}{65\!\cdots\!49}a^{12}-\frac{16\!\cdots\!12}{65\!\cdots\!49}a^{11}+\frac{84\!\cdots\!99}{65\!\cdots\!49}a^{10}+\frac{53\!\cdots\!04}{65\!\cdots\!49}a^{9}-\frac{61\!\cdots\!01}{65\!\cdots\!49}a^{8}+\frac{20\!\cdots\!09}{65\!\cdots\!49}a^{7}-\frac{81\!\cdots\!84}{12\!\cdots\!33}a^{6}-\frac{10\!\cdots\!57}{65\!\cdots\!49}a^{5}+\frac{40\!\cdots\!55}{65\!\cdots\!49}a^{4}+\frac{65\!\cdots\!83}{65\!\cdots\!49}a^{3}+\frac{13\!\cdots\!71}{65\!\cdots\!49}a^{2}+\frac{16\!\cdots\!28}{65\!\cdots\!49}a+\frac{82\!\cdots\!68}{65\!\cdots\!49}$, $\frac{50\!\cdots\!87}{65\!\cdots\!49}a^{26}-\frac{33\!\cdots\!48}{65\!\cdots\!49}a^{25}+\frac{25\!\cdots\!71}{65\!\cdots\!49}a^{24}+\frac{23\!\cdots\!34}{65\!\cdots\!49}a^{23}-\frac{42\!\cdots\!00}{65\!\cdots\!49}a^{22}-\frac{33\!\cdots\!40}{65\!\cdots\!49}a^{21}+\frac{65\!\cdots\!20}{65\!\cdots\!49}a^{20}+\frac{42\!\cdots\!09}{65\!\cdots\!49}a^{19}-\frac{11\!\cdots\!68}{65\!\cdots\!49}a^{18}-\frac{82\!\cdots\!63}{65\!\cdots\!49}a^{17}+\frac{54\!\cdots\!68}{65\!\cdots\!49}a^{16}-\frac{36\!\cdots\!34}{65\!\cdots\!49}a^{15}-\frac{40\!\cdots\!79}{65\!\cdots\!49}a^{14}+\frac{11\!\cdots\!66}{65\!\cdots\!49}a^{13}+\frac{96\!\cdots\!03}{65\!\cdots\!49}a^{12}+\frac{29\!\cdots\!89}{65\!\cdots\!49}a^{11}-\frac{39\!\cdots\!01}{65\!\cdots\!49}a^{10}+\frac{36\!\cdots\!08}{65\!\cdots\!49}a^{9}+\frac{91\!\cdots\!16}{65\!\cdots\!49}a^{8}-\frac{21\!\cdots\!76}{65\!\cdots\!49}a^{7}+\frac{35\!\cdots\!57}{65\!\cdots\!49}a^{6}+\frac{36\!\cdots\!89}{65\!\cdots\!49}a^{5}-\frac{28\!\cdots\!83}{65\!\cdots\!49}a^{4}-\frac{11\!\cdots\!09}{65\!\cdots\!49}a^{3}+\frac{39\!\cdots\!13}{65\!\cdots\!49}a^{2}+\frac{39\!\cdots\!02}{65\!\cdots\!49}a+\frac{34\!\cdots\!31}{65\!\cdots\!49}$, $\frac{17\!\cdots\!83}{65\!\cdots\!49}a^{26}-\frac{15\!\cdots\!91}{65\!\cdots\!49}a^{25}+\frac{46\!\cdots\!34}{65\!\cdots\!49}a^{24}-\frac{40\!\cdots\!24}{65\!\cdots\!49}a^{23}-\frac{19\!\cdots\!88}{65\!\cdots\!49}a^{22}-\frac{10\!\cdots\!08}{65\!\cdots\!49}a^{21}+\frac{47\!\cdots\!34}{65\!\cdots\!49}a^{20}+\frac{26\!\cdots\!94}{65\!\cdots\!49}a^{19}-\frac{41\!\cdots\!74}{65\!\cdots\!49}a^{18}+\frac{70\!\cdots\!79}{65\!\cdots\!49}a^{17}-\frac{33\!\cdots\!27}{65\!\cdots\!49}a^{16}-\frac{65\!\cdots\!28}{65\!\cdots\!49}a^{15}-\frac{32\!\cdots\!19}{65\!\cdots\!49}a^{14}+\frac{12\!\cdots\!04}{65\!\cdots\!49}a^{13}+\frac{29\!\cdots\!21}{65\!\cdots\!49}a^{12}-\frac{57\!\cdots\!32}{65\!\cdots\!49}a^{11}+\frac{19\!\cdots\!02}{65\!\cdots\!49}a^{10}+\frac{34\!\cdots\!57}{65\!\cdots\!49}a^{9}-\frac{25\!\cdots\!64}{65\!\cdots\!49}a^{8}-\frac{68\!\cdots\!53}{65\!\cdots\!49}a^{7}+\frac{34\!\cdots\!55}{65\!\cdots\!49}a^{6}-\frac{45\!\cdots\!76}{65\!\cdots\!49}a^{5}+\frac{12\!\cdots\!30}{65\!\cdots\!49}a^{4}+\frac{34\!\cdots\!07}{65\!\cdots\!49}a^{3}+\frac{18\!\cdots\!90}{65\!\cdots\!49}a^{2}-\frac{11\!\cdots\!70}{65\!\cdots\!49}a+\frac{55\!\cdots\!85}{65\!\cdots\!49}$, $\frac{75\!\cdots\!34}{65\!\cdots\!49}a^{26}-\frac{12\!\cdots\!21}{12\!\cdots\!33}a^{25}+\frac{21\!\cdots\!10}{65\!\cdots\!49}a^{24}-\frac{20\!\cdots\!56}{65\!\cdots\!49}a^{23}-\frac{50\!\cdots\!32}{65\!\cdots\!49}a^{22}-\frac{46\!\cdots\!07}{65\!\cdots\!49}a^{21}+\frac{21\!\cdots\!54}{65\!\cdots\!49}a^{20}+\frac{84\!\cdots\!68}{65\!\cdots\!49}a^{19}-\frac{34\!\cdots\!77}{12\!\cdots\!33}a^{18}+\frac{32\!\cdots\!87}{65\!\cdots\!49}a^{17}-\frac{61\!\cdots\!50}{65\!\cdots\!49}a^{16}-\frac{26\!\cdots\!28}{65\!\cdots\!49}a^{15}+\frac{15\!\cdots\!65}{65\!\cdots\!49}a^{14}+\frac{44\!\cdots\!47}{65\!\cdots\!49}a^{13}+\frac{12\!\cdots\!00}{65\!\cdots\!49}a^{12}-\frac{26\!\cdots\!09}{65\!\cdots\!49}a^{11}+\frac{12\!\cdots\!23}{65\!\cdots\!49}a^{10}+\frac{12\!\cdots\!30}{65\!\cdots\!49}a^{9}-\frac{12\!\cdots\!31}{65\!\cdots\!49}a^{8}+\frac{14\!\cdots\!35}{65\!\cdots\!49}a^{7}+\frac{75\!\cdots\!79}{65\!\cdots\!49}a^{6}-\frac{86\!\cdots\!03}{65\!\cdots\!49}a^{5}+\frac{62\!\cdots\!04}{65\!\cdots\!49}a^{4}-\frac{19\!\cdots\!47}{65\!\cdots\!49}a^{3}+\frac{10\!\cdots\!50}{65\!\cdots\!49}a^{2}-\frac{18\!\cdots\!80}{65\!\cdots\!49}a+\frac{29\!\cdots\!77}{65\!\cdots\!49}$, $\frac{54\!\cdots\!21}{65\!\cdots\!49}a^{26}-\frac{50\!\cdots\!87}{65\!\cdots\!49}a^{25}+\frac{15\!\cdots\!68}{65\!\cdots\!49}a^{24}-\frac{15\!\cdots\!11}{65\!\cdots\!49}a^{23}-\frac{36\!\cdots\!87}{65\!\cdots\!49}a^{22}-\frac{31\!\cdots\!28}{65\!\cdots\!49}a^{21}+\frac{15\!\cdots\!02}{65\!\cdots\!49}a^{20}+\frac{49\!\cdots\!20}{65\!\cdots\!49}a^{19}-\frac{13\!\cdots\!01}{65\!\cdots\!49}a^{18}+\frac{24\!\cdots\!33}{65\!\cdots\!49}a^{17}-\frac{56\!\cdots\!91}{65\!\cdots\!49}a^{16}-\frac{21\!\cdots\!86}{65\!\cdots\!49}a^{15}+\frac{46\!\cdots\!99}{65\!\cdots\!49}a^{14}+\frac{44\!\cdots\!38}{65\!\cdots\!49}a^{13}+\frac{90\!\cdots\!59}{65\!\cdots\!49}a^{12}-\frac{20\!\cdots\!49}{65\!\cdots\!49}a^{11}+\frac{10\!\cdots\!69}{65\!\cdots\!49}a^{10}+\frac{10\!\cdots\!98}{65\!\cdots\!49}a^{9}-\frac{11\!\cdots\!41}{65\!\cdots\!49}a^{8}+\frac{15\!\cdots\!37}{65\!\cdots\!49}a^{7}+\frac{19\!\cdots\!31}{65\!\cdots\!49}a^{6}-\frac{70\!\cdots\!72}{65\!\cdots\!49}a^{5}+\frac{24\!\cdots\!65}{65\!\cdots\!49}a^{4}+\frac{63\!\cdots\!96}{65\!\cdots\!49}a^{3}+\frac{70\!\cdots\!06}{65\!\cdots\!49}a^{2}+\frac{16\!\cdots\!62}{65\!\cdots\!49}a+\frac{55\!\cdots\!93}{65\!\cdots\!49}$, $\frac{51\!\cdots\!60}{65\!\cdots\!49}a^{26}-\frac{45\!\cdots\!30}{65\!\cdots\!49}a^{25}+\frac{13\!\cdots\!83}{65\!\cdots\!49}a^{24}-\frac{10\!\cdots\!60}{65\!\cdots\!49}a^{23}-\frac{50\!\cdots\!08}{65\!\cdots\!49}a^{22}-\frac{34\!\cdots\!60}{65\!\cdots\!49}a^{21}+\frac{13\!\cdots\!10}{65\!\cdots\!49}a^{20}+\frac{89\!\cdots\!32}{65\!\cdots\!49}a^{19}-\frac{11\!\cdots\!37}{65\!\cdots\!49}a^{18}+\frac{19\!\cdots\!37}{65\!\cdots\!49}a^{17}-\frac{32\!\cdots\!50}{65\!\cdots\!49}a^{16}-\frac{16\!\cdots\!51}{65\!\cdots\!49}a^{15}-\frac{34\!\cdots\!48}{65\!\cdots\!49}a^{14}+\frac{29\!\cdots\!67}{65\!\cdots\!49}a^{13}+\frac{87\!\cdots\!27}{65\!\cdots\!49}a^{12}-\frac{15\!\cdots\!59}{65\!\cdots\!49}a^{11}+\frac{50\!\cdots\!53}{65\!\cdots\!49}a^{10}+\frac{81\!\cdots\!45}{65\!\cdots\!49}a^{9}-\frac{53\!\cdots\!94}{65\!\cdots\!49}a^{8}+\frac{61\!\cdots\!65}{65\!\cdots\!49}a^{7}-\frac{33\!\cdots\!50}{65\!\cdots\!49}a^{6}-\frac{18\!\cdots\!38}{65\!\cdots\!49}a^{5}+\frac{65\!\cdots\!41}{65\!\cdots\!49}a^{4}+\frac{16\!\cdots\!66}{65\!\cdots\!49}a^{3}+\frac{77\!\cdots\!49}{65\!\cdots\!49}a^{2}+\frac{95\!\cdots\!01}{65\!\cdots\!49}a+\frac{20\!\cdots\!10}{12\!\cdots\!33}$, $\frac{18\!\cdots\!92}{65\!\cdots\!49}a^{26}-\frac{17\!\cdots\!69}{65\!\cdots\!49}a^{25}+\frac{55\!\cdots\!90}{65\!\cdots\!49}a^{24}-\frac{61\!\cdots\!59}{65\!\cdots\!49}a^{23}-\frac{40\!\cdots\!25}{65\!\cdots\!49}a^{22}-\frac{10\!\cdots\!55}{65\!\cdots\!49}a^{21}+\frac{54\!\cdots\!79}{65\!\cdots\!49}a^{20}+\frac{84\!\cdots\!59}{65\!\cdots\!49}a^{19}-\frac{45\!\cdots\!12}{65\!\cdots\!49}a^{18}+\frac{90\!\cdots\!25}{65\!\cdots\!49}a^{17}-\frac{31\!\cdots\!78}{65\!\cdots\!49}a^{16}-\frac{67\!\cdots\!36}{65\!\cdots\!49}a^{15}+\frac{22\!\cdots\!21}{65\!\cdots\!49}a^{14}+\frac{14\!\cdots\!71}{65\!\cdots\!49}a^{13}+\frac{30\!\cdots\!67}{65\!\cdots\!49}a^{12}-\frac{72\!\cdots\!48}{65\!\cdots\!49}a^{11}+\frac{44\!\cdots\!79}{65\!\cdots\!49}a^{10}+\frac{28\!\cdots\!54}{65\!\cdots\!49}a^{9}-\frac{40\!\cdots\!63}{65\!\cdots\!49}a^{8}+\frac{92\!\cdots\!39}{65\!\cdots\!49}a^{7}+\frac{44\!\cdots\!49}{65\!\cdots\!49}a^{6}-\frac{20\!\cdots\!35}{65\!\cdots\!49}a^{5}+\frac{14\!\cdots\!57}{65\!\cdots\!49}a^{4}-\frac{79\!\cdots\!85}{65\!\cdots\!49}a^{3}-\frac{74\!\cdots\!21}{65\!\cdots\!49}a^{2}-\frac{41\!\cdots\!51}{65\!\cdots\!49}a+\frac{94\!\cdots\!08}{65\!\cdots\!49}$, $\frac{48\!\cdots\!94}{65\!\cdots\!49}a^{26}-\frac{43\!\cdots\!26}{65\!\cdots\!49}a^{25}+\frac{12\!\cdots\!52}{65\!\cdots\!49}a^{24}-\frac{10\!\cdots\!37}{65\!\cdots\!49}a^{23}-\frac{73\!\cdots\!19}{65\!\cdots\!49}a^{22}-\frac{27\!\cdots\!94}{65\!\cdots\!49}a^{21}+\frac{13\!\cdots\!18}{65\!\cdots\!49}a^{20}+\frac{74\!\cdots\!93}{65\!\cdots\!49}a^{19}-\frac{11\!\cdots\!85}{65\!\cdots\!49}a^{18}+\frac{19\!\cdots\!06}{65\!\cdots\!49}a^{17}+\frac{78\!\cdots\!89}{65\!\cdots\!49}a^{16}-\frac{21\!\cdots\!48}{65\!\cdots\!49}a^{15}-\frac{12\!\cdots\!85}{65\!\cdots\!49}a^{14}+\frac{75\!\cdots\!57}{65\!\cdots\!49}a^{13}+\frac{82\!\cdots\!77}{65\!\cdots\!49}a^{12}-\frac{16\!\cdots\!65}{65\!\cdots\!49}a^{11}+\frac{43\!\cdots\!73}{65\!\cdots\!49}a^{10}+\frac{12\!\cdots\!19}{65\!\cdots\!49}a^{9}-\frac{75\!\cdots\!25}{65\!\cdots\!49}a^{8}-\frac{25\!\cdots\!76}{65\!\cdots\!49}a^{7}+\frac{22\!\cdots\!57}{65\!\cdots\!49}a^{6}+\frac{34\!\cdots\!94}{65\!\cdots\!49}a^{5}-\frac{31\!\cdots\!22}{65\!\cdots\!49}a^{4}+\frac{86\!\cdots\!71}{65\!\cdots\!49}a^{3}-\frac{52\!\cdots\!61}{65\!\cdots\!49}a^{2}-\frac{14\!\cdots\!05}{65\!\cdots\!49}a-\frac{19\!\cdots\!17}{65\!\cdots\!49}$, $\frac{13\!\cdots\!79}{65\!\cdots\!49}a^{26}-\frac{11\!\cdots\!72}{65\!\cdots\!49}a^{25}+\frac{35\!\cdots\!82}{65\!\cdots\!49}a^{24}-\frac{29\!\cdots\!59}{65\!\cdots\!49}a^{23}-\frac{14\!\cdots\!74}{65\!\cdots\!49}a^{22}-\frac{83\!\cdots\!52}{65\!\cdots\!49}a^{21}+\frac{35\!\cdots\!03}{65\!\cdots\!49}a^{20}+\frac{21\!\cdots\!59}{65\!\cdots\!49}a^{19}-\frac{31\!\cdots\!49}{65\!\cdots\!49}a^{18}+\frac{52\!\cdots\!44}{65\!\cdots\!49}a^{17}-\frac{12\!\cdots\!14}{65\!\cdots\!49}a^{16}-\frac{48\!\cdots\!65}{65\!\cdots\!49}a^{15}-\frac{42\!\cdots\!42}{65\!\cdots\!49}a^{14}+\frac{69\!\cdots\!06}{65\!\cdots\!49}a^{13}+\frac{22\!\cdots\!04}{65\!\cdots\!49}a^{12}-\frac{42\!\cdots\!60}{65\!\cdots\!49}a^{11}+\frac{13\!\cdots\!60}{65\!\cdots\!49}a^{10}+\frac{25\!\cdots\!57}{65\!\cdots\!49}a^{9}-\frac{18\!\cdots\!38}{65\!\cdots\!49}a^{8}-\frac{31\!\cdots\!77}{65\!\cdots\!49}a^{7}+\frac{35\!\cdots\!61}{12\!\cdots\!33}a^{6}-\frac{54\!\cdots\!22}{65\!\cdots\!49}a^{5}+\frac{12\!\cdots\!43}{65\!\cdots\!49}a^{4}+\frac{26\!\cdots\!91}{65\!\cdots\!49}a^{3}+\frac{15\!\cdots\!70}{65\!\cdots\!49}a^{2}+\frac{30\!\cdots\!17}{65\!\cdots\!49}a+\frac{78\!\cdots\!32}{65\!\cdots\!49}$, $\frac{83\!\cdots\!09}{65\!\cdots\!49}a^{26}-\frac{75\!\cdots\!32}{65\!\cdots\!49}a^{25}+\frac{23\!\cdots\!92}{65\!\cdots\!49}a^{24}-\frac{22\!\cdots\!17}{65\!\cdots\!49}a^{23}-\frac{65\!\cdots\!58}{65\!\cdots\!49}a^{22}-\frac{50\!\cdots\!99}{65\!\cdots\!49}a^{21}+\frac{23\!\cdots\!14}{65\!\cdots\!49}a^{20}+\frac{97\!\cdots\!39}{65\!\cdots\!49}a^{19}-\frac{20\!\cdots\!07}{65\!\cdots\!49}a^{18}+\frac{35\!\cdots\!69}{65\!\cdots\!49}a^{17}-\frac{59\!\cdots\!05}{65\!\cdots\!49}a^{16}-\frac{30\!\cdots\!20}{65\!\cdots\!49}a^{15}+\frac{19\!\cdots\!59}{65\!\cdots\!49}a^{14}+\frac{60\!\cdots\!58}{65\!\cdots\!49}a^{13}+\frac{14\!\cdots\!58}{65\!\cdots\!49}a^{12}-\frac{29\!\cdots\!76}{65\!\cdots\!49}a^{11}+\frac{12\!\cdots\!77}{65\!\cdots\!49}a^{10}+\frac{14\!\cdots\!19}{65\!\cdots\!49}a^{9}-\frac{13\!\cdots\!46}{65\!\cdots\!49}a^{8}+\frac{10\!\cdots\!93}{65\!\cdots\!49}a^{7}+\frac{17\!\cdots\!10}{65\!\cdots\!49}a^{6}-\frac{40\!\cdots\!30}{65\!\cdots\!49}a^{5}+\frac{57\!\cdots\!85}{65\!\cdots\!49}a^{4}+\frac{13\!\cdots\!18}{65\!\cdots\!49}a^{3}+\frac{48\!\cdots\!74}{65\!\cdots\!49}a^{2}+\frac{90\!\cdots\!80}{65\!\cdots\!49}a+\frac{71\!\cdots\!45}{65\!\cdots\!49}$, $\frac{10\!\cdots\!45}{65\!\cdots\!49}a^{26}-\frac{88\!\cdots\!08}{65\!\cdots\!49}a^{25}+\frac{24\!\cdots\!53}{65\!\cdots\!49}a^{24}-\frac{14\!\cdots\!35}{65\!\cdots\!49}a^{23}-\frac{18\!\cdots\!18}{65\!\cdots\!49}a^{22}-\frac{65\!\cdots\!80}{65\!\cdots\!49}a^{21}+\frac{25\!\cdots\!51}{65\!\cdots\!49}a^{20}+\frac{24\!\cdots\!46}{65\!\cdots\!49}a^{19}-\frac{23\!\cdots\!93}{65\!\cdots\!49}a^{18}+\frac{33\!\cdots\!95}{65\!\cdots\!49}a^{17}+\frac{10\!\cdots\!39}{65\!\cdots\!49}a^{16}-\frac{38\!\cdots\!14}{65\!\cdots\!49}a^{15}-\frac{13\!\cdots\!60}{65\!\cdots\!49}a^{14}+\frac{61\!\cdots\!44}{65\!\cdots\!49}a^{13}+\frac{17\!\cdots\!56}{65\!\cdots\!49}a^{12}-\frac{28\!\cdots\!58}{65\!\cdots\!49}a^{11}+\frac{99\!\cdots\!31}{65\!\cdots\!49}a^{10}+\frac{23\!\cdots\!33}{65\!\cdots\!49}a^{9}-\frac{91\!\cdots\!48}{65\!\cdots\!49}a^{8}-\frac{45\!\cdots\!59}{65\!\cdots\!49}a^{7}+\frac{18\!\cdots\!17}{65\!\cdots\!49}a^{6}+\frac{18\!\cdots\!13}{65\!\cdots\!49}a^{5}+\frac{72\!\cdots\!85}{65\!\cdots\!49}a^{4}+\frac{42\!\cdots\!63}{65\!\cdots\!49}a^{3}+\frac{18\!\cdots\!85}{65\!\cdots\!49}a^{2}+\frac{29\!\cdots\!26}{65\!\cdots\!49}a+\frac{36\!\cdots\!51}{65\!\cdots\!49}$, $\frac{10\!\cdots\!60}{65\!\cdots\!49}a^{26}-\frac{99\!\cdots\!63}{65\!\cdots\!49}a^{25}+\frac{31\!\cdots\!55}{65\!\cdots\!49}a^{24}-\frac{33\!\cdots\!22}{65\!\cdots\!49}a^{23}-\frac{32\!\cdots\!09}{65\!\cdots\!49}a^{22}-\frac{63\!\cdots\!46}{65\!\cdots\!49}a^{21}+\frac{31\!\cdots\!81}{65\!\cdots\!49}a^{20}+\frac{74\!\cdots\!77}{65\!\cdots\!49}a^{19}-\frac{26\!\cdots\!64}{65\!\cdots\!49}a^{18}+\frac{50\!\cdots\!05}{65\!\cdots\!49}a^{17}-\frac{15\!\cdots\!58}{65\!\cdots\!49}a^{16}-\frac{37\!\cdots\!73}{65\!\cdots\!49}a^{15}+\frac{10\!\cdots\!77}{65\!\cdots\!49}a^{14}+\frac{54\!\cdots\!49}{65\!\cdots\!49}a^{13}+\frac{18\!\cdots\!91}{65\!\cdots\!49}a^{12}-\frac{40\!\cdots\!62}{65\!\cdots\!49}a^{11}+\frac{23\!\cdots\!09}{65\!\cdots\!49}a^{10}+\frac{15\!\cdots\!42}{65\!\cdots\!49}a^{9}-\frac{21\!\cdots\!65}{65\!\cdots\!49}a^{8}+\frac{57\!\cdots\!89}{65\!\cdots\!49}a^{7}+\frac{13\!\cdots\!95}{65\!\cdots\!49}a^{6}-\frac{10\!\cdots\!59}{65\!\cdots\!49}a^{5}+\frac{11\!\cdots\!63}{65\!\cdots\!49}a^{4}-\frac{18\!\cdots\!10}{12\!\cdots\!33}a^{3}+\frac{10\!\cdots\!75}{65\!\cdots\!49}a^{2}-\frac{34\!\cdots\!95}{65\!\cdots\!49}a+\frac{94\!\cdots\!06}{65\!\cdots\!49}$, $\frac{64\!\cdots\!58}{65\!\cdots\!49}a^{26}-\frac{56\!\cdots\!90}{65\!\cdots\!49}a^{25}+\frac{15\!\cdots\!32}{65\!\cdots\!49}a^{24}-\frac{10\!\cdots\!29}{65\!\cdots\!49}a^{23}-\frac{12\!\cdots\!04}{65\!\cdots\!49}a^{22}-\frac{39\!\cdots\!52}{65\!\cdots\!49}a^{21}+\frac{16\!\cdots\!05}{65\!\cdots\!49}a^{20}+\frac{14\!\cdots\!66}{65\!\cdots\!49}a^{19}-\frac{15\!\cdots\!38}{65\!\cdots\!49}a^{18}+\frac{22\!\cdots\!01}{65\!\cdots\!49}a^{17}+\frac{62\!\cdots\!13}{65\!\cdots\!49}a^{16}-\frac{27\!\cdots\!59}{65\!\cdots\!49}a^{15}-\frac{66\!\cdots\!09}{65\!\cdots\!49}a^{14}+\frac{72\!\cdots\!37}{65\!\cdots\!49}a^{13}+\frac{11\!\cdots\!31}{65\!\cdots\!49}a^{12}-\frac{18\!\cdots\!80}{65\!\cdots\!49}a^{11}+\frac{13\!\cdots\!75}{65\!\cdots\!49}a^{10}+\frac{16\!\cdots\!27}{65\!\cdots\!49}a^{9}-\frac{13\!\cdots\!69}{12\!\cdots\!33}a^{8}-\frac{39\!\cdots\!90}{65\!\cdots\!49}a^{7}+\frac{21\!\cdots\!18}{65\!\cdots\!49}a^{6}+\frac{24\!\cdots\!26}{65\!\cdots\!49}a^{5}+\frac{40\!\cdots\!23}{65\!\cdots\!49}a^{4}+\frac{17\!\cdots\!47}{65\!\cdots\!49}a^{3}-\frac{83\!\cdots\!98}{65\!\cdots\!49}a^{2}+\frac{24\!\cdots\!04}{65\!\cdots\!49}a-\frac{41\!\cdots\!22}{65\!\cdots\!49}$, $\frac{18\!\cdots\!45}{65\!\cdots\!49}a^{26}-\frac{15\!\cdots\!65}{65\!\cdots\!49}a^{25}+\frac{40\!\cdots\!67}{65\!\cdots\!49}a^{24}-\frac{15\!\cdots\!28}{65\!\cdots\!49}a^{23}-\frac{43\!\cdots\!97}{65\!\cdots\!49}a^{22}-\frac{12\!\cdots\!46}{65\!\cdots\!49}a^{21}+\frac{44\!\cdots\!78}{65\!\cdots\!49}a^{20}+\frac{56\!\cdots\!73}{65\!\cdots\!49}a^{19}-\frac{42\!\cdots\!06}{65\!\cdots\!49}a^{18}+\frac{49\!\cdots\!23}{65\!\cdots\!49}a^{17}+\frac{37\!\cdots\!68}{65\!\cdots\!49}a^{16}-\frac{69\!\cdots\!54}{65\!\cdots\!49}a^{15}-\frac{43\!\cdots\!78}{65\!\cdots\!49}a^{14}+\frac{10\!\cdots\!91}{65\!\cdots\!49}a^{13}+\frac{32\!\cdots\!96}{65\!\cdots\!49}a^{12}-\frac{43\!\cdots\!48}{65\!\cdots\!49}a^{11}-\frac{12\!\cdots\!10}{65\!\cdots\!49}a^{10}+\frac{88\!\cdots\!78}{12\!\cdots\!33}a^{9}-\frac{56\!\cdots\!52}{65\!\cdots\!49}a^{8}-\frac{16\!\cdots\!89}{65\!\cdots\!49}a^{7}+\frac{28\!\cdots\!19}{65\!\cdots\!49}a^{6}+\frac{18\!\cdots\!40}{65\!\cdots\!49}a^{5}+\frac{92\!\cdots\!17}{65\!\cdots\!49}a^{4}+\frac{12\!\cdots\!23}{65\!\cdots\!49}a^{3}+\frac{39\!\cdots\!93}{65\!\cdots\!49}a^{2}+\frac{66\!\cdots\!55}{65\!\cdots\!49}a+\frac{16\!\cdots\!62}{65\!\cdots\!49}$, $a$, $\frac{13\!\cdots\!59}{65\!\cdots\!49}a^{26}-\frac{12\!\cdots\!61}{65\!\cdots\!49}a^{25}+\frac{38\!\cdots\!19}{65\!\cdots\!49}a^{24}-\frac{38\!\cdots\!84}{65\!\cdots\!49}a^{23}-\frac{71\!\cdots\!17}{65\!\cdots\!49}a^{22}-\frac{83\!\cdots\!53}{65\!\cdots\!49}a^{21}+\frac{38\!\cdots\!47}{65\!\cdots\!49}a^{20}+\frac{13\!\cdots\!16}{65\!\cdots\!49}a^{19}-\frac{32\!\cdots\!57}{65\!\cdots\!49}a^{18}+\frac{59\!\cdots\!69}{65\!\cdots\!49}a^{17}-\frac{13\!\cdots\!95}{65\!\cdots\!49}a^{16}-\frac{46\!\cdots\!39}{65\!\cdots\!49}a^{15}+\frac{56\!\cdots\!42}{65\!\cdots\!49}a^{14}+\frac{68\!\cdots\!22}{65\!\cdots\!49}a^{13}+\frac{22\!\cdots\!29}{65\!\cdots\!49}a^{12}-\frac{48\!\cdots\!16}{65\!\cdots\!49}a^{11}+\frac{24\!\cdots\!32}{65\!\cdots\!49}a^{10}+\frac{20\!\cdots\!55}{65\!\cdots\!49}a^{9}-\frac{23\!\cdots\!10}{65\!\cdots\!49}a^{8}+\frac{43\!\cdots\!25}{65\!\cdots\!49}a^{7}+\frac{17\!\cdots\!34}{65\!\cdots\!49}a^{6}-\frac{10\!\cdots\!30}{65\!\cdots\!49}a^{5}+\frac{12\!\cdots\!45}{65\!\cdots\!49}a^{4}+\frac{12\!\cdots\!11}{65\!\cdots\!49}a^{3}+\frac{92\!\cdots\!56}{65\!\cdots\!49}a^{2}+\frac{20\!\cdots\!56}{65\!\cdots\!49}a+\frac{18\!\cdots\!19}{65\!\cdots\!49}$, $\frac{13\!\cdots\!85}{65\!\cdots\!49}a^{26}-\frac{11\!\cdots\!48}{65\!\cdots\!49}a^{25}+\frac{34\!\cdots\!42}{65\!\cdots\!49}a^{24}-\frac{29\!\cdots\!94}{65\!\cdots\!49}a^{23}-\frac{13\!\cdots\!54}{65\!\cdots\!49}a^{22}-\frac{82\!\cdots\!49}{65\!\cdots\!49}a^{21}+\frac{35\!\cdots\!80}{65\!\cdots\!49}a^{20}+\frac{20\!\cdots\!18}{65\!\cdots\!49}a^{19}-\frac{30\!\cdots\!56}{65\!\cdots\!49}a^{18}+\frac{51\!\cdots\!83}{65\!\cdots\!49}a^{17}-\frac{20\!\cdots\!87}{65\!\cdots\!49}a^{16}-\frac{46\!\cdots\!58}{65\!\cdots\!49}a^{15}-\frac{54\!\cdots\!48}{65\!\cdots\!49}a^{14}+\frac{79\!\cdots\!15}{65\!\cdots\!49}a^{13}+\frac{22\!\cdots\!52}{65\!\cdots\!49}a^{12}-\frac{42\!\cdots\!04}{65\!\cdots\!49}a^{11}+\frac{14\!\cdots\!62}{65\!\cdots\!49}a^{10}+\frac{24\!\cdots\!83}{65\!\cdots\!49}a^{9}-\frac{16\!\cdots\!19}{65\!\cdots\!49}a^{8}-\frac{10\!\cdots\!13}{65\!\cdots\!49}a^{7}+\frac{33\!\cdots\!18}{12\!\cdots\!33}a^{6}+\frac{97\!\cdots\!99}{65\!\cdots\!49}a^{5}+\frac{87\!\cdots\!01}{65\!\cdots\!49}a^{4}+\frac{30\!\cdots\!74}{65\!\cdots\!49}a^{3}+\frac{15\!\cdots\!90}{65\!\cdots\!49}a^{2}-\frac{16\!\cdots\!47}{65\!\cdots\!49}a+\frac{82\!\cdots\!02}{65\!\cdots\!49}$
| 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: | \( 1816757010.3045344 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{9}\cdot 1816757010.3045344 \cdot 1}{2\cdot\sqrt{656187196700948326335269302720488800256}}\cr\approx \mathstrut & 0.277103332705435 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 + 27*x^25 - 24*x^24 - 9*x^23 - 63*x^22 + 273*x^21 + 144*x^20 - 2376*x^19 + 4050*x^18 - 396*x^17 - 3465*x^16 - 141*x^15 + 441*x^14 + 17019*x^13 - 33225*x^12 + 12852*x^11 + 17379*x^10 - 14277*x^9 + 1188*x^8 + 1143*x^7 - 501*x^6 + 891*x^5 + 216*x^4 + 129*x^3 + 18*x^2 + 9*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 - 9*x^26 + 27*x^25 - 24*x^24 - 9*x^23 - 63*x^22 + 273*x^21 + 144*x^20 - 2376*x^19 + 4050*x^18 - 396*x^17 - 3465*x^16 - 141*x^15 + 441*x^14 + 17019*x^13 - 33225*x^12 + 12852*x^11 + 17379*x^10 - 14277*x^9 + 1188*x^8 + 1143*x^7 - 501*x^6 + 891*x^5 + 216*x^4 + 129*x^3 + 18*x^2 + 9*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 - 9*x^26 + 27*x^25 - 24*x^24 - 9*x^23 - 63*x^22 + 273*x^21 + 144*x^20 - 2376*x^19 + 4050*x^18 - 396*x^17 - 3465*x^16 - 141*x^15 + 441*x^14 + 17019*x^13 - 33225*x^12 + 12852*x^11 + 17379*x^10 - 14277*x^9 + 1188*x^8 + 1143*x^7 - 501*x^6 + 891*x^5 + 216*x^4 + 129*x^3 + 18*x^2 + 9*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 - 9*x^26 + 27*x^25 - 24*x^24 - 9*x^23 - 63*x^22 + 273*x^21 + 144*x^20 - 2376*x^19 + 4050*x^18 - 396*x^17 - 3465*x^16 - 141*x^15 + 441*x^14 + 17019*x^13 - 33225*x^12 + 12852*x^11 + 17379*x^10 - 14277*x^9 + 1188*x^8 + 1143*x^7 - 501*x^6 + 891*x^5 + 216*x^4 + 129*x^3 + 18*x^2 + 9*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
| \(\Q(\zeta_{9})^+\), 3.1.324.1, \(\Q(\zeta_{27})^+\), 9.3.2754990144.1 |
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: | 18.0.258151783382020583032356864.1 |
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 | ${\href{/padicField/5.9.0.1}{9} }^{3}$ | $18{,}\,{\href{/padicField/7.9.0.1}{9} }$ | $18{,}\,{\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{9}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }^{3}$ | $18{,}\,{\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }^{3}$ | $18{,}\,{\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }^{9}$ | ${\href{/padicField/41.9.0.1}{9} }^{3}$ | $18{,}\,{\href{/padicField/43.9.0.1}{9} }$ | $18{,}\,{\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.1.0.1}{1} }^{27}$ | $18{,}\,{\href{/padicField/59.9.0.1}{9} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.9.0.1 | $x^{9} + x^{4} + 1$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |
| 2.18.18.115 | $x^{18} + 18 x^{17} + 162 x^{16} + 960 x^{15} + 4320 x^{14} + 16128 x^{13} + 53696 x^{12} + 165120 x^{11} + 449824 x^{10} + 1006400 x^{9} + 1826368 x^{8} + 2905088 x^{7} + 3317760 x^{6} - 418816 x^{5} - 6684672 x^{4} - 4984832 x^{3} + 2483456 x^{2} + 3566080 x + 1829376$ | $2$ | $9$ | $18$ | $C_{18}$ | $[2]^{9}$ | |
|
\(3\)
| Deg $27$ | $27$ | $1$ | $70$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.36.6t1.b.a | $1$ | $ 2^{2} \cdot 3^{2}$ | 6.0.419904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| 1.36.6t1.b.b | $1$ | $ 2^{2} \cdot 3^{2}$ | 6.0.419904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.27.9t1.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) | $0$ | $1$ |
| 1.108.18t1.a.a | $1$ | $ 2^{2} \cdot 3^{3}$ | 18.0.258151783382020583032356864.7 | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| * | 1.27.9t1.a.c | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| 1.108.18t1.a.b | $1$ | $ 2^{2} \cdot 3^{3}$ | 18.0.258151783382020583032356864.7 | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.108.18t1.a.c | $1$ | $ 2^{2} \cdot 3^{3}$ | 18.0.258151783382020583032356864.7 | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| * | 1.27.9t1.a.d | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| 1.108.18t1.a.d | $1$ | $ 2^{2} \cdot 3^{3}$ | 18.0.258151783382020583032356864.7 | $C_{18}$ (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) | $0$ | $1$ |
| 1.108.18t1.a.e | $1$ | $ 2^{2} \cdot 3^{3}$ | 18.0.258151783382020583032356864.7 | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.108.18t1.a.f | $1$ | $ 2^{2} \cdot 3^{3}$ | 18.0.258151783382020583032356864.7 | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| * | 2.324.3t2.b.a | $2$ | $ 2^{2} \cdot 3^{4}$ | 3.1.324.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.324.6t5.d.a | $2$ | $ 2^{2} \cdot 3^{4}$ | 6.0.419904.3 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.324.6t5.d.b | $2$ | $ 2^{2} \cdot 3^{4}$ | 6.0.419904.3 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.2916.18t16.a.a | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.656187196700948326335269302720488800256.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.a.b | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.656187196700948326335269302720488800256.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.a.c | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.656187196700948326335269302720488800256.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.a.d | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.656187196700948326335269302720488800256.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.a.e | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.656187196700948326335269302720488800256.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.a.f | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.656187196700948326335269302720488800256.1 | $S_3\times C_9$ (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 *.