Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 18*x^25 + 189*x^23 - 63*x^22 - 1062*x^21 + 648*x^20 + 4464*x^19 - 2679*x^18 - 14499*x^17 + 7092*x^16 + 36963*x^15 - 21411*x^14 - 74187*x^13 + 24255*x^12 + 61398*x^11 - 36351*x^10 - 17520*x^9 + 66663*x^8 + 42507*x^7 - 11583*x^6 - 18603*x^5 - 10539*x^4 - 6327*x^3 - 2835*x^2 - 639*x - 53)
gp: K = bnfinit(y^27 - 18*y^25 + 189*y^23 - 63*y^22 - 1062*y^21 + 648*y^20 + 4464*y^19 - 2679*y^18 - 14499*y^17 + 7092*y^16 + 36963*y^15 - 21411*y^14 - 74187*y^13 + 24255*y^12 + 61398*y^11 - 36351*y^10 - 17520*y^9 + 66663*y^8 + 42507*y^7 - 11583*y^6 - 18603*y^5 - 10539*y^4 - 6327*y^3 - 2835*y^2 - 639*y - 53, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 18*x^25 + 189*x^23 - 63*x^22 - 1062*x^21 + 648*x^20 + 4464*x^19 - 2679*x^18 - 14499*x^17 + 7092*x^16 + 36963*x^15 - 21411*x^14 - 74187*x^13 + 24255*x^12 + 61398*x^11 - 36351*x^10 - 17520*x^9 + 66663*x^8 + 42507*x^7 - 11583*x^6 - 18603*x^5 - 10539*x^4 - 6327*x^3 - 2835*x^2 - 639*x - 53);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 18*x^25 + 189*x^23 - 63*x^22 - 1062*x^21 + 648*x^20 + 4464*x^19 - 2679*x^18 - 14499*x^17 + 7092*x^16 + 36963*x^15 - 21411*x^14 - 74187*x^13 + 24255*x^12 + 61398*x^11 - 36351*x^10 - 17520*x^9 + 66663*x^8 + 42507*x^7 - 11583*x^6 - 18603*x^5 - 10539*x^4 - 6327*x^3 - 2835*x^2 - 639*x - 53)
\( x^{27} - 18 x^{25} + 189 x^{23} - 63 x^{22} - 1062 x^{21} + 648 x^{20} + 4464 x^{19} - 2679 x^{18} - 14499 x^{17} + 7092 x^{16} + 36963 x^{15} - 21411 x^{14} - 74187 x^{13} + \cdots - 53 \)
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: |
\(-17717054310925604811052271173453197606912\)
\(\medspace = -\,2^{18}\cdot 3^{73}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.95\) | 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))
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{12}+\frac{1}{3}a^{9}+\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{13}+\frac{1}{3}a^{10}+\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{9}+\frac{1}{3}$, $\frac{1}{3}a^{19}-\frac{1}{3}a^{10}+\frac{1}{3}a$, $\frac{1}{3}a^{20}-\frac{1}{3}a^{11}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{21}-\frac{1}{3}a^{12}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{22}-\frac{1}{3}a^{13}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{23}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{24}-\frac{1}{3}a^{12}+\frac{1}{3}a^{9}-\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{25}-\frac{1}{3}a^{13}+\frac{1}{3}a^{10}-\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{11\!\cdots\!93}a^{26}-\frac{24\!\cdots\!14}{38\!\cdots\!31}a^{25}-\frac{40\!\cdots\!69}{38\!\cdots\!31}a^{24}-\frac{73\!\cdots\!53}{11\!\cdots\!93}a^{23}-\frac{44\!\cdots\!02}{38\!\cdots\!31}a^{22}+\frac{19\!\cdots\!57}{11\!\cdots\!93}a^{21}+\frac{63\!\cdots\!00}{38\!\cdots\!31}a^{20}-\frac{78\!\cdots\!13}{11\!\cdots\!93}a^{19}+\frac{36\!\cdots\!16}{38\!\cdots\!31}a^{18}-\frac{21\!\cdots\!37}{11\!\cdots\!93}a^{17}+\frac{17\!\cdots\!81}{11\!\cdots\!93}a^{16}+\frac{10\!\cdots\!01}{11\!\cdots\!93}a^{15}+\frac{34\!\cdots\!71}{11\!\cdots\!93}a^{14}-\frac{83\!\cdots\!02}{11\!\cdots\!93}a^{13}-\frac{57\!\cdots\!39}{11\!\cdots\!93}a^{12}-\frac{75\!\cdots\!88}{38\!\cdots\!31}a^{11}-\frac{54\!\cdots\!51}{11\!\cdots\!93}a^{10}-\frac{57\!\cdots\!14}{11\!\cdots\!93}a^{9}-\frac{15\!\cdots\!80}{11\!\cdots\!93}a^{8}-\frac{41\!\cdots\!83}{11\!\cdots\!93}a^{7}+\frac{22\!\cdots\!05}{11\!\cdots\!93}a^{6}-\frac{22\!\cdots\!89}{11\!\cdots\!93}a^{5}+\frac{44\!\cdots\!51}{11\!\cdots\!93}a^{4}-\frac{10\!\cdots\!46}{38\!\cdots\!31}a^{3}+\frac{70\!\cdots\!18}{38\!\cdots\!31}a^{2}-\frac{19\!\cdots\!83}{38\!\cdots\!31}a+\frac{15\!\cdots\!69}{11\!\cdots\!93}$
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: |
$\frac{82\!\cdots\!42}{12\!\cdots\!31}a^{26}-\frac{51\!\cdots\!16}{12\!\cdots\!31}a^{25}-\frac{14\!\cdots\!33}{12\!\cdots\!31}a^{24}+\frac{91\!\cdots\!24}{12\!\cdots\!31}a^{23}+\frac{14\!\cdots\!68}{12\!\cdots\!31}a^{22}-\frac{14\!\cdots\!06}{12\!\cdots\!31}a^{21}-\frac{78\!\cdots\!63}{12\!\cdots\!31}a^{20}+\frac{10\!\cdots\!63}{12\!\cdots\!31}a^{19}+\frac{90\!\cdots\!78}{38\!\cdots\!93}a^{18}-\frac{41\!\cdots\!20}{12\!\cdots\!31}a^{17}-\frac{93\!\cdots\!76}{12\!\cdots\!31}a^{16}+\frac{11\!\cdots\!70}{12\!\cdots\!31}a^{15}+\frac{23\!\cdots\!09}{12\!\cdots\!31}a^{14}-\frac{32\!\cdots\!42}{12\!\cdots\!31}a^{13}-\frac{40\!\cdots\!71}{12\!\cdots\!31}a^{12}+\frac{45\!\cdots\!85}{12\!\cdots\!31}a^{11}+\frac{21\!\cdots\!86}{12\!\cdots\!31}a^{10}-\frac{13\!\cdots\!03}{38\!\cdots\!93}a^{9}+\frac{12\!\cdots\!31}{12\!\cdots\!31}a^{8}+\frac{46\!\cdots\!74}{12\!\cdots\!31}a^{7}+\frac{57\!\cdots\!86}{12\!\cdots\!31}a^{6}-\frac{13\!\cdots\!80}{12\!\cdots\!31}a^{5}-\frac{71\!\cdots\!18}{12\!\cdots\!31}a^{4}-\frac{42\!\cdots\!99}{12\!\cdots\!31}a^{3}-\frac{25\!\cdots\!59}{12\!\cdots\!31}a^{2}-\frac{72\!\cdots\!07}{12\!\cdots\!31}a-\frac{21\!\cdots\!19}{38\!\cdots\!93}$, $\frac{25\!\cdots\!68}{12\!\cdots\!31}a^{26}-\frac{15\!\cdots\!22}{12\!\cdots\!31}a^{25}-\frac{45\!\cdots\!29}{12\!\cdots\!31}a^{24}+\frac{27\!\cdots\!49}{12\!\cdots\!31}a^{23}+\frac{46\!\cdots\!40}{12\!\cdots\!31}a^{22}-\frac{44\!\cdots\!32}{12\!\cdots\!31}a^{21}-\frac{24\!\cdots\!00}{12\!\cdots\!31}a^{20}+\frac{31\!\cdots\!08}{12\!\cdots\!31}a^{19}+\frac{28\!\cdots\!11}{38\!\cdots\!93}a^{18}-\frac{12\!\cdots\!24}{12\!\cdots\!31}a^{17}-\frac{29\!\cdots\!41}{12\!\cdots\!31}a^{16}+\frac{36\!\cdots\!99}{12\!\cdots\!31}a^{15}+\frac{72\!\cdots\!00}{12\!\cdots\!31}a^{14}-\frac{99\!\cdots\!45}{12\!\cdots\!31}a^{13}-\frac{12\!\cdots\!77}{12\!\cdots\!31}a^{12}+\frac{14\!\cdots\!96}{12\!\cdots\!31}a^{11}+\frac{69\!\cdots\!00}{12\!\cdots\!31}a^{10}-\frac{40\!\cdots\!76}{38\!\cdots\!93}a^{9}+\frac{38\!\cdots\!80}{12\!\cdots\!31}a^{8}+\frac{14\!\cdots\!61}{12\!\cdots\!31}a^{7}+\frac{18\!\cdots\!27}{12\!\cdots\!31}a^{6}-\frac{40\!\cdots\!99}{12\!\cdots\!31}a^{5}-\frac{22\!\cdots\!32}{12\!\cdots\!31}a^{4}-\frac{13\!\cdots\!12}{12\!\cdots\!31}a^{3}-\frac{80\!\cdots\!52}{12\!\cdots\!31}a^{2}-\frac{23\!\cdots\!68}{12\!\cdots\!31}a-\frac{67\!\cdots\!00}{38\!\cdots\!93}$, $\frac{11\!\cdots\!24}{11\!\cdots\!93}a^{26}-\frac{87\!\cdots\!58}{11\!\cdots\!93}a^{25}-\frac{67\!\cdots\!99}{38\!\cdots\!31}a^{24}+\frac{15\!\cdots\!86}{11\!\cdots\!93}a^{23}+\frac{20\!\cdots\!56}{11\!\cdots\!93}a^{22}-\frac{76\!\cdots\!10}{38\!\cdots\!31}a^{21}-\frac{10\!\cdots\!83}{11\!\cdots\!93}a^{20}+\frac{15\!\cdots\!78}{11\!\cdots\!93}a^{19}+\frac{39\!\cdots\!82}{11\!\cdots\!93}a^{18}-\frac{61\!\cdots\!50}{11\!\cdots\!93}a^{17}-\frac{12\!\cdots\!43}{11\!\cdots\!93}a^{16}+\frac{17\!\cdots\!22}{11\!\cdots\!93}a^{15}+\frac{29\!\cdots\!50}{11\!\cdots\!93}a^{14}-\frac{47\!\cdots\!88}{11\!\cdots\!93}a^{13}-\frac{50\!\cdots\!91}{11\!\cdots\!93}a^{12}+\frac{66\!\cdots\!68}{11\!\cdots\!93}a^{11}+\frac{20\!\cdots\!10}{11\!\cdots\!93}a^{10}-\frac{19\!\cdots\!36}{38\!\cdots\!31}a^{9}+\frac{24\!\cdots\!83}{11\!\cdots\!93}a^{8}+\frac{59\!\cdots\!88}{11\!\cdots\!93}a^{7}+\frac{35\!\cdots\!64}{11\!\cdots\!93}a^{6}-\frac{16\!\cdots\!10}{11\!\cdots\!93}a^{5}-\frac{86\!\cdots\!91}{11\!\cdots\!93}a^{4}-\frac{54\!\cdots\!32}{11\!\cdots\!93}a^{3}-\frac{31\!\cdots\!82}{11\!\cdots\!93}a^{2}-\frac{83\!\cdots\!14}{11\!\cdots\!93}a-\frac{78\!\cdots\!83}{11\!\cdots\!93}$, $\frac{56\!\cdots\!22}{11\!\cdots\!93}a^{26}-\frac{12\!\cdots\!96}{38\!\cdots\!31}a^{25}-\frac{33\!\cdots\!55}{38\!\cdots\!31}a^{24}+\frac{64\!\cdots\!33}{11\!\cdots\!93}a^{23}+\frac{10\!\cdots\!02}{11\!\cdots\!93}a^{22}-\frac{10\!\cdots\!64}{11\!\cdots\!93}a^{21}-\frac{53\!\cdots\!03}{11\!\cdots\!93}a^{20}+\frac{23\!\cdots\!52}{38\!\cdots\!31}a^{19}+\frac{20\!\cdots\!87}{11\!\cdots\!93}a^{18}-\frac{28\!\cdots\!40}{11\!\cdots\!93}a^{17}-\frac{21\!\cdots\!60}{38\!\cdots\!31}a^{16}+\frac{81\!\cdots\!61}{11\!\cdots\!93}a^{15}+\frac{15\!\cdots\!66}{11\!\cdots\!93}a^{14}-\frac{22\!\cdots\!81}{11\!\cdots\!93}a^{13}-\frac{27\!\cdots\!24}{11\!\cdots\!93}a^{12}+\frac{31\!\cdots\!16}{11\!\cdots\!93}a^{11}+\frac{46\!\cdots\!90}{38\!\cdots\!31}a^{10}-\frac{29\!\cdots\!56}{11\!\cdots\!93}a^{9}+\frac{95\!\cdots\!80}{11\!\cdots\!93}a^{8}+\frac{10\!\cdots\!67}{38\!\cdots\!31}a^{7}+\frac{34\!\cdots\!95}{11\!\cdots\!93}a^{6}-\frac{88\!\cdots\!63}{11\!\cdots\!93}a^{5}-\frac{47\!\cdots\!72}{11\!\cdots\!93}a^{4}-\frac{94\!\cdots\!20}{38\!\cdots\!31}a^{3}-\frac{17\!\cdots\!70}{11\!\cdots\!93}a^{2}-\frac{16\!\cdots\!16}{38\!\cdots\!31}a-\frac{15\!\cdots\!73}{38\!\cdots\!31}$, $\frac{49\!\cdots\!60}{11\!\cdots\!93}a^{26}-\frac{30\!\cdots\!18}{11\!\cdots\!93}a^{25}-\frac{29\!\cdots\!33}{38\!\cdots\!31}a^{24}+\frac{54\!\cdots\!67}{11\!\cdots\!93}a^{23}+\frac{30\!\cdots\!52}{38\!\cdots\!31}a^{22}-\frac{87\!\cdots\!02}{11\!\cdots\!93}a^{21}-\frac{47\!\cdots\!35}{11\!\cdots\!93}a^{20}+\frac{61\!\cdots\!43}{11\!\cdots\!93}a^{19}+\frac{61\!\cdots\!69}{38\!\cdots\!31}a^{18}-\frac{24\!\cdots\!40}{11\!\cdots\!93}a^{17}-\frac{56\!\cdots\!73}{11\!\cdots\!93}a^{16}+\frac{70\!\cdots\!61}{11\!\cdots\!93}a^{15}+\frac{14\!\cdots\!29}{11\!\cdots\!93}a^{14}-\frac{64\!\cdots\!99}{38\!\cdots\!31}a^{13}-\frac{83\!\cdots\!87}{38\!\cdots\!31}a^{12}+\frac{27\!\cdots\!33}{11\!\cdots\!93}a^{11}+\frac{13\!\cdots\!30}{11\!\cdots\!93}a^{10}-\frac{26\!\cdots\!43}{11\!\cdots\!93}a^{9}+\frac{76\!\cdots\!27}{11\!\cdots\!93}a^{8}+\frac{28\!\cdots\!29}{11\!\cdots\!93}a^{7}+\frac{36\!\cdots\!41}{11\!\cdots\!93}a^{6}-\frac{79\!\cdots\!19}{11\!\cdots\!93}a^{5}-\frac{14\!\cdots\!86}{38\!\cdots\!31}a^{4}-\frac{25\!\cdots\!83}{11\!\cdots\!93}a^{3}-\frac{15\!\cdots\!79}{11\!\cdots\!93}a^{2}-\frac{44\!\cdots\!27}{11\!\cdots\!93}a-\frac{43\!\cdots\!39}{11\!\cdots\!93}$, $\frac{98\!\cdots\!30}{11\!\cdots\!93}a^{26}-\frac{68\!\cdots\!82}{11\!\cdots\!93}a^{25}-\frac{17\!\cdots\!60}{11\!\cdots\!93}a^{24}+\frac{11\!\cdots\!17}{11\!\cdots\!93}a^{23}+\frac{17\!\cdots\!70}{11\!\cdots\!93}a^{22}-\frac{18\!\cdots\!66}{11\!\cdots\!93}a^{21}-\frac{91\!\cdots\!66}{11\!\cdots\!93}a^{20}+\frac{42\!\cdots\!56}{38\!\cdots\!31}a^{19}+\frac{34\!\cdots\!07}{11\!\cdots\!93}a^{18}-\frac{50\!\cdots\!66}{11\!\cdots\!93}a^{17}-\frac{10\!\cdots\!67}{11\!\cdots\!93}a^{16}+\frac{14\!\cdots\!61}{11\!\cdots\!93}a^{15}+\frac{26\!\cdots\!14}{11\!\cdots\!93}a^{14}-\frac{39\!\cdots\!27}{11\!\cdots\!93}a^{13}-\frac{15\!\cdots\!59}{38\!\cdots\!31}a^{12}+\frac{54\!\cdots\!66}{11\!\cdots\!93}a^{11}+\frac{71\!\cdots\!04}{38\!\cdots\!31}a^{10}-\frac{16\!\cdots\!66}{38\!\cdots\!31}a^{9}+\frac{17\!\cdots\!42}{11\!\cdots\!93}a^{8}+\frac{52\!\cdots\!55}{11\!\cdots\!93}a^{7}+\frac{18\!\cdots\!63}{38\!\cdots\!31}a^{6}-\frac{14\!\cdots\!55}{11\!\cdots\!93}a^{5}-\frac{81\!\cdots\!55}{11\!\cdots\!93}a^{4}-\frac{48\!\cdots\!46}{11\!\cdots\!93}a^{3}-\frac{28\!\cdots\!10}{11\!\cdots\!93}a^{2}-\frac{27\!\cdots\!76}{38\!\cdots\!31}a-\frac{81\!\cdots\!94}{11\!\cdots\!93}$, $\frac{79\!\cdots\!34}{38\!\cdots\!31}a^{26}-\frac{16\!\cdots\!58}{11\!\cdots\!93}a^{25}-\frac{41\!\cdots\!06}{11\!\cdots\!93}a^{24}+\frac{29\!\cdots\!31}{11\!\cdots\!93}a^{23}+\frac{42\!\cdots\!94}{11\!\cdots\!93}a^{22}-\frac{45\!\cdots\!18}{11\!\cdots\!93}a^{21}-\frac{73\!\cdots\!92}{38\!\cdots\!31}a^{20}+\frac{31\!\cdots\!26}{11\!\cdots\!93}a^{19}+\frac{84\!\cdots\!25}{11\!\cdots\!93}a^{18}-\frac{41\!\cdots\!02}{38\!\cdots\!31}a^{17}-\frac{25\!\cdots\!28}{11\!\cdots\!93}a^{16}+\frac{35\!\cdots\!87}{11\!\cdots\!93}a^{15}+\frac{63\!\cdots\!28}{11\!\cdots\!93}a^{14}-\frac{96\!\cdots\!21}{11\!\cdots\!93}a^{13}-\frac{10\!\cdots\!65}{11\!\cdots\!93}a^{12}+\frac{45\!\cdots\!88}{38\!\cdots\!31}a^{11}+\frac{51\!\cdots\!00}{11\!\cdots\!93}a^{10}-\frac{12\!\cdots\!64}{11\!\cdots\!93}a^{9}+\frac{15\!\cdots\!82}{38\!\cdots\!31}a^{8}+\frac{12\!\cdots\!29}{11\!\cdots\!93}a^{7}+\frac{33\!\cdots\!61}{38\!\cdots\!31}a^{6}-\frac{36\!\cdots\!73}{11\!\cdots\!93}a^{5}-\frac{18\!\cdots\!49}{11\!\cdots\!93}a^{4}-\frac{38\!\cdots\!56}{38\!\cdots\!31}a^{3}-\frac{22\!\cdots\!80}{38\!\cdots\!31}a^{2}-\frac{18\!\cdots\!50}{11\!\cdots\!93}a-\frac{55\!\cdots\!64}{38\!\cdots\!31}$, $\frac{29\!\cdots\!98}{38\!\cdots\!31}a^{26}-\frac{76\!\cdots\!32}{11\!\cdots\!93}a^{25}-\frac{15\!\cdots\!90}{11\!\cdots\!93}a^{24}+\frac{13\!\cdots\!82}{11\!\cdots\!93}a^{23}+\frac{52\!\cdots\!70}{38\!\cdots\!31}a^{22}-\frac{64\!\cdots\!62}{38\!\cdots\!31}a^{21}-\frac{26\!\cdots\!90}{38\!\cdots\!31}a^{20}+\frac{12\!\cdots\!46}{11\!\cdots\!93}a^{19}+\frac{29\!\cdots\!28}{11\!\cdots\!93}a^{18}-\frac{16\!\cdots\!78}{38\!\cdots\!31}a^{17}-\frac{88\!\cdots\!01}{11\!\cdots\!93}a^{16}+\frac{47\!\cdots\!24}{38\!\cdots\!31}a^{15}+\frac{21\!\cdots\!96}{11\!\cdots\!93}a^{14}-\frac{12\!\cdots\!04}{38\!\cdots\!31}a^{13}-\frac{34\!\cdots\!46}{11\!\cdots\!93}a^{12}+\frac{17\!\cdots\!68}{38\!\cdots\!31}a^{11}+\frac{10\!\cdots\!48}{11\!\cdots\!93}a^{10}-\frac{44\!\cdots\!52}{11\!\cdots\!93}a^{9}+\frac{76\!\cdots\!97}{38\!\cdots\!31}a^{8}+\frac{42\!\cdots\!16}{11\!\cdots\!93}a^{7}-\frac{20\!\cdots\!44}{11\!\cdots\!93}a^{6}-\frac{12\!\cdots\!42}{11\!\cdots\!93}a^{5}-\frac{19\!\cdots\!41}{38\!\cdots\!31}a^{4}-\frac{38\!\cdots\!24}{11\!\cdots\!93}a^{3}-\frac{71\!\cdots\!60}{38\!\cdots\!31}a^{2}-\frac{52\!\cdots\!90}{11\!\cdots\!93}a-\frac{13\!\cdots\!22}{38\!\cdots\!31}$, $\frac{83\!\cdots\!16}{38\!\cdots\!31}a^{26}-\frac{15\!\cdots\!73}{11\!\cdots\!93}a^{25}-\frac{44\!\cdots\!08}{11\!\cdots\!93}a^{24}+\frac{27\!\cdots\!57}{11\!\cdots\!93}a^{23}+\frac{45\!\cdots\!64}{11\!\cdots\!93}a^{22}-\frac{44\!\cdots\!89}{11\!\cdots\!93}a^{21}-\frac{23\!\cdots\!01}{11\!\cdots\!93}a^{20}+\frac{31\!\cdots\!07}{11\!\cdots\!93}a^{19}+\frac{30\!\cdots\!75}{38\!\cdots\!31}a^{18}-\frac{41\!\cdots\!64}{38\!\cdots\!31}a^{17}-\frac{28\!\cdots\!22}{11\!\cdots\!93}a^{16}+\frac{11\!\cdots\!88}{38\!\cdots\!31}a^{15}+\frac{70\!\cdots\!09}{11\!\cdots\!93}a^{14}-\frac{32\!\cdots\!25}{38\!\cdots\!31}a^{13}-\frac{41\!\cdots\!58}{38\!\cdots\!31}a^{12}+\frac{13\!\cdots\!29}{11\!\cdots\!93}a^{11}+\frac{67\!\cdots\!21}{11\!\cdots\!93}a^{10}-\frac{13\!\cdots\!46}{11\!\cdots\!93}a^{9}+\frac{13\!\cdots\!75}{38\!\cdots\!31}a^{8}+\frac{47\!\cdots\!55}{38\!\cdots\!31}a^{7}+\frac{16\!\cdots\!81}{11\!\cdots\!93}a^{6}-\frac{40\!\cdots\!53}{11\!\cdots\!93}a^{5}-\frac{21\!\cdots\!65}{11\!\cdots\!93}a^{4}-\frac{12\!\cdots\!92}{11\!\cdots\!93}a^{3}-\frac{77\!\cdots\!82}{11\!\cdots\!93}a^{2}-\frac{72\!\cdots\!36}{38\!\cdots\!31}a-\frac{20\!\cdots\!42}{11\!\cdots\!93}$, $\frac{18\!\cdots\!34}{11\!\cdots\!93}a^{26}-\frac{11\!\cdots\!70}{11\!\cdots\!93}a^{25}-\frac{31\!\cdots\!64}{11\!\cdots\!93}a^{24}+\frac{64\!\cdots\!73}{38\!\cdots\!31}a^{23}+\frac{32\!\cdots\!02}{11\!\cdots\!93}a^{22}-\frac{31\!\cdots\!06}{11\!\cdots\!93}a^{21}-\frac{17\!\cdots\!53}{11\!\cdots\!93}a^{20}+\frac{22\!\cdots\!39}{11\!\cdots\!93}a^{19}+\frac{66\!\cdots\!64}{11\!\cdots\!93}a^{18}-\frac{29\!\cdots\!90}{38\!\cdots\!31}a^{17}-\frac{20\!\cdots\!98}{11\!\cdots\!93}a^{16}+\frac{25\!\cdots\!15}{11\!\cdots\!93}a^{15}+\frac{51\!\cdots\!09}{11\!\cdots\!93}a^{14}-\frac{69\!\cdots\!60}{11\!\cdots\!93}a^{13}-\frac{30\!\cdots\!00}{38\!\cdots\!31}a^{12}+\frac{99\!\cdots\!72}{11\!\cdots\!93}a^{11}+\frac{49\!\cdots\!64}{11\!\cdots\!93}a^{10}-\frac{31\!\cdots\!84}{38\!\cdots\!31}a^{9}+\frac{27\!\cdots\!24}{11\!\cdots\!93}a^{8}+\frac{10\!\cdots\!66}{11\!\cdots\!93}a^{7}+\frac{13\!\cdots\!57}{11\!\cdots\!93}a^{6}-\frac{28\!\cdots\!74}{11\!\cdots\!93}a^{5}-\frac{52\!\cdots\!56}{38\!\cdots\!31}a^{4}-\frac{93\!\cdots\!62}{11\!\cdots\!93}a^{3}-\frac{56\!\cdots\!27}{11\!\cdots\!93}a^{2}-\frac{54\!\cdots\!52}{38\!\cdots\!31}a-\frac{15\!\cdots\!69}{11\!\cdots\!93}$, $\frac{64\!\cdots\!58}{11\!\cdots\!93}a^{26}-\frac{14\!\cdots\!38}{38\!\cdots\!31}a^{25}-\frac{37\!\cdots\!78}{38\!\cdots\!31}a^{24}+\frac{74\!\cdots\!23}{11\!\cdots\!93}a^{23}+\frac{11\!\cdots\!41}{11\!\cdots\!93}a^{22}-\frac{11\!\cdots\!36}{11\!\cdots\!93}a^{21}-\frac{61\!\cdots\!10}{11\!\cdots\!93}a^{20}+\frac{81\!\cdots\!73}{11\!\cdots\!93}a^{19}+\frac{78\!\cdots\!56}{38\!\cdots\!31}a^{18}-\frac{32\!\cdots\!93}{11\!\cdots\!93}a^{17}-\frac{24\!\cdots\!09}{38\!\cdots\!31}a^{16}+\frac{31\!\cdots\!85}{38\!\cdots\!31}a^{15}+\frac{59\!\cdots\!35}{38\!\cdots\!31}a^{14}-\frac{25\!\cdots\!72}{11\!\cdots\!93}a^{13}-\frac{10\!\cdots\!37}{38\!\cdots\!31}a^{12}+\frac{36\!\cdots\!65}{11\!\cdots\!93}a^{11}+\frac{16\!\cdots\!64}{11\!\cdots\!93}a^{10}-\frac{33\!\cdots\!13}{11\!\cdots\!93}a^{9}+\frac{35\!\cdots\!13}{38\!\cdots\!31}a^{8}+\frac{12\!\cdots\!06}{38\!\cdots\!31}a^{7}+\frac{13\!\cdots\!17}{38\!\cdots\!31}a^{6}-\frac{10\!\cdots\!12}{11\!\cdots\!93}a^{5}-\frac{18\!\cdots\!08}{38\!\cdots\!31}a^{4}-\frac{32\!\cdots\!89}{11\!\cdots\!93}a^{3}-\frac{19\!\cdots\!00}{11\!\cdots\!93}a^{2}-\frac{18\!\cdots\!20}{38\!\cdots\!31}a-\frac{54\!\cdots\!65}{11\!\cdots\!93}$, $\frac{21\!\cdots\!23}{11\!\cdots\!93}a^{26}-\frac{14\!\cdots\!43}{11\!\cdots\!93}a^{25}-\frac{12\!\cdots\!73}{38\!\cdots\!31}a^{24}+\frac{83\!\cdots\!38}{38\!\cdots\!31}a^{23}+\frac{12\!\cdots\!09}{38\!\cdots\!31}a^{22}-\frac{39\!\cdots\!40}{11\!\cdots\!93}a^{21}-\frac{19\!\cdots\!48}{11\!\cdots\!93}a^{20}+\frac{27\!\cdots\!45}{11\!\cdots\!93}a^{19}+\frac{25\!\cdots\!77}{38\!\cdots\!31}a^{18}-\frac{36\!\cdots\!38}{38\!\cdots\!31}a^{17}-\frac{77\!\cdots\!22}{38\!\cdots\!31}a^{16}+\frac{30\!\cdots\!35}{11\!\cdots\!93}a^{15}+\frac{57\!\cdots\!71}{11\!\cdots\!93}a^{14}-\frac{28\!\cdots\!03}{38\!\cdots\!31}a^{13}-\frac{10\!\cdots\!25}{11\!\cdots\!93}a^{12}+\frac{11\!\cdots\!59}{11\!\cdots\!93}a^{11}+\frac{49\!\cdots\!45}{11\!\cdots\!93}a^{10}-\frac{10\!\cdots\!04}{11\!\cdots\!93}a^{9}+\frac{37\!\cdots\!92}{11\!\cdots\!93}a^{8}+\frac{11\!\cdots\!96}{11\!\cdots\!93}a^{7}+\frac{12\!\cdots\!36}{11\!\cdots\!93}a^{6}-\frac{32\!\cdots\!54}{11\!\cdots\!93}a^{5}-\frac{58\!\cdots\!32}{38\!\cdots\!31}a^{4}-\frac{10\!\cdots\!67}{11\!\cdots\!93}a^{3}-\frac{63\!\cdots\!64}{11\!\cdots\!93}a^{2}-\frac{59\!\cdots\!50}{38\!\cdots\!31}a-\frac{17\!\cdots\!65}{11\!\cdots\!93}$, $\frac{11\!\cdots\!43}{11\!\cdots\!93}a^{26}-\frac{12\!\cdots\!82}{38\!\cdots\!31}a^{25}-\frac{20\!\cdots\!65}{11\!\cdots\!93}a^{24}+\frac{22\!\cdots\!56}{38\!\cdots\!31}a^{23}+\frac{21\!\cdots\!38}{11\!\cdots\!93}a^{22}-\frac{14\!\cdots\!32}{11\!\cdots\!93}a^{21}-\frac{40\!\cdots\!29}{38\!\cdots\!31}a^{20}+\frac{39\!\cdots\!86}{38\!\cdots\!31}a^{19}+\frac{49\!\cdots\!49}{11\!\cdots\!93}a^{18}-\frac{49\!\cdots\!23}{11\!\cdots\!93}a^{17}-\frac{52\!\cdots\!44}{38\!\cdots\!31}a^{16}+\frac{14\!\cdots\!51}{11\!\cdots\!93}a^{15}+\frac{39\!\cdots\!96}{11\!\cdots\!93}a^{14}-\frac{13\!\cdots\!19}{38\!\cdots\!31}a^{13}-\frac{25\!\cdots\!04}{38\!\cdots\!31}a^{12}+\frac{59\!\cdots\!10}{11\!\cdots\!93}a^{11}+\frac{55\!\cdots\!88}{11\!\cdots\!93}a^{10}-\frac{69\!\cdots\!79}{11\!\cdots\!93}a^{9}+\frac{38\!\cdots\!67}{11\!\cdots\!93}a^{8}+\frac{27\!\cdots\!36}{38\!\cdots\!31}a^{7}+\frac{17\!\cdots\!07}{11\!\cdots\!93}a^{6}-\frac{22\!\cdots\!99}{11\!\cdots\!93}a^{5}-\frac{12\!\cdots\!24}{11\!\cdots\!93}a^{4}-\frac{71\!\cdots\!32}{11\!\cdots\!93}a^{3}-\frac{46\!\cdots\!29}{11\!\cdots\!93}a^{2}-\frac{13\!\cdots\!15}{11\!\cdots\!93}a-\frac{46\!\cdots\!63}{38\!\cdots\!31}$, $\frac{35\!\cdots\!46}{11\!\cdots\!93}a^{26}-\frac{19\!\cdots\!54}{11\!\cdots\!93}a^{25}-\frac{20\!\cdots\!00}{38\!\cdots\!31}a^{24}+\frac{34\!\cdots\!27}{11\!\cdots\!93}a^{23}+\frac{65\!\cdots\!65}{11\!\cdots\!93}a^{22}-\frac{58\!\cdots\!60}{11\!\cdots\!93}a^{21}-\frac{34\!\cdots\!88}{11\!\cdots\!93}a^{20}+\frac{14\!\cdots\!62}{38\!\cdots\!31}a^{19}+\frac{13\!\cdots\!39}{11\!\cdots\!93}a^{18}-\frac{17\!\cdots\!05}{11\!\cdots\!93}a^{17}-\frac{42\!\cdots\!79}{11\!\cdots\!93}a^{16}+\frac{48\!\cdots\!09}{11\!\cdots\!93}a^{15}+\frac{10\!\cdots\!67}{11\!\cdots\!93}a^{14}-\frac{44\!\cdots\!65}{38\!\cdots\!31}a^{13}-\frac{19\!\cdots\!26}{11\!\cdots\!93}a^{12}+\frac{63\!\cdots\!94}{38\!\cdots\!31}a^{11}+\frac{37\!\cdots\!79}{38\!\cdots\!31}a^{10}-\frac{63\!\cdots\!51}{38\!\cdots\!31}a^{9}+\frac{43\!\cdots\!30}{11\!\cdots\!93}a^{8}+\frac{71\!\cdots\!26}{38\!\cdots\!31}a^{7}+\frac{33\!\cdots\!17}{11\!\cdots\!93}a^{6}-\frac{19\!\cdots\!13}{38\!\cdots\!31}a^{5}-\frac{33\!\cdots\!03}{11\!\cdots\!93}a^{4}-\frac{19\!\cdots\!41}{11\!\cdots\!93}a^{3}-\frac{11\!\cdots\!30}{11\!\cdots\!93}a^{2}-\frac{11\!\cdots\!03}{38\!\cdots\!31}a-\frac{34\!\cdots\!71}{11\!\cdots\!93}$, $\frac{31\!\cdots\!25}{11\!\cdots\!93}a^{26}-\frac{22\!\cdots\!28}{11\!\cdots\!93}a^{25}-\frac{55\!\cdots\!92}{11\!\cdots\!93}a^{24}+\frac{39\!\cdots\!60}{11\!\cdots\!93}a^{23}+\frac{56\!\cdots\!93}{11\!\cdots\!93}a^{22}-\frac{20\!\cdots\!84}{38\!\cdots\!31}a^{21}-\frac{29\!\cdots\!04}{11\!\cdots\!93}a^{20}+\frac{13\!\cdots\!44}{38\!\cdots\!31}a^{19}+\frac{11\!\cdots\!02}{11\!\cdots\!93}a^{18}-\frac{16\!\cdots\!51}{11\!\cdots\!93}a^{17}-\frac{34\!\cdots\!92}{11\!\cdots\!93}a^{16}+\frac{46\!\cdots\!36}{11\!\cdots\!93}a^{15}+\frac{83\!\cdots\!13}{11\!\cdots\!93}a^{14}-\frac{12\!\cdots\!12}{11\!\cdots\!93}a^{13}-\frac{14\!\cdots\!08}{11\!\cdots\!93}a^{12}+\frac{60\!\cdots\!49}{38\!\cdots\!31}a^{11}+\frac{65\!\cdots\!88}{11\!\cdots\!93}a^{10}-\frac{16\!\cdots\!44}{11\!\cdots\!93}a^{9}+\frac{20\!\cdots\!66}{38\!\cdots\!31}a^{8}+\frac{55\!\cdots\!11}{38\!\cdots\!31}a^{7}+\frac{40\!\cdots\!03}{38\!\cdots\!31}a^{6}-\frac{46\!\cdots\!85}{11\!\cdots\!93}a^{5}-\frac{24\!\cdots\!59}{11\!\cdots\!93}a^{4}-\frac{15\!\cdots\!88}{11\!\cdots\!93}a^{3}-\frac{88\!\cdots\!70}{11\!\cdots\!93}a^{2}-\frac{24\!\cdots\!00}{11\!\cdots\!93}a-\frac{21\!\cdots\!75}{11\!\cdots\!93}$, $\frac{13\!\cdots\!42}{38\!\cdots\!31}a^{26}-\frac{28\!\cdots\!29}{11\!\cdots\!93}a^{25}-\frac{23\!\cdots\!94}{38\!\cdots\!31}a^{24}+\frac{49\!\cdots\!96}{11\!\cdots\!93}a^{23}+\frac{73\!\cdots\!47}{11\!\cdots\!93}a^{22}-\frac{76\!\cdots\!75}{11\!\cdots\!93}a^{21}-\frac{37\!\cdots\!90}{11\!\cdots\!93}a^{20}+\frac{52\!\cdots\!22}{11\!\cdots\!93}a^{19}+\frac{14\!\cdots\!25}{11\!\cdots\!93}a^{18}-\frac{21\!\cdots\!28}{11\!\cdots\!93}a^{17}-\frac{44\!\cdots\!70}{11\!\cdots\!93}a^{16}+\frac{19\!\cdots\!40}{38\!\cdots\!31}a^{15}+\frac{10\!\cdots\!38}{11\!\cdots\!93}a^{14}-\frac{54\!\cdots\!47}{38\!\cdots\!31}a^{13}-\frac{18\!\cdots\!75}{11\!\cdots\!93}a^{12}+\frac{23\!\cdots\!45}{11\!\cdots\!93}a^{11}+\frac{29\!\cdots\!71}{38\!\cdots\!31}a^{10}-\frac{21\!\cdots\!25}{11\!\cdots\!93}a^{9}+\frac{76\!\cdots\!10}{11\!\cdots\!93}a^{8}+\frac{21\!\cdots\!92}{11\!\cdots\!93}a^{7}+\frac{66\!\cdots\!17}{38\!\cdots\!31}a^{6}-\frac{61\!\cdots\!32}{11\!\cdots\!93}a^{5}-\frac{32\!\cdots\!83}{11\!\cdots\!93}a^{4}-\frac{66\!\cdots\!07}{38\!\cdots\!31}a^{3}-\frac{39\!\cdots\!24}{38\!\cdots\!31}a^{2}-\frac{32\!\cdots\!35}{11\!\cdots\!93}a-\frac{10\!\cdots\!71}{38\!\cdots\!31}$, $\frac{18\!\cdots\!23}{11\!\cdots\!93}a^{26}-\frac{11\!\cdots\!78}{11\!\cdots\!93}a^{25}-\frac{11\!\cdots\!67}{38\!\cdots\!31}a^{24}+\frac{67\!\cdots\!13}{38\!\cdots\!31}a^{23}+\frac{34\!\cdots\!28}{11\!\cdots\!93}a^{22}-\frac{10\!\cdots\!95}{38\!\cdots\!31}a^{21}-\frac{17\!\cdots\!82}{11\!\cdots\!93}a^{20}+\frac{23\!\cdots\!10}{11\!\cdots\!93}a^{19}+\frac{69\!\cdots\!35}{11\!\cdots\!93}a^{18}-\frac{92\!\cdots\!50}{11\!\cdots\!93}a^{17}-\frac{71\!\cdots\!33}{38\!\cdots\!31}a^{16}+\frac{26\!\cdots\!02}{11\!\cdots\!93}a^{15}+\frac{17\!\cdots\!83}{38\!\cdots\!31}a^{14}-\frac{72\!\cdots\!50}{11\!\cdots\!93}a^{13}-\frac{94\!\cdots\!74}{11\!\cdots\!93}a^{12}+\frac{10\!\cdots\!38}{11\!\cdots\!93}a^{11}+\frac{52\!\cdots\!09}{11\!\cdots\!93}a^{10}-\frac{10\!\cdots\!55}{11\!\cdots\!93}a^{9}+\frac{94\!\cdots\!83}{38\!\cdots\!31}a^{8}+\frac{10\!\cdots\!35}{11\!\cdots\!93}a^{7}+\frac{13\!\cdots\!28}{11\!\cdots\!93}a^{6}-\frac{10\!\cdots\!08}{38\!\cdots\!31}a^{5}-\frac{16\!\cdots\!36}{11\!\cdots\!93}a^{4}-\frac{96\!\cdots\!25}{11\!\cdots\!93}a^{3}-\frac{19\!\cdots\!70}{38\!\cdots\!31}a^{2}-\frac{55\!\cdots\!61}{38\!\cdots\!31}a-\frac{15\!\cdots\!30}{11\!\cdots\!93}$
| 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: | \( 6360080771.530314 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{9}\cdot 6360080771.530314 \cdot 1}{2\cdot\sqrt{17717054310925604811052271173453197606912}}\cr\approx \mathstrut & 0.186691976425618 \end{aligned}\] (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 + 189*x^23 - 63*x^22 - 1062*x^21 + 648*x^20 + 4464*x^19 - 2679*x^18 - 14499*x^17 + 7092*x^16 + 36963*x^15 - 21411*x^14 - 74187*x^13 + 24255*x^12 + 61398*x^11 - 36351*x^10 - 17520*x^9 + 66663*x^8 + 42507*x^7 - 11583*x^6 - 18603*x^5 - 10539*x^4 - 6327*x^3 - 2835*x^2 - 639*x - 53)
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 + 189*x^23 - 63*x^22 - 1062*x^21 + 648*x^20 + 4464*x^19 - 2679*x^18 - 14499*x^17 + 7092*x^16 + 36963*x^15 - 21411*x^14 - 74187*x^13 + 24255*x^12 + 61398*x^11 - 36351*x^10 - 17520*x^9 + 66663*x^8 + 42507*x^7 - 11583*x^6 - 18603*x^5 - 10539*x^4 - 6327*x^3 - 2835*x^2 - 639*x - 53, 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 + 189*x^23 - 63*x^22 - 1062*x^21 + 648*x^20 + 4464*x^19 - 2679*x^18 - 14499*x^17 + 7092*x^16 + 36963*x^15 - 21411*x^14 - 74187*x^13 + 24255*x^12 + 61398*x^11 - 36351*x^10 - 17520*x^9 + 66663*x^8 + 42507*x^7 - 11583*x^6 - 18603*x^5 - 10539*x^4 - 6327*x^3 - 2835*x^2 - 639*x - 53);
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 + 189*x^23 - 63*x^22 - 1062*x^21 + 648*x^20 + 4464*x^19 - 2679*x^18 - 14499*x^17 + 7092*x^16 + 36963*x^15 - 21411*x^14 - 74187*x^13 + 24255*x^12 + 61398*x^11 - 36351*x^10 - 17520*x^9 + 66663*x^8 + 42507*x^7 - 11583*x^6 - 18603*x^5 - 10539*x^4 - 6327*x^3 - 2835*x^2 - 639*x - 53);
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 $S_3\times C_9$ is not computed |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 3.1.972.1, \(\Q(\zeta_{27})^+\), 9.3.74384733888.5 |
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 |
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 $27$ | $27$ | $1$ | $73$ |
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) | $1$ | $-1$ | |
| 1.9.6t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| 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) | $0$ | $1$ |
| * | 1.27.9t1.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) | $0$ | $1$ |
| 1.27.18t1.a.a | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 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) | $0$ | $-1$ | |
| * | 1.27.9t1.a.d | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.27.9t1.a.e | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| 1.27.18t1.a.d | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| * | 1.27.9t1.a.f | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| 1.27.18t1.a.e | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.18t1.a.f | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| * | 2.972.3t2.c.a | $2$ | $ 2^{2} \cdot 3^{5}$ | 3.1.972.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.972.6t5.c.a | $2$ | $ 2^{2} \cdot 3^{5}$ | 6.0.2834352.3 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.972.6t5.c.b | $2$ | $ 2^{2} \cdot 3^{5}$ | 6.0.2834352.3 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.2916.18t16.c.a | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.17717054310925604811052271173453197606912.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.c.b | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.17717054310925604811052271173453197606912.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.c.c | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.17717054310925604811052271173453197606912.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.c.d | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.17717054310925604811052271173453197606912.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.c.e | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.17717054310925604811052271173453197606912.1 | $S_3\times C_9$ (as 27T12) | $0$ | $0$ |
| * | 2.2916.18t16.c.f | $2$ | $ 2^{2} \cdot 3^{6}$ | 27.9.17717054310925604811052271173453197606912.1 | $S_3\times C_9$ (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 *.