Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 8*x^26 + 46*x^25 - 158*x^24 + 412*x^23 - 651*x^22 + 948*x^21 - 1384*x^20 + 7521*x^19 - 22249*x^18 + 61004*x^17 - 97499*x^16 + 110127*x^15 + 98440*x^14 - 785239*x^13 + 2068135*x^12 - 3012058*x^11 + 444932*x^10 + 9236470*x^9 - 28098698*x^8 + 53493798*x^7 - 76988059*x^6 + 87553666*x^5 - 79021445*x^4 + 56695661*x^3 - 30768538*x^2 + 10754601*x - 1715933)
gp: K = bnfinit(y^27 - 8*y^26 + 46*y^25 - 158*y^24 + 412*y^23 - 651*y^22 + 948*y^21 - 1384*y^20 + 7521*y^19 - 22249*y^18 + 61004*y^17 - 97499*y^16 + 110127*y^15 + 98440*y^14 - 785239*y^13 + 2068135*y^12 - 3012058*y^11 + 444932*y^10 + 9236470*y^9 - 28098698*y^8 + 53493798*y^7 - 76988059*y^6 + 87553666*y^5 - 79021445*y^4 + 56695661*y^3 - 30768538*y^2 + 10754601*y - 1715933, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 8*x^26 + 46*x^25 - 158*x^24 + 412*x^23 - 651*x^22 + 948*x^21 - 1384*x^20 + 7521*x^19 - 22249*x^18 + 61004*x^17 - 97499*x^16 + 110127*x^15 + 98440*x^14 - 785239*x^13 + 2068135*x^12 - 3012058*x^11 + 444932*x^10 + 9236470*x^9 - 28098698*x^8 + 53493798*x^7 - 76988059*x^6 + 87553666*x^5 - 79021445*x^4 + 56695661*x^3 - 30768538*x^2 + 10754601*x - 1715933);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 8*x^26 + 46*x^25 - 158*x^24 + 412*x^23 - 651*x^22 + 948*x^21 - 1384*x^20 + 7521*x^19 - 22249*x^18 + 61004*x^17 - 97499*x^16 + 110127*x^15 + 98440*x^14 - 785239*x^13 + 2068135*x^12 - 3012058*x^11 + 444932*x^10 + 9236470*x^9 - 28098698*x^8 + 53493798*x^7 - 76988059*x^6 + 87553666*x^5 - 79021445*x^4 + 56695661*x^3 - 30768538*x^2 + 10754601*x - 1715933)
\( x^{27} - 8 x^{26} + 46 x^{25} - 158 x^{24} + 412 x^{23} - 651 x^{22} + 948 x^{21} - 1384 x^{20} + \cdots - 1715933 \)
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: | $[1, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-21988570612019400506053514781537922706637489911\)
\(\medspace = -\,3671^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(52.04\) | 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: | $3671^{1/2}\approx 60.58877783880444$ | ||
| Ramified primes: |
\(3671\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3671}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{1417}a^{24}-\frac{111}{1417}a^{23}+\frac{505}{1417}a^{22}-\frac{28}{109}a^{21}-\frac{437}{1417}a^{20}-\frac{444}{1417}a^{19}-\frac{45}{109}a^{18}+\frac{30}{109}a^{17}+\frac{326}{1417}a^{16}+\frac{424}{1417}a^{15}+\frac{196}{1417}a^{14}+\frac{54}{1417}a^{13}-\frac{349}{1417}a^{12}-\frac{190}{1417}a^{11}+\frac{215}{1417}a^{10}+\frac{605}{1417}a^{9}-\frac{600}{1417}a^{8}-\frac{183}{1417}a^{7}-\frac{461}{1417}a^{6}+\frac{454}{1417}a^{5}+\frac{561}{1417}a^{4}-\frac{617}{1417}a^{3}-\frac{437}{1417}a^{2}+\frac{404}{1417}a-\frac{447}{1417}$, $\frac{1}{58097}a^{25}+\frac{1}{58097}a^{24}+\frac{24915}{58097}a^{23}-\frac{1901}{58097}a^{22}+\frac{12641}{58097}a^{21}+\frac{7292}{58097}a^{20}-\frac{6386}{58097}a^{19}+\frac{222}{4469}a^{18}-\frac{1338}{58097}a^{17}+\frac{7179}{58097}a^{16}-\frac{1019}{4469}a^{15}-\frac{19087}{58097}a^{14}-\frac{9888}{58097}a^{13}+\frac{28738}{58097}a^{12}-\frac{19648}{58097}a^{11}-\frac{27744}{58097}a^{10}-\frac{16443}{58097}a^{9}+\frac{50}{1417}a^{8}+\frac{7383}{58097}a^{7}+\frac{14004}{58097}a^{6}+\frac{15984}{58097}a^{5}+\frac{135}{1417}a^{4}+\frac{19730}{58097}a^{3}+\frac{10974}{58097}a^{2}+\frac{125}{1417}a-\frac{11805}{58097}$, $\frac{1}{30\!\cdots\!57}a^{26}+\frac{57\!\cdots\!07}{30\!\cdots\!57}a^{25}-\frac{86\!\cdots\!72}{30\!\cdots\!57}a^{24}-\frac{77\!\cdots\!39}{30\!\cdots\!57}a^{23}+\frac{39\!\cdots\!90}{30\!\cdots\!57}a^{22}+\frac{95\!\cdots\!23}{30\!\cdots\!57}a^{21}-\frac{42\!\cdots\!01}{30\!\cdots\!57}a^{20}+\frac{20\!\cdots\!07}{30\!\cdots\!57}a^{19}-\frac{28\!\cdots\!51}{30\!\cdots\!57}a^{18}+\frac{16\!\cdots\!66}{30\!\cdots\!57}a^{17}+\frac{39\!\cdots\!86}{81\!\cdots\!61}a^{16}+\frac{61\!\cdots\!53}{30\!\cdots\!57}a^{15}-\frac{12\!\cdots\!61}{30\!\cdots\!57}a^{14}+\frac{79\!\cdots\!45}{30\!\cdots\!57}a^{13}-\frac{13\!\cdots\!76}{30\!\cdots\!57}a^{12}+\frac{10\!\cdots\!42}{56\!\cdots\!29}a^{11}+\frac{97\!\cdots\!88}{30\!\cdots\!57}a^{10}+\frac{13\!\cdots\!98}{30\!\cdots\!57}a^{9}-\frac{55\!\cdots\!29}{30\!\cdots\!57}a^{8}-\frac{11\!\cdots\!75}{73\!\cdots\!77}a^{7}-\frac{39\!\cdots\!70}{30\!\cdots\!57}a^{6}-\frac{92\!\cdots\!57}{30\!\cdots\!57}a^{5}+\frac{12\!\cdots\!76}{30\!\cdots\!57}a^{4}+\frac{53\!\cdots\!57}{30\!\cdots\!57}a^{3}+\frac{11\!\cdots\!42}{30\!\cdots\!57}a^{2}+\frac{17\!\cdots\!54}{23\!\cdots\!89}a+\frac{67\!\cdots\!20}{30\!\cdots\!57}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $13$ | 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: |
$\frac{86\!\cdots\!83}{81\!\cdots\!61}a^{26}-\frac{63\!\cdots\!41}{81\!\cdots\!61}a^{25}+\frac{35\!\cdots\!50}{81\!\cdots\!61}a^{24}-\frac{11\!\cdots\!39}{81\!\cdots\!61}a^{23}+\frac{28\!\cdots\!93}{81\!\cdots\!61}a^{22}-\frac{38\!\cdots\!50}{81\!\cdots\!61}a^{21}+\frac{57\!\cdots\!16}{81\!\cdots\!61}a^{20}-\frac{82\!\cdots\!94}{81\!\cdots\!61}a^{19}+\frac{59\!\cdots\!46}{81\!\cdots\!61}a^{18}-\frac{15\!\cdots\!09}{81\!\cdots\!61}a^{17}+\frac{42\!\cdots\!58}{81\!\cdots\!61}a^{16}-\frac{56\!\cdots\!77}{81\!\cdots\!61}a^{15}+\frac{45\!\cdots\!19}{62\!\cdots\!97}a^{14}+\frac{12\!\cdots\!76}{81\!\cdots\!61}a^{13}-\frac{59\!\cdots\!17}{81\!\cdots\!61}a^{12}+\frac{14\!\cdots\!06}{81\!\cdots\!61}a^{11}-\frac{17\!\cdots\!95}{81\!\cdots\!61}a^{10}-\frac{48\!\cdots\!58}{57\!\cdots\!33}a^{9}+\frac{75\!\cdots\!28}{81\!\cdots\!61}a^{8}-\frac{19\!\cdots\!47}{81\!\cdots\!61}a^{7}+\frac{33\!\cdots\!21}{81\!\cdots\!61}a^{6}-\frac{45\!\cdots\!23}{81\!\cdots\!61}a^{5}+\frac{47\!\cdots\!05}{81\!\cdots\!61}a^{4}-\frac{38\!\cdots\!27}{81\!\cdots\!61}a^{3}+\frac{24\!\cdots\!19}{81\!\cdots\!61}a^{2}-\frac{10\!\cdots\!13}{81\!\cdots\!61}a+\frac{22\!\cdots\!43}{81\!\cdots\!61}$, $\frac{48\!\cdots\!52}{30\!\cdots\!57}a^{26}-\frac{35\!\cdots\!35}{30\!\cdots\!57}a^{25}+\frac{20\!\cdots\!00}{30\!\cdots\!57}a^{24}-\frac{63\!\cdots\!80}{30\!\cdots\!57}a^{23}+\frac{12\!\cdots\!60}{23\!\cdots\!89}a^{22}-\frac{21\!\cdots\!20}{30\!\cdots\!57}a^{21}+\frac{32\!\cdots\!80}{30\!\cdots\!57}a^{20}-\frac{46\!\cdots\!18}{30\!\cdots\!57}a^{19}+\frac{33\!\cdots\!91}{30\!\cdots\!57}a^{18}-\frac{86\!\cdots\!52}{30\!\cdots\!57}a^{17}+\frac{50\!\cdots\!83}{62\!\cdots\!97}a^{16}-\frac{31\!\cdots\!10}{30\!\cdots\!57}a^{15}+\frac{32\!\cdots\!12}{30\!\cdots\!57}a^{14}+\frac{69\!\cdots\!71}{30\!\cdots\!57}a^{13}-\frac{33\!\cdots\!24}{30\!\cdots\!57}a^{12}+\frac{78\!\cdots\!35}{30\!\cdots\!57}a^{11}-\frac{94\!\cdots\!74}{30\!\cdots\!57}a^{10}-\frac{41\!\cdots\!19}{30\!\cdots\!57}a^{9}+\frac{32\!\cdots\!74}{23\!\cdots\!89}a^{8}-\frac{10\!\cdots\!99}{30\!\cdots\!57}a^{7}+\frac{18\!\cdots\!86}{30\!\cdots\!57}a^{6}-\frac{25\!\cdots\!29}{30\!\cdots\!57}a^{5}+\frac{26\!\cdots\!80}{30\!\cdots\!57}a^{4}-\frac{21\!\cdots\!82}{30\!\cdots\!57}a^{3}+\frac{13\!\cdots\!88}{30\!\cdots\!57}a^{2}-\frac{60\!\cdots\!09}{30\!\cdots\!57}a+\frac{12\!\cdots\!45}{30\!\cdots\!57}$, $\frac{16\!\cdots\!90}{30\!\cdots\!57}a^{26}-\frac{12\!\cdots\!33}{30\!\cdots\!57}a^{25}+\frac{69\!\cdots\!89}{30\!\cdots\!57}a^{24}-\frac{22\!\cdots\!37}{30\!\cdots\!57}a^{23}+\frac{54\!\cdots\!40}{30\!\cdots\!57}a^{22}-\frac{72\!\cdots\!46}{30\!\cdots\!57}a^{21}+\frac{11\!\cdots\!24}{30\!\cdots\!57}a^{20}-\frac{15\!\cdots\!16}{30\!\cdots\!57}a^{19}+\frac{11\!\cdots\!64}{30\!\cdots\!57}a^{18}-\frac{29\!\cdots\!36}{30\!\cdots\!57}a^{17}+\frac{22\!\cdots\!01}{81\!\cdots\!61}a^{16}-\frac{10\!\cdots\!48}{30\!\cdots\!57}a^{15}+\frac{27\!\cdots\!08}{73\!\cdots\!77}a^{14}+\frac{24\!\cdots\!48}{30\!\cdots\!57}a^{13}-\frac{89\!\cdots\!64}{23\!\cdots\!89}a^{12}+\frac{27\!\cdots\!73}{30\!\cdots\!57}a^{11}-\frac{32\!\cdots\!36}{30\!\cdots\!57}a^{10}-\frac{14\!\cdots\!11}{30\!\cdots\!57}a^{9}+\frac{14\!\cdots\!16}{30\!\cdots\!57}a^{8}-\frac{37\!\cdots\!43}{30\!\cdots\!57}a^{7}+\frac{64\!\cdots\!05}{30\!\cdots\!57}a^{6}-\frac{85\!\cdots\!58}{30\!\cdots\!57}a^{5}+\frac{89\!\cdots\!19}{30\!\cdots\!57}a^{4}-\frac{72\!\cdots\!58}{30\!\cdots\!57}a^{3}+\frac{45\!\cdots\!48}{30\!\cdots\!57}a^{2}-\frac{20\!\cdots\!32}{30\!\cdots\!57}a+\frac{38\!\cdots\!92}{27\!\cdots\!73}$, $\frac{85\!\cdots\!57}{23\!\cdots\!89}a^{26}-\frac{80\!\cdots\!06}{30\!\cdots\!57}a^{25}+\frac{45\!\cdots\!81}{30\!\cdots\!57}a^{24}-\frac{14\!\cdots\!82}{30\!\cdots\!57}a^{23}+\frac{35\!\cdots\!14}{30\!\cdots\!57}a^{22}-\frac{46\!\cdots\!09}{30\!\cdots\!57}a^{21}+\frac{70\!\cdots\!48}{30\!\cdots\!57}a^{20}-\frac{10\!\cdots\!23}{30\!\cdots\!57}a^{19}+\frac{58\!\cdots\!11}{23\!\cdots\!89}a^{18}-\frac{19\!\cdots\!02}{30\!\cdots\!57}a^{17}+\frac{14\!\cdots\!91}{81\!\cdots\!61}a^{16}-\frac{16\!\cdots\!81}{73\!\cdots\!77}a^{15}+\frac{70\!\cdots\!31}{30\!\cdots\!57}a^{14}+\frac{16\!\cdots\!34}{30\!\cdots\!57}a^{13}-\frac{58\!\cdots\!82}{23\!\cdots\!89}a^{12}+\frac{17\!\cdots\!77}{30\!\cdots\!57}a^{11}-\frac{20\!\cdots\!73}{30\!\cdots\!57}a^{10}-\frac{81\!\cdots\!24}{23\!\cdots\!89}a^{9}+\frac{95\!\cdots\!05}{30\!\cdots\!57}a^{8}-\frac{24\!\cdots\!65}{30\!\cdots\!57}a^{7}+\frac{41\!\cdots\!38}{30\!\cdots\!57}a^{6}-\frac{54\!\cdots\!07}{30\!\cdots\!57}a^{5}+\frac{56\!\cdots\!24}{30\!\cdots\!57}a^{4}-\frac{45\!\cdots\!05}{30\!\cdots\!57}a^{3}+\frac{28\!\cdots\!67}{30\!\cdots\!57}a^{2}-\frac{12\!\cdots\!28}{30\!\cdots\!57}a+\frac{24\!\cdots\!18}{30\!\cdots\!57}$, $\frac{68\!\cdots\!90}{30\!\cdots\!57}a^{26}-\frac{52\!\cdots\!72}{30\!\cdots\!57}a^{25}+\frac{29\!\cdots\!76}{30\!\cdots\!57}a^{24}-\frac{98\!\cdots\!95}{30\!\cdots\!57}a^{23}+\frac{24\!\cdots\!50}{30\!\cdots\!57}a^{22}-\frac{35\!\cdots\!56}{30\!\cdots\!57}a^{21}+\frac{50\!\cdots\!19}{30\!\cdots\!57}a^{20}-\frac{77\!\cdots\!86}{30\!\cdots\!57}a^{19}+\frac{48\!\cdots\!53}{30\!\cdots\!57}a^{18}-\frac{13\!\cdots\!88}{30\!\cdots\!57}a^{17}+\frac{98\!\cdots\!77}{81\!\cdots\!61}a^{16}-\frac{54\!\cdots\!74}{30\!\cdots\!57}a^{15}+\frac{52\!\cdots\!56}{30\!\cdots\!57}a^{14}+\frac{85\!\cdots\!03}{30\!\cdots\!57}a^{13}-\frac{52\!\cdots\!66}{30\!\cdots\!57}a^{12}+\frac{86\!\cdots\!59}{21\!\cdots\!21}a^{11}-\frac{16\!\cdots\!10}{30\!\cdots\!57}a^{10}-\frac{36\!\cdots\!04}{30\!\cdots\!57}a^{9}+\frac{63\!\cdots\!41}{30\!\cdots\!57}a^{8}-\frac{17\!\cdots\!03}{30\!\cdots\!57}a^{7}+\frac{30\!\cdots\!28}{30\!\cdots\!57}a^{6}-\frac{41\!\cdots\!83}{30\!\cdots\!57}a^{5}+\frac{33\!\cdots\!04}{23\!\cdots\!89}a^{4}-\frac{36\!\cdots\!06}{30\!\cdots\!57}a^{3}+\frac{23\!\cdots\!46}{30\!\cdots\!57}a^{2}-\frac{10\!\cdots\!22}{30\!\cdots\!57}a+\frac{23\!\cdots\!86}{30\!\cdots\!57}$, $\frac{10\!\cdots\!25}{30\!\cdots\!57}a^{26}-\frac{77\!\cdots\!93}{30\!\cdots\!57}a^{25}+\frac{43\!\cdots\!88}{30\!\cdots\!57}a^{24}-\frac{14\!\cdots\!62}{30\!\cdots\!57}a^{23}+\frac{34\!\cdots\!44}{30\!\cdots\!57}a^{22}-\frac{47\!\cdots\!85}{30\!\cdots\!57}a^{21}+\frac{68\!\cdots\!93}{30\!\cdots\!57}a^{20}-\frac{10\!\cdots\!32}{30\!\cdots\!57}a^{19}+\frac{71\!\cdots\!36}{30\!\cdots\!57}a^{18}-\frac{19\!\cdots\!22}{30\!\cdots\!57}a^{17}+\frac{14\!\cdots\!50}{81\!\cdots\!61}a^{16}-\frac{17\!\cdots\!47}{73\!\cdots\!77}a^{15}+\frac{69\!\cdots\!60}{30\!\cdots\!57}a^{14}+\frac{14\!\cdots\!86}{30\!\cdots\!57}a^{13}-\frac{74\!\cdots\!42}{30\!\cdots\!57}a^{12}+\frac{17\!\cdots\!54}{30\!\cdots\!57}a^{11}-\frac{21\!\cdots\!72}{30\!\cdots\!57}a^{10}-\frac{90\!\cdots\!81}{30\!\cdots\!57}a^{9}+\frac{93\!\cdots\!87}{30\!\cdots\!57}a^{8}-\frac{23\!\cdots\!44}{30\!\cdots\!57}a^{7}+\frac{41\!\cdots\!19}{30\!\cdots\!57}a^{6}-\frac{41\!\cdots\!78}{23\!\cdots\!89}a^{5}+\frac{56\!\cdots\!24}{30\!\cdots\!57}a^{4}-\frac{46\!\cdots\!84}{30\!\cdots\!57}a^{3}+\frac{29\!\cdots\!49}{30\!\cdots\!57}a^{2}-\frac{13\!\cdots\!39}{30\!\cdots\!57}a+\frac{29\!\cdots\!03}{30\!\cdots\!57}$, $\frac{19\!\cdots\!04}{30\!\cdots\!57}a^{26}-\frac{13\!\cdots\!59}{30\!\cdots\!57}a^{25}+\frac{76\!\cdots\!68}{30\!\cdots\!57}a^{24}-\frac{22\!\cdots\!86}{30\!\cdots\!57}a^{23}+\frac{55\!\cdots\!54}{30\!\cdots\!57}a^{22}-\frac{64\!\cdots\!26}{30\!\cdots\!57}a^{21}+\frac{10\!\cdots\!69}{30\!\cdots\!57}a^{20}-\frac{14\!\cdots\!65}{30\!\cdots\!57}a^{19}+\frac{13\!\cdots\!53}{30\!\cdots\!57}a^{18}-\frac{29\!\cdots\!88}{30\!\cdots\!57}a^{17}+\frac{23\!\cdots\!34}{81\!\cdots\!61}a^{16}-\frac{93\!\cdots\!20}{30\!\cdots\!57}a^{15}+\frac{79\!\cdots\!57}{23\!\cdots\!89}a^{14}+\frac{32\!\cdots\!29}{30\!\cdots\!57}a^{13}-\frac{12\!\cdots\!12}{30\!\cdots\!57}a^{12}+\frac{27\!\cdots\!63}{30\!\cdots\!57}a^{11}-\frac{28\!\cdots\!58}{30\!\cdots\!57}a^{10}-\frac{19\!\cdots\!33}{23\!\cdots\!89}a^{9}+\frac{15\!\cdots\!60}{30\!\cdots\!57}a^{8}-\frac{37\!\cdots\!51}{30\!\cdots\!57}a^{7}+\frac{62\!\cdots\!81}{30\!\cdots\!57}a^{6}-\frac{79\!\cdots\!35}{30\!\cdots\!57}a^{5}+\frac{78\!\cdots\!90}{30\!\cdots\!57}a^{4}-\frac{60\!\cdots\!32}{30\!\cdots\!57}a^{3}+\frac{36\!\cdots\!88}{30\!\cdots\!57}a^{2}-\frac{14\!\cdots\!95}{30\!\cdots\!57}a+\frac{21\!\cdots\!11}{30\!\cdots\!57}$, $\frac{33\!\cdots\!25}{30\!\cdots\!57}a^{26}-\frac{22\!\cdots\!79}{30\!\cdots\!57}a^{25}+\frac{12\!\cdots\!91}{30\!\cdots\!57}a^{24}-\frac{36\!\cdots\!54}{30\!\cdots\!57}a^{23}+\frac{87\!\cdots\!56}{30\!\cdots\!57}a^{22}-\frac{92\!\cdots\!61}{30\!\cdots\!57}a^{21}+\frac{13\!\cdots\!49}{23\!\cdots\!89}a^{20}-\frac{21\!\cdots\!10}{30\!\cdots\!57}a^{19}+\frac{22\!\cdots\!54}{30\!\cdots\!57}a^{18}-\frac{45\!\cdots\!59}{30\!\cdots\!57}a^{17}+\frac{34\!\cdots\!53}{74\!\cdots\!29}a^{16}-\frac{13\!\cdots\!55}{30\!\cdots\!57}a^{15}+\frac{15\!\cdots\!85}{30\!\cdots\!57}a^{14}+\frac{56\!\cdots\!08}{30\!\cdots\!57}a^{13}-\frac{19\!\cdots\!04}{30\!\cdots\!57}a^{12}+\frac{41\!\cdots\!06}{30\!\cdots\!57}a^{11}-\frac{39\!\cdots\!19}{30\!\cdots\!57}a^{10}-\frac{50\!\cdots\!42}{30\!\cdots\!57}a^{9}+\frac{25\!\cdots\!26}{30\!\cdots\!57}a^{8}-\frac{58\!\cdots\!45}{30\!\cdots\!57}a^{7}+\frac{72\!\cdots\!76}{23\!\cdots\!89}a^{6}-\frac{11\!\cdots\!96}{30\!\cdots\!57}a^{5}+\frac{11\!\cdots\!87}{30\!\cdots\!57}a^{4}-\frac{85\!\cdots\!91}{30\!\cdots\!57}a^{3}+\frac{50\!\cdots\!44}{30\!\cdots\!57}a^{2}-\frac{17\!\cdots\!43}{30\!\cdots\!57}a+\frac{18\!\cdots\!27}{30\!\cdots\!57}$, $\frac{37\!\cdots\!36}{30\!\cdots\!57}a^{26}-\frac{27\!\cdots\!08}{30\!\cdots\!57}a^{25}+\frac{36\!\cdots\!58}{73\!\cdots\!77}a^{24}-\frac{47\!\cdots\!95}{30\!\cdots\!57}a^{23}+\frac{11\!\cdots\!46}{30\!\cdots\!57}a^{22}-\frac{15\!\cdots\!74}{30\!\cdots\!57}a^{21}+\frac{23\!\cdots\!28}{30\!\cdots\!57}a^{20}-\frac{34\!\cdots\!78}{30\!\cdots\!57}a^{19}+\frac{25\!\cdots\!19}{30\!\cdots\!57}a^{18}-\frac{64\!\cdots\!65}{30\!\cdots\!57}a^{17}+\frac{48\!\cdots\!09}{81\!\cdots\!61}a^{16}-\frac{23\!\cdots\!82}{30\!\cdots\!57}a^{15}+\frac{24\!\cdots\!89}{30\!\cdots\!57}a^{14}+\frac{41\!\cdots\!81}{23\!\cdots\!89}a^{13}-\frac{25\!\cdots\!21}{30\!\cdots\!57}a^{12}+\frac{58\!\cdots\!61}{30\!\cdots\!57}a^{11}-\frac{53\!\cdots\!80}{23\!\cdots\!89}a^{10}-\frac{34\!\cdots\!90}{30\!\cdots\!57}a^{9}+\frac{31\!\cdots\!35}{30\!\cdots\!57}a^{8}-\frac{81\!\cdots\!81}{30\!\cdots\!57}a^{7}+\frac{13\!\cdots\!93}{30\!\cdots\!57}a^{6}-\frac{18\!\cdots\!66}{30\!\cdots\!57}a^{5}+\frac{19\!\cdots\!93}{30\!\cdots\!57}a^{4}-\frac{15\!\cdots\!19}{30\!\cdots\!57}a^{3}+\frac{98\!\cdots\!53}{30\!\cdots\!57}a^{2}-\frac{42\!\cdots\!44}{30\!\cdots\!57}a+\frac{86\!\cdots\!95}{30\!\cdots\!57}$, $\frac{50\!\cdots\!44}{30\!\cdots\!57}a^{26}-\frac{36\!\cdots\!11}{30\!\cdots\!57}a^{25}+\frac{20\!\cdots\!36}{30\!\cdots\!57}a^{24}-\frac{65\!\cdots\!48}{30\!\cdots\!57}a^{23}+\frac{16\!\cdots\!25}{30\!\cdots\!57}a^{22}-\frac{16\!\cdots\!65}{23\!\cdots\!89}a^{21}+\frac{32\!\cdots\!12}{30\!\cdots\!57}a^{20}-\frac{36\!\cdots\!55}{23\!\cdots\!89}a^{19}+\frac{34\!\cdots\!09}{30\!\cdots\!57}a^{18}-\frac{88\!\cdots\!94}{30\!\cdots\!57}a^{17}+\frac{66\!\cdots\!90}{81\!\cdots\!61}a^{16}-\frac{32\!\cdots\!93}{30\!\cdots\!57}a^{15}+\frac{33\!\cdots\!24}{30\!\cdots\!57}a^{14}+\frac{72\!\cdots\!93}{30\!\cdots\!57}a^{13}-\frac{34\!\cdots\!23}{30\!\cdots\!57}a^{12}+\frac{80\!\cdots\!25}{30\!\cdots\!57}a^{11}-\frac{96\!\cdots\!43}{30\!\cdots\!57}a^{10}-\frac{10\!\cdots\!41}{73\!\cdots\!77}a^{9}+\frac{43\!\cdots\!46}{30\!\cdots\!57}a^{8}-\frac{11\!\cdots\!45}{30\!\cdots\!57}a^{7}+\frac{19\!\cdots\!93}{30\!\cdots\!57}a^{6}-\frac{47\!\cdots\!19}{56\!\cdots\!29}a^{5}+\frac{20\!\cdots\!80}{23\!\cdots\!89}a^{4}-\frac{21\!\cdots\!32}{30\!\cdots\!57}a^{3}+\frac{33\!\cdots\!70}{73\!\cdots\!77}a^{2}-\frac{60\!\cdots\!02}{30\!\cdots\!57}a+\frac{23\!\cdots\!78}{56\!\cdots\!29}$, $\frac{93\!\cdots\!61}{30\!\cdots\!57}a^{26}-\frac{68\!\cdots\!88}{30\!\cdots\!57}a^{25}+\frac{38\!\cdots\!93}{30\!\cdots\!57}a^{24}-\frac{12\!\cdots\!80}{30\!\cdots\!57}a^{23}+\frac{29\!\cdots\!35}{30\!\cdots\!57}a^{22}-\frac{39\!\cdots\!80}{30\!\cdots\!57}a^{21}+\frac{59\!\cdots\!15}{30\!\cdots\!57}a^{20}-\frac{65\!\cdots\!26}{23\!\cdots\!89}a^{19}+\frac{15\!\cdots\!28}{73\!\cdots\!77}a^{18}-\frac{16\!\cdots\!60}{30\!\cdots\!57}a^{17}+\frac{94\!\cdots\!63}{62\!\cdots\!97}a^{16}-\frac{58\!\cdots\!99}{30\!\cdots\!57}a^{15}+\frac{60\!\cdots\!45}{30\!\cdots\!57}a^{14}+\frac{13\!\cdots\!32}{30\!\cdots\!57}a^{13}-\frac{63\!\cdots\!06}{30\!\cdots\!57}a^{12}+\frac{14\!\cdots\!91}{30\!\cdots\!57}a^{11}-\frac{17\!\cdots\!46}{30\!\cdots\!57}a^{10}-\frac{87\!\cdots\!56}{30\!\cdots\!57}a^{9}+\frac{80\!\cdots\!55}{30\!\cdots\!57}a^{8}-\frac{20\!\cdots\!88}{30\!\cdots\!57}a^{7}+\frac{35\!\cdots\!40}{30\!\cdots\!57}a^{6}-\frac{46\!\cdots\!11}{30\!\cdots\!57}a^{5}+\frac{47\!\cdots\!64}{30\!\cdots\!57}a^{4}-\frac{38\!\cdots\!58}{30\!\cdots\!57}a^{3}+\frac{24\!\cdots\!50}{30\!\cdots\!57}a^{2}-\frac{10\!\cdots\!61}{30\!\cdots\!57}a+\frac{16\!\cdots\!07}{23\!\cdots\!89}$, $\frac{42\!\cdots\!27}{30\!\cdots\!57}a^{26}-\frac{12\!\cdots\!31}{30\!\cdots\!57}a^{25}-\frac{12\!\cdots\!36}{30\!\cdots\!57}a^{24}+\frac{46\!\cdots\!15}{30\!\cdots\!57}a^{23}-\frac{13\!\cdots\!99}{30\!\cdots\!57}a^{22}+\frac{47\!\cdots\!40}{30\!\cdots\!57}a^{21}-\frac{26\!\cdots\!70}{30\!\cdots\!57}a^{20}+\frac{91\!\cdots\!15}{30\!\cdots\!57}a^{19}+\frac{16\!\cdots\!32}{30\!\cdots\!57}a^{18}+\frac{11\!\cdots\!48}{30\!\cdots\!57}a^{17}-\frac{28\!\cdots\!44}{81\!\cdots\!61}a^{16}+\frac{85\!\cdots\!98}{30\!\cdots\!57}a^{15}-\frac{35\!\cdots\!77}{30\!\cdots\!57}a^{14}+\frac{18\!\cdots\!42}{30\!\cdots\!57}a^{13}+\frac{29\!\cdots\!10}{30\!\cdots\!57}a^{12}-\frac{71\!\cdots\!04}{30\!\cdots\!57}a^{11}+\frac{24\!\cdots\!53}{30\!\cdots\!57}a^{10}-\frac{23\!\cdots\!98}{30\!\cdots\!57}a^{9}-\frac{13\!\cdots\!66}{30\!\cdots\!57}a^{8}+\frac{11\!\cdots\!05}{30\!\cdots\!57}a^{7}-\frac{27\!\cdots\!44}{30\!\cdots\!57}a^{6}+\frac{33\!\cdots\!51}{23\!\cdots\!89}a^{5}-\frac{55\!\cdots\!68}{30\!\cdots\!57}a^{4}+\frac{50\!\cdots\!62}{30\!\cdots\!57}a^{3}-\frac{37\!\cdots\!84}{30\!\cdots\!57}a^{2}+\frac{21\!\cdots\!83}{30\!\cdots\!57}a-\frac{59\!\cdots\!72}{30\!\cdots\!57}$, $\frac{81\!\cdots\!93}{81\!\cdots\!61}a^{26}-\frac{58\!\cdots\!31}{81\!\cdots\!61}a^{25}+\frac{32\!\cdots\!55}{81\!\cdots\!61}a^{24}-\frac{10\!\cdots\!40}{81\!\cdots\!61}a^{23}+\frac{24\!\cdots\!82}{81\!\cdots\!61}a^{22}-\frac{31\!\cdots\!97}{81\!\cdots\!61}a^{21}+\frac{50\!\cdots\!80}{81\!\cdots\!61}a^{20}-\frac{69\!\cdots\!39}{81\!\cdots\!61}a^{19}+\frac{55\!\cdots\!31}{81\!\cdots\!61}a^{18}-\frac{13\!\cdots\!88}{81\!\cdots\!61}a^{17}+\frac{38\!\cdots\!26}{81\!\cdots\!61}a^{16}-\frac{46\!\cdots\!31}{81\!\cdots\!61}a^{15}+\frac{49\!\cdots\!43}{81\!\cdots\!61}a^{14}+\frac{12\!\cdots\!62}{81\!\cdots\!61}a^{13}-\frac{53\!\cdots\!25}{81\!\cdots\!61}a^{12}+\frac{12\!\cdots\!85}{81\!\cdots\!61}a^{11}-\frac{14\!\cdots\!01}{81\!\cdots\!61}a^{10}-\frac{83\!\cdots\!94}{81\!\cdots\!61}a^{9}+\frac{68\!\cdots\!24}{81\!\cdots\!61}a^{8}-\frac{17\!\cdots\!47}{81\!\cdots\!61}a^{7}+\frac{22\!\cdots\!48}{62\!\cdots\!97}a^{6}-\frac{37\!\cdots\!12}{81\!\cdots\!61}a^{5}+\frac{38\!\cdots\!09}{81\!\cdots\!61}a^{4}-\frac{31\!\cdots\!98}{81\!\cdots\!61}a^{3}+\frac{19\!\cdots\!69}{81\!\cdots\!61}a^{2}-\frac{83\!\cdots\!22}{81\!\cdots\!61}a+\frac{16\!\cdots\!76}{81\!\cdots\!61}$
| 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: | \( 1347745809122.2559 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 1347745809122.2559 \cdot 1}{2\cdot\sqrt{21988570612019400506053514781537922706637489911}}\cr\approx \mathstrut & 0.216196062191980 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 - 8*x^26 + 46*x^25 - 158*x^24 + 412*x^23 - 651*x^22 + 948*x^21 - 1384*x^20 + 7521*x^19 - 22249*x^18 + 61004*x^17 - 97499*x^16 + 110127*x^15 + 98440*x^14 - 785239*x^13 + 2068135*x^12 - 3012058*x^11 + 444932*x^10 + 9236470*x^9 - 28098698*x^8 + 53493798*x^7 - 76988059*x^6 + 87553666*x^5 - 79021445*x^4 + 56695661*x^3 - 30768538*x^2 + 10754601*x - 1715933)
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 - 8*x^26 + 46*x^25 - 158*x^24 + 412*x^23 - 651*x^22 + 948*x^21 - 1384*x^20 + 7521*x^19 - 22249*x^18 + 61004*x^17 - 97499*x^16 + 110127*x^15 + 98440*x^14 - 785239*x^13 + 2068135*x^12 - 3012058*x^11 + 444932*x^10 + 9236470*x^9 - 28098698*x^8 + 53493798*x^7 - 76988059*x^6 + 87553666*x^5 - 79021445*x^4 + 56695661*x^3 - 30768538*x^2 + 10754601*x - 1715933, 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 - 8*x^26 + 46*x^25 - 158*x^24 + 412*x^23 - 651*x^22 + 948*x^21 - 1384*x^20 + 7521*x^19 - 22249*x^18 + 61004*x^17 - 97499*x^16 + 110127*x^15 + 98440*x^14 - 785239*x^13 + 2068135*x^12 - 3012058*x^11 + 444932*x^10 + 9236470*x^9 - 28098698*x^8 + 53493798*x^7 - 76988059*x^6 + 87553666*x^5 - 79021445*x^4 + 56695661*x^3 - 30768538*x^2 + 10754601*x - 1715933);
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 - 8*x^26 + 46*x^25 - 158*x^24 + 412*x^23 - 651*x^22 + 948*x^21 - 1384*x^20 + 7521*x^19 - 22249*x^18 + 61004*x^17 - 97499*x^16 + 110127*x^15 + 98440*x^14 - 785239*x^13 + 2068135*x^12 - 3012058*x^11 + 444932*x^10 + 9236470*x^9 - 28098698*x^8 + 53493798*x^7 - 76988059*x^6 + 87553666*x^5 - 79021445*x^4 + 56695661*x^3 - 30768538*x^2 + 10754601*x - 1715933);
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
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 15 conjugacy class representatives for $D_{27}$ |
| Character table for $D_{27}$ |
Intermediate fields
| 3.1.3671.1, 9.1.181609071490081.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $27$ | $27$ | $27$ | $27$ | ${\href{/padicField/11.9.0.1}{9} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{13}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $27$ | ${\href{/padicField/19.2.0.1}{2} }^{13}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $27$ | ${\href{/padicField/29.2.0.1}{2} }^{13}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $27$ | ${\href{/padicField/37.2.0.1}{2} }^{13}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{13}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{13}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $27$ | ${\href{/padicField/53.2.0.1}{2} }^{13}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $27$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3671\)
| $\Q_{3671}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
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 *.