Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 + 63*x^25 - 309*x^24 + 1278*x^23 - 4374*x^22 + 13038*x^21 - 33687*x^20 + 76365*x^19 - 150976*x^18 + 256509*x^17 - 363843*x^16 + 387384*x^15 - 193563*x^14 - 398313*x^13 + 1478712*x^12 - 3028527*x^11 + 4723794*x^10 - 6156003*x^9 + 6804108*x^8 - 6491583*x^7 + 5295510*x^6 - 3678588*x^5 + 2126304*x^4 - 986832*x^3 + 348192*x^2 - 79488*x + 8064)
gp: K = bnfinit(y^27 - 9*y^26 + 63*y^25 - 309*y^24 + 1278*y^23 - 4374*y^22 + 13038*y^21 - 33687*y^20 + 76365*y^19 - 150976*y^18 + 256509*y^17 - 363843*y^16 + 387384*y^15 - 193563*y^14 - 398313*y^13 + 1478712*y^12 - 3028527*y^11 + 4723794*y^10 - 6156003*y^9 + 6804108*y^8 - 6491583*y^7 + 5295510*y^6 - 3678588*y^5 + 2126304*y^4 - 986832*y^3 + 348192*y^2 - 79488*y + 8064, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 9*x^26 + 63*x^25 - 309*x^24 + 1278*x^23 - 4374*x^22 + 13038*x^21 - 33687*x^20 + 76365*x^19 - 150976*x^18 + 256509*x^17 - 363843*x^16 + 387384*x^15 - 193563*x^14 - 398313*x^13 + 1478712*x^12 - 3028527*x^11 + 4723794*x^10 - 6156003*x^9 + 6804108*x^8 - 6491583*x^7 + 5295510*x^6 - 3678588*x^5 + 2126304*x^4 - 986832*x^3 + 348192*x^2 - 79488*x + 8064);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^27 - 9*x^26 + 63*x^25 - 309*x^24 + 1278*x^23 - 4374*x^22 + 13038*x^21 - 33687*x^20 + 76365*x^19 - 150976*x^18 + 256509*x^17 - 363843*x^16 + 387384*x^15 - 193563*x^14 - 398313*x^13 + 1478712*x^12 - 3028527*x^11 + 4723794*x^10 - 6156003*x^9 + 6804108*x^8 - 6491583*x^7 + 5295510*x^6 - 3678588*x^5 + 2126304*x^4 - 986832*x^3 + 348192*x^2 - 79488*x + 8064)
\( x^{27} - 9 x^{26} + 63 x^{25} - 309 x^{24} + 1278 x^{23} - 4374 x^{22} + 13038 x^{21} - 33687 x^{20} + \cdots + 8064 \)
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: | $[3, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(3465994417848590593957315027713775634765625\)
\(\medspace = 3^{54}\cdot 5^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(37.63\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $3^{37/18}5^{8/9}\approx 39.999624680904844$ | ||
| Ramified primes: |
\(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{12}+\frac{1}{3}a^{9}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{13}+\frac{1}{3}a^{10}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}$, $\frac{1}{9}a^{18}+\frac{1}{3}a^{12}+\frac{2}{9}a^{9}+\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{18}a^{19}-\frac{1}{6}a^{16}-\frac{1}{6}a^{15}-\frac{1}{2}a^{13}+\frac{1}{3}a^{12}+\frac{4}{9}a^{10}-\frac{1}{6}a^{9}+\frac{1}{6}a^{7}+\frac{1}{3}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a$, $\frac{1}{18}a^{20}-\frac{1}{6}a^{17}-\frac{1}{6}a^{16}-\frac{1}{6}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{9}a^{11}-\frac{1}{2}a^{10}-\frac{1}{3}a^{9}+\frac{1}{6}a^{8}+\frac{1}{3}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{6}a^{2}$, $\frac{1}{18}a^{21}-\frac{1}{18}a^{18}-\frac{1}{6}a^{17}-\frac{1}{6}a^{15}-\frac{1}{3}a^{13}-\frac{2}{9}a^{12}+\frac{1}{6}a^{11}+\frac{1}{3}a^{10}+\frac{1}{18}a^{9}-\frac{1}{3}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{6}a^{3}+\frac{1}{3}$, $\frac{1}{36}a^{22}-\frac{1}{36}a^{21}-\frac{1}{36}a^{20}-\frac{1}{36}a^{19}-\frac{1}{18}a^{18}-\frac{1}{6}a^{17}-\frac{1}{6}a^{16}-\frac{1}{12}a^{15}-\frac{1}{12}a^{14}-\frac{1}{9}a^{13}-\frac{11}{36}a^{12}+\frac{13}{36}a^{11}+\frac{1}{9}a^{10}+\frac{5}{36}a^{9}+\frac{5}{12}a^{8}+\frac{1}{3}a^{7}-\frac{1}{12}a^{6}-\frac{1}{6}a^{5}-\frac{1}{12}a^{4}+\frac{1}{3}a^{3}-\frac{1}{12}a^{2}+\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{72}a^{23}-\frac{1}{72}a^{22}-\frac{1}{72}a^{21}-\frac{1}{72}a^{20}-\frac{1}{36}a^{19}+\frac{1}{36}a^{18}-\frac{1}{12}a^{17}-\frac{1}{24}a^{16}-\frac{1}{24}a^{15}-\frac{1}{18}a^{14}-\frac{11}{72}a^{13}-\frac{35}{72}a^{12}-\frac{4}{9}a^{11}-\frac{31}{72}a^{10}+\frac{31}{72}a^{9}+\frac{1}{6}a^{8}+\frac{11}{24}a^{7}+\frac{1}{4}a^{6}-\frac{1}{24}a^{5}-\frac{1}{3}a^{4}-\frac{3}{8}a^{3}-\frac{5}{12}a^{2}+\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{432}a^{24}-\frac{1}{144}a^{23}-\frac{1}{144}a^{22}+\frac{1}{48}a^{21}-\frac{1}{36}a^{20}-\frac{1}{72}a^{19}-\frac{1}{72}a^{18}-\frac{1}{16}a^{17}-\frac{5}{48}a^{16}-\frac{35}{216}a^{15}+\frac{19}{144}a^{14}+\frac{25}{144}a^{13}-\frac{1}{72}a^{12}-\frac{53}{144}a^{11}+\frac{47}{144}a^{10}+\frac{13}{72}a^{9}-\frac{19}{48}a^{8}-\frac{1}{6}a^{7}+\frac{31}{144}a^{6}-\frac{1}{8}a^{5}-\frac{11}{48}a^{4}+\frac{1}{4}a^{3}+\frac{5}{12}a^{2}+\frac{1}{3}a$, $\frac{1}{864}a^{25}-\frac{1}{864}a^{24}+\frac{1}{288}a^{23}-\frac{1}{96}a^{22}-\frac{1}{144}a^{21}-\frac{1}{48}a^{20}+\frac{1}{144}a^{19}+\frac{11}{288}a^{18}-\frac{11}{96}a^{17}-\frac{31}{216}a^{16}-\frac{119}{864}a^{15}-\frac{25}{288}a^{14}-\frac{11}{72}a^{13}-\frac{53}{288}a^{12}-\frac{3}{32}a^{11}-\frac{11}{72}a^{10}-\frac{89}{288}a^{9}+\frac{13}{48}a^{8}+\frac{19}{288}a^{7}-\frac{31}{72}a^{6}+\frac{37}{96}a^{5}-\frac{17}{48}a^{4}-\frac{1}{3}a^{3}+\frac{1}{4}a^{2}-\frac{1}{3}a$, $\frac{1}{17\!\cdots\!84}a^{26}-\frac{18\!\cdots\!49}{59\!\cdots\!28}a^{25}-\frac{19\!\cdots\!27}{17\!\cdots\!84}a^{24}+\frac{19\!\cdots\!55}{66\!\cdots\!92}a^{23}-\frac{81\!\cdots\!63}{74\!\cdots\!16}a^{22}-\frac{77\!\cdots\!99}{99\!\cdots\!88}a^{21}-\frac{56\!\cdots\!37}{29\!\cdots\!64}a^{20}-\frac{13\!\cdots\!05}{59\!\cdots\!28}a^{19}+\frac{68\!\cdots\!17}{59\!\cdots\!28}a^{18}+\frac{99\!\cdots\!95}{89\!\cdots\!92}a^{17}-\frac{88\!\cdots\!37}{59\!\cdots\!28}a^{16}-\frac{14\!\cdots\!01}{17\!\cdots\!84}a^{15}-\frac{20\!\cdots\!63}{29\!\cdots\!64}a^{14}+\frac{11\!\cdots\!03}{59\!\cdots\!28}a^{13}+\frac{38\!\cdots\!71}{59\!\cdots\!28}a^{12}-\frac{79\!\cdots\!85}{29\!\cdots\!64}a^{11}-\frac{25\!\cdots\!97}{59\!\cdots\!28}a^{10}-\frac{65\!\cdots\!07}{14\!\cdots\!32}a^{9}-\frac{51\!\cdots\!01}{59\!\cdots\!28}a^{8}-\frac{88\!\cdots\!63}{33\!\cdots\!96}a^{7}-\frac{63\!\cdots\!57}{59\!\cdots\!28}a^{6}-\frac{70\!\cdots\!19}{41\!\cdots\!12}a^{5}-\frac{13\!\cdots\!67}{49\!\cdots\!44}a^{4}-\frac{69\!\cdots\!79}{12\!\cdots\!36}a^{3}-\frac{89\!\cdots\!83}{20\!\cdots\!06}a^{2}-\frac{47\!\cdots\!09}{62\!\cdots\!18}a+\frac{69\!\cdots\!25}{31\!\cdots\!59}$
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
$C_{2}$, which has order $2$ (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: | $14$ | 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: |
$\frac{19\!\cdots\!43}{29\!\cdots\!64}a^{26}-\frac{57\!\cdots\!29}{82\!\cdots\!24}a^{25}+\frac{11\!\cdots\!21}{24\!\cdots\!72}a^{24}-\frac{91\!\cdots\!95}{37\!\cdots\!08}a^{23}+\frac{30\!\cdots\!89}{29\!\cdots\!64}a^{22}-\frac{51\!\cdots\!01}{14\!\cdots\!32}a^{21}+\frac{94\!\cdots\!60}{93\!\cdots\!77}a^{20}-\frac{77\!\cdots\!43}{29\!\cdots\!64}a^{19}+\frac{28\!\cdots\!91}{49\!\cdots\!44}a^{18}-\frac{32\!\cdots\!83}{29\!\cdots\!64}a^{17}+\frac{17\!\cdots\!95}{99\!\cdots\!88}a^{16}-\frac{38\!\cdots\!45}{16\!\cdots\!48}a^{15}+\frac{62\!\cdots\!01}{29\!\cdots\!64}a^{14}-\frac{25\!\cdots\!67}{29\!\cdots\!64}a^{13}-\frac{34\!\cdots\!01}{74\!\cdots\!16}a^{12}+\frac{36\!\cdots\!59}{29\!\cdots\!64}a^{11}-\frac{65\!\cdots\!79}{29\!\cdots\!64}a^{10}+\frac{31\!\cdots\!55}{99\!\cdots\!88}a^{9}-\frac{36\!\cdots\!43}{99\!\cdots\!88}a^{8}+\frac{12\!\cdots\!07}{33\!\cdots\!96}a^{7}-\frac{31\!\cdots\!37}{99\!\cdots\!88}a^{6}+\frac{75\!\cdots\!17}{33\!\cdots\!96}a^{5}-\frac{11\!\cdots\!69}{82\!\cdots\!24}a^{4}+\frac{15\!\cdots\!53}{24\!\cdots\!72}a^{3}-\frac{28\!\cdots\!97}{12\!\cdots\!36}a^{2}+\frac{31\!\cdots\!27}{62\!\cdots\!18}a-\frac{47\!\cdots\!67}{10\!\cdots\!53}$, $\frac{18\!\cdots\!09}{19\!\cdots\!76}a^{26}+\frac{43\!\cdots\!01}{59\!\cdots\!28}a^{25}-\frac{29\!\cdots\!39}{59\!\cdots\!28}a^{24}+\frac{44\!\cdots\!91}{19\!\cdots\!76}a^{23}-\frac{14\!\cdots\!55}{16\!\cdots\!48}a^{22}+\frac{27\!\cdots\!89}{99\!\cdots\!88}a^{21}-\frac{23\!\cdots\!11}{29\!\cdots\!64}a^{20}+\frac{11\!\cdots\!41}{59\!\cdots\!28}a^{19}-\frac{26\!\cdots\!77}{66\!\cdots\!92}a^{18}+\frac{24\!\cdots\!97}{33\!\cdots\!96}a^{17}-\frac{66\!\cdots\!35}{59\!\cdots\!28}a^{16}+\frac{81\!\cdots\!95}{59\!\cdots\!28}a^{15}-\frac{34\!\cdots\!15}{33\!\cdots\!96}a^{14}-\frac{23\!\cdots\!03}{66\!\cdots\!92}a^{13}+\frac{68\!\cdots\!79}{19\!\cdots\!76}a^{12}-\frac{23\!\cdots\!57}{29\!\cdots\!64}a^{11}+\frac{82\!\cdots\!69}{59\!\cdots\!28}a^{10}-\frac{94\!\cdots\!35}{49\!\cdots\!44}a^{9}+\frac{44\!\cdots\!75}{19\!\cdots\!76}a^{8}-\frac{74\!\cdots\!53}{33\!\cdots\!96}a^{7}+\frac{39\!\cdots\!79}{19\!\cdots\!76}a^{6}-\frac{18\!\cdots\!35}{12\!\cdots\!36}a^{5}+\frac{23\!\cdots\!87}{24\!\cdots\!72}a^{4}-\frac{39\!\cdots\!67}{82\!\cdots\!24}a^{3}+\frac{12\!\cdots\!95}{62\!\cdots\!18}a^{2}-\frac{35\!\cdots\!17}{62\!\cdots\!18}a+\frac{83\!\cdots\!10}{10\!\cdots\!53}$, $\frac{27\!\cdots\!91}{17\!\cdots\!84}a^{26}+\frac{10\!\cdots\!59}{17\!\cdots\!84}a^{25}-\frac{45\!\cdots\!57}{17\!\cdots\!84}a^{24}-\frac{34\!\cdots\!27}{59\!\cdots\!28}a^{23}+\frac{96\!\cdots\!21}{29\!\cdots\!64}a^{22}-\frac{70\!\cdots\!11}{29\!\cdots\!64}a^{21}+\frac{29\!\cdots\!69}{29\!\cdots\!64}a^{20}-\frac{20\!\cdots\!57}{59\!\cdots\!28}a^{19}+\frac{56\!\cdots\!07}{59\!\cdots\!28}a^{18}-\frac{24\!\cdots\!39}{11\!\cdots\!24}a^{17}+\frac{79\!\cdots\!89}{17\!\cdots\!84}a^{16}-\frac{13\!\cdots\!67}{17\!\cdots\!84}a^{15}+\frac{25\!\cdots\!41}{24\!\cdots\!72}a^{14}-\frac{62\!\cdots\!09}{66\!\cdots\!92}a^{13}+\frac{93\!\cdots\!05}{59\!\cdots\!28}a^{12}+\frac{68\!\cdots\!19}{37\!\cdots\!08}a^{11}-\frac{30\!\cdots\!73}{59\!\cdots\!28}a^{10}+\frac{27\!\cdots\!53}{29\!\cdots\!64}a^{9}-\frac{75\!\cdots\!45}{59\!\cdots\!28}a^{8}+\frac{11\!\cdots\!43}{74\!\cdots\!16}a^{7}-\frac{86\!\cdots\!25}{59\!\cdots\!28}a^{6}+\frac{39\!\cdots\!83}{33\!\cdots\!96}a^{5}-\frac{20\!\cdots\!43}{24\!\cdots\!72}a^{4}+\frac{11\!\cdots\!13}{24\!\cdots\!72}a^{3}-\frac{24\!\cdots\!79}{12\!\cdots\!36}a^{2}+\frac{59\!\cdots\!93}{10\!\cdots\!53}a-\frac{76\!\cdots\!91}{10\!\cdots\!53}$, $\frac{28\!\cdots\!69}{17\!\cdots\!84}a^{26}+\frac{24\!\cdots\!29}{17\!\cdots\!84}a^{25}-\frac{16\!\cdots\!79}{17\!\cdots\!84}a^{24}+\frac{86\!\cdots\!41}{19\!\cdots\!76}a^{23}-\frac{51\!\cdots\!95}{29\!\cdots\!64}a^{22}+\frac{16\!\cdots\!47}{29\!\cdots\!64}a^{21}-\frac{48\!\cdots\!49}{29\!\cdots\!64}a^{20}+\frac{23\!\cdots\!57}{59\!\cdots\!28}a^{19}-\frac{51\!\cdots\!99}{59\!\cdots\!28}a^{18}+\frac{71\!\cdots\!59}{44\!\cdots\!96}a^{17}-\frac{44\!\cdots\!01}{17\!\cdots\!84}a^{16}+\frac{55\!\cdots\!47}{17\!\cdots\!84}a^{15}-\frac{36\!\cdots\!65}{14\!\cdots\!32}a^{14}-\frac{42\!\cdots\!07}{59\!\cdots\!28}a^{13}+\frac{45\!\cdots\!63}{59\!\cdots\!28}a^{12}-\frac{27\!\cdots\!53}{14\!\cdots\!32}a^{11}+\frac{18\!\cdots\!33}{59\!\cdots\!28}a^{10}-\frac{12\!\cdots\!39}{29\!\cdots\!64}a^{9}+\frac{29\!\cdots\!05}{59\!\cdots\!28}a^{8}-\frac{71\!\cdots\!57}{14\!\cdots\!32}a^{7}+\frac{24\!\cdots\!53}{59\!\cdots\!28}a^{6}-\frac{94\!\cdots\!09}{33\!\cdots\!96}a^{5}+\frac{51\!\cdots\!54}{31\!\cdots\!59}a^{4}-\frac{19\!\cdots\!65}{24\!\cdots\!72}a^{3}+\frac{11\!\cdots\!97}{41\!\cdots\!12}a^{2}-\frac{12\!\cdots\!87}{20\!\cdots\!06}a+\frac{67\!\cdots\!74}{10\!\cdots\!53}$, $\frac{20\!\cdots\!43}{17\!\cdots\!84}a^{26}+\frac{12\!\cdots\!19}{17\!\cdots\!84}a^{25}-\frac{12\!\cdots\!09}{17\!\cdots\!84}a^{24}+\frac{35\!\cdots\!97}{59\!\cdots\!28}a^{23}-\frac{21\!\cdots\!93}{74\!\cdots\!16}a^{22}+\frac{39\!\cdots\!31}{33\!\cdots\!96}a^{21}-\frac{13\!\cdots\!15}{33\!\cdots\!96}a^{20}+\frac{67\!\cdots\!89}{59\!\cdots\!28}a^{19}-\frac{55\!\cdots\!51}{19\!\cdots\!76}a^{18}+\frac{52\!\cdots\!29}{89\!\cdots\!92}a^{17}-\frac{18\!\cdots\!05}{17\!\cdots\!84}a^{16}+\frac{28\!\cdots\!05}{17\!\cdots\!84}a^{15}-\frac{18\!\cdots\!35}{99\!\cdots\!88}a^{14}+\frac{71\!\cdots\!21}{66\!\cdots\!92}a^{13}+\frac{77\!\cdots\!37}{59\!\cdots\!28}a^{12}-\frac{60\!\cdots\!69}{99\!\cdots\!88}a^{11}+\frac{74\!\cdots\!49}{59\!\cdots\!28}a^{10}-\frac{16\!\cdots\!89}{82\!\cdots\!24}a^{9}+\frac{15\!\cdots\!13}{59\!\cdots\!28}a^{8}-\frac{81\!\cdots\!27}{29\!\cdots\!64}a^{7}+\frac{14\!\cdots\!61}{59\!\cdots\!28}a^{6}-\frac{93\!\cdots\!75}{49\!\cdots\!44}a^{5}+\frac{36\!\cdots\!97}{31\!\cdots\!59}a^{4}-\frac{18\!\cdots\!73}{31\!\cdots\!59}a^{3}+\frac{28\!\cdots\!57}{12\!\cdots\!36}a^{2}-\frac{58\!\cdots\!97}{10\!\cdots\!53}a+\frac{18\!\cdots\!65}{31\!\cdots\!59}$, $\frac{12\!\cdots\!45}{99\!\cdots\!88}a^{26}-\frac{35\!\cdots\!89}{29\!\cdots\!64}a^{25}+\frac{79\!\cdots\!09}{99\!\cdots\!88}a^{24}-\frac{12\!\cdots\!81}{33\!\cdots\!96}a^{23}+\frac{58\!\cdots\!81}{37\!\cdots\!08}a^{22}-\frac{77\!\cdots\!51}{14\!\cdots\!32}a^{21}+\frac{22\!\cdots\!05}{14\!\cdots\!32}a^{20}-\frac{11\!\cdots\!09}{29\!\cdots\!64}a^{19}+\frac{81\!\cdots\!15}{99\!\cdots\!88}a^{18}-\frac{25\!\cdots\!73}{16\!\cdots\!48}a^{17}+\frac{73\!\cdots\!71}{29\!\cdots\!64}a^{16}-\frac{31\!\cdots\!05}{99\!\cdots\!88}a^{15}+\frac{13\!\cdots\!85}{49\!\cdots\!44}a^{14}+\frac{24\!\cdots\!79}{29\!\cdots\!64}a^{13}-\frac{19\!\cdots\!09}{29\!\cdots\!64}a^{12}+\frac{25\!\cdots\!45}{14\!\cdots\!32}a^{11}-\frac{91\!\cdots\!81}{29\!\cdots\!64}a^{10}+\frac{35\!\cdots\!73}{82\!\cdots\!24}a^{9}-\frac{50\!\cdots\!79}{99\!\cdots\!88}a^{8}+\frac{25\!\cdots\!63}{49\!\cdots\!44}a^{7}-\frac{44\!\cdots\!99}{99\!\cdots\!88}a^{6}+\frac{40\!\cdots\!35}{12\!\cdots\!36}a^{5}-\frac{48\!\cdots\!51}{24\!\cdots\!72}a^{4}+\frac{23\!\cdots\!13}{24\!\cdots\!72}a^{3}-\frac{44\!\cdots\!59}{12\!\cdots\!36}a^{2}+\frac{27\!\cdots\!37}{31\!\cdots\!59}a-\frac{10\!\cdots\!67}{10\!\cdots\!53}$, $\frac{18\!\cdots\!83}{17\!\cdots\!84}a^{26}-\frac{14\!\cdots\!19}{17\!\cdots\!84}a^{25}+\frac{94\!\cdots\!49}{17\!\cdots\!84}a^{24}-\frac{14\!\cdots\!41}{59\!\cdots\!28}a^{23}+\frac{27\!\cdots\!57}{29\!\cdots\!64}a^{22}-\frac{87\!\cdots\!75}{29\!\cdots\!64}a^{21}+\frac{80\!\cdots\!45}{99\!\cdots\!88}a^{20}-\frac{11\!\cdots\!51}{59\!\cdots\!28}a^{19}+\frac{23\!\cdots\!69}{59\!\cdots\!28}a^{18}-\frac{30\!\cdots\!27}{44\!\cdots\!96}a^{17}+\frac{17\!\cdots\!91}{17\!\cdots\!84}a^{16}-\frac{18\!\cdots\!01}{17\!\cdots\!84}a^{15}+\frac{21\!\cdots\!75}{49\!\cdots\!44}a^{14}+\frac{24\!\cdots\!43}{19\!\cdots\!76}a^{13}-\frac{26\!\cdots\!89}{59\!\cdots\!28}a^{12}+\frac{41\!\cdots\!73}{49\!\cdots\!44}a^{11}-\frac{75\!\cdots\!63}{59\!\cdots\!28}a^{10}+\frac{45\!\cdots\!47}{29\!\cdots\!64}a^{9}-\frac{93\!\cdots\!55}{59\!\cdots\!28}a^{8}+\frac{97\!\cdots\!71}{74\!\cdots\!16}a^{7}-\frac{53\!\cdots\!27}{59\!\cdots\!28}a^{6}+\frac{46\!\cdots\!99}{99\!\cdots\!88}a^{5}-\frac{81\!\cdots\!21}{49\!\cdots\!44}a^{4}+\frac{36\!\cdots\!77}{24\!\cdots\!72}a^{3}+\frac{32\!\cdots\!85}{12\!\cdots\!36}a^{2}-\frac{29\!\cdots\!99}{20\!\cdots\!06}a+\frac{22\!\cdots\!87}{10\!\cdots\!53}$, $\frac{43\!\cdots\!45}{59\!\cdots\!28}a^{26}-\frac{42\!\cdots\!37}{59\!\cdots\!28}a^{25}+\frac{12\!\cdots\!93}{17\!\cdots\!84}a^{24}-\frac{27\!\cdots\!01}{59\!\cdots\!28}a^{23}+\frac{82\!\cdots\!77}{37\!\cdots\!08}a^{22}-\frac{87\!\cdots\!39}{99\!\cdots\!88}a^{21}+\frac{86\!\cdots\!31}{29\!\cdots\!64}a^{20}-\frac{49\!\cdots\!41}{59\!\cdots\!28}a^{19}+\frac{12\!\cdots\!65}{59\!\cdots\!28}a^{18}-\frac{12\!\cdots\!07}{29\!\cdots\!64}a^{17}+\frac{46\!\cdots\!71}{59\!\cdots\!28}a^{16}-\frac{21\!\cdots\!73}{17\!\cdots\!84}a^{15}+\frac{42\!\cdots\!41}{29\!\cdots\!64}a^{14}-\frac{59\!\cdots\!81}{59\!\cdots\!28}a^{13}-\frac{41\!\cdots\!73}{59\!\cdots\!28}a^{12}+\frac{12\!\cdots\!51}{29\!\cdots\!64}a^{11}-\frac{55\!\cdots\!49}{59\!\cdots\!28}a^{10}+\frac{22\!\cdots\!59}{14\!\cdots\!32}a^{9}-\frac{13\!\cdots\!45}{66\!\cdots\!92}a^{8}+\frac{72\!\cdots\!65}{33\!\cdots\!96}a^{7}-\frac{12\!\cdots\!21}{59\!\cdots\!28}a^{6}+\frac{40\!\cdots\!15}{24\!\cdots\!72}a^{5}-\frac{18\!\cdots\!17}{16\!\cdots\!48}a^{4}+\frac{72\!\cdots\!79}{12\!\cdots\!36}a^{3}-\frac{30\!\cdots\!93}{12\!\cdots\!36}a^{2}+\frac{42\!\cdots\!33}{62\!\cdots\!18}a-\frac{26\!\cdots\!37}{31\!\cdots\!59}$, $\frac{48\!\cdots\!39}{99\!\cdots\!88}a^{26}+\frac{45\!\cdots\!43}{44\!\cdots\!96}a^{25}-\frac{33\!\cdots\!57}{44\!\cdots\!96}a^{24}+\frac{70\!\cdots\!31}{14\!\cdots\!32}a^{23}-\frac{60\!\cdots\!17}{29\!\cdots\!64}a^{22}+\frac{37\!\cdots\!93}{49\!\cdots\!44}a^{21}-\frac{57\!\cdots\!17}{24\!\cdots\!72}a^{20}+\frac{18\!\cdots\!49}{29\!\cdots\!64}a^{19}-\frac{86\!\cdots\!83}{62\!\cdots\!18}a^{18}+\frac{27\!\cdots\!89}{99\!\cdots\!88}a^{17}-\frac{40\!\cdots\!69}{89\!\cdots\!92}a^{16}+\frac{13\!\cdots\!83}{22\!\cdots\!48}a^{15}-\frac{16\!\cdots\!69}{29\!\cdots\!64}a^{14}+\frac{15\!\cdots\!05}{29\!\cdots\!64}a^{13}+\frac{16\!\cdots\!29}{14\!\cdots\!32}a^{12}-\frac{10\!\cdots\!03}{33\!\cdots\!96}a^{11}+\frac{16\!\cdots\!69}{29\!\cdots\!64}a^{10}-\frac{25\!\cdots\!61}{33\!\cdots\!96}a^{9}+\frac{91\!\cdots\!65}{99\!\cdots\!88}a^{8}-\frac{27\!\cdots\!59}{29\!\cdots\!64}a^{7}+\frac{22\!\cdots\!37}{29\!\cdots\!64}a^{6}-\frac{54\!\cdots\!55}{99\!\cdots\!88}a^{5}+\frac{78\!\cdots\!41}{24\!\cdots\!72}a^{4}-\frac{36\!\cdots\!93}{24\!\cdots\!72}a^{3}+\frac{10\!\cdots\!13}{20\!\cdots\!06}a^{2}-\frac{68\!\cdots\!55}{62\!\cdots\!18}a+\frac{38\!\cdots\!18}{31\!\cdots\!59}$, $\frac{21\!\cdots\!17}{17\!\cdots\!84}a^{26}+\frac{19\!\cdots\!47}{17\!\cdots\!84}a^{25}-\frac{13\!\cdots\!33}{17\!\cdots\!84}a^{24}+\frac{73\!\cdots\!31}{19\!\cdots\!76}a^{23}-\frac{55\!\cdots\!89}{37\!\cdots\!08}a^{22}+\frac{14\!\cdots\!41}{29\!\cdots\!64}a^{21}-\frac{43\!\cdots\!63}{29\!\cdots\!64}a^{20}+\frac{24\!\cdots\!29}{66\!\cdots\!92}a^{19}-\frac{47\!\cdots\!29}{59\!\cdots\!28}a^{18}+\frac{13\!\cdots\!25}{89\!\cdots\!92}a^{17}-\frac{43\!\cdots\!37}{17\!\cdots\!84}a^{16}+\frac{56\!\cdots\!29}{17\!\cdots\!84}a^{15}-\frac{84\!\cdots\!81}{29\!\cdots\!64}a^{14}+\frac{34\!\cdots\!39}{19\!\cdots\!76}a^{13}+\frac{37\!\cdots\!97}{59\!\cdots\!28}a^{12}-\frac{49\!\cdots\!63}{29\!\cdots\!64}a^{11}+\frac{59\!\cdots\!67}{19\!\cdots\!76}a^{10}-\frac{63\!\cdots\!17}{14\!\cdots\!32}a^{9}+\frac{30\!\cdots\!85}{59\!\cdots\!28}a^{8}-\frac{15\!\cdots\!13}{29\!\cdots\!64}a^{7}+\frac{27\!\cdots\!01}{59\!\cdots\!28}a^{6}-\frac{10\!\cdots\!29}{31\!\cdots\!59}a^{5}+\frac{10\!\cdots\!45}{49\!\cdots\!44}a^{4}-\frac{26\!\cdots\!83}{24\!\cdots\!72}a^{3}+\frac{51\!\cdots\!81}{12\!\cdots\!36}a^{2}-\frac{68\!\cdots\!23}{62\!\cdots\!18}a+\frac{13\!\cdots\!44}{10\!\cdots\!53}$, $\frac{95\!\cdots\!75}{49\!\cdots\!44}a^{26}+\frac{52\!\cdots\!65}{29\!\cdots\!64}a^{25}-\frac{36\!\cdots\!97}{29\!\cdots\!64}a^{24}+\frac{17\!\cdots\!15}{29\!\cdots\!64}a^{23}-\frac{78\!\cdots\!39}{33\!\cdots\!96}a^{22}+\frac{11\!\cdots\!89}{14\!\cdots\!32}a^{21}-\frac{11\!\cdots\!09}{49\!\cdots\!44}a^{20}+\frac{21\!\cdots\!31}{37\!\cdots\!08}a^{19}-\frac{12\!\cdots\!29}{99\!\cdots\!88}a^{18}+\frac{81\!\cdots\!51}{33\!\cdots\!96}a^{17}-\frac{59\!\cdots\!83}{14\!\cdots\!32}a^{16}+\frac{15\!\cdots\!45}{29\!\cdots\!64}a^{15}-\frac{14\!\cdots\!93}{29\!\cdots\!64}a^{14}+\frac{29\!\cdots\!35}{49\!\cdots\!44}a^{13}+\frac{29\!\cdots\!37}{29\!\cdots\!64}a^{12}-\frac{26\!\cdots\!57}{99\!\cdots\!88}a^{11}+\frac{73\!\cdots\!49}{14\!\cdots\!32}a^{10}-\frac{70\!\cdots\!91}{99\!\cdots\!88}a^{9}+\frac{10\!\cdots\!79}{12\!\cdots\!36}a^{8}-\frac{29\!\cdots\!05}{33\!\cdots\!96}a^{7}+\frac{38\!\cdots\!93}{49\!\cdots\!44}a^{6}-\frac{57\!\cdots\!19}{99\!\cdots\!88}a^{5}+\frac{18\!\cdots\!89}{49\!\cdots\!44}a^{4}-\frac{45\!\cdots\!41}{24\!\cdots\!72}a^{3}+\frac{29\!\cdots\!31}{41\!\cdots\!12}a^{2}-\frac{58\!\cdots\!05}{31\!\cdots\!59}a+\frac{23\!\cdots\!41}{10\!\cdots\!53}$, $\frac{32\!\cdots\!37}{17\!\cdots\!84}a^{26}-\frac{27\!\cdots\!49}{17\!\cdots\!84}a^{25}+\frac{18\!\cdots\!23}{17\!\cdots\!84}a^{24}-\frac{29\!\cdots\!19}{59\!\cdots\!28}a^{23}+\frac{58\!\cdots\!03}{29\!\cdots\!64}a^{22}-\frac{21\!\cdots\!29}{33\!\cdots\!96}a^{21}+\frac{54\!\cdots\!61}{29\!\cdots\!64}a^{20}-\frac{26\!\cdots\!97}{59\!\cdots\!28}a^{19}+\frac{57\!\cdots\!75}{59\!\cdots\!28}a^{18}-\frac{80\!\cdots\!43}{44\!\cdots\!96}a^{17}+\frac{50\!\cdots\!77}{17\!\cdots\!84}a^{16}-\frac{62\!\cdots\!19}{17\!\cdots\!84}a^{15}+\frac{41\!\cdots\!57}{14\!\cdots\!32}a^{14}+\frac{45\!\cdots\!43}{59\!\cdots\!28}a^{13}-\frac{51\!\cdots\!99}{59\!\cdots\!28}a^{12}+\frac{30\!\cdots\!83}{14\!\cdots\!32}a^{11}-\frac{20\!\cdots\!65}{59\!\cdots\!28}a^{10}+\frac{14\!\cdots\!21}{29\!\cdots\!64}a^{9}-\frac{33\!\cdots\!29}{59\!\cdots\!28}a^{8}+\frac{20\!\cdots\!41}{37\!\cdots\!08}a^{7}-\frac{27\!\cdots\!65}{59\!\cdots\!28}a^{6}+\frac{32\!\cdots\!09}{99\!\cdots\!88}a^{5}-\frac{94\!\cdots\!73}{49\!\cdots\!44}a^{4}+\frac{11\!\cdots\!19}{12\!\cdots\!36}a^{3}-\frac{65\!\cdots\!47}{20\!\cdots\!06}a^{2}+\frac{76\!\cdots\!00}{10\!\cdots\!53}a-\frac{24\!\cdots\!22}{31\!\cdots\!59}$, $\frac{96\!\cdots\!01}{66\!\cdots\!92}a^{26}+\frac{16\!\cdots\!67}{17\!\cdots\!84}a^{25}-\frac{33\!\cdots\!03}{59\!\cdots\!28}a^{24}+\frac{12\!\cdots\!29}{59\!\cdots\!28}a^{23}-\frac{22\!\cdots\!37}{29\!\cdots\!64}a^{22}+\frac{60\!\cdots\!33}{29\!\cdots\!64}a^{21}-\frac{14\!\cdots\!63}{29\!\cdots\!64}a^{20}+\frac{58\!\cdots\!71}{66\!\cdots\!92}a^{19}-\frac{82\!\cdots\!81}{66\!\cdots\!92}a^{18}+\frac{46\!\cdots\!39}{49\!\cdots\!44}a^{17}+\frac{27\!\cdots\!05}{17\!\cdots\!84}a^{16}-\frac{44\!\cdots\!09}{59\!\cdots\!28}a^{15}+\frac{27\!\cdots\!83}{14\!\cdots\!32}a^{14}-\frac{18\!\cdots\!85}{59\!\cdots\!28}a^{13}+\frac{24\!\cdots\!85}{59\!\cdots\!28}a^{12}-\frac{48\!\cdots\!29}{14\!\cdots\!32}a^{11}-\frac{38\!\cdots\!15}{19\!\cdots\!76}a^{10}+\frac{22\!\cdots\!87}{33\!\cdots\!96}a^{9}-\frac{29\!\cdots\!59}{19\!\cdots\!76}a^{8}+\frac{32\!\cdots\!45}{14\!\cdots\!32}a^{7}-\frac{16\!\cdots\!25}{66\!\cdots\!92}a^{6}+\frac{75\!\cdots\!09}{33\!\cdots\!96}a^{5}-\frac{70\!\cdots\!17}{41\!\cdots\!12}a^{4}+\frac{12\!\cdots\!75}{12\!\cdots\!36}a^{3}-\frac{54\!\cdots\!41}{12\!\cdots\!36}a^{2}+\frac{81\!\cdots\!53}{62\!\cdots\!18}a-\frac{17\!\cdots\!49}{10\!\cdots\!53}$, $\frac{11\!\cdots\!01}{44\!\cdots\!96}a^{26}-\frac{19\!\cdots\!65}{89\!\cdots\!92}a^{25}+\frac{43\!\cdots\!61}{29\!\cdots\!64}a^{24}-\frac{20\!\cdots\!11}{29\!\cdots\!64}a^{23}+\frac{26\!\cdots\!49}{99\!\cdots\!88}a^{22}-\frac{16\!\cdots\!05}{18\!\cdots\!54}a^{21}+\frac{37\!\cdots\!39}{14\!\cdots\!32}a^{20}-\frac{45\!\cdots\!59}{74\!\cdots\!16}a^{19}+\frac{38\!\cdots\!65}{29\!\cdots\!64}a^{18}-\frac{21\!\cdots\!33}{89\!\cdots\!92}a^{17}+\frac{82\!\cdots\!05}{22\!\cdots\!48}a^{16}-\frac{13\!\cdots\!79}{29\!\cdots\!64}a^{15}+\frac{32\!\cdots\!67}{99\!\cdots\!88}a^{14}+\frac{11\!\cdots\!21}{74\!\cdots\!16}a^{13}-\frac{35\!\cdots\!59}{29\!\cdots\!64}a^{12}+\frac{81\!\cdots\!99}{29\!\cdots\!64}a^{11}-\frac{17\!\cdots\!75}{37\!\cdots\!08}a^{10}+\frac{18\!\cdots\!15}{29\!\cdots\!64}a^{9}-\frac{10\!\cdots\!07}{14\!\cdots\!32}a^{8}+\frac{20\!\cdots\!19}{29\!\cdots\!64}a^{7}-\frac{71\!\cdots\!27}{12\!\cdots\!36}a^{6}+\frac{39\!\cdots\!09}{99\!\cdots\!88}a^{5}-\frac{14\!\cdots\!91}{62\!\cdots\!18}a^{4}+\frac{43\!\cdots\!09}{41\!\cdots\!12}a^{3}-\frac{72\!\cdots\!69}{20\!\cdots\!06}a^{2}+\frac{41\!\cdots\!19}{62\!\cdots\!18}a-\frac{15\!\cdots\!21}{31\!\cdots\!59}$
| 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: | \( 102814738585.40065 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{12}\cdot 102814738585.40065 \cdot 2}{2\cdot\sqrt{3465994417848590593957315027713775634765625}}\cr\approx \mathstrut & 1.67259137697075 \end{aligned}\] (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 + 63*x^25 - 309*x^24 + 1278*x^23 - 4374*x^22 + 13038*x^21 - 33687*x^20 + 76365*x^19 - 150976*x^18 + 256509*x^17 - 363843*x^16 + 387384*x^15 - 193563*x^14 - 398313*x^13 + 1478712*x^12 - 3028527*x^11 + 4723794*x^10 - 6156003*x^9 + 6804108*x^8 - 6491583*x^7 + 5295510*x^6 - 3678588*x^5 + 2126304*x^4 - 986832*x^3 + 348192*x^2 - 79488*x + 8064)
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 + 63*x^25 - 309*x^24 + 1278*x^23 - 4374*x^22 + 13038*x^21 - 33687*x^20 + 76365*x^19 - 150976*x^18 + 256509*x^17 - 363843*x^16 + 387384*x^15 - 193563*x^14 - 398313*x^13 + 1478712*x^12 - 3028527*x^11 + 4723794*x^10 - 6156003*x^9 + 6804108*x^8 - 6491583*x^7 + 5295510*x^6 - 3678588*x^5 + 2126304*x^4 - 986832*x^3 + 348192*x^2 - 79488*x + 8064, 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 + 63*x^25 - 309*x^24 + 1278*x^23 - 4374*x^22 + 13038*x^21 - 33687*x^20 + 76365*x^19 - 150976*x^18 + 256509*x^17 - 363843*x^16 + 387384*x^15 - 193563*x^14 - 398313*x^13 + 1478712*x^12 - 3028527*x^11 + 4723794*x^10 - 6156003*x^9 + 6804108*x^8 - 6491583*x^7 + 5295510*x^6 - 3678588*x^5 + 2126304*x^4 - 986832*x^3 + 348192*x^2 - 79488*x + 8064);
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 + 63*x^25 - 309*x^24 + 1278*x^23 - 4374*x^22 + 13038*x^21 - 33687*x^20 + 76365*x^19 - 150976*x^18 + 256509*x^17 - 363843*x^16 + 387384*x^15 - 193563*x^14 - 398313*x^13 + 1478712*x^12 - 3028527*x^11 + 4723794*x^10 - 6156003*x^9 + 6804108*x^8 - 6491583*x^7 + 5295510*x^6 - 3678588*x^5 + 2126304*x^4 - 986832*x^3 + 348192*x^2 - 79488*x + 8064);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_3^3:S_4$ (as 27T211):
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 648 |
| The 14 conjugacy class representatives for $C_3^3:S_4$ |
| Character table for $C_3^3:S_4$ |
Intermediate fields
| 3.1.6075.2 |
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 9 sibling: | data not computed |
| Degree 12 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 9.1.151336128515625.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | R | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.9.0.1}{9} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{9}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.9.0.1}{9} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }^{5}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.9.0.1}{9} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.9.18.2 | $x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$ | $9$ | $1$ | $18$ | $C_3^2:C_2$ | $[3/2, 5/2]_{2}$ |
| 3.9.18.2 | $x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$ | $9$ | $1$ | $18$ | $C_3^2:C_2$ | $[3/2, 5/2]_{2}$ | |
| 3.9.18.2 | $x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$ | $9$ | $1$ | $18$ | $C_3^2:C_2$ | $[3/2, 5/2]_{2}$ | |
|
\(5\)
| Deg $27$ | $9$ | $3$ | $24$ |