Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 10*x^26 + 38*x^25 - 52*x^24 - 84*x^23 + 458*x^22 - 715*x^21 + 40*x^20 + 1948*x^19 - 3756*x^18 + 1448*x^17 + 4952*x^16 - 7109*x^15 + 94*x^14 + 4978*x^13 - 2352*x^12 - 1300*x^11 + 2422*x^10 + 639*x^9 - 3848*x^8 - 28*x^7 + 1728*x^6 + 572*x^5 - 368*x^4 - 244*x^3 + 48*x^2 + 32*x - 16)
gp: K = bnfinit(y^27 - 10*y^26 + 38*y^25 - 52*y^24 - 84*y^23 + 458*y^22 - 715*y^21 + 40*y^20 + 1948*y^19 - 3756*y^18 + 1448*y^17 + 4952*y^16 - 7109*y^15 + 94*y^14 + 4978*y^13 - 2352*y^12 - 1300*y^11 + 2422*y^10 + 639*y^9 - 3848*y^8 - 28*y^7 + 1728*y^6 + 572*y^5 - 368*y^4 - 244*y^3 + 48*y^2 + 32*y - 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 10*x^26 + 38*x^25 - 52*x^24 - 84*x^23 + 458*x^22 - 715*x^21 + 40*x^20 + 1948*x^19 - 3756*x^18 + 1448*x^17 + 4952*x^16 - 7109*x^15 + 94*x^14 + 4978*x^13 - 2352*x^12 - 1300*x^11 + 2422*x^10 + 639*x^9 - 3848*x^8 - 28*x^7 + 1728*x^6 + 572*x^5 - 368*x^4 - 244*x^3 + 48*x^2 + 32*x - 16);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 10*x^26 + 38*x^25 - 52*x^24 - 84*x^23 + 458*x^22 - 715*x^21 + 40*x^20 + 1948*x^19 - 3756*x^18 + 1448*x^17 + 4952*x^16 - 7109*x^15 + 94*x^14 + 4978*x^13 - 2352*x^12 - 1300*x^11 + 2422*x^10 + 639*x^9 - 3848*x^8 - 28*x^7 + 1728*x^6 + 572*x^5 - 368*x^4 - 244*x^3 + 48*x^2 + 32*x - 16)
\( x^{27} - 10 x^{26} + 38 x^{25} - 52 x^{24} - 84 x^{23} + 458 x^{22} - 715 x^{21} + 40 x^{20} + 1948 x^{19} + \cdots - 16 \)
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: |
\(-3216045767164746225347277064349747511296\)
\(\medspace = -\,2^{18}\cdot 419^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.06\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2^{2/3}419^{1/2}\approx 32.49328915040618$ | ||
| Ramified primes: |
\(2\), \(419\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-419}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{7}$, $\frac{1}{4}a^{20}-\frac{1}{4}a^{17}+\frac{1}{4}a^{8}+\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{21}-\frac{1}{4}a^{18}+\frac{1}{4}a^{9}+\frac{1}{4}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{22}-\frac{1}{4}a^{19}+\frac{1}{4}a^{10}+\frac{1}{4}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{23}-\frac{1}{4}a^{17}+\frac{1}{4}a^{11}-\frac{1}{2}a^{8}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{304}a^{24}+\frac{17}{152}a^{23}-\frac{5}{76}a^{22}-\frac{17}{152}a^{21}+\frac{1}{76}a^{20}-\frac{37}{152}a^{19}+\frac{5}{304}a^{18}-\frac{7}{76}a^{17}-\frac{29}{152}a^{16}-\frac{13}{152}a^{15}-\frac{15}{76}a^{14}+\frac{5}{38}a^{13}-\frac{17}{304}a^{12}+\frac{73}{152}a^{11}-\frac{25}{76}a^{10}+\frac{15}{152}a^{9}+\frac{35}{76}a^{8}+\frac{33}{152}a^{7}+\frac{135}{304}a^{6}+\frac{13}{76}a^{5}+\frac{41}{152}a^{4}-\frac{3}{8}a^{3}+\frac{1}{19}a^{2}+\frac{15}{38}a+\frac{1}{38}$, $\frac{1}{1401136}a^{25}+\frac{455}{700568}a^{24}-\frac{1303}{31844}a^{23}-\frac{31653}{700568}a^{22}+\frac{490}{87571}a^{21}+\frac{38727}{700568}a^{20}+\frac{309557}{1401136}a^{19}-\frac{58515}{350284}a^{18}+\frac{3453}{36872}a^{17}-\frac{25721}{700568}a^{16}+\frac{1283}{350284}a^{15}-\frac{28909}{175142}a^{14}-\frac{7993}{1401136}a^{13}-\frac{91809}{700568}a^{12}-\frac{140001}{350284}a^{11}-\frac{29649}{700568}a^{10}-\frac{7996}{87571}a^{9}+\frac{42277}{700568}a^{8}-\frac{261553}{1401136}a^{7}+\frac{5893}{31844}a^{6}+\frac{19397}{700568}a^{5}-\frac{108921}{700568}a^{4}+\frac{27647}{175142}a^{3}+\frac{3059}{9218}a^{2}+\frac{72079}{175142}a+\frac{43492}{87571}$, $\frac{1}{20\!\cdots\!36}a^{26}-\frac{14\!\cdots\!47}{10\!\cdots\!68}a^{25}+\frac{72\!\cdots\!71}{10\!\cdots\!68}a^{24}+\frac{47\!\cdots\!41}{10\!\cdots\!68}a^{23}+\frac{77\!\cdots\!79}{25\!\cdots\!42}a^{22}-\frac{65\!\cdots\!77}{10\!\cdots\!68}a^{21}+\frac{14\!\cdots\!93}{20\!\cdots\!36}a^{20}+\frac{49\!\cdots\!71}{50\!\cdots\!84}a^{19}+\frac{25\!\cdots\!81}{50\!\cdots\!84}a^{18}+\frac{14\!\cdots\!39}{10\!\cdots\!68}a^{17}-\frac{93\!\cdots\!20}{12\!\cdots\!21}a^{16}+\frac{83\!\cdots\!43}{50\!\cdots\!84}a^{15}+\frac{48\!\cdots\!99}{20\!\cdots\!36}a^{14}+\frac{22\!\cdots\!15}{91\!\cdots\!88}a^{13}+\frac{49\!\cdots\!53}{10\!\cdots\!68}a^{12}-\frac{16\!\cdots\!71}{10\!\cdots\!68}a^{11}-\frac{89\!\cdots\!82}{18\!\cdots\!63}a^{10}+\frac{21\!\cdots\!25}{10\!\cdots\!68}a^{9}-\frac{62\!\cdots\!85}{20\!\cdots\!36}a^{8}-\frac{99\!\cdots\!37}{50\!\cdots\!84}a^{7}+\frac{21\!\cdots\!95}{50\!\cdots\!84}a^{6}-\frac{44\!\cdots\!01}{10\!\cdots\!68}a^{5}-\frac{20\!\cdots\!01}{50\!\cdots\!84}a^{4}+\frac{18\!\cdots\!21}{50\!\cdots\!84}a^{3}-\frac{79\!\cdots\!89}{25\!\cdots\!42}a^{2}+\frac{65\!\cdots\!40}{12\!\cdots\!21}a-\frac{19\!\cdots\!61}{12\!\cdots\!21}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
$\frac{45\!\cdots\!55}{20\!\cdots\!36}a^{26}-\frac{57\!\cdots\!87}{18\!\cdots\!76}a^{25}+\frac{22\!\cdots\!89}{12\!\cdots\!21}a^{24}-\frac{50\!\cdots\!83}{10\!\cdots\!68}a^{23}+\frac{39\!\cdots\!35}{10\!\cdots\!68}a^{22}+\frac{17\!\cdots\!79}{10\!\cdots\!68}a^{21}-\frac{67\!\cdots\!37}{10\!\cdots\!44}a^{20}+\frac{16\!\cdots\!77}{20\!\cdots\!36}a^{19}+\frac{13\!\cdots\!37}{45\!\cdots\!44}a^{18}-\frac{14\!\cdots\!47}{50\!\cdots\!84}a^{17}+\frac{25\!\cdots\!25}{53\!\cdots\!72}a^{16}-\frac{13\!\cdots\!07}{12\!\cdots\!21}a^{15}-\frac{14\!\cdots\!63}{20\!\cdots\!36}a^{14}+\frac{10\!\cdots\!27}{10\!\cdots\!44}a^{13}-\frac{41\!\cdots\!85}{50\!\cdots\!84}a^{12}-\frac{82\!\cdots\!59}{10\!\cdots\!68}a^{11}+\frac{86\!\cdots\!21}{14\!\cdots\!04}a^{10}+\frac{11\!\cdots\!49}{10\!\cdots\!68}a^{9}-\frac{83\!\cdots\!45}{20\!\cdots\!36}a^{8}+\frac{59\!\cdots\!23}{20\!\cdots\!36}a^{7}+\frac{11\!\cdots\!73}{26\!\cdots\!36}a^{6}-\frac{36\!\cdots\!99}{25\!\cdots\!42}a^{5}-\frac{25\!\cdots\!31}{10\!\cdots\!68}a^{4}+\frac{31\!\cdots\!77}{50\!\cdots\!84}a^{3}+\frac{60\!\cdots\!43}{25\!\cdots\!42}a^{2}+\frac{22\!\cdots\!31}{25\!\cdots\!42}a-\frac{28\!\cdots\!78}{12\!\cdots\!21}$, $\frac{13\!\cdots\!77}{10\!\cdots\!68}a^{26}-\frac{25\!\cdots\!31}{20\!\cdots\!36}a^{25}+\frac{46\!\cdots\!29}{10\!\cdots\!44}a^{24}-\frac{48\!\cdots\!61}{10\!\cdots\!68}a^{23}-\frac{12\!\cdots\!89}{10\!\cdots\!68}a^{22}+\frac{52\!\cdots\!83}{10\!\cdots\!68}a^{21}-\frac{16\!\cdots\!43}{25\!\cdots\!42}a^{20}-\frac{43\!\cdots\!87}{18\!\cdots\!76}a^{19}+\frac{45\!\cdots\!29}{20\!\cdots\!36}a^{18}-\frac{35\!\cdots\!37}{10\!\cdots\!68}a^{17}+\frac{21\!\cdots\!13}{50\!\cdots\!84}a^{16}+\frac{60\!\cdots\!81}{10\!\cdots\!68}a^{15}-\frac{51\!\cdots\!59}{10\!\cdots\!68}a^{14}-\frac{73\!\cdots\!77}{20\!\cdots\!36}a^{13}+\frac{90\!\cdots\!93}{20\!\cdots\!36}a^{12}+\frac{12\!\cdots\!71}{10\!\cdots\!68}a^{11}-\frac{27\!\cdots\!21}{78\!\cdots\!16}a^{10}+\frac{14\!\cdots\!71}{10\!\cdots\!68}a^{9}+\frac{79\!\cdots\!53}{25\!\cdots\!42}a^{8}-\frac{84\!\cdots\!71}{20\!\cdots\!36}a^{7}-\frac{51\!\cdots\!49}{20\!\cdots\!36}a^{6}+\frac{16\!\cdots\!09}{10\!\cdots\!68}a^{5}+\frac{21\!\cdots\!75}{12\!\cdots\!21}a^{4}-\frac{85\!\cdots\!61}{10\!\cdots\!68}a^{3}-\frac{13\!\cdots\!27}{25\!\cdots\!42}a^{2}+\frac{18\!\cdots\!53}{12\!\cdots\!21}a+\frac{44\!\cdots\!83}{25\!\cdots\!42}$, $\frac{21\!\cdots\!29}{20\!\cdots\!36}a^{26}-\frac{97\!\cdots\!45}{10\!\cdots\!68}a^{25}+\frac{31\!\cdots\!57}{96\!\cdots\!04}a^{24}-\frac{15\!\cdots\!41}{50\!\cdots\!84}a^{23}-\frac{27\!\cdots\!37}{25\!\cdots\!42}a^{22}+\frac{19\!\cdots\!93}{50\!\cdots\!84}a^{21}-\frac{90\!\cdots\!47}{20\!\cdots\!36}a^{20}-\frac{30\!\cdots\!13}{10\!\cdots\!68}a^{19}+\frac{36\!\cdots\!15}{20\!\cdots\!36}a^{18}-\frac{25\!\cdots\!91}{10\!\cdots\!68}a^{17}-\frac{46\!\cdots\!59}{10\!\cdots\!68}a^{16}+\frac{49\!\cdots\!39}{10\!\cdots\!68}a^{15}-\frac{73\!\cdots\!17}{20\!\cdots\!36}a^{14}-\frac{29\!\cdots\!73}{10\!\cdots\!68}a^{13}+\frac{32\!\cdots\!53}{10\!\cdots\!44}a^{12}+\frac{12\!\cdots\!31}{50\!\cdots\!84}a^{11}-\frac{27\!\cdots\!81}{18\!\cdots\!63}a^{10}+\frac{17\!\cdots\!19}{12\!\cdots\!21}a^{9}+\frac{38\!\cdots\!59}{20\!\cdots\!36}a^{8}-\frac{25\!\cdots\!77}{91\!\cdots\!88}a^{7}-\frac{43\!\cdots\!95}{20\!\cdots\!36}a^{6}+\frac{33\!\cdots\!77}{10\!\cdots\!68}a^{5}+\frac{10\!\cdots\!69}{10\!\cdots\!68}a^{4}+\frac{31\!\cdots\!95}{10\!\cdots\!68}a^{3}-\frac{20\!\cdots\!47}{25\!\cdots\!42}a^{2}-\frac{64\!\cdots\!49}{25\!\cdots\!42}a+\frac{65\!\cdots\!05}{22\!\cdots\!22}$, $\frac{40\!\cdots\!41}{10\!\cdots\!68}a^{26}-\frac{72\!\cdots\!23}{20\!\cdots\!36}a^{25}+\frac{11\!\cdots\!05}{10\!\cdots\!68}a^{24}-\frac{38\!\cdots\!99}{50\!\cdots\!84}a^{23}-\frac{45\!\cdots\!41}{10\!\cdots\!68}a^{22}+\frac{34\!\cdots\!97}{25\!\cdots\!42}a^{21}-\frac{61\!\cdots\!37}{50\!\cdots\!84}a^{20}-\frac{33\!\cdots\!95}{20\!\cdots\!36}a^{19}+\frac{15\!\cdots\!93}{25\!\cdots\!42}a^{18}-\frac{38\!\cdots\!13}{53\!\cdots\!72}a^{17}-\frac{46\!\cdots\!05}{10\!\cdots\!68}a^{16}+\frac{85\!\cdots\!89}{50\!\cdots\!84}a^{15}-\frac{63\!\cdots\!97}{10\!\cdots\!68}a^{14}-\frac{31\!\cdots\!45}{20\!\cdots\!36}a^{13}+\frac{50\!\cdots\!97}{10\!\cdots\!68}a^{12}+\frac{41\!\cdots\!41}{50\!\cdots\!84}a^{11}-\frac{60\!\cdots\!53}{13\!\cdots\!64}a^{10}-\frac{65\!\cdots\!99}{25\!\cdots\!42}a^{9}+\frac{51\!\cdots\!57}{50\!\cdots\!84}a^{8}-\frac{14\!\cdots\!65}{20\!\cdots\!36}a^{7}-\frac{16\!\cdots\!76}{12\!\cdots\!21}a^{6}-\frac{17\!\cdots\!41}{10\!\cdots\!68}a^{5}+\frac{66\!\cdots\!23}{10\!\cdots\!68}a^{4}+\frac{30\!\cdots\!85}{12\!\cdots\!21}a^{3}-\frac{24\!\cdots\!35}{25\!\cdots\!42}a^{2}-\frac{70\!\cdots\!57}{25\!\cdots\!42}a+\frac{22\!\cdots\!27}{12\!\cdots\!21}$, $\frac{96\!\cdots\!37}{20\!\cdots\!36}a^{26}-\frac{89\!\cdots\!87}{20\!\cdots\!36}a^{25}+\frac{30\!\cdots\!23}{20\!\cdots\!36}a^{24}-\frac{73\!\cdots\!31}{50\!\cdots\!84}a^{23}-\frac{50\!\cdots\!93}{10\!\cdots\!68}a^{22}+\frac{84\!\cdots\!93}{45\!\cdots\!44}a^{21}-\frac{43\!\cdots\!41}{20\!\cdots\!36}a^{20}-\frac{25\!\cdots\!71}{20\!\cdots\!36}a^{19}+\frac{16\!\cdots\!51}{20\!\cdots\!36}a^{18}-\frac{14\!\cdots\!96}{11\!\cdots\!11}a^{17}-\frac{61\!\cdots\!29}{45\!\cdots\!44}a^{16}+\frac{23\!\cdots\!17}{10\!\cdots\!68}a^{15}-\frac{37\!\cdots\!45}{20\!\cdots\!36}a^{14}-\frac{24\!\cdots\!61}{20\!\cdots\!36}a^{13}+\frac{28\!\cdots\!67}{18\!\cdots\!76}a^{12}-\frac{48\!\cdots\!43}{50\!\cdots\!84}a^{11}-\frac{99\!\cdots\!51}{14\!\cdots\!04}a^{10}+\frac{89\!\cdots\!84}{12\!\cdots\!21}a^{9}+\frac{15\!\cdots\!21}{20\!\cdots\!36}a^{8}-\frac{26\!\cdots\!57}{20\!\cdots\!36}a^{7}-\frac{17\!\cdots\!95}{20\!\cdots\!36}a^{6}+\frac{11\!\cdots\!69}{50\!\cdots\!84}a^{5}+\frac{20\!\cdots\!59}{50\!\cdots\!84}a^{4}+\frac{10\!\cdots\!35}{10\!\cdots\!68}a^{3}-\frac{12\!\cdots\!45}{25\!\cdots\!42}a^{2}-\frac{60\!\cdots\!36}{12\!\cdots\!21}a+\frac{30\!\cdots\!35}{25\!\cdots\!42}$, $\frac{10\!\cdots\!41}{10\!\cdots\!44}a^{26}-\frac{19\!\cdots\!01}{20\!\cdots\!36}a^{25}+\frac{68\!\cdots\!01}{20\!\cdots\!36}a^{24}-\frac{17\!\cdots\!11}{50\!\cdots\!84}a^{23}-\frac{10\!\cdots\!19}{10\!\cdots\!68}a^{22}+\frac{10\!\cdots\!31}{25\!\cdots\!42}a^{21}-\frac{10\!\cdots\!31}{20\!\cdots\!36}a^{20}-\frac{50\!\cdots\!01}{20\!\cdots\!36}a^{19}+\frac{38\!\cdots\!93}{20\!\cdots\!36}a^{18}-\frac{71\!\cdots\!37}{25\!\cdots\!42}a^{17}-\frac{60\!\cdots\!39}{50\!\cdots\!84}a^{16}+\frac{52\!\cdots\!39}{10\!\cdots\!68}a^{15}-\frac{92\!\cdots\!71}{20\!\cdots\!36}a^{14}-\frac{52\!\cdots\!99}{20\!\cdots\!36}a^{13}+\frac{81\!\cdots\!63}{20\!\cdots\!36}a^{12}-\frac{10\!\cdots\!87}{50\!\cdots\!84}a^{11}-\frac{25\!\cdots\!65}{14\!\cdots\!04}a^{10}+\frac{42\!\cdots\!45}{26\!\cdots\!36}a^{9}+\frac{32\!\cdots\!15}{20\!\cdots\!36}a^{8}-\frac{63\!\cdots\!99}{20\!\cdots\!36}a^{7}-\frac{36\!\cdots\!21}{20\!\cdots\!36}a^{6}+\frac{49\!\cdots\!49}{50\!\cdots\!84}a^{5}+\frac{59\!\cdots\!55}{50\!\cdots\!84}a^{4}+\frac{96\!\cdots\!05}{10\!\cdots\!68}a^{3}-\frac{69\!\cdots\!65}{25\!\cdots\!42}a^{2}-\frac{80\!\cdots\!37}{11\!\cdots\!11}a+\frac{12\!\cdots\!57}{25\!\cdots\!42}$, $\frac{73\!\cdots\!71}{20\!\cdots\!36}a^{26}-\frac{64\!\cdots\!37}{20\!\cdots\!36}a^{25}+\frac{19\!\cdots\!97}{20\!\cdots\!36}a^{24}-\frac{98\!\cdots\!35}{25\!\cdots\!42}a^{23}-\frac{47\!\cdots\!31}{10\!\cdots\!68}a^{22}+\frac{62\!\cdots\!05}{50\!\cdots\!84}a^{21}-\frac{16\!\cdots\!03}{20\!\cdots\!36}a^{20}-\frac{45\!\cdots\!29}{20\!\cdots\!36}a^{19}+\frac{12\!\cdots\!05}{20\!\cdots\!36}a^{18}-\frac{29\!\cdots\!69}{50\!\cdots\!84}a^{17}-\frac{39\!\cdots\!67}{50\!\cdots\!84}a^{16}+\frac{19\!\cdots\!11}{10\!\cdots\!68}a^{15}-\frac{10\!\cdots\!77}{18\!\cdots\!76}a^{14}-\frac{42\!\cdots\!03}{20\!\cdots\!36}a^{13}+\frac{24\!\cdots\!91}{20\!\cdots\!36}a^{12}+\frac{15\!\cdots\!27}{25\!\cdots\!42}a^{11}-\frac{12\!\cdots\!69}{14\!\cdots\!04}a^{10}+\frac{90\!\cdots\!33}{25\!\cdots\!42}a^{9}+\frac{19\!\cdots\!87}{20\!\cdots\!36}a^{8}-\frac{16\!\cdots\!59}{20\!\cdots\!36}a^{7}-\frac{27\!\cdots\!77}{20\!\cdots\!36}a^{6}+\frac{12\!\cdots\!77}{13\!\cdots\!18}a^{5}+\frac{22\!\cdots\!37}{45\!\cdots\!44}a^{4}+\frac{20\!\cdots\!69}{10\!\cdots\!68}a^{3}-\frac{62\!\cdots\!07}{11\!\cdots\!11}a^{2}-\frac{17\!\cdots\!80}{66\!\cdots\!59}a+\frac{29\!\cdots\!81}{25\!\cdots\!42}$, $\frac{58\!\cdots\!25}{50\!\cdots\!84}a^{26}-\frac{13\!\cdots\!03}{12\!\cdots\!21}a^{25}+\frac{90\!\cdots\!09}{25\!\cdots\!42}a^{24}-\frac{83\!\cdots\!51}{25\!\cdots\!42}a^{23}-\frac{62\!\cdots\!45}{50\!\cdots\!84}a^{22}+\frac{55\!\cdots\!47}{12\!\cdots\!21}a^{21}-\frac{25\!\cdots\!29}{50\!\cdots\!84}a^{20}-\frac{17\!\cdots\!81}{50\!\cdots\!84}a^{19}+\frac{51\!\cdots\!25}{25\!\cdots\!42}a^{18}-\frac{36\!\cdots\!53}{12\!\cdots\!21}a^{17}-\frac{13\!\cdots\!31}{25\!\cdots\!42}a^{16}+\frac{70\!\cdots\!98}{12\!\cdots\!21}a^{15}-\frac{21\!\cdots\!75}{50\!\cdots\!84}a^{14}-\frac{36\!\cdots\!46}{11\!\cdots\!11}a^{13}+\frac{96\!\cdots\!89}{25\!\cdots\!42}a^{12}-\frac{18\!\cdots\!87}{12\!\cdots\!21}a^{11}-\frac{12\!\cdots\!91}{74\!\cdots\!52}a^{10}+\frac{21\!\cdots\!02}{12\!\cdots\!21}a^{9}+\frac{97\!\cdots\!07}{50\!\cdots\!84}a^{8}-\frac{15\!\cdots\!03}{50\!\cdots\!84}a^{7}-\frac{58\!\cdots\!19}{25\!\cdots\!42}a^{6}+\frac{64\!\cdots\!63}{13\!\cdots\!18}a^{5}+\frac{12\!\cdots\!60}{12\!\cdots\!21}a^{4}+\frac{33\!\cdots\!77}{12\!\cdots\!21}a^{3}-\frac{12\!\cdots\!91}{12\!\cdots\!21}a^{2}-\frac{66\!\cdots\!11}{66\!\cdots\!59}a+\frac{31\!\cdots\!67}{12\!\cdots\!21}$, $\frac{14\!\cdots\!10}{12\!\cdots\!21}a^{26}-\frac{27\!\cdots\!17}{25\!\cdots\!42}a^{25}+\frac{76\!\cdots\!59}{20\!\cdots\!36}a^{24}-\frac{42\!\cdots\!55}{10\!\cdots\!68}a^{23}-\frac{13\!\cdots\!87}{12\!\cdots\!21}a^{22}+\frac{46\!\cdots\!03}{10\!\cdots\!68}a^{21}-\frac{15\!\cdots\!31}{26\!\cdots\!36}a^{20}-\frac{19\!\cdots\!61}{10\!\cdots\!68}a^{19}+\frac{40\!\cdots\!91}{20\!\cdots\!36}a^{18}-\frac{82\!\cdots\!41}{25\!\cdots\!42}a^{17}+\frac{25\!\cdots\!53}{10\!\cdots\!68}a^{16}+\frac{28\!\cdots\!43}{53\!\cdots\!72}a^{15}-\frac{26\!\cdots\!79}{50\!\cdots\!84}a^{14}-\frac{54\!\cdots\!67}{25\!\cdots\!42}a^{13}+\frac{81\!\cdots\!17}{20\!\cdots\!36}a^{12}-\frac{38\!\cdots\!95}{10\!\cdots\!68}a^{11}-\frac{59\!\cdots\!53}{37\!\cdots\!26}a^{10}+\frac{17\!\cdots\!55}{10\!\cdots\!68}a^{9}+\frac{83\!\cdots\!17}{50\!\cdots\!84}a^{8}-\frac{34\!\cdots\!91}{10\!\cdots\!68}a^{7}-\frac{33\!\cdots\!27}{20\!\cdots\!36}a^{6}+\frac{12\!\cdots\!98}{12\!\cdots\!21}a^{5}+\frac{12\!\cdots\!23}{10\!\cdots\!68}a^{4}+\frac{86\!\cdots\!69}{10\!\cdots\!68}a^{3}-\frac{58\!\cdots\!03}{25\!\cdots\!42}a^{2}-\frac{17\!\cdots\!53}{22\!\cdots\!22}a+\frac{40\!\cdots\!71}{13\!\cdots\!18}$, $\frac{15\!\cdots\!94}{12\!\cdots\!21}a^{26}-\frac{57\!\cdots\!07}{50\!\cdots\!84}a^{25}+\frac{37\!\cdots\!11}{10\!\cdots\!68}a^{24}-\frac{15\!\cdots\!77}{50\!\cdots\!84}a^{23}-\frac{71\!\cdots\!79}{50\!\cdots\!84}a^{22}+\frac{23\!\cdots\!45}{50\!\cdots\!84}a^{21}-\frac{59\!\cdots\!89}{12\!\cdots\!21}a^{20}-\frac{21\!\cdots\!45}{45\!\cdots\!44}a^{19}+\frac{22\!\cdots\!51}{10\!\cdots\!68}a^{18}-\frac{35\!\cdots\!84}{12\!\cdots\!21}a^{17}-\frac{56\!\cdots\!97}{50\!\cdots\!84}a^{16}+\frac{31\!\cdots\!85}{50\!\cdots\!84}a^{15}-\frac{48\!\cdots\!54}{12\!\cdots\!21}a^{14}-\frac{22\!\cdots\!77}{50\!\cdots\!84}a^{13}+\frac{41\!\cdots\!89}{10\!\cdots\!68}a^{12}+\frac{27\!\cdots\!05}{50\!\cdots\!84}a^{11}-\frac{16\!\cdots\!53}{74\!\cdots\!52}a^{10}+\frac{88\!\cdots\!85}{50\!\cdots\!84}a^{9}+\frac{63\!\cdots\!53}{25\!\cdots\!42}a^{8}-\frac{16\!\cdots\!97}{50\!\cdots\!84}a^{7}-\frac{30\!\cdots\!55}{10\!\cdots\!68}a^{6}+\frac{10\!\cdots\!43}{25\!\cdots\!42}a^{5}+\frac{60\!\cdots\!47}{50\!\cdots\!84}a^{4}+\frac{15\!\cdots\!33}{50\!\cdots\!84}a^{3}-\frac{98\!\cdots\!40}{12\!\cdots\!21}a^{2}-\frac{29\!\cdots\!99}{12\!\cdots\!21}a+\frac{29\!\cdots\!20}{12\!\cdots\!21}$, $\frac{76\!\cdots\!09}{50\!\cdots\!84}a^{26}-\frac{17\!\cdots\!63}{12\!\cdots\!21}a^{25}+\frac{96\!\cdots\!55}{20\!\cdots\!36}a^{24}-\frac{46\!\cdots\!27}{10\!\cdots\!68}a^{23}-\frac{79\!\cdots\!13}{50\!\cdots\!84}a^{22}+\frac{58\!\cdots\!95}{10\!\cdots\!68}a^{21}-\frac{34\!\cdots\!49}{50\!\cdots\!84}a^{20}-\frac{36\!\cdots\!37}{91\!\cdots\!88}a^{19}+\frac{52\!\cdots\!19}{20\!\cdots\!36}a^{18}-\frac{19\!\cdots\!23}{50\!\cdots\!84}a^{17}-\frac{41\!\cdots\!43}{10\!\cdots\!68}a^{16}+\frac{71\!\cdots\!89}{10\!\cdots\!68}a^{15}-\frac{14\!\cdots\!93}{25\!\cdots\!42}a^{14}-\frac{93\!\cdots\!49}{25\!\cdots\!42}a^{13}+\frac{95\!\cdots\!85}{20\!\cdots\!36}a^{12}-\frac{33\!\cdots\!71}{10\!\cdots\!68}a^{11}-\frac{14\!\cdots\!19}{74\!\cdots\!52}a^{10}+\frac{22\!\cdots\!47}{10\!\cdots\!68}a^{9}+\frac{12\!\cdots\!21}{50\!\cdots\!84}a^{8}-\frac{40\!\cdots\!17}{10\!\cdots\!68}a^{7}-\frac{56\!\cdots\!11}{20\!\cdots\!36}a^{6}+\frac{25\!\cdots\!07}{50\!\cdots\!84}a^{5}+\frac{11\!\cdots\!59}{10\!\cdots\!68}a^{4}+\frac{35\!\cdots\!93}{10\!\cdots\!68}a^{3}-\frac{11\!\cdots\!42}{12\!\cdots\!21}a^{2}+\frac{59\!\cdots\!01}{25\!\cdots\!42}a+\frac{87\!\cdots\!61}{25\!\cdots\!42}$, $\frac{86\!\cdots\!65}{50\!\cdots\!84}a^{26}-\frac{85\!\cdots\!09}{50\!\cdots\!84}a^{25}+\frac{12\!\cdots\!87}{20\!\cdots\!36}a^{24}-\frac{91\!\cdots\!25}{10\!\cdots\!68}a^{23}-\frac{32\!\cdots\!49}{25\!\cdots\!42}a^{22}+\frac{75\!\cdots\!49}{10\!\cdots\!68}a^{21}-\frac{15\!\cdots\!71}{12\!\cdots\!21}a^{20}+\frac{23\!\cdots\!19}{91\!\cdots\!88}a^{19}+\frac{59\!\cdots\!67}{20\!\cdots\!36}a^{18}-\frac{31\!\cdots\!37}{50\!\cdots\!84}a^{17}+\frac{33\!\cdots\!25}{10\!\cdots\!68}a^{16}+\frac{66\!\cdots\!73}{10\!\cdots\!68}a^{15}-\frac{28\!\cdots\!97}{25\!\cdots\!42}a^{14}+\frac{14\!\cdots\!53}{50\!\cdots\!84}a^{13}+\frac{51\!\cdots\!71}{10\!\cdots\!44}a^{12}-\frac{45\!\cdots\!61}{10\!\cdots\!68}a^{11}+\frac{78\!\cdots\!64}{18\!\cdots\!63}a^{10}+\frac{31\!\cdots\!25}{10\!\cdots\!68}a^{9}+\frac{66\!\cdots\!56}{12\!\cdots\!21}a^{8}-\frac{52\!\cdots\!09}{10\!\cdots\!68}a^{7}+\frac{96\!\cdots\!49}{20\!\cdots\!36}a^{6}+\frac{32\!\cdots\!91}{50\!\cdots\!84}a^{5}+\frac{31\!\cdots\!23}{10\!\cdots\!68}a^{4}+\frac{19\!\cdots\!61}{10\!\cdots\!68}a^{3}-\frac{37\!\cdots\!22}{65\!\cdots\!59}a^{2}+\frac{57\!\cdots\!73}{25\!\cdots\!42}a-\frac{16\!\cdots\!63}{25\!\cdots\!42}$, $\frac{35\!\cdots\!05}{25\!\cdots\!42}a^{26}-\frac{17\!\cdots\!10}{12\!\cdots\!21}a^{25}+\frac{95\!\cdots\!23}{20\!\cdots\!36}a^{24}-\frac{52\!\cdots\!93}{10\!\cdots\!68}a^{23}-\frac{69\!\cdots\!89}{50\!\cdots\!84}a^{22}+\frac{57\!\cdots\!59}{10\!\cdots\!68}a^{21}-\frac{37\!\cdots\!07}{50\!\cdots\!84}a^{20}-\frac{19\!\cdots\!01}{91\!\cdots\!88}a^{19}+\frac{49\!\cdots\!43}{20\!\cdots\!36}a^{18}-\frac{20\!\cdots\!75}{50\!\cdots\!84}a^{17}+\frac{42\!\cdots\!89}{10\!\cdots\!68}a^{16}+\frac{63\!\cdots\!13}{10\!\cdots\!68}a^{15}-\frac{32\!\cdots\!57}{50\!\cdots\!84}a^{14}-\frac{52\!\cdots\!39}{25\!\cdots\!42}a^{13}+\frac{86\!\cdots\!45}{20\!\cdots\!36}a^{12}-\frac{98\!\cdots\!89}{10\!\cdots\!68}a^{11}-\frac{10\!\cdots\!63}{74\!\cdots\!52}a^{10}+\frac{21\!\cdots\!55}{10\!\cdots\!68}a^{9}+\frac{51\!\cdots\!87}{26\!\cdots\!36}a^{8}-\frac{39\!\cdots\!13}{10\!\cdots\!68}a^{7}-\frac{39\!\cdots\!39}{20\!\cdots\!36}a^{6}+\frac{28\!\cdots\!35}{50\!\cdots\!84}a^{5}+\frac{10\!\cdots\!39}{10\!\cdots\!68}a^{4}+\frac{20\!\cdots\!09}{10\!\cdots\!68}a^{3}-\frac{13\!\cdots\!02}{12\!\cdots\!21}a^{2}+\frac{76\!\cdots\!85}{25\!\cdots\!42}a+\frac{54\!\cdots\!41}{25\!\cdots\!42}$
| 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: | \( 2806458409.197778 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 2806458409.197778 \cdot 1}{2\cdot\sqrt{3216045767164746225347277064349747511296}}\cr\approx \mathstrut & 1.17716079305573 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 - 10*x^26 + 38*x^25 - 52*x^24 - 84*x^23 + 458*x^22 - 715*x^21 + 40*x^20 + 1948*x^19 - 3756*x^18 + 1448*x^17 + 4952*x^16 - 7109*x^15 + 94*x^14 + 4978*x^13 - 2352*x^12 - 1300*x^11 + 2422*x^10 + 639*x^9 - 3848*x^8 - 28*x^7 + 1728*x^6 + 572*x^5 - 368*x^4 - 244*x^3 + 48*x^2 + 32*x - 16)
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 - 10*x^26 + 38*x^25 - 52*x^24 - 84*x^23 + 458*x^22 - 715*x^21 + 40*x^20 + 1948*x^19 - 3756*x^18 + 1448*x^17 + 4952*x^16 - 7109*x^15 + 94*x^14 + 4978*x^13 - 2352*x^12 - 1300*x^11 + 2422*x^10 + 639*x^9 - 3848*x^8 - 28*x^7 + 1728*x^6 + 572*x^5 - 368*x^4 - 244*x^3 + 48*x^2 + 32*x - 16, 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 - 10*x^26 + 38*x^25 - 52*x^24 - 84*x^23 + 458*x^22 - 715*x^21 + 40*x^20 + 1948*x^19 - 3756*x^18 + 1448*x^17 + 4952*x^16 - 7109*x^15 + 94*x^14 + 4978*x^13 - 2352*x^12 - 1300*x^11 + 2422*x^10 + 639*x^9 - 3848*x^8 - 28*x^7 + 1728*x^6 + 572*x^5 - 368*x^4 - 244*x^3 + 48*x^2 + 32*x - 16);
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 - 10*x^26 + 38*x^25 - 52*x^24 - 84*x^23 + 458*x^22 - 715*x^21 + 40*x^20 + 1948*x^19 - 3756*x^18 + 1448*x^17 + 4952*x^16 - 7109*x^15 + 94*x^14 + 4978*x^13 - 2352*x^12 - 1300*x^11 + 2422*x^10 + 639*x^9 - 3848*x^8 - 28*x^7 + 1728*x^6 + 572*x^5 - 368*x^4 - 244*x^3 + 48*x^2 + 32*x - 16);
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
| 3.1.419.1, 9.1.30821664721.1 |
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 | R | $27$ | ${\href{/padicField/5.9.0.1}{9} }^{3}$ | $27$ | ${\href{/padicField/11.2.0.1}{2} }^{13}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $27$ | ${\href{/padicField/17.2.0.1}{2} }^{13}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{13}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $27$ | $27$ | ${\href{/padicField/31.2.0.1}{2} }^{13}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $27$ | $27$ | ${\href{/padicField/43.9.0.1}{9} }^{3}$ | $27$ | ${\href{/padicField/53.2.0.1}{2} }^{13}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $27$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(419\)
| $\Q_{419}$ | $x$ | $1$ | $1$ | $0$ | 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 *.