Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 3*x^26 - 15*x^25 + 45*x^24 + 78*x^23 - 210*x^22 - 198*x^21 + 306*x^20 + 231*x^19 - 493*x^18 + 1887*x^17 + 3*x^16 - 8556*x^15 + 12804*x^14 - 7536*x^13 - 6672*x^12 + 20184*x^11 - 10560*x^10 - 11760*x^9 + 30192*x^8 - 53700*x^7 + 65796*x^6 - 58956*x^5 + 51156*x^4 - 36306*x^3 + 18390*x^2 - 8994*x + 2870)
gp: K = bnfinit(y^27 - 3*y^26 - 15*y^25 + 45*y^24 + 78*y^23 - 210*y^22 - 198*y^21 + 306*y^20 + 231*y^19 - 493*y^18 + 1887*y^17 + 3*y^16 - 8556*y^15 + 12804*y^14 - 7536*y^13 - 6672*y^12 + 20184*y^11 - 10560*y^10 - 11760*y^9 + 30192*y^8 - 53700*y^7 + 65796*y^6 - 58956*y^5 + 51156*y^4 - 36306*y^3 + 18390*y^2 - 8994*y + 2870, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 3*x^26 - 15*x^25 + 45*x^24 + 78*x^23 - 210*x^22 - 198*x^21 + 306*x^20 + 231*x^19 - 493*x^18 + 1887*x^17 + 3*x^16 - 8556*x^15 + 12804*x^14 - 7536*x^13 - 6672*x^12 + 20184*x^11 - 10560*x^10 - 11760*x^9 + 30192*x^8 - 53700*x^7 + 65796*x^6 - 58956*x^5 + 51156*x^4 - 36306*x^3 + 18390*x^2 - 8994*x + 2870);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 3*x^26 - 15*x^25 + 45*x^24 + 78*x^23 - 210*x^22 - 198*x^21 + 306*x^20 + 231*x^19 - 493*x^18 + 1887*x^17 + 3*x^16 - 8556*x^15 + 12804*x^14 - 7536*x^13 - 6672*x^12 + 20184*x^11 - 10560*x^10 - 11760*x^9 + 30192*x^8 - 53700*x^7 + 65796*x^6 - 58956*x^5 + 51156*x^4 - 36306*x^3 + 18390*x^2 - 8994*x + 2870)
\( x^{27} - 3 x^{26} - 15 x^{25} + 45 x^{24} + 78 x^{23} - 210 x^{22} - 198 x^{21} + 306 x^{20} + \cdots + 2870 \)
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: |
\(181453279996183512417624142594449705271296\)
\(\medspace = 2^{80}\cdot 3^{36}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.74\) | 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: | not computed | ||
| 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: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{16}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{17}$, $\frac{1}{4}a^{22}-\frac{1}{4}a^{20}+\frac{1}{4}a^{18}-\frac{1}{4}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{23}-\frac{1}{4}a^{21}+\frac{1}{4}a^{19}-\frac{1}{4}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{24}+\frac{1}{4}a^{16}-\frac{1}{2}$, $\frac{1}{8}a^{25}-\frac{1}{8}a^{24}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{3}{8}a^{17}+\frac{3}{8}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{64\!\cdots\!64}a^{26}+\frac{49\!\cdots\!15}{16\!\cdots\!66}a^{25}-\frac{50\!\cdots\!23}{64\!\cdots\!64}a^{24}-\frac{31\!\cdots\!09}{80\!\cdots\!33}a^{23}+\frac{37\!\cdots\!55}{32\!\cdots\!32}a^{22}-\frac{10\!\cdots\!56}{80\!\cdots\!33}a^{21}+\frac{64\!\cdots\!83}{32\!\cdots\!32}a^{20}-\frac{36\!\cdots\!94}{80\!\cdots\!33}a^{19}+\frac{17\!\cdots\!15}{64\!\cdots\!64}a^{18}+\frac{41\!\cdots\!01}{16\!\cdots\!66}a^{17}-\frac{86\!\cdots\!77}{64\!\cdots\!64}a^{16}+\frac{47\!\cdots\!74}{80\!\cdots\!33}a^{15}+\frac{15\!\cdots\!99}{16\!\cdots\!66}a^{14}+\frac{18\!\cdots\!16}{80\!\cdots\!33}a^{13}-\frac{53\!\cdots\!78}{80\!\cdots\!33}a^{12}+\frac{31\!\cdots\!46}{80\!\cdots\!33}a^{11}-\frac{33\!\cdots\!61}{80\!\cdots\!33}a^{10}-\frac{30\!\cdots\!93}{80\!\cdots\!33}a^{9}+\frac{37\!\cdots\!07}{80\!\cdots\!33}a^{8}-\frac{28\!\cdots\!32}{80\!\cdots\!33}a^{7}+\frac{68\!\cdots\!13}{16\!\cdots\!66}a^{6}+\frac{37\!\cdots\!70}{80\!\cdots\!33}a^{5}-\frac{32\!\cdots\!55}{16\!\cdots\!66}a^{4}-\frac{28\!\cdots\!10}{80\!\cdots\!33}a^{3}-\frac{99\!\cdots\!73}{32\!\cdots\!32}a^{2}-\frac{19\!\cdots\!36}{80\!\cdots\!33}a-\frac{11\!\cdots\!21}{32\!\cdots\!32}$
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: | $14$ | 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: |
$\frac{15\!\cdots\!49}{64\!\cdots\!64}a^{26}+\frac{13\!\cdots\!33}{32\!\cdots\!32}a^{25}+\frac{25\!\cdots\!79}{64\!\cdots\!64}a^{24}-\frac{90\!\cdots\!23}{16\!\cdots\!66}a^{23}-\frac{39\!\cdots\!91}{16\!\cdots\!66}a^{22}+\frac{27\!\cdots\!63}{16\!\cdots\!66}a^{21}+\frac{53\!\cdots\!97}{80\!\cdots\!33}a^{20}+\frac{37\!\cdots\!95}{16\!\cdots\!66}a^{19}-\frac{20\!\cdots\!37}{64\!\cdots\!64}a^{18}+\frac{13\!\cdots\!47}{32\!\cdots\!32}a^{17}-\frac{24\!\cdots\!49}{64\!\cdots\!64}a^{16}-\frac{32\!\cdots\!79}{80\!\cdots\!33}a^{15}+\frac{11\!\cdots\!40}{80\!\cdots\!33}a^{14}-\frac{98\!\cdots\!06}{80\!\cdots\!33}a^{13}+\frac{11\!\cdots\!75}{16\!\cdots\!66}a^{12}+\frac{11\!\cdots\!69}{80\!\cdots\!33}a^{11}-\frac{38\!\cdots\!49}{16\!\cdots\!66}a^{10}+\frac{17\!\cdots\!75}{80\!\cdots\!33}a^{9}+\frac{21\!\cdots\!87}{16\!\cdots\!66}a^{8}-\frac{34\!\cdots\!43}{80\!\cdots\!33}a^{7}+\frac{68\!\cdots\!24}{80\!\cdots\!33}a^{6}-\frac{64\!\cdots\!40}{80\!\cdots\!33}a^{5}+\frac{57\!\cdots\!06}{80\!\cdots\!33}a^{4}-\frac{55\!\cdots\!95}{80\!\cdots\!33}a^{3}+\frac{11\!\cdots\!03}{32\!\cdots\!32}a^{2}-\frac{34\!\cdots\!05}{16\!\cdots\!66}a+\frac{34\!\cdots\!63}{32\!\cdots\!32}$, $\frac{38\!\cdots\!39}{80\!\cdots\!33}a^{26}-\frac{82\!\cdots\!99}{64\!\cdots\!64}a^{25}-\frac{49\!\cdots\!79}{64\!\cdots\!64}a^{24}+\frac{62\!\cdots\!01}{32\!\cdots\!32}a^{23}+\frac{14\!\cdots\!71}{32\!\cdots\!32}a^{22}-\frac{30\!\cdots\!89}{32\!\cdots\!32}a^{21}-\frac{42\!\cdots\!89}{32\!\cdots\!32}a^{20}+\frac{47\!\cdots\!39}{32\!\cdots\!32}a^{19}+\frac{53\!\cdots\!45}{32\!\cdots\!32}a^{18}-\frac{17\!\cdots\!85}{64\!\cdots\!64}a^{17}+\frac{51\!\cdots\!23}{64\!\cdots\!64}a^{16}+\frac{32\!\cdots\!92}{80\!\cdots\!33}a^{15}-\frac{34\!\cdots\!17}{80\!\cdots\!33}a^{14}+\frac{39\!\cdots\!92}{80\!\cdots\!33}a^{13}-\frac{56\!\cdots\!26}{80\!\cdots\!33}a^{12}-\frac{97\!\cdots\!03}{16\!\cdots\!66}a^{11}+\frac{14\!\cdots\!75}{16\!\cdots\!66}a^{10}-\frac{15\!\cdots\!79}{80\!\cdots\!33}a^{9}-\frac{81\!\cdots\!07}{80\!\cdots\!33}a^{8}+\frac{21\!\cdots\!03}{16\!\cdots\!66}a^{7}-\frac{28\!\cdots\!75}{16\!\cdots\!66}a^{6}+\frac{30\!\cdots\!21}{16\!\cdots\!66}a^{5}-\frac{23\!\cdots\!77}{16\!\cdots\!66}a^{4}+\frac{16\!\cdots\!43}{16\!\cdots\!66}a^{3}-\frac{95\!\cdots\!03}{16\!\cdots\!66}a^{2}+\frac{50\!\cdots\!55}{32\!\cdots\!32}a-\frac{50\!\cdots\!21}{32\!\cdots\!32}$, $\frac{12\!\cdots\!05}{64\!\cdots\!64}a^{26}+\frac{33\!\cdots\!31}{80\!\cdots\!33}a^{25}+\frac{20\!\cdots\!77}{64\!\cdots\!64}a^{24}-\frac{47\!\cdots\!03}{80\!\cdots\!33}a^{23}-\frac{16\!\cdots\!44}{80\!\cdots\!33}a^{22}+\frac{18\!\cdots\!98}{80\!\cdots\!33}a^{21}+\frac{93\!\cdots\!55}{16\!\cdots\!66}a^{20}-\frac{67\!\cdots\!92}{80\!\cdots\!33}a^{19}-\frac{33\!\cdots\!77}{64\!\cdots\!64}a^{18}+\frac{39\!\cdots\!80}{80\!\cdots\!33}a^{17}-\frac{20\!\cdots\!59}{64\!\cdots\!64}a^{16}-\frac{22\!\cdots\!97}{80\!\cdots\!33}a^{15}+\frac{11\!\cdots\!51}{80\!\cdots\!33}a^{14}-\frac{99\!\cdots\!47}{80\!\cdots\!33}a^{13}+\frac{60\!\cdots\!77}{16\!\cdots\!66}a^{12}+\frac{12\!\cdots\!92}{80\!\cdots\!33}a^{11}-\frac{40\!\cdots\!89}{16\!\cdots\!66}a^{10}-\frac{11\!\cdots\!44}{80\!\cdots\!33}a^{9}+\frac{34\!\cdots\!71}{16\!\cdots\!66}a^{8}-\frac{31\!\cdots\!74}{80\!\cdots\!33}a^{7}+\frac{55\!\cdots\!72}{80\!\cdots\!33}a^{6}-\frac{53\!\cdots\!09}{80\!\cdots\!33}a^{5}+\frac{44\!\cdots\!32}{80\!\cdots\!33}a^{4}-\frac{40\!\cdots\!44}{80\!\cdots\!33}a^{3}+\frac{84\!\cdots\!19}{32\!\cdots\!32}a^{2}-\frac{10\!\cdots\!87}{80\!\cdots\!33}a+\frac{20\!\cdots\!81}{32\!\cdots\!32}$, $\frac{60\!\cdots\!77}{16\!\cdots\!66}a^{26}-\frac{22\!\cdots\!55}{32\!\cdots\!32}a^{25}-\frac{10\!\cdots\!43}{16\!\cdots\!66}a^{24}+\frac{77\!\cdots\!43}{80\!\cdots\!33}a^{23}+\frac{64\!\cdots\!39}{16\!\cdots\!66}a^{22}-\frac{51\!\cdots\!65}{16\!\cdots\!66}a^{21}-\frac{90\!\cdots\!94}{80\!\cdots\!33}a^{20}-\frac{17\!\cdots\!56}{80\!\cdots\!33}a^{19}+\frac{58\!\cdots\!17}{80\!\cdots\!33}a^{18}-\frac{23\!\cdots\!29}{32\!\cdots\!32}a^{17}+\frac{98\!\cdots\!67}{16\!\cdots\!66}a^{16}+\frac{51\!\cdots\!10}{80\!\cdots\!33}a^{15}-\frac{19\!\cdots\!37}{80\!\cdots\!33}a^{14}+\frac{15\!\cdots\!84}{80\!\cdots\!33}a^{13}-\frac{67\!\cdots\!77}{80\!\cdots\!33}a^{12}-\frac{20\!\cdots\!71}{80\!\cdots\!33}a^{11}+\frac{31\!\cdots\!83}{80\!\cdots\!33}a^{10}+\frac{18\!\cdots\!10}{80\!\cdots\!33}a^{9}-\frac{22\!\cdots\!46}{80\!\cdots\!33}a^{8}+\frac{55\!\cdots\!26}{80\!\cdots\!33}a^{7}-\frac{10\!\cdots\!03}{80\!\cdots\!33}a^{6}+\frac{99\!\cdots\!09}{80\!\cdots\!33}a^{5}-\frac{88\!\cdots\!38}{80\!\cdots\!33}a^{4}+\frac{81\!\cdots\!50}{80\!\cdots\!33}a^{3}-\frac{42\!\cdots\!91}{80\!\cdots\!33}a^{2}+\frac{45\!\cdots\!19}{16\!\cdots\!66}a-\frac{12\!\cdots\!32}{80\!\cdots\!33}$, $\frac{20\!\cdots\!95}{32\!\cdots\!32}a^{26}+\frac{89\!\cdots\!95}{64\!\cdots\!64}a^{25}+\frac{66\!\cdots\!53}{64\!\cdots\!64}a^{24}-\frac{32\!\cdots\!29}{16\!\cdots\!66}a^{23}-\frac{50\!\cdots\!57}{80\!\cdots\!33}a^{22}+\frac{12\!\cdots\!59}{16\!\cdots\!66}a^{21}+\frac{29\!\cdots\!47}{16\!\cdots\!66}a^{20}-\frac{33\!\cdots\!51}{80\!\cdots\!33}a^{19}-\frac{52\!\cdots\!61}{32\!\cdots\!32}a^{18}+\frac{10\!\cdots\!55}{64\!\cdots\!64}a^{17}-\frac{68\!\cdots\!47}{64\!\cdots\!64}a^{16}-\frac{12\!\cdots\!27}{16\!\cdots\!66}a^{15}+\frac{74\!\cdots\!97}{16\!\cdots\!66}a^{14}-\frac{71\!\cdots\!23}{16\!\cdots\!66}a^{13}+\frac{27\!\cdots\!81}{16\!\cdots\!66}a^{12}+\frac{39\!\cdots\!93}{80\!\cdots\!33}a^{11}-\frac{70\!\cdots\!90}{80\!\cdots\!33}a^{10}+\frac{85\!\cdots\!91}{16\!\cdots\!66}a^{9}+\frac{10\!\cdots\!97}{16\!\cdots\!66}a^{8}-\frac{10\!\cdots\!34}{80\!\cdots\!33}a^{7}+\frac{19\!\cdots\!08}{80\!\cdots\!33}a^{6}-\frac{19\!\cdots\!73}{80\!\cdots\!33}a^{5}+\frac{16\!\cdots\!49}{80\!\cdots\!33}a^{4}-\frac{14\!\cdots\!57}{80\!\cdots\!33}a^{3}+\frac{16\!\cdots\!49}{16\!\cdots\!66}a^{2}-\frac{15\!\cdots\!01}{32\!\cdots\!32}a+\frac{80\!\cdots\!69}{32\!\cdots\!32}$, $\frac{15\!\cdots\!79}{64\!\cdots\!64}a^{26}+\frac{34\!\cdots\!09}{64\!\cdots\!64}a^{25}+\frac{65\!\cdots\!79}{16\!\cdots\!66}a^{24}-\frac{62\!\cdots\!06}{80\!\cdots\!33}a^{23}-\frac{19\!\cdots\!06}{80\!\cdots\!33}a^{22}+\frac{25\!\cdots\!01}{80\!\cdots\!33}a^{21}+\frac{53\!\cdots\!20}{80\!\cdots\!33}a^{20}-\frac{28\!\cdots\!51}{16\!\cdots\!66}a^{19}-\frac{34\!\cdots\!75}{64\!\cdots\!64}a^{18}+\frac{49\!\cdots\!13}{64\!\cdots\!64}a^{17}-\frac{67\!\cdots\!03}{16\!\cdots\!66}a^{16}-\frac{49\!\cdots\!47}{16\!\cdots\!66}a^{15}+\frac{28\!\cdots\!19}{16\!\cdots\!66}a^{14}-\frac{29\!\cdots\!87}{16\!\cdots\!66}a^{13}+\frac{61\!\cdots\!86}{80\!\cdots\!33}a^{12}+\frac{16\!\cdots\!36}{80\!\cdots\!33}a^{11}-\frac{56\!\cdots\!57}{16\!\cdots\!66}a^{10}+\frac{61\!\cdots\!69}{16\!\cdots\!66}a^{9}+\frac{22\!\cdots\!28}{80\!\cdots\!33}a^{8}-\frac{44\!\cdots\!99}{80\!\cdots\!33}a^{7}+\frac{76\!\cdots\!86}{80\!\cdots\!33}a^{6}-\frac{76\!\cdots\!25}{80\!\cdots\!33}a^{5}+\frac{65\!\cdots\!17}{80\!\cdots\!33}a^{4}-\frac{58\!\cdots\!42}{80\!\cdots\!33}a^{3}+\frac{12\!\cdots\!89}{32\!\cdots\!32}a^{2}-\frac{64\!\cdots\!95}{32\!\cdots\!32}a+\frac{75\!\cdots\!53}{80\!\cdots\!33}$, $\frac{59\!\cdots\!59}{64\!\cdots\!64}a^{26}-\frac{34\!\cdots\!47}{16\!\cdots\!66}a^{25}-\frac{98\!\cdots\!57}{64\!\cdots\!64}a^{24}+\frac{24\!\cdots\!78}{80\!\cdots\!33}a^{23}+\frac{15\!\cdots\!67}{16\!\cdots\!66}a^{22}-\frac{10\!\cdots\!80}{80\!\cdots\!33}a^{21}-\frac{22\!\cdots\!04}{80\!\cdots\!33}a^{20}+\frac{75\!\cdots\!76}{80\!\cdots\!33}a^{19}+\frac{19\!\cdots\!11}{64\!\cdots\!64}a^{18}-\frac{39\!\cdots\!67}{16\!\cdots\!66}a^{17}+\frac{99\!\cdots\!91}{64\!\cdots\!64}a^{16}+\frac{88\!\cdots\!70}{80\!\cdots\!33}a^{15}-\frac{57\!\cdots\!11}{80\!\cdots\!33}a^{14}+\frac{53\!\cdots\!33}{80\!\cdots\!33}a^{13}-\frac{30\!\cdots\!09}{16\!\cdots\!66}a^{12}-\frac{62\!\cdots\!82}{80\!\cdots\!33}a^{11}+\frac{21\!\cdots\!13}{16\!\cdots\!66}a^{10}+\frac{66\!\cdots\!17}{80\!\cdots\!33}a^{9}-\frac{18\!\cdots\!13}{16\!\cdots\!66}a^{8}+\frac{15\!\cdots\!71}{80\!\cdots\!33}a^{7}-\frac{28\!\cdots\!68}{80\!\cdots\!33}a^{6}+\frac{28\!\cdots\!15}{80\!\cdots\!33}a^{5}-\frac{22\!\cdots\!86}{80\!\cdots\!33}a^{4}+\frac{20\!\cdots\!81}{80\!\cdots\!33}a^{3}-\frac{45\!\cdots\!57}{32\!\cdots\!32}a^{2}+\frac{49\!\cdots\!00}{80\!\cdots\!33}a-\frac{11\!\cdots\!97}{32\!\cdots\!32}$, $\frac{27\!\cdots\!59}{32\!\cdots\!32}a^{26}-\frac{55\!\cdots\!49}{32\!\cdots\!32}a^{25}-\frac{11\!\cdots\!23}{80\!\cdots\!33}a^{24}+\frac{77\!\cdots\!69}{32\!\cdots\!32}a^{23}+\frac{29\!\cdots\!55}{32\!\cdots\!32}a^{22}-\frac{28\!\cdots\!43}{32\!\cdots\!32}a^{21}-\frac{83\!\cdots\!25}{32\!\cdots\!32}a^{20}-\frac{13\!\cdots\!95}{32\!\cdots\!32}a^{19}+\frac{31\!\cdots\!93}{16\!\cdots\!66}a^{18}-\frac{16\!\cdots\!75}{80\!\cdots\!33}a^{17}+\frac{46\!\cdots\!19}{32\!\cdots\!32}a^{16}+\frac{22\!\cdots\!07}{16\!\cdots\!66}a^{15}-\frac{95\!\cdots\!65}{16\!\cdots\!66}a^{14}+\frac{81\!\cdots\!67}{16\!\cdots\!66}a^{13}-\frac{26\!\cdots\!03}{16\!\cdots\!66}a^{12}-\frac{11\!\cdots\!29}{16\!\cdots\!66}a^{11}+\frac{16\!\cdots\!15}{16\!\cdots\!66}a^{10}+\frac{11\!\cdots\!15}{16\!\cdots\!66}a^{9}-\frac{13\!\cdots\!97}{16\!\cdots\!66}a^{8}+\frac{28\!\cdots\!11}{16\!\cdots\!66}a^{7}-\frac{48\!\cdots\!93}{16\!\cdots\!66}a^{6}+\frac{44\!\cdots\!15}{16\!\cdots\!66}a^{5}-\frac{37\!\cdots\!45}{16\!\cdots\!66}a^{4}+\frac{33\!\cdots\!21}{16\!\cdots\!66}a^{3}-\frac{83\!\cdots\!27}{80\!\cdots\!33}a^{2}+\frac{42\!\cdots\!58}{80\!\cdots\!33}a-\frac{39\!\cdots\!17}{16\!\cdots\!66}$, $\frac{55\!\cdots\!87}{64\!\cdots\!64}a^{26}+\frac{11\!\cdots\!39}{32\!\cdots\!32}a^{25}+\frac{74\!\cdots\!67}{64\!\cdots\!64}a^{24}-\frac{17\!\cdots\!85}{32\!\cdots\!32}a^{23}-\frac{15\!\cdots\!37}{32\!\cdots\!32}a^{22}+\frac{94\!\cdots\!65}{32\!\cdots\!32}a^{21}+\frac{32\!\cdots\!19}{32\!\cdots\!32}a^{20}-\frac{19\!\cdots\!77}{32\!\cdots\!32}a^{19}-\frac{14\!\cdots\!45}{64\!\cdots\!64}a^{18}+\frac{61\!\cdots\!10}{80\!\cdots\!33}a^{17}-\frac{12\!\cdots\!19}{64\!\cdots\!64}a^{16}+\frac{18\!\cdots\!39}{16\!\cdots\!66}a^{15}+\frac{15\!\cdots\!17}{16\!\cdots\!66}a^{14}-\frac{27\!\cdots\!69}{16\!\cdots\!66}a^{13}+\frac{69\!\cdots\!10}{80\!\cdots\!33}a^{12}+\frac{16\!\cdots\!19}{16\!\cdots\!66}a^{11}-\frac{23\!\cdots\!69}{80\!\cdots\!33}a^{10}+\frac{25\!\cdots\!25}{16\!\cdots\!66}a^{9}+\frac{17\!\cdots\!12}{80\!\cdots\!33}a^{8}-\frac{68\!\cdots\!39}{16\!\cdots\!66}a^{7}+\frac{94\!\cdots\!03}{16\!\cdots\!66}a^{6}-\frac{11\!\cdots\!79}{16\!\cdots\!66}a^{5}+\frac{89\!\cdots\!91}{16\!\cdots\!66}a^{4}-\frac{70\!\cdots\!41}{16\!\cdots\!66}a^{3}+\frac{94\!\cdots\!63}{32\!\cdots\!32}a^{2}-\frac{76\!\cdots\!26}{80\!\cdots\!33}a+\frac{86\!\cdots\!13}{32\!\cdots\!32}$, $\frac{20\!\cdots\!51}{16\!\cdots\!66}a^{26}+\frac{78\!\cdots\!03}{32\!\cdots\!32}a^{25}+\frac{17\!\cdots\!85}{80\!\cdots\!33}a^{24}-\frac{53\!\cdots\!37}{16\!\cdots\!66}a^{23}-\frac{22\!\cdots\!59}{16\!\cdots\!66}a^{22}+\frac{18\!\cdots\!01}{16\!\cdots\!66}a^{21}+\frac{62\!\cdots\!65}{16\!\cdots\!66}a^{20}+\frac{77\!\cdots\!35}{16\!\cdots\!66}a^{19}-\frac{19\!\cdots\!92}{80\!\cdots\!33}a^{18}+\frac{10\!\cdots\!37}{32\!\cdots\!32}a^{17}-\frac{33\!\cdots\!77}{16\!\cdots\!66}a^{16}-\frac{18\!\cdots\!81}{80\!\cdots\!33}a^{15}+\frac{67\!\cdots\!72}{80\!\cdots\!33}a^{14}-\frac{56\!\cdots\!36}{80\!\cdots\!33}a^{13}+\frac{16\!\cdots\!62}{80\!\cdots\!33}a^{12}+\frac{85\!\cdots\!71}{80\!\cdots\!33}a^{11}-\frac{11\!\cdots\!40}{80\!\cdots\!33}a^{10}-\frac{15\!\cdots\!40}{80\!\cdots\!33}a^{9}+\frac{99\!\cdots\!81}{80\!\cdots\!33}a^{8}-\frac{20\!\cdots\!61}{80\!\cdots\!33}a^{7}+\frac{33\!\cdots\!76}{80\!\cdots\!33}a^{6}-\frac{30\!\cdots\!77}{80\!\cdots\!33}a^{5}+\frac{27\!\cdots\!00}{80\!\cdots\!33}a^{4}-\frac{23\!\cdots\!56}{80\!\cdots\!33}a^{3}+\frac{11\!\cdots\!54}{80\!\cdots\!33}a^{2}-\frac{12\!\cdots\!67}{16\!\cdots\!66}a+\frac{27\!\cdots\!22}{80\!\cdots\!33}$, $\frac{13\!\cdots\!71}{64\!\cdots\!64}a^{26}-\frac{75\!\cdots\!65}{64\!\cdots\!64}a^{25}-\frac{63\!\cdots\!57}{32\!\cdots\!32}a^{24}+\frac{60\!\cdots\!15}{32\!\cdots\!32}a^{23}-\frac{66\!\cdots\!27}{32\!\cdots\!32}a^{22}-\frac{33\!\cdots\!15}{32\!\cdots\!32}a^{21}+\frac{15\!\cdots\!55}{32\!\cdots\!32}a^{20}+\frac{84\!\cdots\!45}{32\!\cdots\!32}a^{19}-\frac{56\!\cdots\!07}{64\!\cdots\!64}a^{18}-\frac{26\!\cdots\!91}{64\!\cdots\!64}a^{17}+\frac{10\!\cdots\!39}{16\!\cdots\!66}a^{16}-\frac{47\!\cdots\!19}{80\!\cdots\!33}a^{15}-\frac{20\!\cdots\!31}{80\!\cdots\!33}a^{14}+\frac{59\!\cdots\!53}{80\!\cdots\!33}a^{13}-\frac{83\!\cdots\!89}{16\!\cdots\!66}a^{12}-\frac{69\!\cdots\!65}{16\!\cdots\!66}a^{11}+\frac{91\!\cdots\!70}{80\!\cdots\!33}a^{10}-\frac{73\!\cdots\!09}{80\!\cdots\!33}a^{9}-\frac{11\!\cdots\!43}{16\!\cdots\!66}a^{8}+\frac{31\!\cdots\!19}{16\!\cdots\!66}a^{7}-\frac{31\!\cdots\!69}{16\!\cdots\!66}a^{6}+\frac{42\!\cdots\!69}{16\!\cdots\!66}a^{5}-\frac{40\!\cdots\!25}{16\!\cdots\!66}a^{4}+\frac{22\!\cdots\!53}{16\!\cdots\!66}a^{3}-\frac{33\!\cdots\!15}{32\!\cdots\!32}a^{2}+\frac{15\!\cdots\!49}{32\!\cdots\!32}a-\frac{29\!\cdots\!68}{80\!\cdots\!33}$, $\frac{60\!\cdots\!10}{80\!\cdots\!33}a^{26}-\frac{10\!\cdots\!47}{64\!\cdots\!64}a^{25}-\frac{82\!\cdots\!81}{64\!\cdots\!64}a^{24}+\frac{37\!\cdots\!41}{16\!\cdots\!66}a^{23}+\frac{12\!\cdots\!75}{16\!\cdots\!66}a^{22}-\frac{77\!\cdots\!42}{80\!\cdots\!33}a^{21}-\frac{36\!\cdots\!79}{16\!\cdots\!66}a^{20}+\frac{63\!\cdots\!50}{80\!\cdots\!33}a^{19}+\frac{18\!\cdots\!43}{80\!\cdots\!33}a^{18}-\frac{20\!\cdots\!23}{64\!\cdots\!64}a^{17}+\frac{70\!\cdots\!43}{64\!\cdots\!64}a^{16}+\frac{17\!\cdots\!13}{16\!\cdots\!66}a^{15}-\frac{90\!\cdots\!65}{16\!\cdots\!66}a^{14}+\frac{84\!\cdots\!37}{16\!\cdots\!66}a^{13}+\frac{69\!\cdots\!59}{16\!\cdots\!66}a^{12}-\frac{59\!\cdots\!15}{80\!\cdots\!33}a^{11}+\frac{75\!\cdots\!13}{80\!\cdots\!33}a^{10}+\frac{29\!\cdots\!17}{16\!\cdots\!66}a^{9}-\frac{18\!\cdots\!53}{16\!\cdots\!66}a^{8}+\frac{11\!\cdots\!96}{80\!\cdots\!33}a^{7}-\frac{18\!\cdots\!03}{80\!\cdots\!33}a^{6}+\frac{20\!\cdots\!71}{80\!\cdots\!33}a^{5}-\frac{14\!\cdots\!60}{80\!\cdots\!33}a^{4}+\frac{12\!\cdots\!94}{80\!\cdots\!33}a^{3}-\frac{76\!\cdots\!79}{80\!\cdots\!33}a^{2}+\frac{95\!\cdots\!85}{32\!\cdots\!32}a-\frac{64\!\cdots\!77}{32\!\cdots\!32}$, $\frac{40\!\cdots\!61}{64\!\cdots\!64}a^{26}-\frac{15\!\cdots\!29}{80\!\cdots\!33}a^{25}-\frac{59\!\cdots\!21}{64\!\cdots\!64}a^{24}+\frac{23\!\cdots\!52}{80\!\cdots\!33}a^{23}+\frac{75\!\cdots\!59}{16\!\cdots\!66}a^{22}-\frac{10\!\cdots\!82}{80\!\cdots\!33}a^{21}-\frac{89\!\cdots\!91}{80\!\cdots\!33}a^{20}+\frac{15\!\cdots\!73}{80\!\cdots\!33}a^{19}+\frac{58\!\cdots\!25}{64\!\cdots\!64}a^{18}-\frac{23\!\cdots\!06}{80\!\cdots\!33}a^{17}+\frac{88\!\cdots\!31}{64\!\cdots\!64}a^{16}+\frac{13\!\cdots\!99}{80\!\cdots\!33}a^{15}-\frac{44\!\cdots\!60}{80\!\cdots\!33}a^{14}+\frac{67\!\cdots\!43}{80\!\cdots\!33}a^{13}-\frac{94\!\cdots\!17}{16\!\cdots\!66}a^{12}-\frac{38\!\cdots\!09}{80\!\cdots\!33}a^{11}+\frac{24\!\cdots\!67}{16\!\cdots\!66}a^{10}-\frac{61\!\cdots\!02}{80\!\cdots\!33}a^{9}-\frac{12\!\cdots\!19}{16\!\cdots\!66}a^{8}+\frac{18\!\cdots\!18}{80\!\cdots\!33}a^{7}-\frac{29\!\cdots\!08}{80\!\cdots\!33}a^{6}+\frac{32\!\cdots\!86}{80\!\cdots\!33}a^{5}-\frac{29\!\cdots\!01}{80\!\cdots\!33}a^{4}+\frac{25\!\cdots\!11}{80\!\cdots\!33}a^{3}-\frac{59\!\cdots\!11}{32\!\cdots\!32}a^{2}+\frac{78\!\cdots\!97}{80\!\cdots\!33}a-\frac{14\!\cdots\!21}{32\!\cdots\!32}$, $\frac{42\!\cdots\!17}{64\!\cdots\!64}a^{26}+\frac{84\!\cdots\!11}{64\!\cdots\!64}a^{25}+\frac{36\!\cdots\!29}{32\!\cdots\!32}a^{24}-\frac{58\!\cdots\!61}{32\!\cdots\!32}a^{23}-\frac{23\!\cdots\!53}{32\!\cdots\!32}a^{22}+\frac{21\!\cdots\!35}{32\!\cdots\!32}a^{21}+\frac{67\!\cdots\!91}{32\!\cdots\!32}a^{20}-\frac{48\!\cdots\!27}{32\!\cdots\!32}a^{19}-\frac{11\!\cdots\!99}{64\!\cdots\!64}a^{18}+\frac{10\!\cdots\!45}{64\!\cdots\!64}a^{17}-\frac{84\!\cdots\!50}{80\!\cdots\!33}a^{16}-\frac{92\!\cdots\!37}{80\!\cdots\!33}a^{15}+\frac{37\!\cdots\!50}{80\!\cdots\!33}a^{14}-\frac{29\!\cdots\!91}{80\!\cdots\!33}a^{13}+\frac{77\!\cdots\!35}{16\!\cdots\!66}a^{12}+\frac{93\!\cdots\!49}{16\!\cdots\!66}a^{11}-\frac{59\!\cdots\!65}{80\!\cdots\!33}a^{10}-\frac{14\!\cdots\!91}{80\!\cdots\!33}a^{9}+\frac{11\!\cdots\!03}{16\!\cdots\!66}a^{8}-\frac{19\!\cdots\!91}{16\!\cdots\!66}a^{7}+\frac{34\!\cdots\!61}{16\!\cdots\!66}a^{6}-\frac{31\!\cdots\!31}{16\!\cdots\!66}a^{5}+\frac{25\!\cdots\!63}{16\!\cdots\!66}a^{4}-\frac{22\!\cdots\!21}{16\!\cdots\!66}a^{3}+\frac{22\!\cdots\!05}{32\!\cdots\!32}a^{2}-\frac{10\!\cdots\!67}{32\!\cdots\!32}a+\frac{12\!\cdots\!09}{80\!\cdots\!33}$
| 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: | \( 226969346121.17557 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{12}\cdot 226969346121.17557 \cdot 1}{2\cdot\sqrt{181453279996183512417624142594449705271296}}\cr\approx \mathstrut & 8.06869327842704 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 - 3*x^26 - 15*x^25 + 45*x^24 + 78*x^23 - 210*x^22 - 198*x^21 + 306*x^20 + 231*x^19 - 493*x^18 + 1887*x^17 + 3*x^16 - 8556*x^15 + 12804*x^14 - 7536*x^13 - 6672*x^12 + 20184*x^11 - 10560*x^10 - 11760*x^9 + 30192*x^8 - 53700*x^7 + 65796*x^6 - 58956*x^5 + 51156*x^4 - 36306*x^3 + 18390*x^2 - 8994*x + 2870)
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 - 3*x^26 - 15*x^25 + 45*x^24 + 78*x^23 - 210*x^22 - 198*x^21 + 306*x^20 + 231*x^19 - 493*x^18 + 1887*x^17 + 3*x^16 - 8556*x^15 + 12804*x^14 - 7536*x^13 - 6672*x^12 + 20184*x^11 - 10560*x^10 - 11760*x^9 + 30192*x^8 - 53700*x^7 + 65796*x^6 - 58956*x^5 + 51156*x^4 - 36306*x^3 + 18390*x^2 - 8994*x + 2870, 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 - 3*x^26 - 15*x^25 + 45*x^24 + 78*x^23 - 210*x^22 - 198*x^21 + 306*x^20 + 231*x^19 - 493*x^18 + 1887*x^17 + 3*x^16 - 8556*x^15 + 12804*x^14 - 7536*x^13 - 6672*x^12 + 20184*x^11 - 10560*x^10 - 11760*x^9 + 30192*x^8 - 53700*x^7 + 65796*x^6 - 58956*x^5 + 51156*x^4 - 36306*x^3 + 18390*x^2 - 8994*x + 2870);
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 - 3*x^26 - 15*x^25 + 45*x^24 + 78*x^23 - 210*x^22 - 198*x^21 + 306*x^20 + 231*x^19 - 493*x^18 + 1887*x^17 + 3*x^16 - 8556*x^15 + 12804*x^14 - 7536*x^13 - 6672*x^12 + 20184*x^11 - 10560*x^10 - 11760*x^9 + 30192*x^8 - 53700*x^7 + 65796*x^6 - 58956*x^5 + 51156*x^4 - 36306*x^3 + 18390*x^2 - 8994*x + 2870);
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
$\SO(5,3)$ (as 27T1161):
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 non-solvable group of order 51840 |
| The 25 conjugacy class representatives for $\SO(5,3)$ |
| Character table for $\SO(5,3)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 36 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Degree 45 sibling: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.5.0.1}{5} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{5}$ | ${\href{/padicField/13.5.0.1}{5} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{5}$ | ${\href{/padicField/19.9.0.1}{9} }^{3}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }^{4}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.9.0.1}{9} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.12.0.1}{12} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.8.24.145 | $x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 4 x^{2} + 8 x + 18$ | $8$ | $1$ | $24$ | $C_2 \wr S_4$ | $[2, 8/3, 8/3, 10/3, 10/3, 4]_{3}^{2}$ | |
| Deg $16$ | $16$ | $1$ | $54$ | ||||
|
\(3\)
| 3.9.12.21 | $x^{9} + 6 x^{4} + 3$ | $9$ | $1$ | $12$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ |
| Deg $18$ | $9$ | $2$ | $24$ |