Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 8*x^14 + 160*x^13 - 400*x^12 - 1282*x^11 + 11189*x^10 - 35603*x^9 + 58718*x^8 - 16818*x^7 - 136633*x^6 + 401269*x^5 - 424715*x^4 + 202432*x^3 + 912840*x^2 - 1240424*x + 1871824)
gp: K = bnfinit(y^16 - 6*y^15 - 8*y^14 + 160*y^13 - 400*y^12 - 1282*y^11 + 11189*y^10 - 35603*y^9 + 58718*y^8 - 16818*y^7 - 136633*y^6 + 401269*y^5 - 424715*y^4 + 202432*y^3 + 912840*y^2 - 1240424*y + 1871824, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 - 8*x^14 + 160*x^13 - 400*x^12 - 1282*x^11 + 11189*x^10 - 35603*x^9 + 58718*x^8 - 16818*x^7 - 136633*x^6 + 401269*x^5 - 424715*x^4 + 202432*x^3 + 912840*x^2 - 1240424*x + 1871824);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 - 8*x^14 + 160*x^13 - 400*x^12 - 1282*x^11 + 11189*x^10 - 35603*x^9 + 58718*x^8 - 16818*x^7 - 136633*x^6 + 401269*x^5 - 424715*x^4 + 202432*x^3 + 912840*x^2 - 1240424*x + 1871824)
\( x^{16} - 6 x^{15} - 8 x^{14} + 160 x^{13} - 400 x^{12} - 1282 x^{11} + 11189 x^{10} - 35603 x^{9} + \cdots + 1871824 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(525162048993676835600871662401\)
\(\medspace = 41^{14}\cdot 61^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(72.03\) | 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: | $41^{7/8}61^{1/2}\approx 201.30325259286246$ | ||
| Ramified primes: |
\(41\), \(61\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{187912}a^{14}+\frac{15913}{187912}a^{13}-\frac{18591}{187912}a^{12}+\frac{35157}{187912}a^{11}-\frac{25087}{187912}a^{10}-\frac{47149}{187912}a^{9}+\frac{8471}{23489}a^{8}+\frac{26383}{187912}a^{7}-\frac{24269}{187912}a^{6}+\frac{88209}{187912}a^{5}+\frac{1108}{23489}a^{4}-\frac{1901}{187912}a^{3}-\frac{24213}{93956}a^{2}-\frac{1051}{23489}a+\frac{2661}{23489}$, $\frac{1}{48\!\cdots\!68}a^{15}+\frac{52\!\cdots\!53}{12\!\cdots\!42}a^{14}+\frac{31\!\cdots\!81}{24\!\cdots\!84}a^{13}+\frac{21\!\cdots\!67}{24\!\cdots\!84}a^{12}-\frac{14\!\cdots\!75}{24\!\cdots\!84}a^{11}-\frac{70\!\cdots\!53}{12\!\cdots\!42}a^{10}-\frac{81\!\cdots\!49}{48\!\cdots\!68}a^{9}+\frac{23\!\cdots\!37}{48\!\cdots\!68}a^{8}+\frac{50\!\cdots\!33}{12\!\cdots\!42}a^{7}-\frac{24\!\cdots\!97}{12\!\cdots\!42}a^{6}-\frac{21\!\cdots\!19}{48\!\cdots\!68}a^{5}-\frac{14\!\cdots\!87}{48\!\cdots\!68}a^{4}+\frac{19\!\cdots\!59}{48\!\cdots\!68}a^{3}+\frac{15\!\cdots\!69}{61\!\cdots\!71}a^{2}-\frac{21\!\cdots\!21}{61\!\cdots\!71}a-\frac{25\!\cdots\!44}{61\!\cdots\!71}$
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
$C_{41}$, which has order $41$ (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: | $7$ | 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: |
$\frac{24\!\cdots\!79}{41\!\cdots\!86}a^{15}-\frac{53\!\cdots\!43}{41\!\cdots\!86}a^{14}-\frac{54\!\cdots\!53}{41\!\cdots\!86}a^{13}+\frac{88\!\cdots\!99}{20\!\cdots\!93}a^{12}+\frac{27\!\cdots\!87}{41\!\cdots\!86}a^{11}-\frac{14\!\cdots\!62}{20\!\cdots\!93}a^{10}+\frac{38\!\cdots\!52}{20\!\cdots\!93}a^{9}-\frac{99\!\cdots\!10}{20\!\cdots\!93}a^{8}+\frac{66\!\cdots\!89}{41\!\cdots\!86}a^{7}-\frac{18\!\cdots\!79}{41\!\cdots\!86}a^{6}+\frac{45\!\cdots\!43}{41\!\cdots\!86}a^{5}+\frac{17\!\cdots\!05}{41\!\cdots\!86}a^{4}-\frac{12\!\cdots\!91}{41\!\cdots\!86}a^{3}+\frac{50\!\cdots\!27}{41\!\cdots\!86}a^{2}-\frac{24\!\cdots\!39}{20\!\cdots\!93}a+\frac{23\!\cdots\!03}{20\!\cdots\!93}$, $\frac{10\!\cdots\!49}{83\!\cdots\!72}a^{15}-\frac{75\!\cdots\!31}{83\!\cdots\!72}a^{14}-\frac{11\!\cdots\!81}{83\!\cdots\!72}a^{13}+\frac{10\!\cdots\!95}{41\!\cdots\!86}a^{12}-\frac{40\!\cdots\!25}{83\!\cdots\!72}a^{11}-\frac{53\!\cdots\!87}{20\!\cdots\!93}a^{10}+\frac{66\!\cdots\!63}{41\!\cdots\!86}a^{9}-\frac{83\!\cdots\!51}{20\!\cdots\!93}a^{8}+\frac{42\!\cdots\!41}{83\!\cdots\!72}a^{7}+\frac{13\!\cdots\!49}{83\!\cdots\!72}a^{6}-\frac{16\!\cdots\!63}{83\!\cdots\!72}a^{5}+\frac{31\!\cdots\!11}{83\!\cdots\!72}a^{4}-\frac{30\!\cdots\!61}{83\!\cdots\!72}a^{3}-\frac{10\!\cdots\!27}{83\!\cdots\!72}a^{2}+\frac{20\!\cdots\!23}{41\!\cdots\!86}a-\frac{27\!\cdots\!63}{20\!\cdots\!93}$, $\frac{11\!\cdots\!69}{24\!\cdots\!84}a^{15}-\frac{19\!\cdots\!81}{12\!\cdots\!42}a^{14}-\frac{25\!\cdots\!43}{24\!\cdots\!84}a^{13}+\frac{74\!\cdots\!07}{12\!\cdots\!42}a^{12}+\frac{21\!\cdots\!97}{24\!\cdots\!84}a^{11}-\frac{58\!\cdots\!44}{61\!\cdots\!71}a^{10}+\frac{38\!\cdots\!99}{12\!\cdots\!42}a^{9}-\frac{10\!\cdots\!11}{24\!\cdots\!84}a^{8}-\frac{31\!\cdots\!83}{61\!\cdots\!71}a^{7}+\frac{87\!\cdots\!11}{24\!\cdots\!84}a^{6}-\frac{12\!\cdots\!61}{24\!\cdots\!84}a^{5}+\frac{36\!\cdots\!73}{61\!\cdots\!71}a^{4}+\frac{11\!\cdots\!26}{61\!\cdots\!71}a^{3}-\frac{17\!\cdots\!70}{61\!\cdots\!71}a^{2}+\frac{37\!\cdots\!56}{61\!\cdots\!71}a-\frac{40\!\cdots\!69}{61\!\cdots\!71}$, $\frac{82\!\cdots\!11}{83\!\cdots\!72}a^{15}+\frac{16\!\cdots\!15}{83\!\cdots\!72}a^{14}-\frac{30\!\cdots\!23}{83\!\cdots\!72}a^{13}+\frac{23\!\cdots\!45}{41\!\cdots\!86}a^{12}+\frac{43\!\cdots\!89}{83\!\cdots\!72}a^{11}-\frac{46\!\cdots\!00}{20\!\cdots\!93}a^{10}-\frac{13\!\cdots\!21}{41\!\cdots\!86}a^{9}+\frac{43\!\cdots\!73}{20\!\cdots\!93}a^{8}-\frac{49\!\cdots\!13}{83\!\cdots\!72}a^{7}+\frac{45\!\cdots\!47}{83\!\cdots\!72}a^{6}+\frac{12\!\cdots\!75}{83\!\cdots\!72}a^{5}-\frac{22\!\cdots\!19}{83\!\cdots\!72}a^{4}+\frac{39\!\cdots\!73}{83\!\cdots\!72}a^{3}+\frac{39\!\cdots\!99}{83\!\cdots\!72}a^{2}-\frac{14\!\cdots\!75}{41\!\cdots\!86}a+\frac{21\!\cdots\!53}{20\!\cdots\!93}$, $\frac{12\!\cdots\!21}{24\!\cdots\!84}a^{15}-\frac{85\!\cdots\!97}{61\!\cdots\!71}a^{14}-\frac{32\!\cdots\!35}{24\!\cdots\!84}a^{13}+\frac{71\!\cdots\!59}{12\!\cdots\!42}a^{12}+\frac{17\!\cdots\!53}{24\!\cdots\!84}a^{11}-\frac{65\!\cdots\!58}{61\!\cdots\!71}a^{10}+\frac{31\!\cdots\!19}{12\!\cdots\!42}a^{9}-\frac{21\!\cdots\!49}{24\!\cdots\!84}a^{8}-\frac{10\!\cdots\!29}{12\!\cdots\!42}a^{7}+\frac{53\!\cdots\!59}{24\!\cdots\!84}a^{6}+\frac{28\!\cdots\!67}{24\!\cdots\!84}a^{5}-\frac{95\!\cdots\!27}{12\!\cdots\!42}a^{4}+\frac{11\!\cdots\!59}{12\!\cdots\!42}a^{3}+\frac{57\!\cdots\!77}{12\!\cdots\!42}a^{2}+\frac{47\!\cdots\!27}{61\!\cdots\!71}a+\frac{82\!\cdots\!99}{61\!\cdots\!71}$, $\frac{33\!\cdots\!91}{29\!\cdots\!48}a^{15}-\frac{12\!\cdots\!11}{29\!\cdots\!48}a^{14}-\frac{80\!\cdots\!63}{29\!\cdots\!48}a^{13}+\frac{21\!\cdots\!51}{14\!\cdots\!74}a^{12}+\frac{47\!\cdots\!25}{29\!\cdots\!48}a^{11}-\frac{18\!\cdots\!65}{73\!\cdots\!37}a^{10}+\frac{73\!\cdots\!07}{14\!\cdots\!74}a^{9}+\frac{27\!\cdots\!09}{14\!\cdots\!74}a^{8}+\frac{17\!\cdots\!09}{29\!\cdots\!48}a^{7}-\frac{28\!\cdots\!73}{29\!\cdots\!48}a^{6}+\frac{90\!\cdots\!31}{29\!\cdots\!48}a^{5}-\frac{60\!\cdots\!53}{29\!\cdots\!48}a^{4}-\frac{23\!\cdots\!57}{29\!\cdots\!48}a^{3}+\frac{83\!\cdots\!61}{29\!\cdots\!48}a^{2}-\frac{54\!\cdots\!13}{14\!\cdots\!74}a+\frac{19\!\cdots\!66}{73\!\cdots\!37}$, $\frac{76\!\cdots\!47}{24\!\cdots\!84}a^{15}-\frac{10\!\cdots\!79}{12\!\cdots\!42}a^{14}-\frac{18\!\cdots\!57}{24\!\cdots\!84}a^{13}+\frac{42\!\cdots\!99}{12\!\cdots\!42}a^{12}+\frac{76\!\cdots\!35}{24\!\cdots\!84}a^{11}-\frac{13\!\cdots\!29}{21\!\cdots\!37}a^{10}+\frac{20\!\cdots\!75}{12\!\cdots\!42}a^{9}-\frac{32\!\cdots\!09}{24\!\cdots\!84}a^{8}-\frac{25\!\cdots\!83}{61\!\cdots\!71}a^{7}+\frac{39\!\cdots\!97}{24\!\cdots\!84}a^{6}-\frac{29\!\cdots\!11}{24\!\cdots\!84}a^{5}+\frac{66\!\cdots\!92}{61\!\cdots\!71}a^{4}+\frac{45\!\cdots\!14}{61\!\cdots\!71}a^{3}-\frac{33\!\cdots\!42}{61\!\cdots\!71}a^{2}+\frac{10\!\cdots\!12}{61\!\cdots\!71}a+\frac{19\!\cdots\!35}{61\!\cdots\!71}$
| 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: | \( 2911738.86032 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2911738.86032 \cdot 41}{2\cdot\sqrt{525162048993676835600871662401}}\cr\approx \mathstrut & 0.200077643421 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 8*x^14 + 160*x^13 - 400*x^12 - 1282*x^11 + 11189*x^10 - 35603*x^9 + 58718*x^8 - 16818*x^7 - 136633*x^6 + 401269*x^5 - 424715*x^4 + 202432*x^3 + 912840*x^2 - 1240424*x + 1871824)
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^16 - 6*x^15 - 8*x^14 + 160*x^13 - 400*x^12 - 1282*x^11 + 11189*x^10 - 35603*x^9 + 58718*x^8 - 16818*x^7 - 136633*x^6 + 401269*x^5 - 424715*x^4 + 202432*x^3 + 912840*x^2 - 1240424*x + 1871824, 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^16 - 6*x^15 - 8*x^14 + 160*x^13 - 400*x^12 - 1282*x^11 + 11189*x^10 - 35603*x^9 + 58718*x^8 - 16818*x^7 - 136633*x^6 + 401269*x^5 - 424715*x^4 + 202432*x^3 + 912840*x^2 - 1240424*x + 1871824);
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^16 - 6*x^15 - 8*x^14 + 160*x^13 - 400*x^12 - 1282*x^11 + 11189*x^10 - 35603*x^9 + 58718*x^8 - 16818*x^7 - 136633*x^6 + 401269*x^5 - 424715*x^4 + 202432*x^3 + 912840*x^2 - 1240424*x + 1871824);
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
$C_2^2:C_8$ (as 16T24):
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 32 |
| The 20 conjugacy class representatives for $C_2^2 : C_8$ |
| Character table for $C_2^2 : C_8$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.4.68921.1, 4.4.102541.1, 4.4.4204181.1, 8.0.724680653111201.1, 8.0.194754273881.1, 8.8.17675137880761.1 |
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]
Sibling fields
| Galois closure: | deg 32 |
| Degree 16 sibling: | 16.0.7271310229600659471112808499009924241.2 |
| 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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(41\)
| 41.16.14.1 | $x^{16} + 304 x^{15} + 40480 x^{14} + 3085600 x^{13} + 147416080 x^{12} + 4529584192 x^{11} + 87831092608 x^{10} + 996302227840 x^{9} + 5391168776882 x^{8} + 5977813379504 x^{7} + 3161920977824 x^{6} + 978514601120 x^{5} + 196936323920 x^{4} + 202153692608 x^{3} + 3372805705856 x^{2} + 36445904670848 x + 172395305267889$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |
|
\(61\)
| 61.4.0.1 | $x^{4} + 3 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 61.4.0.1 | $x^{4} + 3 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 61.8.4.1 | $x^{8} + 9516 x^{7} + 33958096 x^{6} + 53858928086 x^{5} + 32035798457059 x^{4} + 3448323535094 x^{3} + 98379480266246 x^{2} + 1286299880976982 x + 96975348777163$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |