Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 18*x^14 + 18*x^13 + 163*x^12 - 172*x^11 - 552*x^10 + 1347*x^9 + 1266*x^8 - 6786*x^7 + 2547*x^6 + 11246*x^5 - 11672*x^4 - 9291*x^3 + 33042*x^2 - 29141*x + 16381)
gp: K = bnfinit(y^16 - 2*y^15 - 18*y^14 + 18*y^13 + 163*y^12 - 172*y^11 - 552*y^10 + 1347*y^9 + 1266*y^8 - 6786*y^7 + 2547*y^6 + 11246*y^5 - 11672*y^4 - 9291*y^3 + 33042*y^2 - 29141*y + 16381, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 18*x^14 + 18*x^13 + 163*x^12 - 172*x^11 - 552*x^10 + 1347*x^9 + 1266*x^8 - 6786*x^7 + 2547*x^6 + 11246*x^5 - 11672*x^4 - 9291*x^3 + 33042*x^2 - 29141*x + 16381);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 18*x^14 + 18*x^13 + 163*x^12 - 172*x^11 - 552*x^10 + 1347*x^9 + 1266*x^8 - 6786*x^7 + 2547*x^6 + 11246*x^5 - 11672*x^4 - 9291*x^3 + 33042*x^2 - 29141*x + 16381)
\( x^{16} - 2 x^{15} - 18 x^{14} + 18 x^{13} + 163 x^{12} - 172 x^{11} - 552 x^{10} + 1347 x^{9} + \cdots + 16381 \)
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: |
\(483823969655762431707489\)
\(\medspace = 3^{8}\cdot 97^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.22\) | 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: | $3^{1/2}97^{3/4}\approx 53.535199825186666$ | ||
| Ramified primes: |
\(3\), \(97\)
| 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 not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{14}a^{14}+\frac{1}{7}a^{13}-\frac{1}{14}a^{12}-\frac{3}{14}a^{11}+\frac{1}{7}a^{10}+\frac{5}{14}a^{9}-\frac{3}{14}a^{8}-\frac{3}{7}a^{7}-\frac{3}{14}a^{6}+\frac{1}{14}a^{5}+\frac{2}{7}a^{4}+\frac{1}{14}a^{3}-\frac{5}{14}a^{2}-\frac{3}{7}a+\frac{3}{14}$, $\frac{1}{65\!\cdots\!98}a^{15}+\frac{20\!\cdots\!59}{32\!\cdots\!49}a^{14}+\frac{25\!\cdots\!64}{32\!\cdots\!49}a^{13}+\frac{77\!\cdots\!91}{32\!\cdots\!49}a^{12}-\frac{14\!\cdots\!02}{32\!\cdots\!49}a^{11}-\frac{79\!\cdots\!91}{32\!\cdots\!49}a^{10}+\frac{30\!\cdots\!41}{32\!\cdots\!49}a^{9}-\frac{11\!\cdots\!36}{32\!\cdots\!49}a^{8}-\frac{19\!\cdots\!33}{32\!\cdots\!49}a^{7}+\frac{57\!\cdots\!82}{32\!\cdots\!49}a^{6}+\frac{13\!\cdots\!84}{32\!\cdots\!49}a^{5}-\frac{40\!\cdots\!95}{32\!\cdots\!49}a^{4}-\frac{15\!\cdots\!80}{32\!\cdots\!49}a^{3}+\frac{15\!\cdots\!90}{32\!\cdots\!49}a^{2}+\frac{14\!\cdots\!02}{32\!\cdots\!49}a-\frac{93\!\cdots\!51}{94\!\cdots\!14}$
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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{1635044887083493}{25925487728830265582} a^{15} - \frac{2266218916099537}{25925487728830265582} a^{14} - \frac{28982100472406337}{25925487728830265582} a^{13} + \frac{3380228533810795}{12962743864415132791} a^{12} + \frac{243174892866235377}{25925487728830265582} a^{11} - \frac{83579489680520449}{25925487728830265582} a^{10} - \frac{368938399675886743}{12962743864415132791} a^{9} + \frac{1377627882526655127}{25925487728830265582} a^{8} + \frac{2402718408110796251}{25925487728830265582} a^{7} - \frac{3853528833812264616}{12962743864415132791} a^{6} + \frac{248620025616991945}{25925487728830265582} a^{5} + \frac{13011015059914101035}{25925487728830265582} a^{4} - \frac{3673169606597379325}{12962743864415132791} a^{3} - \frac{14729991011466854863}{25925487728830265582} a^{2} + \frac{45273357898236017427}{25925487728830265582} a - \frac{12615760837496252921}{25925487728830265582} \)
(order $6$)
| 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: |
$\frac{34\!\cdots\!33}{94\!\cdots\!14}a^{15}-\frac{53\!\cdots\!47}{65\!\cdots\!98}a^{14}-\frac{37\!\cdots\!61}{65\!\cdots\!98}a^{13}+\frac{28\!\cdots\!76}{32\!\cdots\!49}a^{12}+\frac{29\!\cdots\!71}{65\!\cdots\!98}a^{11}-\frac{67\!\cdots\!13}{65\!\cdots\!98}a^{10}-\frac{36\!\cdots\!43}{32\!\cdots\!49}a^{9}+\frac{44\!\cdots\!51}{65\!\cdots\!98}a^{8}+\frac{20\!\cdots\!51}{65\!\cdots\!98}a^{7}-\frac{74\!\cdots\!48}{32\!\cdots\!49}a^{6}+\frac{16\!\cdots\!95}{65\!\cdots\!98}a^{5}+\frac{24\!\cdots\!69}{65\!\cdots\!98}a^{4}-\frac{11\!\cdots\!12}{32\!\cdots\!49}a^{3}-\frac{41\!\cdots\!21}{65\!\cdots\!98}a^{2}+\frac{83\!\cdots\!71}{65\!\cdots\!98}a-\frac{67\!\cdots\!57}{65\!\cdots\!98}$, $\frac{32\!\cdots\!71}{94\!\cdots\!14}a^{15}-\frac{66\!\cdots\!85}{32\!\cdots\!49}a^{14}-\frac{34\!\cdots\!77}{65\!\cdots\!98}a^{13}+\frac{10\!\cdots\!65}{32\!\cdots\!49}a^{12}+\frac{20\!\cdots\!11}{32\!\cdots\!49}a^{11}-\frac{19\!\cdots\!05}{65\!\cdots\!98}a^{10}-\frac{74\!\cdots\!06}{32\!\cdots\!49}a^{9}+\frac{46\!\cdots\!02}{32\!\cdots\!49}a^{8}-\frac{29\!\cdots\!15}{65\!\cdots\!98}a^{7}-\frac{19\!\cdots\!03}{32\!\cdots\!49}a^{6}+\frac{21\!\cdots\!04}{32\!\cdots\!49}a^{5}+\frac{70\!\cdots\!79}{65\!\cdots\!98}a^{4}-\frac{59\!\cdots\!63}{32\!\cdots\!49}a^{3}-\frac{22\!\cdots\!78}{32\!\cdots\!49}a^{2}+\frac{20\!\cdots\!09}{65\!\cdots\!98}a-\frac{12\!\cdots\!83}{65\!\cdots\!98}$, $\frac{36\!\cdots\!90}{47\!\cdots\!07}a^{15}-\frac{42\!\cdots\!50}{47\!\cdots\!07}a^{14}-\frac{15\!\cdots\!59}{94\!\cdots\!14}a^{13}+\frac{16\!\cdots\!81}{47\!\cdots\!07}a^{12}+\frac{74\!\cdots\!82}{47\!\cdots\!07}a^{11}-\frac{29\!\cdots\!99}{94\!\cdots\!14}a^{10}-\frac{32\!\cdots\!85}{47\!\cdots\!07}a^{9}+\frac{36\!\cdots\!23}{47\!\cdots\!07}a^{8}+\frac{21\!\cdots\!87}{94\!\cdots\!14}a^{7}-\frac{26\!\cdots\!42}{47\!\cdots\!07}a^{6}-\frac{18\!\cdots\!86}{47\!\cdots\!07}a^{5}+\frac{13\!\cdots\!85}{94\!\cdots\!14}a^{4}-\frac{22\!\cdots\!57}{47\!\cdots\!07}a^{3}-\frac{11\!\cdots\!29}{47\!\cdots\!07}a^{2}+\frac{21\!\cdots\!73}{94\!\cdots\!14}a-\frac{70\!\cdots\!70}{47\!\cdots\!07}$, $\frac{15\!\cdots\!47}{94\!\cdots\!14}a^{15}-\frac{20\!\cdots\!52}{47\!\cdots\!07}a^{14}-\frac{26\!\cdots\!99}{94\!\cdots\!14}a^{13}+\frac{22\!\cdots\!18}{47\!\cdots\!07}a^{12}+\frac{12\!\cdots\!12}{47\!\cdots\!07}a^{11}-\frac{41\!\cdots\!79}{94\!\cdots\!14}a^{10}-\frac{44\!\cdots\!86}{47\!\cdots\!07}a^{9}+\frac{12\!\cdots\!70}{47\!\cdots\!07}a^{8}+\frac{17\!\cdots\!47}{94\!\cdots\!14}a^{7}-\frac{62\!\cdots\!55}{47\!\cdots\!07}a^{6}+\frac{31\!\cdots\!97}{47\!\cdots\!07}a^{5}+\frac{22\!\cdots\!93}{94\!\cdots\!14}a^{4}-\frac{10\!\cdots\!50}{47\!\cdots\!07}a^{3}-\frac{13\!\cdots\!30}{47\!\cdots\!07}a^{2}+\frac{59\!\cdots\!25}{94\!\cdots\!14}a-\frac{28\!\cdots\!09}{94\!\cdots\!14}$, $\frac{13\!\cdots\!37}{47\!\cdots\!07}a^{15}-\frac{12\!\cdots\!98}{32\!\cdots\!49}a^{14}+\frac{49\!\cdots\!04}{32\!\cdots\!49}a^{13}+\frac{16\!\cdots\!30}{32\!\cdots\!49}a^{12}+\frac{12\!\cdots\!40}{32\!\cdots\!49}a^{11}-\frac{12\!\cdots\!05}{32\!\cdots\!49}a^{10}-\frac{81\!\cdots\!70}{32\!\cdots\!49}a^{9}+\frac{30\!\cdots\!89}{32\!\cdots\!49}a^{8}+\frac{17\!\cdots\!35}{32\!\cdots\!49}a^{7}-\frac{19\!\cdots\!18}{32\!\cdots\!49}a^{6}+\frac{11\!\cdots\!42}{32\!\cdots\!49}a^{5}+\frac{59\!\cdots\!27}{32\!\cdots\!49}a^{4}-\frac{18\!\cdots\!94}{32\!\cdots\!49}a^{3}-\frac{12\!\cdots\!97}{32\!\cdots\!49}a^{2}+\frac{12\!\cdots\!20}{32\!\cdots\!49}a-\frac{12\!\cdots\!15}{32\!\cdots\!49}$, $\frac{53\!\cdots\!72}{32\!\cdots\!49}a^{15}+\frac{51\!\cdots\!03}{65\!\cdots\!98}a^{14}-\frac{92\!\cdots\!19}{32\!\cdots\!49}a^{13}-\frac{10\!\cdots\!25}{32\!\cdots\!49}a^{12}+\frac{11\!\cdots\!27}{65\!\cdots\!98}a^{11}+\frac{52\!\cdots\!18}{32\!\cdots\!49}a^{10}-\frac{12\!\cdots\!38}{32\!\cdots\!49}a^{9}+\frac{55\!\cdots\!11}{65\!\cdots\!98}a^{8}+\frac{10\!\cdots\!83}{32\!\cdots\!49}a^{7}-\frac{51\!\cdots\!17}{32\!\cdots\!49}a^{6}-\frac{18\!\cdots\!97}{65\!\cdots\!98}a^{5}+\frac{20\!\cdots\!74}{32\!\cdots\!49}a^{4}+\frac{39\!\cdots\!01}{32\!\cdots\!49}a^{3}-\frac{71\!\cdots\!07}{65\!\cdots\!98}a^{2}+\frac{30\!\cdots\!00}{32\!\cdots\!49}a-\frac{22\!\cdots\!56}{32\!\cdots\!49}$, $\frac{24\!\cdots\!41}{65\!\cdots\!98}a^{15}-\frac{40\!\cdots\!36}{47\!\cdots\!07}a^{14}-\frac{35\!\cdots\!43}{65\!\cdots\!98}a^{13}+\frac{28\!\cdots\!37}{32\!\cdots\!49}a^{12}+\frac{13\!\cdots\!00}{32\!\cdots\!49}a^{11}-\frac{53\!\cdots\!61}{65\!\cdots\!98}a^{10}-\frac{19\!\cdots\!66}{32\!\cdots\!49}a^{9}+\frac{17\!\cdots\!13}{47\!\cdots\!07}a^{8}-\frac{13\!\cdots\!27}{65\!\cdots\!98}a^{7}-\frac{55\!\cdots\!94}{47\!\cdots\!07}a^{6}+\frac{89\!\cdots\!16}{32\!\cdots\!49}a^{5}-\frac{19\!\cdots\!53}{94\!\cdots\!14}a^{4}+\frac{85\!\cdots\!70}{47\!\cdots\!07}a^{3}+\frac{46\!\cdots\!07}{32\!\cdots\!49}a^{2}-\frac{78\!\cdots\!93}{65\!\cdots\!98}a+\frac{31\!\cdots\!59}{65\!\cdots\!98}$
| 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: | \( 329160.9111 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 329160.9111 \cdot 1}{6\cdot\sqrt{483823969655762431707489}}\cr\approx \mathstrut & 0.1915809233 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 18*x^14 + 18*x^13 + 163*x^12 - 172*x^11 - 552*x^10 + 1347*x^9 + 1266*x^8 - 6786*x^7 + 2547*x^6 + 11246*x^5 - 11672*x^4 - 9291*x^3 + 33042*x^2 - 29141*x + 16381)
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 - 2*x^15 - 18*x^14 + 18*x^13 + 163*x^12 - 172*x^11 - 552*x^10 + 1347*x^9 + 1266*x^8 - 6786*x^7 + 2547*x^6 + 11246*x^5 - 11672*x^4 - 9291*x^3 + 33042*x^2 - 29141*x + 16381, 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 - 2*x^15 - 18*x^14 + 18*x^13 + 163*x^12 - 172*x^11 - 552*x^10 + 1347*x^9 + 1266*x^8 - 6786*x^7 + 2547*x^6 + 11246*x^5 - 11672*x^4 - 9291*x^3 + 33042*x^2 - 29141*x + 16381);
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 - 2*x^15 - 18*x^14 + 18*x^13 + 163*x^12 - 172*x^11 - 552*x^10 + 1347*x^9 + 1266*x^8 - 6786*x^7 + 2547*x^6 + 11246*x^5 - 11672*x^4 - 9291*x^3 + 33042*x^2 - 29141*x + 16381);
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
$C_4\wr C_2$ (as 16T42):
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 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{97}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-291}) \), 4.2.28227.1 x2, 4.0.873.1 x2, \(\Q(\sqrt{-3}, \sqrt{97})\), 8.0.73926513.1, 8.0.695574560817.1, 8.0.7170871761.1 |
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 8 siblings: | 8.0.73926513.1, 8.0.695574560817.1 |
| Degree 16 sibling: | 16.4.505811081165674302215084889.1 |
| Minimal sibling: | 8.0.73926513.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(97\)
| 97.8.6.2 | $x^{8} + 8536 x^{4} - 3415467$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 97.8.4.1 | $x^{8} + 20564 x^{7} + 158579686 x^{6} + 543510242244 x^{5} + 698570486224711 x^{4} + 55994198721100 x^{3} + 4236267101096262 x^{2} + 56043996888369552 x + 4548521252040853$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |