Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 7*x^16 - 11*x^15 + 37*x^14 - 22*x^13 + 71*x^12 + 33*x^11 + 204*x^10 + 118*x^9 + 305*x^8 + 169*x^7 + 216*x^6 + 59*x^5 + 84*x^4 + 8*x^3 + 12*x^2 + x + 1)
gp: K = bnfinit(y^18 - 4*y^17 + 7*y^16 - 11*y^15 + 37*y^14 - 22*y^13 + 71*y^12 + 33*y^11 + 204*y^10 + 118*y^9 + 305*y^8 + 169*y^7 + 216*y^6 + 59*y^5 + 84*y^4 + 8*y^3 + 12*y^2 + y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 4*x^17 + 7*x^16 - 11*x^15 + 37*x^14 - 22*x^13 + 71*x^12 + 33*x^11 + 204*x^10 + 118*x^9 + 305*x^8 + 169*x^7 + 216*x^6 + 59*x^5 + 84*x^4 + 8*x^3 + 12*x^2 + x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 4*x^17 + 7*x^16 - 11*x^15 + 37*x^14 - 22*x^13 + 71*x^12 + 33*x^11 + 204*x^10 + 118*x^9 + 305*x^8 + 169*x^7 + 216*x^6 + 59*x^5 + 84*x^4 + 8*x^3 + 12*x^2 + x + 1)
\( x^{18} - 4 x^{17} + 7 x^{16} - 11 x^{15} + 37 x^{14} - 22 x^{13} + 71 x^{12} + 33 x^{11} + 204 x^{10} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-4466413101188628826579\)
\(\medspace = -\,7^{12}\cdot 19^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.95\) | 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: | $7^{2/3}19^{1/2}\approx 15.950543793511486$ | ||
| Ramified primes: |
\(7\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13}a^{16}-\frac{1}{13}a^{15}-\frac{6}{13}a^{14}-\frac{6}{13}a^{13}+\frac{1}{13}a^{12}+\frac{2}{13}a^{11}+\frac{2}{13}a^{10}+\frac{6}{13}a^{9}-\frac{6}{13}a^{8}+\frac{1}{13}a^{7}+\frac{4}{13}a^{6}+\frac{2}{13}a^{5}+\frac{6}{13}a^{2}+\frac{4}{13}$, $\frac{1}{39\!\cdots\!39}a^{17}-\frac{14\!\cdots\!35}{39\!\cdots\!39}a^{16}+\frac{12\!\cdots\!80}{39\!\cdots\!39}a^{15}-\frac{32\!\cdots\!48}{39\!\cdots\!39}a^{14}+\frac{69\!\cdots\!66}{39\!\cdots\!39}a^{13}-\frac{16\!\cdots\!63}{39\!\cdots\!39}a^{12}-\frac{18\!\cdots\!35}{39\!\cdots\!39}a^{11}+\frac{18\!\cdots\!37}{39\!\cdots\!39}a^{10}+\frac{13\!\cdots\!15}{39\!\cdots\!39}a^{9}-\frac{95\!\cdots\!35}{39\!\cdots\!39}a^{8}-\frac{41\!\cdots\!35}{39\!\cdots\!39}a^{7}+\frac{17\!\cdots\!59}{39\!\cdots\!39}a^{6}-\frac{87\!\cdots\!72}{30\!\cdots\!03}a^{5}+\frac{51\!\cdots\!91}{30\!\cdots\!03}a^{4}-\frac{17\!\cdots\!35}{39\!\cdots\!39}a^{3}+\frac{55\!\cdots\!21}{30\!\cdots\!03}a^{2}-\frac{16\!\cdots\!51}{39\!\cdots\!39}a-\frac{12\!\cdots\!08}{30\!\cdots\!03}$
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
Trivial group, which has order $1$
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: | $8$ | 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: |
$\frac{2994530600305}{6617445668093}a^{17}-\frac{12872771791889}{6617445668093}a^{16}+\frac{23821455544059}{6617445668093}a^{15}-\frac{35835389602039}{6617445668093}a^{14}+\frac{113760548396564}{6617445668093}a^{13}-\frac{88105020793119}{6617445668093}a^{12}+\frac{201014156652084}{6617445668093}a^{11}+\frac{68179752956481}{6617445668093}a^{10}+\frac{524230239090920}{6617445668093}a^{9}+\frac{178547836878025}{6617445668093}a^{8}+\frac{685807639492640}{6617445668093}a^{7}+\frac{246677530797124}{6617445668093}a^{6}+\frac{363342862666188}{6617445668093}a^{5}+\frac{865159726717}{509034282161}a^{4}+\frac{149158399103891}{6617445668093}a^{3}+\frac{2878669247460}{6617445668093}a^{2}+\frac{7162740444614}{6617445668093}a+\frac{7852178339126}{6617445668093}$, $\frac{2994530600305}{6617445668093}a^{17}-\frac{12872771791889}{6617445668093}a^{16}+\frac{23821455544059}{6617445668093}a^{15}-\frac{35835389602039}{6617445668093}a^{14}+\frac{113760548396564}{6617445668093}a^{13}-\frac{88105020793119}{6617445668093}a^{12}+\frac{201014156652084}{6617445668093}a^{11}+\frac{68179752956481}{6617445668093}a^{10}+\frac{524230239090920}{6617445668093}a^{9}+\frac{178547836878025}{6617445668093}a^{8}+\frac{685807639492640}{6617445668093}a^{7}+\frac{246677530797124}{6617445668093}a^{6}+\frac{363342862666188}{6617445668093}a^{5}+\frac{865159726717}{509034282161}a^{4}+\frac{149158399103891}{6617445668093}a^{3}+\frac{2878669247460}{6617445668093}a^{2}+\frac{7162740444614}{6617445668093}a+\frac{1234732671033}{6617445668093}$, $\frac{90\!\cdots\!15}{39\!\cdots\!39}a^{17}-\frac{43\!\cdots\!16}{39\!\cdots\!39}a^{16}+\frac{88\!\cdots\!40}{39\!\cdots\!39}a^{15}-\frac{10\!\cdots\!56}{30\!\cdots\!03}a^{14}+\frac{38\!\cdots\!83}{39\!\cdots\!39}a^{13}-\frac{42\!\cdots\!83}{39\!\cdots\!39}a^{12}+\frac{51\!\cdots\!94}{30\!\cdots\!03}a^{11}-\frac{10\!\cdots\!32}{39\!\cdots\!39}a^{10}+\frac{13\!\cdots\!41}{39\!\cdots\!39}a^{9}-\frac{41\!\cdots\!12}{39\!\cdots\!39}a^{8}+\frac{13\!\cdots\!60}{39\!\cdots\!39}a^{7}-\frac{77\!\cdots\!99}{39\!\cdots\!39}a^{6}+\frac{85\!\cdots\!18}{39\!\cdots\!39}a^{5}-\frac{85\!\cdots\!64}{30\!\cdots\!03}a^{4}+\frac{11\!\cdots\!61}{39\!\cdots\!39}a^{3}-\frac{29\!\cdots\!54}{39\!\cdots\!39}a^{2}+\frac{35\!\cdots\!41}{39\!\cdots\!39}a-\frac{13\!\cdots\!82}{39\!\cdots\!39}$, $\frac{90\!\cdots\!15}{39\!\cdots\!39}a^{17}-\frac{43\!\cdots\!16}{39\!\cdots\!39}a^{16}+\frac{88\!\cdots\!40}{39\!\cdots\!39}a^{15}-\frac{10\!\cdots\!56}{30\!\cdots\!03}a^{14}+\frac{38\!\cdots\!83}{39\!\cdots\!39}a^{13}-\frac{42\!\cdots\!83}{39\!\cdots\!39}a^{12}+\frac{51\!\cdots\!94}{30\!\cdots\!03}a^{11}-\frac{10\!\cdots\!32}{39\!\cdots\!39}a^{10}+\frac{13\!\cdots\!41}{39\!\cdots\!39}a^{9}-\frac{41\!\cdots\!12}{39\!\cdots\!39}a^{8}+\frac{13\!\cdots\!60}{39\!\cdots\!39}a^{7}-\frac{77\!\cdots\!99}{39\!\cdots\!39}a^{6}+\frac{85\!\cdots\!18}{39\!\cdots\!39}a^{5}-\frac{85\!\cdots\!64}{30\!\cdots\!03}a^{4}+\frac{11\!\cdots\!61}{39\!\cdots\!39}a^{3}-\frac{29\!\cdots\!54}{39\!\cdots\!39}a^{2}+\frac{35\!\cdots\!41}{39\!\cdots\!39}a+\frac{25\!\cdots\!57}{39\!\cdots\!39}$, $\frac{13\!\cdots\!42}{39\!\cdots\!39}a^{17}-\frac{42\!\cdots\!73}{39\!\cdots\!39}a^{16}+\frac{46\!\cdots\!03}{39\!\cdots\!39}a^{15}-\frac{69\!\cdots\!89}{39\!\cdots\!39}a^{14}+\frac{39\!\cdots\!88}{39\!\cdots\!39}a^{13}+\frac{13\!\cdots\!70}{39\!\cdots\!39}a^{12}+\frac{76\!\cdots\!54}{39\!\cdots\!39}a^{11}+\frac{12\!\cdots\!65}{39\!\cdots\!39}a^{10}+\frac{33\!\cdots\!91}{39\!\cdots\!39}a^{9}+\frac{39\!\cdots\!86}{39\!\cdots\!39}a^{8}+\frac{56\!\cdots\!48}{39\!\cdots\!39}a^{7}+\frac{55\!\cdots\!95}{39\!\cdots\!39}a^{6}+\frac{47\!\cdots\!21}{39\!\cdots\!39}a^{5}+\frac{20\!\cdots\!95}{30\!\cdots\!03}a^{4}+\frac{13\!\cdots\!29}{39\!\cdots\!39}a^{3}+\frac{61\!\cdots\!21}{39\!\cdots\!39}a^{2}+\frac{11\!\cdots\!68}{39\!\cdots\!39}a+\frac{32\!\cdots\!62}{39\!\cdots\!39}$, $\frac{10\!\cdots\!41}{39\!\cdots\!39}a^{17}-\frac{72\!\cdots\!04}{39\!\cdots\!39}a^{16}+\frac{20\!\cdots\!86}{39\!\cdots\!39}a^{15}-\frac{33\!\cdots\!32}{39\!\cdots\!39}a^{14}+\frac{69\!\cdots\!44}{39\!\cdots\!39}a^{13}-\frac{13\!\cdots\!08}{39\!\cdots\!39}a^{12}+\frac{14\!\cdots\!57}{39\!\cdots\!39}a^{11}-\frac{16\!\cdots\!59}{39\!\cdots\!39}a^{10}+\frac{94\!\cdots\!20}{39\!\cdots\!39}a^{9}-\frac{47\!\cdots\!52}{39\!\cdots\!39}a^{8}+\frac{45\!\cdots\!30}{39\!\cdots\!39}a^{7}-\frac{61\!\cdots\!27}{39\!\cdots\!39}a^{6}-\frac{16\!\cdots\!88}{39\!\cdots\!39}a^{5}-\frac{33\!\cdots\!29}{30\!\cdots\!03}a^{4}+\frac{21\!\cdots\!60}{39\!\cdots\!39}a^{3}-\frac{24\!\cdots\!90}{39\!\cdots\!39}a^{2}-\frac{17\!\cdots\!14}{39\!\cdots\!39}a-\frac{51\!\cdots\!27}{39\!\cdots\!39}$, $\frac{12\!\cdots\!79}{39\!\cdots\!39}a^{17}-\frac{94\!\cdots\!79}{39\!\cdots\!39}a^{16}-\frac{10\!\cdots\!37}{39\!\cdots\!39}a^{15}+\frac{24\!\cdots\!50}{39\!\cdots\!39}a^{14}-\frac{12\!\cdots\!02}{39\!\cdots\!39}a^{13}+\frac{14\!\cdots\!54}{39\!\cdots\!39}a^{12}-\frac{90\!\cdots\!20}{39\!\cdots\!39}a^{11}+\frac{34\!\cdots\!24}{39\!\cdots\!39}a^{10}+\frac{21\!\cdots\!63}{39\!\cdots\!39}a^{9}+\frac{80\!\cdots\!45}{39\!\cdots\!39}a^{8}+\frac{26\!\cdots\!32}{39\!\cdots\!39}a^{7}+\frac{93\!\cdots\!21}{39\!\cdots\!39}a^{6}+\frac{84\!\cdots\!14}{30\!\cdots\!03}a^{5}+\frac{25\!\cdots\!05}{30\!\cdots\!03}a^{4}-\frac{21\!\cdots\!59}{39\!\cdots\!39}a^{3}+\frac{13\!\cdots\!73}{30\!\cdots\!03}a^{2}-\frac{90\!\cdots\!60}{39\!\cdots\!39}a+\frac{22\!\cdots\!75}{30\!\cdots\!03}$, $\frac{63\!\cdots\!88}{39\!\cdots\!39}a^{17}-\frac{28\!\cdots\!23}{39\!\cdots\!39}a^{16}+\frac{52\!\cdots\!09}{39\!\cdots\!39}a^{15}-\frac{79\!\cdots\!04}{39\!\cdots\!39}a^{14}+\frac{24\!\cdots\!23}{39\!\cdots\!39}a^{13}-\frac{21\!\cdots\!69}{39\!\cdots\!39}a^{12}+\frac{42\!\cdots\!72}{39\!\cdots\!39}a^{11}+\frac{77\!\cdots\!44}{39\!\cdots\!39}a^{10}+\frac{10\!\cdots\!28}{39\!\cdots\!39}a^{9}+\frac{14\!\cdots\!01}{39\!\cdots\!39}a^{8}+\frac{91\!\cdots\!40}{30\!\cdots\!03}a^{7}+\frac{28\!\cdots\!83}{39\!\cdots\!39}a^{6}+\frac{28\!\cdots\!53}{39\!\cdots\!39}a^{5}-\frac{44\!\cdots\!74}{30\!\cdots\!03}a^{4}-\frac{73\!\cdots\!72}{39\!\cdots\!39}a^{3}-\frac{31\!\cdots\!41}{39\!\cdots\!39}a^{2}-\frac{74\!\cdots\!00}{39\!\cdots\!39}a-\frac{14\!\cdots\!11}{39\!\cdots\!39}$
| 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: | \( 1060.85049787 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 1060.85049787 \cdot 1}{2\cdot\sqrt{4466413101188628826579}}\cr\approx \mathstrut & 0.121133303062 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 7*x^16 - 11*x^15 + 37*x^14 - 22*x^13 + 71*x^12 + 33*x^11 + 204*x^10 + 118*x^9 + 305*x^8 + 169*x^7 + 216*x^6 + 59*x^5 + 84*x^4 + 8*x^3 + 12*x^2 + x + 1)
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^18 - 4*x^17 + 7*x^16 - 11*x^15 + 37*x^14 - 22*x^13 + 71*x^12 + 33*x^11 + 204*x^10 + 118*x^9 + 305*x^8 + 169*x^7 + 216*x^6 + 59*x^5 + 84*x^4 + 8*x^3 + 12*x^2 + x + 1, 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^18 - 4*x^17 + 7*x^16 - 11*x^15 + 37*x^14 - 22*x^13 + 71*x^12 + 33*x^11 + 204*x^10 + 118*x^9 + 305*x^8 + 169*x^7 + 216*x^6 + 59*x^5 + 84*x^4 + 8*x^3 + 12*x^2 + x + 1);
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^18 - 4*x^17 + 7*x^16 - 11*x^15 + 37*x^14 - 22*x^13 + 71*x^12 + 33*x^11 + 204*x^10 + 118*x^9 + 305*x^8 + 169*x^7 + 216*x^6 + 59*x^5 + 84*x^4 + 8*x^3 + 12*x^2 + x + 1);
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
$C_3\times S_3$ (as 18T3):
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 18 |
| The 9 conjugacy class representatives for $S_3 \times C_3$ |
| Character table for $S_3 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{-19}) \), 3.1.931.1 x3, \(\Q(\zeta_{7})^+\), 6.0.16468459.2, 6.0.16468459.1, 6.0.336091.1 x2, 9.3.806954491.1 x3 |
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 6 sibling: | 6.0.336091.1 |
| Degree 9 sibling: | 9.3.806954491.1 |
| Minimal sibling: | 6.0.336091.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | ${\href{/padicField/13.2.0.1}{2} }^{9}$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
|
\(19\)
| 19.6.3.2 | $x^{6} + 65 x^{4} + 34 x^{3} + 1099 x^{2} - 1802 x + 4564$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 19.6.3.2 | $x^{6} + 65 x^{4} + 34 x^{3} + 1099 x^{2} - 1802 x + 4564$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 19.6.3.2 | $x^{6} + 65 x^{4} + 34 x^{3} + 1099 x^{2} - 1802 x + 4564$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |