Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 14*x^12 + 7*x^11 - 49*x^10 + 14*x^9 + 77*x^8 + 355*x^7 - 1421*x^6 + 12110*x^5 - 26831*x^4 + 47047*x^3 - 44366*x^2 + 24185*x + 31295)
gp: K = bnfinit(y^14 - 7*y^13 + 14*y^12 + 7*y^11 - 49*y^10 + 14*y^9 + 77*y^8 + 355*y^7 - 1421*y^6 + 12110*y^5 - 26831*y^4 + 47047*y^3 - 44366*y^2 + 24185*y + 31295, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 7*x^13 + 14*x^12 + 7*x^11 - 49*x^10 + 14*x^9 + 77*x^8 + 355*x^7 - 1421*x^6 + 12110*x^5 - 26831*x^4 + 47047*x^3 - 44366*x^2 + 24185*x + 31295);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 7*x^13 + 14*x^12 + 7*x^11 - 49*x^10 + 14*x^9 + 77*x^8 + 355*x^7 - 1421*x^6 + 12110*x^5 - 26831*x^4 + 47047*x^3 - 44366*x^2 + 24185*x + 31295)
\( x^{14} - 7 x^{13} + 14 x^{12} + 7 x^{11} - 49 x^{10} + 14 x^{9} + 77 x^{8} + 355 x^{7} - 1421 x^{6} + \cdots + 31295 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(115339375378542530880000000\)
\(\medspace = 2^{12}\cdot 3^{12}\cdot 5^{7}\cdot 7^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(72.71\) | 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: | $2^{6/7}3^{6/7}5^{1/2}7^{47/42}\approx 91.6593425328703$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{5}-\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{8}$, $\frac{1}{8}a^{9}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{8}a^{7}+\frac{1}{16}a^{6}+\frac{1}{16}a^{5}+\frac{1}{16}a^{4}+\frac{1}{8}a^{3}+\frac{3}{16}a^{2}+\frac{3}{16}a+\frac{1}{16}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{8}-\frac{1}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{3}{16}a^{4}-\frac{3}{16}a^{3}-\frac{1}{8}a^{2}-\frac{1}{8}a-\frac{1}{16}$, $\frac{1}{32}a^{12}-\frac{1}{32}a^{10}-\frac{1}{16}a^{8}+\frac{3}{32}a^{6}-\frac{1}{4}a^{5}-\frac{1}{16}a^{4}-\frac{1}{4}a^{3}-\frac{1}{32}a^{2}-\frac{1}{4}a-\frac{7}{32}$, $\frac{1}{21\!\cdots\!44}a^{13}+\frac{80\!\cdots\!91}{10\!\cdots\!72}a^{12}-\frac{38\!\cdots\!79}{21\!\cdots\!44}a^{11}+\frac{52\!\cdots\!71}{10\!\cdots\!72}a^{10}+\frac{57\!\cdots\!07}{10\!\cdots\!72}a^{9}+\frac{25\!\cdots\!57}{10\!\cdots\!72}a^{8}-\frac{26\!\cdots\!35}{21\!\cdots\!44}a^{7}-\frac{71\!\cdots\!23}{10\!\cdots\!72}a^{6}+\frac{18\!\cdots\!19}{10\!\cdots\!72}a^{5}+\frac{25\!\cdots\!19}{10\!\cdots\!72}a^{4}+\frac{57\!\cdots\!57}{21\!\cdots\!44}a^{3}-\frac{46\!\cdots\!19}{10\!\cdots\!72}a^{2}-\frac{22\!\cdots\!55}{21\!\cdots\!44}a-\frac{11\!\cdots\!09}{54\!\cdots\!36}$
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 $1$ (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: |
$\frac{19\!\cdots\!80}{68\!\cdots\!17}a^{13}-\frac{10\!\cdots\!70}{68\!\cdots\!17}a^{12}+\frac{79\!\cdots\!60}{68\!\cdots\!17}a^{11}+\frac{39\!\cdots\!40}{68\!\cdots\!17}a^{10}-\frac{54\!\cdots\!00}{68\!\cdots\!17}a^{9}-\frac{16\!\cdots\!73}{68\!\cdots\!17}a^{8}-\frac{11\!\cdots\!08}{68\!\cdots\!17}a^{7}+\frac{13\!\cdots\!04}{68\!\cdots\!17}a^{6}-\frac{91\!\cdots\!84}{68\!\cdots\!17}a^{5}+\frac{18\!\cdots\!90}{68\!\cdots\!17}a^{4}-\frac{26\!\cdots\!36}{68\!\cdots\!17}a^{3}+\frac{36\!\cdots\!84}{68\!\cdots\!17}a^{2}-\frac{56\!\cdots\!88}{68\!\cdots\!17}a+\frac{68\!\cdots\!91}{68\!\cdots\!17}$, $\frac{13\!\cdots\!83}{27\!\cdots\!68}a^{13}-\frac{79\!\cdots\!01}{21\!\cdots\!44}a^{12}+\frac{91\!\cdots\!31}{10\!\cdots\!72}a^{11}+\frac{12\!\cdots\!47}{21\!\cdots\!44}a^{10}-\frac{27\!\cdots\!99}{10\!\cdots\!72}a^{9}+\frac{20\!\cdots\!19}{10\!\cdots\!72}a^{8}+\frac{33\!\cdots\!75}{10\!\cdots\!72}a^{7}+\frac{35\!\cdots\!63}{21\!\cdots\!44}a^{6}-\frac{90\!\cdots\!21}{10\!\cdots\!72}a^{5}+\frac{70\!\cdots\!23}{10\!\cdots\!72}a^{4}-\frac{17\!\cdots\!07}{10\!\cdots\!72}a^{3}+\frac{62\!\cdots\!35}{21\!\cdots\!44}a^{2}-\frac{38\!\cdots\!85}{10\!\cdots\!72}a+\frac{51\!\cdots\!27}{21\!\cdots\!44}$, $\frac{22\!\cdots\!81}{54\!\cdots\!36}a^{13}-\frac{62\!\cdots\!71}{21\!\cdots\!44}a^{12}+\frac{53\!\cdots\!93}{10\!\cdots\!72}a^{11}+\frac{15\!\cdots\!35}{21\!\cdots\!44}a^{10}-\frac{15\!\cdots\!68}{68\!\cdots\!17}a^{9}-\frac{93\!\cdots\!63}{54\!\cdots\!36}a^{8}+\frac{49\!\cdots\!11}{10\!\cdots\!72}a^{7}+\frac{57\!\cdots\!51}{21\!\cdots\!44}a^{6}-\frac{17\!\cdots\!55}{27\!\cdots\!68}a^{5}+\frac{11\!\cdots\!89}{27\!\cdots\!68}a^{4}-\frac{10\!\cdots\!99}{10\!\cdots\!72}a^{3}+\frac{28\!\cdots\!35}{21\!\cdots\!44}a^{2}-\frac{24\!\cdots\!05}{27\!\cdots\!68}a+\frac{26\!\cdots\!91}{21\!\cdots\!44}$, $\frac{80\!\cdots\!77}{68\!\cdots\!17}a^{13}-\frac{22\!\cdots\!57}{27\!\cdots\!68}a^{12}+\frac{10\!\cdots\!90}{68\!\cdots\!17}a^{11}+\frac{24\!\cdots\!83}{13\!\cdots\!34}a^{10}-\frac{18\!\cdots\!91}{27\!\cdots\!68}a^{9}-\frac{22\!\cdots\!49}{68\!\cdots\!17}a^{8}+\frac{96\!\cdots\!67}{68\!\cdots\!17}a^{7}+\frac{18\!\cdots\!21}{27\!\cdots\!68}a^{6}-\frac{26\!\cdots\!95}{13\!\cdots\!34}a^{5}+\frac{87\!\cdots\!95}{68\!\cdots\!17}a^{4}-\frac{81\!\cdots\!63}{27\!\cdots\!68}a^{3}+\frac{56\!\cdots\!29}{13\!\cdots\!34}a^{2}-\frac{42\!\cdots\!45}{13\!\cdots\!34}a+\frac{37\!\cdots\!13}{13\!\cdots\!34}$, $\frac{51\!\cdots\!25}{27\!\cdots\!68}a^{13}-\frac{71\!\cdots\!53}{10\!\cdots\!72}a^{12}-\frac{22\!\cdots\!13}{10\!\cdots\!72}a^{11}+\frac{93\!\cdots\!07}{10\!\cdots\!72}a^{10}+\frac{14\!\cdots\!05}{27\!\cdots\!68}a^{9}-\frac{22\!\cdots\!09}{10\!\cdots\!72}a^{8}-\frac{27\!\cdots\!23}{10\!\cdots\!72}a^{7}+\frac{25\!\cdots\!61}{10\!\cdots\!72}a^{6}-\frac{29\!\cdots\!91}{27\!\cdots\!68}a^{5}+\frac{16\!\cdots\!85}{10\!\cdots\!72}a^{4}+\frac{37\!\cdots\!55}{10\!\cdots\!72}a^{3}-\frac{40\!\cdots\!23}{10\!\cdots\!72}a^{2}+\frac{74\!\cdots\!99}{54\!\cdots\!36}a-\frac{10\!\cdots\!32}{68\!\cdots\!17}$, $\frac{57\!\cdots\!15}{10\!\cdots\!72}a^{13}-\frac{86\!\cdots\!83}{21\!\cdots\!44}a^{12}+\frac{28\!\cdots\!83}{27\!\cdots\!68}a^{11}-\frac{32\!\cdots\!13}{21\!\cdots\!44}a^{10}+\frac{28\!\cdots\!17}{54\!\cdots\!36}a^{9}-\frac{13\!\cdots\!08}{68\!\cdots\!17}a^{8}+\frac{24\!\cdots\!95}{54\!\cdots\!36}a^{7}-\frac{19\!\cdots\!13}{21\!\cdots\!44}a^{6}+\frac{14\!\cdots\!79}{54\!\cdots\!36}a^{5}-\frac{12\!\cdots\!53}{54\!\cdots\!36}a^{4}+\frac{45\!\cdots\!57}{13\!\cdots\!34}a^{3}-\frac{13\!\cdots\!33}{21\!\cdots\!44}a^{2}+\frac{80\!\cdots\!45}{10\!\cdots\!72}a+\frac{12\!\cdots\!59}{21\!\cdots\!44}$, $\frac{72\!\cdots\!05}{10\!\cdots\!72}a^{13}-\frac{10\!\cdots\!29}{21\!\cdots\!44}a^{12}+\frac{11\!\cdots\!83}{10\!\cdots\!72}a^{11}+\frac{48\!\cdots\!15}{21\!\cdots\!44}a^{10}-\frac{31\!\cdots\!73}{10\!\cdots\!72}a^{9}+\frac{57\!\cdots\!75}{68\!\cdots\!17}a^{8}+\frac{40\!\cdots\!93}{10\!\cdots\!72}a^{7}+\frac{57\!\cdots\!71}{21\!\cdots\!44}a^{6}-\frac{10\!\cdots\!07}{10\!\cdots\!72}a^{5}+\frac{44\!\cdots\!41}{54\!\cdots\!36}a^{4}-\frac{21\!\cdots\!63}{10\!\cdots\!72}a^{3}+\frac{77\!\cdots\!43}{21\!\cdots\!44}a^{2}-\frac{22\!\cdots\!07}{54\!\cdots\!36}a+\frac{79\!\cdots\!41}{21\!\cdots\!44}$
| 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: | \( 38929705.374924675 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 38929705.374924675 \cdot 1}{2\cdot\sqrt{115339375378542530880000000}}\cr\approx \mathstrut & 0.446068468569026 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 14*x^12 + 7*x^11 - 49*x^10 + 14*x^9 + 77*x^8 + 355*x^7 - 1421*x^6 + 12110*x^5 - 26831*x^4 + 47047*x^3 - 44366*x^2 + 24185*x + 31295)
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^14 - 7*x^13 + 14*x^12 + 7*x^11 - 49*x^10 + 14*x^9 + 77*x^8 + 355*x^7 - 1421*x^6 + 12110*x^5 - 26831*x^4 + 47047*x^3 - 44366*x^2 + 24185*x + 31295, 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^14 - 7*x^13 + 14*x^12 + 7*x^11 - 49*x^10 + 14*x^9 + 77*x^8 + 355*x^7 - 1421*x^6 + 12110*x^5 - 26831*x^4 + 47047*x^3 - 44366*x^2 + 24185*x + 31295);
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^14 - 7*x^13 + 14*x^12 + 7*x^11 - 49*x^10 + 14*x^9 + 77*x^8 + 355*x^7 - 1421*x^6 + 12110*x^5 - 26831*x^4 + 47047*x^3 - 44366*x^2 + 24185*x + 31295);
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
$C_2\times F_7$ (as 14T7):
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 84 |
| The 14 conjugacy class representatives for $F_7 \times C_2$ |
| Character table for $F_7 \times C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 7.1.38423222208.4 |
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 14 sibling: | data not computed |
| Degree 28 sibling: | data not computed |
| Degree 42 siblings: | 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 | R | R | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.14.12.1 | $x^{14} + 7 x^{13} + 28 x^{12} + 77 x^{11} + 161 x^{10} + 266 x^{9} + 357 x^{8} + 397 x^{7} + 371 x^{6} + 224 x^{5} + 21 x^{4} + 7 x^{3} + 70 x^{2} + 35 x + 7$ | $7$ | $2$ | $12$ | $(C_7:C_3) \times C_2$ | $[\ ]_{7}^{6}$ |
|
\(3\)
| 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(7\)
| 7.14.14.21 | $x^{14} - 14 x^{9} + 14 x^{8} + 14 x^{7} - 1127 x^{4} - 98 x^{3} - 49 x^{2} + 98 x + 49$ | $7$ | $2$ | $14$ | $F_7 \times C_2$ | $[7/6]_{6}^{2}$ |