Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 9*x^14 - 3*x^13 + 25*x^12 + 75*x^11 + 111*x^10 - 360*x^9 - 501*x^8 + 1266*x^7 + 1104*x^6 - 2487*x^5 - 629*x^4 + 2982*x^3 - 420*x^2 - 1176*x + 784)
gp: K = bnfinit(y^16 - 9*y^14 - 3*y^13 + 25*y^12 + 75*y^11 + 111*y^10 - 360*y^9 - 501*y^8 + 1266*y^7 + 1104*y^6 - 2487*y^5 - 629*y^4 + 2982*y^3 - 420*y^2 - 1176*y + 784, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 9*x^14 - 3*x^13 + 25*x^12 + 75*x^11 + 111*x^10 - 360*x^9 - 501*x^8 + 1266*x^7 + 1104*x^6 - 2487*x^5 - 629*x^4 + 2982*x^3 - 420*x^2 - 1176*x + 784);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 9*x^14 - 3*x^13 + 25*x^12 + 75*x^11 + 111*x^10 - 360*x^9 - 501*x^8 + 1266*x^7 + 1104*x^6 - 2487*x^5 - 629*x^4 + 2982*x^3 - 420*x^2 - 1176*x + 784)
\( x^{16} - 9 x^{14} - 3 x^{13} + 25 x^{12} + 75 x^{11} + 111 x^{10} - 360 x^{9} - 501 x^{8} + 1266 x^{7} + \cdots + 784 \)
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: |
\(2499114735283240471761\)
\(\medspace = 3^{12}\cdot 7^{8}\cdot 13^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.75\) | 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: | $3^{3/4}7^{1/2}13^{1/2}\approx 21.745111415357325$ | ||
| Ramified primes: |
\(3\), \(7\), \(13\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{14}a^{12}-\frac{1}{14}a^{11}-\frac{3}{14}a^{10}+\frac{1}{7}a^{9}+\frac{1}{14}a^{8}-\frac{3}{14}a^{6}-\frac{5}{14}a^{4}-\frac{3}{14}a^{3}-\frac{3}{14}a^{2}$, $\frac{1}{14}a^{13}+\frac{3}{14}a^{11}-\frac{1}{14}a^{10}+\frac{3}{14}a^{9}+\frac{1}{14}a^{8}-\frac{3}{14}a^{7}+\frac{2}{7}a^{6}-\frac{5}{14}a^{5}+\frac{3}{7}a^{4}-\frac{3}{7}a^{3}-\frac{3}{14}a^{2}-\frac{1}{2}a$, $\frac{1}{364}a^{14}-\frac{1}{182}a^{13}+\frac{1}{52}a^{12}-\frac{67}{364}a^{11}+\frac{5}{52}a^{10}+\frac{59}{364}a^{9}-\frac{29}{364}a^{8}+\frac{47}{182}a^{7}+\frac{3}{364}a^{6}-\frac{41}{182}a^{5}-\frac{89}{182}a^{4}+\frac{109}{364}a^{3}-\frac{97}{364}a^{2}+\frac{6}{13}a+\frac{2}{13}$, $\frac{1}{73\!\cdots\!72}a^{15}-\frac{30\!\cdots\!97}{36\!\cdots\!36}a^{14}-\frac{25\!\cdots\!01}{73\!\cdots\!72}a^{13}-\frac{50\!\cdots\!99}{38\!\cdots\!88}a^{12}-\frac{50\!\cdots\!57}{73\!\cdots\!72}a^{11}-\frac{16\!\cdots\!59}{73\!\cdots\!72}a^{10}-\frac{47\!\cdots\!99}{73\!\cdots\!72}a^{9}-\frac{15\!\cdots\!87}{36\!\cdots\!36}a^{8}+\frac{28\!\cdots\!51}{56\!\cdots\!44}a^{7}-\frac{67\!\cdots\!73}{26\!\cdots\!74}a^{6}+\frac{18\!\cdots\!15}{91\!\cdots\!09}a^{5}+\frac{95\!\cdots\!97}{73\!\cdots\!72}a^{4}+\frac{82\!\cdots\!25}{73\!\cdots\!72}a^{3}-\frac{39\!\cdots\!95}{91\!\cdots\!09}a^{2}+\frac{47\!\cdots\!75}{71\!\cdots\!02}a+\frac{48\!\cdots\!85}{13\!\cdots\!87}$
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
$C_{4}$, which has order $4$ (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: |
\( \frac{3990085306109}{1150263891112264} a^{15} - \frac{110537005045}{287565972778066} a^{14} - \frac{32036728053753}{1150263891112264} a^{13} - \frac{10886467117747}{1150263891112264} a^{12} + \frac{67796389981069}{1150263891112264} a^{11} + \frac{295482012452391}{1150263891112264} a^{10} + \frac{497758045943555}{1150263891112264} a^{9} - \frac{152093074380366}{143782986389033} a^{8} - \frac{1538612586141973}{1150263891112264} a^{7} + \frac{1790605943101659}{575131945556132} a^{6} + \frac{644722568357335}{287565972778066} a^{5} - \frac{5866594218312963}{1150263891112264} a^{4} - \frac{234976527859437}{1150263891112264} a^{3} + \frac{2357643201278155}{575131945556132} a^{2} - \frac{143212850294402}{143782986389033} a + \frac{22369318065284}{143782986389033} \)
(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: |
$\frac{37\!\cdots\!69}{36\!\cdots\!36}a^{15}+\frac{10\!\cdots\!51}{36\!\cdots\!36}a^{14}-\frac{53\!\cdots\!13}{36\!\cdots\!36}a^{13}-\frac{24\!\cdots\!43}{96\!\cdots\!22}a^{12}-\frac{28\!\cdots\!20}{70\!\cdots\!93}a^{11}+\frac{12\!\cdots\!73}{18\!\cdots\!18}a^{10}+\frac{20\!\cdots\!83}{91\!\cdots\!09}a^{9}+\frac{11\!\cdots\!69}{36\!\cdots\!36}a^{8}-\frac{38\!\cdots\!97}{36\!\cdots\!36}a^{7}-\frac{42\!\cdots\!17}{36\!\cdots\!36}a^{6}+\frac{55\!\cdots\!93}{18\!\cdots\!18}a^{5}+\frac{63\!\cdots\!37}{36\!\cdots\!36}a^{4}-\frac{96\!\cdots\!47}{18\!\cdots\!18}a^{3}+\frac{12\!\cdots\!11}{36\!\cdots\!36}a^{2}+\frac{28\!\cdots\!13}{71\!\cdots\!02}a-\frac{17\!\cdots\!10}{13\!\cdots\!87}$, $\frac{25\!\cdots\!13}{19\!\cdots\!56}a^{15}-\frac{191994923431172}{24\!\cdots\!57}a^{14}-\frac{27\!\cdots\!89}{19\!\cdots\!56}a^{13}+\frac{10199476379959}{10\!\cdots\!24}a^{12}+\frac{75\!\cdots\!77}{15\!\cdots\!12}a^{11}+\frac{22\!\cdots\!47}{19\!\cdots\!56}a^{10}+\frac{16\!\cdots\!39}{19\!\cdots\!56}a^{9}-\frac{38\!\cdots\!27}{49\!\cdots\!14}a^{8}-\frac{18\!\cdots\!29}{19\!\cdots\!56}a^{7}+\frac{20\!\cdots\!59}{99\!\cdots\!28}a^{6}+\frac{48\!\cdots\!30}{24\!\cdots\!57}a^{5}-\frac{10\!\cdots\!41}{28\!\cdots\!08}a^{4}-\frac{36\!\cdots\!21}{19\!\cdots\!56}a^{3}+\frac{33\!\cdots\!73}{14\!\cdots\!04}a^{2}-\frac{33\!\cdots\!46}{35\!\cdots\!51}a-\frac{15\!\cdots\!58}{35\!\cdots\!51}$, $\frac{19\!\cdots\!75}{18\!\cdots\!18}a^{15}+\frac{11\!\cdots\!64}{91\!\cdots\!09}a^{14}-\frac{16\!\cdots\!93}{14\!\cdots\!86}a^{13}-\frac{12\!\cdots\!77}{96\!\cdots\!22}a^{12}-\frac{17\!\cdots\!47}{18\!\cdots\!18}a^{11}+\frac{85\!\cdots\!71}{18\!\cdots\!18}a^{10}+\frac{21\!\cdots\!23}{18\!\cdots\!18}a^{9}+\frac{10\!\cdots\!28}{91\!\cdots\!09}a^{8}-\frac{11\!\cdots\!05}{18\!\cdots\!18}a^{7}-\frac{53\!\cdots\!63}{70\!\cdots\!93}a^{6}+\frac{16\!\cdots\!63}{91\!\cdots\!09}a^{5}+\frac{20\!\cdots\!17}{18\!\cdots\!18}a^{4}-\frac{56\!\cdots\!65}{18\!\cdots\!18}a^{3}-\frac{59\!\cdots\!55}{13\!\cdots\!87}a^{2}+\frac{51\!\cdots\!48}{35\!\cdots\!51}a-\frac{97\!\cdots\!83}{13\!\cdots\!87}$, $\frac{33\!\cdots\!65}{28\!\cdots\!72}a^{15}-\frac{12\!\cdots\!73}{36\!\cdots\!36}a^{14}-\frac{21\!\cdots\!59}{36\!\cdots\!36}a^{13}-\frac{19\!\cdots\!65}{96\!\cdots\!22}a^{12}-\frac{16\!\cdots\!79}{18\!\cdots\!18}a^{11}+\frac{64\!\cdots\!49}{91\!\cdots\!09}a^{10}+\frac{17\!\cdots\!30}{91\!\cdots\!09}a^{9}-\frac{27\!\cdots\!47}{36\!\cdots\!36}a^{8}+\frac{42\!\cdots\!29}{36\!\cdots\!36}a^{7}-\frac{35\!\cdots\!31}{36\!\cdots\!36}a^{6}-\frac{22\!\cdots\!71}{18\!\cdots\!18}a^{5}+\frac{20\!\cdots\!95}{36\!\cdots\!36}a^{4}+\frac{64\!\cdots\!99}{18\!\cdots\!18}a^{3}-\frac{67\!\cdots\!43}{52\!\cdots\!48}a^{2}-\frac{87\!\cdots\!17}{71\!\cdots\!02}a+\frac{19\!\cdots\!57}{13\!\cdots\!87}$, $\frac{39\!\cdots\!18}{91\!\cdots\!09}a^{15}+\frac{60\!\cdots\!77}{52\!\cdots\!48}a^{14}-\frac{49\!\cdots\!14}{91\!\cdots\!09}a^{13}-\frac{25\!\cdots\!59}{19\!\cdots\!44}a^{12}+\frac{46\!\cdots\!47}{28\!\cdots\!72}a^{11}+\frac{39\!\cdots\!33}{52\!\cdots\!48}a^{10}+\frac{71\!\cdots\!75}{52\!\cdots\!48}a^{9}-\frac{51\!\cdots\!39}{36\!\cdots\!36}a^{8}-\frac{17\!\cdots\!97}{18\!\cdots\!18}a^{7}-\frac{53\!\cdots\!13}{52\!\cdots\!48}a^{6}+\frac{45\!\cdots\!21}{18\!\cdots\!18}a^{5}+\frac{95\!\cdots\!73}{91\!\cdots\!09}a^{4}-\frac{14\!\cdots\!85}{36\!\cdots\!36}a^{3}+\frac{12\!\cdots\!71}{36\!\cdots\!36}a^{2}+\frac{20\!\cdots\!99}{71\!\cdots\!02}a+\frac{24\!\cdots\!01}{13\!\cdots\!87}$, $\frac{16\!\cdots\!09}{73\!\cdots\!72}a^{15}-\frac{20\!\cdots\!15}{36\!\cdots\!36}a^{14}-\frac{16\!\cdots\!25}{73\!\cdots\!72}a^{13}-\frac{21\!\cdots\!51}{38\!\cdots\!88}a^{12}+\frac{52\!\cdots\!07}{73\!\cdots\!72}a^{11}+\frac{14\!\cdots\!93}{73\!\cdots\!72}a^{10}+\frac{14\!\cdots\!17}{73\!\cdots\!72}a^{9}-\frac{39\!\cdots\!93}{36\!\cdots\!36}a^{8}-\frac{10\!\cdots\!37}{73\!\cdots\!72}a^{7}+\frac{28\!\cdots\!09}{91\!\cdots\!09}a^{6}+\frac{36\!\cdots\!14}{91\!\cdots\!09}a^{5}-\frac{36\!\cdots\!63}{73\!\cdots\!72}a^{4}-\frac{33\!\cdots\!95}{73\!\cdots\!72}a^{3}+\frac{77\!\cdots\!79}{18\!\cdots\!18}a^{2}+\frac{15\!\cdots\!69}{71\!\cdots\!02}a-\frac{11\!\cdots\!19}{13\!\cdots\!87}$, $\frac{99\!\cdots\!87}{36\!\cdots\!36}a^{15}-\frac{96\!\cdots\!53}{36\!\cdots\!36}a^{14}-\frac{47\!\cdots\!27}{36\!\cdots\!36}a^{13}+\frac{10\!\cdots\!23}{48\!\cdots\!11}a^{12}-\frac{44\!\cdots\!10}{91\!\cdots\!09}a^{11}+\frac{56\!\cdots\!13}{91\!\cdots\!09}a^{10}+\frac{34\!\cdots\!19}{18\!\cdots\!18}a^{9}-\frac{15\!\cdots\!23}{36\!\cdots\!36}a^{8}+\frac{85\!\cdots\!19}{52\!\cdots\!48}a^{7}+\frac{16\!\cdots\!03}{52\!\cdots\!48}a^{6}-\frac{79\!\cdots\!03}{18\!\cdots\!18}a^{5}-\frac{12\!\cdots\!85}{36\!\cdots\!36}a^{4}+\frac{11\!\cdots\!24}{10\!\cdots\!99}a^{3}+\frac{46\!\cdots\!79}{36\!\cdots\!36}a^{2}-\frac{23\!\cdots\!81}{35\!\cdots\!51}a+\frac{54\!\cdots\!52}{13\!\cdots\!87}$
| 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: | \( 15235.3835913 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 15235.3835913 \cdot 4}{6\cdot\sqrt{2499114735283240471761}}\cr\approx \mathstrut & 0.493523666160 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 9*x^14 - 3*x^13 + 25*x^12 + 75*x^11 + 111*x^10 - 360*x^9 - 501*x^8 + 1266*x^7 + 1104*x^6 - 2487*x^5 - 629*x^4 + 2982*x^3 - 420*x^2 - 1176*x + 784)
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 - 9*x^14 - 3*x^13 + 25*x^12 + 75*x^11 + 111*x^10 - 360*x^9 - 501*x^8 + 1266*x^7 + 1104*x^6 - 2487*x^5 - 629*x^4 + 2982*x^3 - 420*x^2 - 1176*x + 784, 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 - 9*x^14 - 3*x^13 + 25*x^12 + 75*x^11 + 111*x^10 - 360*x^9 - 501*x^8 + 1266*x^7 + 1104*x^6 - 2487*x^5 - 629*x^4 + 2982*x^3 - 420*x^2 - 1176*x + 784);
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 - 9*x^14 - 3*x^13 + 25*x^12 + 75*x^11 + 111*x^10 - 360*x^9 - 501*x^8 + 1266*x^7 + 1104*x^6 - 2487*x^5 - 629*x^4 + 2982*x^3 - 420*x^2 - 1176*x + 784);
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
$C_2^2\wr C_2$ (as 16T39):
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 $C_2^2\wr C_2$ |
| Character table for $C_2^2\wr C_2$ |
Intermediate fields
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
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |
| 3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
|
\(7\)
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(13\)
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |