Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 10*x^14 - 6*x^13 + 54*x^12 + 6*x^11 + 274*x^10 + 12*x^9 + 106*x^8 - 1062*x^7 + 402*x^6 + 3798*x^5 + 10509*x^4 + 11844*x^3 + 7524*x^2 + 1728*x + 144)
gp: K = bnfinit(y^16 + 10*y^14 - 6*y^13 + 54*y^12 + 6*y^11 + 274*y^10 + 12*y^9 + 106*y^8 - 1062*y^7 + 402*y^6 + 3798*y^5 + 10509*y^4 + 11844*y^3 + 7524*y^2 + 1728*y + 144, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 10*x^14 - 6*x^13 + 54*x^12 + 6*x^11 + 274*x^10 + 12*x^9 + 106*x^8 - 1062*x^7 + 402*x^6 + 3798*x^5 + 10509*x^4 + 11844*x^3 + 7524*x^2 + 1728*x + 144);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 10*x^14 - 6*x^13 + 54*x^12 + 6*x^11 + 274*x^10 + 12*x^9 + 106*x^8 - 1062*x^7 + 402*x^6 + 3798*x^5 + 10509*x^4 + 11844*x^3 + 7524*x^2 + 1728*x + 144)
\( x^{16} + 10 x^{14} - 6 x^{13} + 54 x^{12} + 6 x^{11} + 274 x^{10} + 12 x^{9} + 106 x^{8} - 1062 x^{7} + \cdots + 144 \)
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: |
\(2939612246589864738816\)
\(\medspace = 2^{16}\cdot 3^{12}\cdot 7^{8}\cdot 11^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.97\) | 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: | $2\cdot 3^{3/4}7^{1/2}11^{1/2}\approx 40.0051864911744$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(11\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{12}a^{12}+\frac{1}{12}a^{10}-\frac{1}{4}a^{8}+\frac{1}{12}a^{6}-\frac{1}{2}a^{5}+\frac{1}{12}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{12}a^{13}+\frac{1}{12}a^{11}-\frac{1}{4}a^{9}+\frac{1}{12}a^{7}-\frac{5}{12}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{24}a^{14}-\frac{1}{24}a^{13}-\frac{1}{24}a^{12}-\frac{1}{24}a^{11}-\frac{5}{24}a^{10}+\frac{1}{8}a^{9}-\frac{5}{24}a^{8}+\frac{5}{24}a^{7}+\frac{5}{24}a^{6}-\frac{7}{24}a^{5}-\frac{11}{24}a^{4}+\frac{3}{8}a^{3}$, $\frac{1}{79\!\cdots\!88}a^{15}+\frac{21\!\cdots\!36}{11\!\cdots\!79}a^{14}-\frac{44\!\cdots\!85}{19\!\cdots\!22}a^{13}+\frac{13\!\cdots\!67}{44\!\cdots\!16}a^{12}+\frac{38\!\cdots\!55}{66\!\cdots\!74}a^{11}+\frac{10\!\cdots\!69}{10\!\cdots\!96}a^{10}+\frac{12\!\cdots\!54}{99\!\cdots\!11}a^{9}-\frac{71\!\cdots\!36}{33\!\cdots\!37}a^{8}+\frac{13\!\cdots\!97}{19\!\cdots\!22}a^{7}+\frac{12\!\cdots\!25}{60\!\cdots\!88}a^{6}-\frac{24\!\cdots\!91}{66\!\cdots\!74}a^{5}-\frac{61\!\cdots\!23}{13\!\cdots\!48}a^{4}-\frac{10\!\cdots\!47}{26\!\cdots\!96}a^{3}-\frac{47\!\cdots\!77}{11\!\cdots\!79}a^{2}-\frac{51\!\cdots\!80}{11\!\cdots\!79}a-\frac{52\!\cdots\!41}{11\!\cdots\!79}$
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
$C_{2}$, which has order $2$
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: |
\( -\frac{27302313476409733}{12751527056814524268} a^{15} + \frac{1039913549057365}{2833672679292116504} a^{14} - \frac{551414893110221003}{25503054113629048536} a^{13} + \frac{142857092190824021}{8501018037876349512} a^{12} - \frac{1023570653828363477}{8501018037876349512} a^{11} + \frac{32276130160798427}{2833672679292116504} a^{10} - \frac{15338918340552247967}{25503054113629048536} a^{9} + \frac{771624942240919249}{8501018037876349512} a^{8} - \frac{7372081397680448597}{25503054113629048536} a^{7} + \frac{1173554771039141855}{500059884580961736} a^{6} - \frac{3553224559698395033}{2833672679292116504} a^{5} - \frac{67625297101260956489}{8501018037876349512} a^{4} - \frac{179007631268273496041}{8501018037876349512} a^{3} - \frac{15868463067558149595}{708418169823029126} a^{2} - \frac{8907776329245402813}{708418169823029126} a - \frac{884955794228596309}{354209084911514563} \)
(order $12$)
| 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: |
$\frac{22\!\cdots\!86}{99\!\cdots\!11}a^{15}-\frac{18\!\cdots\!14}{33\!\cdots\!37}a^{14}+\frac{23\!\cdots\!98}{99\!\cdots\!11}a^{13}-\frac{94\!\cdots\!37}{44\!\cdots\!16}a^{12}+\frac{91\!\cdots\!43}{66\!\cdots\!74}a^{11}-\frac{49\!\cdots\!61}{10\!\cdots\!96}a^{10}+\frac{71\!\cdots\!72}{99\!\cdots\!11}a^{9}-\frac{12\!\cdots\!97}{44\!\cdots\!16}a^{8}+\frac{12\!\cdots\!79}{19\!\cdots\!22}a^{7}-\frac{18\!\cdots\!27}{60\!\cdots\!88}a^{6}+\frac{70\!\cdots\!96}{33\!\cdots\!37}a^{5}+\frac{32\!\cdots\!73}{44\!\cdots\!16}a^{4}+\frac{15\!\cdots\!31}{66\!\cdots\!74}a^{3}+\frac{97\!\cdots\!87}{44\!\cdots\!16}a^{2}+\frac{14\!\cdots\!76}{11\!\cdots\!79}a+\frac{12\!\cdots\!00}{11\!\cdots\!79}$, $\frac{25\!\cdots\!27}{12\!\cdots\!68}a^{15}+\frac{45\!\cdots\!07}{85\!\cdots\!12}a^{14}+\frac{49\!\cdots\!41}{25\!\cdots\!36}a^{13}-\frac{20\!\cdots\!83}{28\!\cdots\!04}a^{12}+\frac{28\!\cdots\!37}{28\!\cdots\!04}a^{11}+\frac{11\!\cdots\!01}{28\!\cdots\!04}a^{10}+\frac{13\!\cdots\!49}{25\!\cdots\!36}a^{9}+\frac{97\!\cdots\!87}{85\!\cdots\!12}a^{8}+\frac{20\!\cdots\!39}{25\!\cdots\!36}a^{7}-\frac{11\!\cdots\!87}{50\!\cdots\!36}a^{6}+\frac{47\!\cdots\!01}{85\!\cdots\!12}a^{5}+\frac{72\!\cdots\!29}{85\!\cdots\!12}a^{4}+\frac{19\!\cdots\!63}{85\!\cdots\!12}a^{3}+\frac{34\!\cdots\!35}{14\!\cdots\!52}a^{2}+\frac{46\!\cdots\!61}{35\!\cdots\!63}a-\frac{63\!\cdots\!90}{35\!\cdots\!63}$, $\frac{212794406616731}{11\!\cdots\!52}a^{15}-\frac{392340621479207}{67\!\cdots\!12}a^{14}+\frac{11\!\cdots\!23}{67\!\cdots\!12}a^{13}-\frac{10\!\cdots\!49}{67\!\cdots\!12}a^{12}+\frac{62\!\cdots\!31}{67\!\cdots\!12}a^{11}+\frac{13052032308385}{17\!\cdots\!08}a^{10}+\frac{99\!\cdots\!65}{22\!\cdots\!04}a^{9}-\frac{75\!\cdots\!85}{67\!\cdots\!12}a^{8}-\frac{25\!\cdots\!51}{67\!\cdots\!12}a^{7}-\frac{58\!\cdots\!31}{30\!\cdots\!72}a^{6}+\frac{11\!\cdots\!89}{67\!\cdots\!12}a^{5}+\frac{16\!\cdots\!43}{22\!\cdots\!04}a^{4}+\frac{36\!\cdots\!41}{22\!\cdots\!04}a^{3}+\frac{31\!\cdots\!29}{28\!\cdots\!63}a^{2}+\frac{14\!\cdots\!93}{56\!\cdots\!26}a-\frac{55\!\cdots\!97}{28\!\cdots\!63}$, $\frac{45\!\cdots\!79}{39\!\cdots\!44}a^{15}-\frac{10\!\cdots\!21}{26\!\cdots\!96}a^{14}+\frac{91\!\cdots\!65}{79\!\cdots\!88}a^{13}-\frac{28\!\cdots\!43}{26\!\cdots\!96}a^{12}+\frac{57\!\cdots\!85}{88\!\cdots\!32}a^{11}-\frac{27\!\cdots\!43}{20\!\cdots\!92}a^{10}+\frac{24\!\cdots\!85}{79\!\cdots\!88}a^{9}-\frac{21\!\cdots\!23}{26\!\cdots\!96}a^{8}+\frac{10\!\cdots\!07}{79\!\cdots\!88}a^{7}-\frac{14\!\cdots\!61}{12\!\cdots\!76}a^{6}+\frac{21\!\cdots\!05}{26\!\cdots\!96}a^{5}+\frac{36\!\cdots\!73}{88\!\cdots\!32}a^{4}+\frac{27\!\cdots\!43}{26\!\cdots\!96}a^{3}+\frac{11\!\cdots\!16}{11\!\cdots\!79}a^{2}+\frac{12\!\cdots\!39}{22\!\cdots\!58}a+\frac{36\!\cdots\!97}{11\!\cdots\!79}$, $\frac{58\!\cdots\!33}{66\!\cdots\!74}a^{15}+\frac{47\!\cdots\!99}{13\!\cdots\!48}a^{14}+\frac{13\!\cdots\!59}{13\!\cdots\!48}a^{13}+\frac{47\!\cdots\!47}{13\!\cdots\!48}a^{12}-\frac{49\!\cdots\!63}{13\!\cdots\!48}a^{11}+\frac{26\!\cdots\!13}{10\!\cdots\!96}a^{10}-\frac{26\!\cdots\!69}{13\!\cdots\!48}a^{9}+\frac{10\!\cdots\!77}{13\!\cdots\!48}a^{8}-\frac{10\!\cdots\!05}{13\!\cdots\!48}a^{7}-\frac{52\!\cdots\!75}{60\!\cdots\!88}a^{6}-\frac{14\!\cdots\!97}{13\!\cdots\!48}a^{5}+\frac{98\!\cdots\!67}{13\!\cdots\!48}a^{4}+\frac{77\!\cdots\!77}{44\!\cdots\!16}a^{3}+\frac{17\!\cdots\!23}{11\!\cdots\!79}a^{2}+\frac{27\!\cdots\!85}{22\!\cdots\!58}a+\frac{23\!\cdots\!49}{11\!\cdots\!79}$, $\frac{97\!\cdots\!23}{39\!\cdots\!44}a^{15}-\frac{19\!\cdots\!63}{66\!\cdots\!74}a^{14}-\frac{94\!\cdots\!07}{39\!\cdots\!44}a^{13}+\frac{85\!\cdots\!63}{33\!\cdots\!37}a^{12}-\frac{33\!\cdots\!33}{13\!\cdots\!48}a^{11}+\frac{10\!\cdots\!56}{25\!\cdots\!49}a^{10}-\frac{66\!\cdots\!03}{39\!\cdots\!44}a^{9}+\frac{66\!\cdots\!21}{66\!\cdots\!74}a^{8}-\frac{20\!\cdots\!05}{39\!\cdots\!44}a^{7}+\frac{65\!\cdots\!91}{10\!\cdots\!98}a^{6}+\frac{30\!\cdots\!67}{13\!\cdots\!48}a^{5}+\frac{45\!\cdots\!05}{66\!\cdots\!74}a^{4}-\frac{30\!\cdots\!85}{66\!\cdots\!74}a^{3}-\frac{15\!\cdots\!85}{22\!\cdots\!58}a^{2}-\frac{12\!\cdots\!65}{22\!\cdots\!58}a-\frac{96\!\cdots\!80}{11\!\cdots\!79}$, $\frac{36\!\cdots\!24}{99\!\cdots\!11}a^{15}+\frac{36\!\cdots\!19}{26\!\cdots\!96}a^{14}+\frac{22\!\cdots\!95}{79\!\cdots\!88}a^{13}+\frac{76\!\cdots\!43}{26\!\cdots\!96}a^{12}+\frac{95\!\cdots\!47}{88\!\cdots\!32}a^{11}+\frac{15\!\cdots\!49}{68\!\cdots\!64}a^{10}+\frac{43\!\cdots\!47}{79\!\cdots\!88}a^{9}+\frac{63\!\cdots\!85}{88\!\cdots\!32}a^{8}-\frac{96\!\cdots\!91}{79\!\cdots\!88}a^{7}-\frac{98\!\cdots\!73}{40\!\cdots\!92}a^{6}+\frac{18\!\cdots\!59}{26\!\cdots\!96}a^{5}+\frac{48\!\cdots\!77}{26\!\cdots\!96}a^{4}+\frac{94\!\cdots\!87}{26\!\cdots\!96}a^{3}+\frac{14\!\cdots\!21}{44\!\cdots\!16}a^{2}+\frac{28\!\cdots\!43}{22\!\cdots\!58}a+\frac{15\!\cdots\!93}{11\!\cdots\!79}$
| 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: | \( 69862.88502677123 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 69862.88502677123 \cdot 2}{12\cdot\sqrt{2939612246589864738816}}\cr\approx \mathstrut & 0.521661983783720 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 10*x^14 - 6*x^13 + 54*x^12 + 6*x^11 + 274*x^10 + 12*x^9 + 106*x^8 - 1062*x^7 + 402*x^6 + 3798*x^5 + 10509*x^4 + 11844*x^3 + 7524*x^2 + 1728*x + 144)
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 + 10*x^14 - 6*x^13 + 54*x^12 + 6*x^11 + 274*x^10 + 12*x^9 + 106*x^8 - 1062*x^7 + 402*x^6 + 3798*x^5 + 10509*x^4 + 11844*x^3 + 7524*x^2 + 1728*x + 144, 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 + 10*x^14 - 6*x^13 + 54*x^12 + 6*x^11 + 274*x^10 + 12*x^9 + 106*x^8 - 1062*x^7 + 402*x^6 + 3798*x^5 + 10509*x^4 + 11844*x^3 + 7524*x^2 + 1728*x + 144);
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 + 10*x^14 - 6*x^13 + 54*x^12 + 6*x^11 + 274*x^10 + 12*x^9 + 106*x^8 - 1062*x^7 + 402*x^6 + 3798*x^5 + 10509*x^4 + 11844*x^3 + 7524*x^2 + 1728*x + 144);
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
$C_2^2\times D_4$ (as 16T25):
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 $C_2^2 \times D_4$ |
| Character table for $C_2^2 \times D_4$ |
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
| Galois closure: | deg 32 |
| Degree 16 siblings: | deg 16, deg 16, deg 16, deg 16, deg 16, deg 16, deg 16 |
| 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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | R | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
|
\(3\)
| 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 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
|
\(7\)
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(11\)
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |