Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 - 24*x^13 + 528*x^12 + 696*x^11 - 4340*x^10 - 8976*x^9 + 20776*x^8 + 65760*x^7 - 11160*x^6 - 191712*x^5 + 39996*x^4 + 850104*x^3 + 1807452*x^2 + 1890576*x + 1377009)
gp: K = bnfinit(y^16 - 32*y^14 - 24*y^13 + 528*y^12 + 696*y^11 - 4340*y^10 - 8976*y^9 + 20776*y^8 + 65760*y^7 - 11160*y^6 - 191712*y^5 + 39996*y^4 + 850104*y^3 + 1807452*y^2 + 1890576*y + 1377009, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 32*x^14 - 24*x^13 + 528*x^12 + 696*x^11 - 4340*x^10 - 8976*x^9 + 20776*x^8 + 65760*x^7 - 11160*x^6 - 191712*x^5 + 39996*x^4 + 850104*x^3 + 1807452*x^2 + 1890576*x + 1377009);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 32*x^14 - 24*x^13 + 528*x^12 + 696*x^11 - 4340*x^10 - 8976*x^9 + 20776*x^8 + 65760*x^7 - 11160*x^6 - 191712*x^5 + 39996*x^4 + 850104*x^3 + 1807452*x^2 + 1890576*x + 1377009)
\( x^{16} - 32 x^{14} - 24 x^{13} + 528 x^{12} + 696 x^{11} - 4340 x^{10} - 8976 x^{9} + 20776 x^{8} + \cdots + 1377009 \)
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: |
\(22144295318673432094540038144\)
\(\medspace = 2^{48}\cdot 3^{12}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(59.10\) | 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: | $2^{3}3^{3/4}23^{1/2}\approx 87.45705441086021$ | ||
| Ramified primes: |
\(2\), \(3\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}$, $\frac{1}{28977325491}a^{14}+\frac{630825893}{9659108497}a^{13}-\frac{808754470}{28977325491}a^{12}-\frac{2224212499}{9659108497}a^{11}+\frac{309255262}{28977325491}a^{10}+\frac{2753437559}{9659108497}a^{9}-\frac{227534638}{1259883717}a^{8}+\frac{2227491032}{9659108497}a^{7}+\frac{2012211509}{28977325491}a^{6}+\frac{391602657}{9659108497}a^{5}+\frac{429674876}{1259883717}a^{4}-\frac{3459708764}{9659108497}a^{3}+\frac{1449190320}{9659108497}a^{2}-\frac{1560940578}{9659108497}a+\frac{2126013471}{9659108497}$, $\frac{1}{35\!\cdots\!53}a^{15}-\frac{35\!\cdots\!95}{35\!\cdots\!53}a^{14}+\frac{17\!\cdots\!22}{35\!\cdots\!53}a^{13}+\frac{53\!\cdots\!72}{32\!\cdots\!23}a^{12}-\frac{53\!\cdots\!73}{10\!\cdots\!41}a^{11}-\frac{20\!\cdots\!35}{11\!\cdots\!51}a^{10}+\frac{50\!\cdots\!15}{35\!\cdots\!53}a^{9}-\frac{77\!\cdots\!32}{35\!\cdots\!53}a^{8}+\frac{15\!\cdots\!42}{35\!\cdots\!53}a^{7}+\frac{35\!\cdots\!94}{32\!\cdots\!23}a^{6}+\frac{47\!\cdots\!15}{11\!\cdots\!51}a^{5}+\frac{11\!\cdots\!40}{11\!\cdots\!51}a^{4}+\frac{14\!\cdots\!29}{11\!\cdots\!51}a^{3}-\frac{44\!\cdots\!24}{11\!\cdots\!51}a^{2}-\frac{51\!\cdots\!49}{11\!\cdots\!51}a-\frac{37\!\cdots\!60}{11\!\cdots\!51}$
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
$C_{2}\times C_{2}\times C_{4}\times C_{8}$, which has order $128$ (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: |
$\frac{18\!\cdots\!98}{21\!\cdots\!69}a^{15}-\frac{43\!\cdots\!59}{21\!\cdots\!69}a^{14}-\frac{51\!\cdots\!02}{21\!\cdots\!69}a^{13}+\frac{90\!\cdots\!68}{19\!\cdots\!79}a^{12}+\frac{72\!\cdots\!82}{19\!\cdots\!79}a^{11}-\frac{11\!\cdots\!08}{21\!\cdots\!69}a^{10}-\frac{60\!\cdots\!56}{21\!\cdots\!69}a^{9}+\frac{56\!\cdots\!79}{21\!\cdots\!69}a^{8}+\frac{27\!\cdots\!22}{21\!\cdots\!69}a^{7}-\frac{61\!\cdots\!51}{19\!\cdots\!79}a^{6}-\frac{33\!\cdots\!62}{21\!\cdots\!69}a^{5}+\frac{37\!\cdots\!25}{21\!\cdots\!69}a^{4}+\frac{87\!\cdots\!06}{71\!\cdots\!23}a^{3}+\frac{99\!\cdots\!76}{71\!\cdots\!23}a^{2}+\frac{13\!\cdots\!06}{71\!\cdots\!23}a+\frac{37\!\cdots\!91}{71\!\cdots\!23}$, $\frac{12\!\cdots\!52}{85\!\cdots\!71}a^{15}-\frac{15\!\cdots\!80}{25\!\cdots\!13}a^{14}-\frac{86\!\cdots\!20}{25\!\cdots\!13}a^{13}+\frac{10\!\cdots\!09}{77\!\cdots\!61}a^{12}+\frac{11\!\cdots\!08}{23\!\cdots\!83}a^{11}-\frac{45\!\cdots\!22}{25\!\cdots\!13}a^{10}-\frac{29\!\cdots\!06}{85\!\cdots\!71}a^{9}+\frac{26\!\cdots\!06}{25\!\cdots\!13}a^{8}+\frac{52\!\cdots\!60}{25\!\cdots\!13}a^{7}-\frac{31\!\cdots\!50}{77\!\cdots\!61}a^{6}-\frac{10\!\cdots\!88}{25\!\cdots\!13}a^{5}+\frac{11\!\cdots\!55}{25\!\cdots\!13}a^{4}+\frac{22\!\cdots\!12}{85\!\cdots\!71}a^{3}+\frac{10\!\cdots\!82}{85\!\cdots\!71}a^{2}+\frac{21\!\cdots\!62}{85\!\cdots\!71}a-\frac{57\!\cdots\!11}{85\!\cdots\!71}$, $\frac{67\!\cdots\!12}{21\!\cdots\!69}a^{15}-\frac{26\!\cdots\!84}{21\!\cdots\!69}a^{14}-\frac{55\!\cdots\!92}{71\!\cdots\!23}a^{13}+\frac{57\!\cdots\!29}{19\!\cdots\!79}a^{12}+\frac{23\!\cdots\!36}{19\!\cdots\!79}a^{11}-\frac{83\!\cdots\!28}{21\!\cdots\!69}a^{10}-\frac{18\!\cdots\!98}{21\!\cdots\!69}a^{9}+\frac{50\!\cdots\!73}{21\!\cdots\!69}a^{8}+\frac{37\!\cdots\!96}{71\!\cdots\!23}a^{7}-\frac{16\!\cdots\!82}{19\!\cdots\!79}a^{6}-\frac{28\!\cdots\!00}{21\!\cdots\!69}a^{5}+\frac{12\!\cdots\!91}{21\!\cdots\!69}a^{4}+\frac{43\!\cdots\!28}{71\!\cdots\!23}a^{3}+\frac{38\!\cdots\!80}{71\!\cdots\!23}a^{2}+\frac{59\!\cdots\!46}{71\!\cdots\!23}a-\frac{24\!\cdots\!64}{71\!\cdots\!23}$, $\frac{41\!\cdots\!80}{35\!\cdots\!53}a^{15}-\frac{45\!\cdots\!72}{35\!\cdots\!53}a^{14}-\frac{13\!\cdots\!66}{35\!\cdots\!53}a^{13}+\frac{40\!\cdots\!31}{32\!\cdots\!23}a^{12}+\frac{20\!\cdots\!12}{32\!\cdots\!23}a^{11}+\frac{66\!\cdots\!93}{35\!\cdots\!53}a^{10}-\frac{20\!\cdots\!05}{35\!\cdots\!53}a^{9}-\frac{20\!\cdots\!01}{35\!\cdots\!53}a^{8}+\frac{11\!\cdots\!66}{35\!\cdots\!53}a^{7}+\frac{18\!\cdots\!08}{32\!\cdots\!23}a^{6}-\frac{25\!\cdots\!19}{35\!\cdots\!53}a^{5}-\frac{83\!\cdots\!74}{35\!\cdots\!53}a^{4}+\frac{27\!\cdots\!94}{11\!\cdots\!51}a^{3}+\frac{12\!\cdots\!82}{11\!\cdots\!51}a^{2}+\frac{14\!\cdots\!52}{11\!\cdots\!51}a-\frac{13\!\cdots\!46}{11\!\cdots\!51}$, $\frac{18\!\cdots\!31}{35\!\cdots\!53}a^{15}-\frac{82\!\cdots\!41}{35\!\cdots\!53}a^{14}-\frac{63\!\cdots\!54}{35\!\cdots\!53}a^{13}-\frac{71\!\cdots\!14}{32\!\cdots\!23}a^{12}+\frac{98\!\cdots\!89}{32\!\cdots\!23}a^{11}+\frac{68\!\cdots\!11}{35\!\cdots\!53}a^{10}-\frac{98\!\cdots\!36}{35\!\cdots\!53}a^{9}-\frac{11\!\cdots\!73}{35\!\cdots\!53}a^{8}+\frac{53\!\cdots\!63}{35\!\cdots\!53}a^{7}+\frac{97\!\cdots\!14}{32\!\cdots\!23}a^{6}-\frac{12\!\cdots\!57}{35\!\cdots\!53}a^{5}-\frac{40\!\cdots\!93}{35\!\cdots\!53}a^{4}+\frac{12\!\cdots\!95}{11\!\cdots\!51}a^{3}+\frac{63\!\cdots\!31}{11\!\cdots\!51}a^{2}+\frac{70\!\cdots\!07}{11\!\cdots\!51}a+\frac{13\!\cdots\!94}{11\!\cdots\!51}$, $\frac{10\!\cdots\!81}{35\!\cdots\!53}a^{15}+\frac{10\!\cdots\!53}{35\!\cdots\!53}a^{14}-\frac{20\!\cdots\!27}{11\!\cdots\!51}a^{13}-\frac{10\!\cdots\!36}{10\!\cdots\!41}a^{12}+\frac{10\!\cdots\!15}{32\!\cdots\!23}a^{11}+\frac{59\!\cdots\!96}{35\!\cdots\!53}a^{10}-\frac{12\!\cdots\!11}{35\!\cdots\!53}a^{9}-\frac{54\!\cdots\!78}{35\!\cdots\!53}a^{8}+\frac{21\!\cdots\!51}{11\!\cdots\!51}a^{7}+\frac{10\!\cdots\!34}{10\!\cdots\!41}a^{6}-\frac{12\!\cdots\!66}{35\!\cdots\!53}a^{5}-\frac{10\!\cdots\!67}{35\!\cdots\!53}a^{4}+\frac{60\!\cdots\!69}{11\!\cdots\!51}a^{3}+\frac{12\!\cdots\!98}{11\!\cdots\!51}a^{2}+\frac{13\!\cdots\!34}{11\!\cdots\!51}a+\frac{22\!\cdots\!61}{11\!\cdots\!51}$, $\frac{29\!\cdots\!22}{35\!\cdots\!53}a^{15}-\frac{90\!\cdots\!51}{35\!\cdots\!53}a^{14}-\frac{78\!\cdots\!63}{35\!\cdots\!53}a^{13}+\frac{64\!\cdots\!89}{10\!\cdots\!41}a^{12}+\frac{36\!\cdots\!80}{10\!\cdots\!41}a^{11}-\frac{26\!\cdots\!29}{35\!\cdots\!53}a^{10}-\frac{93\!\cdots\!22}{35\!\cdots\!53}a^{9}+\frac{15\!\cdots\!06}{35\!\cdots\!53}a^{8}+\frac{50\!\cdots\!02}{35\!\cdots\!53}a^{7}-\frac{11\!\cdots\!71}{10\!\cdots\!41}a^{6}-\frac{35\!\cdots\!58}{11\!\cdots\!51}a^{5}+\frac{16\!\cdots\!38}{35\!\cdots\!53}a^{4}+\frac{17\!\cdots\!12}{11\!\cdots\!51}a^{3}+\frac{22\!\cdots\!64}{11\!\cdots\!51}a^{2}+\frac{30\!\cdots\!91}{11\!\cdots\!51}a+\frac{11\!\cdots\!19}{11\!\cdots\!51}$
| 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: | \( 475824.014517 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 475824.014517 \cdot 128}{2\cdot\sqrt{22144295318673432094540038144}}\cr\approx \mathstrut & 0.497089220370 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 - 24*x^13 + 528*x^12 + 696*x^11 - 4340*x^10 - 8976*x^9 + 20776*x^8 + 65760*x^7 - 11160*x^6 - 191712*x^5 + 39996*x^4 + 850104*x^3 + 1807452*x^2 + 1890576*x + 1377009)
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 - 32*x^14 - 24*x^13 + 528*x^12 + 696*x^11 - 4340*x^10 - 8976*x^9 + 20776*x^8 + 65760*x^7 - 11160*x^6 - 191712*x^5 + 39996*x^4 + 850104*x^3 + 1807452*x^2 + 1890576*x + 1377009, 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 - 32*x^14 - 24*x^13 + 528*x^12 + 696*x^11 - 4340*x^10 - 8976*x^9 + 20776*x^8 + 65760*x^7 - 11160*x^6 - 191712*x^5 + 39996*x^4 + 850104*x^3 + 1807452*x^2 + 1890576*x + 1377009);
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 - 32*x^14 - 24*x^13 + 528*x^12 + 696*x^11 - 4340*x^10 - 8976*x^9 + 20776*x^8 + 65760*x^7 - 11160*x^6 - 191712*x^5 + 39996*x^4 + 850104*x^3 + 1807452*x^2 + 1890576*x + 1377009);
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
$C_4^2:C_2$ (as 16T30):
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 $C_4^2:C_2$ |
| Character table for $C_4^2:C_2$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.0.105984.1, 4.0.105984.2, \(\Q(\sqrt{2}, \sqrt{3})\), 8.0.404373897216.34, 8.0.179721732096.5, 8.8.1617495588864.1 |
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 16 sibling: | 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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.48.13 | $x^{16} + 4 x^{12} + 4 x^{10} + 10 x^{8} + 8 x^{6} + 8 x^{2} + 8 x + 14$ | $16$ | $1$ | $48$ | 16T30 | $[2, 3, 3, 7/2]^{2}$ |
|
\(3\)
| 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
| 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ | |
|
\(23\)
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |