Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 54*x^14 - 162*x^13 + 800*x^12 - 1524*x^11 + 5317*x^10 - 5127*x^9 + 20353*x^8 - 1644*x^7 + 49532*x^6 + 15954*x^5 + 43350*x^4 - 5499*x^3 + 2977*x^2 - 438*x + 963)
gp: K = bnfinit(y^16 - 6*y^15 + 54*y^14 - 162*y^13 + 800*y^12 - 1524*y^11 + 5317*y^10 - 5127*y^9 + 20353*y^8 - 1644*y^7 + 49532*y^6 + 15954*y^5 + 43350*y^4 - 5499*y^3 + 2977*y^2 - 438*y + 963, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 54*x^14 - 162*x^13 + 800*x^12 - 1524*x^11 + 5317*x^10 - 5127*x^9 + 20353*x^8 - 1644*x^7 + 49532*x^6 + 15954*x^5 + 43350*x^4 - 5499*x^3 + 2977*x^2 - 438*x + 963);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 54*x^14 - 162*x^13 + 800*x^12 - 1524*x^11 + 5317*x^10 - 5127*x^9 + 20353*x^8 - 1644*x^7 + 49532*x^6 + 15954*x^5 + 43350*x^4 - 5499*x^3 + 2977*x^2 - 438*x + 963)
\( x^{16} - 6 x^{15} + 54 x^{14} - 162 x^{13} + 800 x^{12} - 1524 x^{11} + 5317 x^{10} - 5127 x^{9} + \cdots + 963 \)
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: |
\(51952739971213319841055446889\)
\(\medspace = 7^{8}\cdot 37^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(62.33\) | 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: | $7^{1/2}37^{7/8}\approx 62.334076345165585$ | ||
| Ramified primes: |
\(7\), \(37\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{21}a^{12}-\frac{1}{7}a^{11}-\frac{2}{21}a^{10}-\frac{1}{21}a^{9}-\frac{1}{21}a^{8}+\frac{8}{21}a^{6}-\frac{1}{7}a^{5}-\frac{8}{21}a^{4}-\frac{1}{7}a^{3}+\frac{1}{3}a^{2}+\frac{10}{21}a+\frac{3}{7}$, $\frac{1}{21}a^{13}+\frac{1}{7}a^{11}+\frac{1}{7}a^{9}-\frac{1}{7}a^{8}+\frac{1}{21}a^{7}-\frac{1}{7}a^{5}-\frac{2}{7}a^{4}-\frac{2}{21}a^{3}+\frac{1}{7}a^{2}+\frac{4}{21}a+\frac{2}{7}$, $\frac{1}{153279}a^{14}+\frac{200}{17031}a^{13}+\frac{1835}{153279}a^{12}+\frac{5554}{51093}a^{11}-\frac{295}{2433}a^{10}-\frac{767}{51093}a^{9}-\frac{2123}{21897}a^{8}+\frac{299}{51093}a^{7}+\frac{10874}{51093}a^{6}-\frac{7288}{51093}a^{5}-\frac{33067}{153279}a^{4}+\frac{355}{7299}a^{3}-\frac{8558}{21897}a^{2}+\frac{2512}{51093}a+\frac{4061}{17031}$, $\frac{1}{21\!\cdots\!87}a^{15}-\frac{34\!\cdots\!96}{70\!\cdots\!29}a^{14}-\frac{46\!\cdots\!02}{21\!\cdots\!87}a^{13}+\frac{10\!\cdots\!78}{70\!\cdots\!29}a^{12}-\frac{35\!\cdots\!66}{23\!\cdots\!43}a^{11}-\frac{31\!\cdots\!83}{70\!\cdots\!29}a^{10}-\frac{22\!\cdots\!33}{21\!\cdots\!87}a^{9}-\frac{10\!\cdots\!84}{70\!\cdots\!29}a^{8}+\frac{46\!\cdots\!94}{70\!\cdots\!29}a^{7}-\frac{11\!\cdots\!86}{10\!\cdots\!47}a^{6}-\frac{58\!\cdots\!38}{21\!\cdots\!87}a^{5}-\frac{15\!\cdots\!02}{70\!\cdots\!29}a^{4}-\frac{24\!\cdots\!64}{21\!\cdots\!87}a^{3}+\frac{57\!\cdots\!86}{23\!\cdots\!43}a^{2}-\frac{22\!\cdots\!85}{77\!\cdots\!81}a-\frac{30\!\cdots\!00}{11\!\cdots\!83}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{2}$, which has order $2$ (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: |
$\frac{13378008234960}{67\!\cdots\!83}a^{15}-\frac{21\!\cdots\!78}{18\!\cdots\!41}a^{14}+\frac{21\!\cdots\!90}{20\!\cdots\!49}a^{13}-\frac{55\!\cdots\!30}{18\!\cdots\!41}a^{12}+\frac{93\!\cdots\!82}{60\!\cdots\!47}a^{11}-\frac{18\!\cdots\!60}{67\!\cdots\!83}a^{10}+\frac{61\!\cdots\!72}{60\!\cdots\!47}a^{9}-\frac{15\!\cdots\!60}{18\!\cdots\!41}a^{8}+\frac{23\!\cdots\!30}{60\!\cdots\!47}a^{7}+\frac{10\!\cdots\!80}{60\!\cdots\!47}a^{6}+\frac{57\!\cdots\!28}{60\!\cdots\!47}a^{5}+\frac{74\!\cdots\!30}{18\!\cdots\!41}a^{4}+\frac{48\!\cdots\!62}{60\!\cdots\!47}a^{3}-\frac{18\!\cdots\!48}{18\!\cdots\!41}a^{2}-\frac{40\!\cdots\!06}{60\!\cdots\!47}a-\frac{11\!\cdots\!55}{20\!\cdots\!49}$, $\frac{29\!\cdots\!81}{50\!\cdots\!51}a^{15}-\frac{46\!\cdots\!47}{10\!\cdots\!71}a^{14}+\frac{13\!\cdots\!66}{35\!\cdots\!57}a^{13}-\frac{14\!\cdots\!87}{10\!\cdots\!71}a^{12}+\frac{21\!\cdots\!01}{35\!\cdots\!57}a^{11}-\frac{17\!\cdots\!74}{11\!\cdots\!19}a^{10}+\frac{21\!\cdots\!38}{50\!\cdots\!51}a^{9}-\frac{69\!\cdots\!63}{10\!\cdots\!71}a^{8}+\frac{49\!\cdots\!38}{35\!\cdots\!57}a^{7}-\frac{55\!\cdots\!58}{50\!\cdots\!51}a^{6}+\frac{10\!\cdots\!04}{50\!\cdots\!51}a^{5}-\frac{77\!\cdots\!63}{10\!\cdots\!71}a^{4}-\frac{96\!\cdots\!09}{43\!\cdots\!87}a^{3}+\frac{25\!\cdots\!10}{10\!\cdots\!71}a^{2}+\frac{70\!\cdots\!83}{35\!\cdots\!57}a+\frac{62\!\cdots\!07}{11\!\cdots\!19}$, $\frac{42\!\cdots\!36}{70\!\cdots\!29}a^{15}-\frac{43\!\cdots\!92}{10\!\cdots\!47}a^{14}+\frac{25\!\cdots\!26}{70\!\cdots\!29}a^{13}-\frac{13\!\cdots\!16}{10\!\cdots\!47}a^{12}+\frac{18\!\cdots\!72}{33\!\cdots\!49}a^{11}-\frac{30\!\cdots\!22}{23\!\cdots\!43}a^{10}+\frac{25\!\cdots\!94}{70\!\cdots\!29}a^{9}-\frac{33\!\cdots\!72}{70\!\cdots\!29}a^{8}+\frac{84\!\cdots\!70}{77\!\cdots\!81}a^{7}-\frac{47\!\cdots\!18}{77\!\cdots\!81}a^{6}+\frac{10\!\cdots\!82}{70\!\cdots\!29}a^{5}-\frac{27\!\cdots\!56}{70\!\cdots\!29}a^{4}-\frac{85\!\cdots\!40}{70\!\cdots\!29}a^{3}-\frac{11\!\cdots\!43}{70\!\cdots\!29}a^{2}+\frac{25\!\cdots\!01}{77\!\cdots\!81}a+\frac{50\!\cdots\!79}{77\!\cdots\!81}$, $\frac{42\!\cdots\!35}{21\!\cdots\!87}a^{15}-\frac{32\!\cdots\!88}{21\!\cdots\!87}a^{14}+\frac{26\!\cdots\!70}{21\!\cdots\!87}a^{13}-\frac{98\!\cdots\!40}{21\!\cdots\!87}a^{12}+\frac{13\!\cdots\!51}{70\!\cdots\!29}a^{11}-\frac{31\!\cdots\!74}{70\!\cdots\!29}a^{10}+\frac{24\!\cdots\!01}{21\!\cdots\!87}a^{9}-\frac{43\!\cdots\!79}{30\!\cdots\!41}a^{8}+\frac{19\!\cdots\!47}{70\!\cdots\!29}a^{7}-\frac{66\!\cdots\!30}{70\!\cdots\!29}a^{6}+\frac{90\!\cdots\!87}{30\!\cdots\!41}a^{5}-\frac{37\!\cdots\!53}{21\!\cdots\!87}a^{4}+\frac{95\!\cdots\!44}{21\!\cdots\!87}a^{3}-\frac{28\!\cdots\!64}{21\!\cdots\!87}a^{2}+\frac{77\!\cdots\!90}{70\!\cdots\!29}a-\frac{62\!\cdots\!30}{23\!\cdots\!43}$, $\frac{41\!\cdots\!39}{34\!\cdots\!43}a^{15}-\frac{37\!\cdots\!12}{49\!\cdots\!49}a^{14}+\frac{23\!\cdots\!13}{34\!\cdots\!43}a^{13}-\frac{73\!\cdots\!81}{34\!\cdots\!43}a^{12}+\frac{11\!\cdots\!12}{11\!\cdots\!81}a^{11}-\frac{23\!\cdots\!51}{11\!\cdots\!81}a^{10}+\frac{32\!\cdots\!24}{49\!\cdots\!49}a^{9}-\frac{26\!\cdots\!55}{34\!\cdots\!43}a^{8}+\frac{28\!\cdots\!89}{11\!\cdots\!81}a^{7}-\frac{98\!\cdots\!46}{11\!\cdots\!81}a^{6}+\frac{25\!\cdots\!07}{49\!\cdots\!49}a^{5}-\frac{90\!\cdots\!52}{34\!\cdots\!43}a^{4}+\frac{87\!\cdots\!99}{34\!\cdots\!43}a^{3}-\frac{12\!\cdots\!73}{34\!\cdots\!43}a^{2}-\frac{17\!\cdots\!99}{11\!\cdots\!81}a-\frac{26\!\cdots\!15}{38\!\cdots\!27}$, $\frac{28\!\cdots\!32}{21\!\cdots\!87}a^{15}-\frac{27\!\cdots\!89}{21\!\cdots\!87}a^{14}+\frac{27\!\cdots\!04}{30\!\cdots\!41}a^{13}-\frac{83\!\cdots\!21}{21\!\cdots\!87}a^{12}+\frac{86\!\cdots\!78}{70\!\cdots\!29}a^{11}-\frac{24\!\cdots\!69}{70\!\cdots\!29}a^{10}+\frac{14\!\cdots\!33}{21\!\cdots\!87}a^{9}-\frac{26\!\cdots\!46}{21\!\cdots\!87}a^{8}+\frac{83\!\cdots\!00}{70\!\cdots\!29}a^{7}-\frac{26\!\cdots\!31}{70\!\cdots\!29}a^{6}-\frac{14\!\cdots\!05}{21\!\cdots\!87}a^{5}-\frac{83\!\cdots\!38}{21\!\cdots\!87}a^{4}+\frac{18\!\cdots\!51}{21\!\cdots\!87}a^{3}-\frac{56\!\cdots\!12}{21\!\cdots\!87}a^{2}+\frac{15\!\cdots\!45}{70\!\cdots\!29}a-\frac{20\!\cdots\!08}{23\!\cdots\!43}$, $\frac{45\!\cdots\!11}{21\!\cdots\!87}a^{15}-\frac{46\!\cdots\!11}{30\!\cdots\!41}a^{14}+\frac{27\!\cdots\!29}{21\!\cdots\!87}a^{13}-\frac{10\!\cdots\!47}{21\!\cdots\!87}a^{12}+\frac{15\!\cdots\!33}{70\!\cdots\!29}a^{11}-\frac{38\!\cdots\!97}{70\!\cdots\!29}a^{10}+\frac{34\!\cdots\!58}{21\!\cdots\!87}a^{9}-\frac{56\!\cdots\!18}{21\!\cdots\!87}a^{8}+\frac{64\!\cdots\!69}{10\!\cdots\!47}a^{7}-\frac{46\!\cdots\!75}{70\!\cdots\!29}a^{6}+\frac{42\!\cdots\!41}{30\!\cdots\!41}a^{5}-\frac{24\!\cdots\!62}{21\!\cdots\!87}a^{4}+\frac{24\!\cdots\!02}{21\!\cdots\!87}a^{3}-\frac{30\!\cdots\!50}{21\!\cdots\!87}a^{2}+\frac{46\!\cdots\!50}{70\!\cdots\!29}a-\frac{34\!\cdots\!76}{23\!\cdots\!43}$
| 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: | \( 87423196.4181 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 87423196.4181 \cdot 2}{2\cdot\sqrt{51952739971213319841055446889}}\cr\approx \mathstrut & 0.931668473411 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 54*x^14 - 162*x^13 + 800*x^12 - 1524*x^11 + 5317*x^10 - 5127*x^9 + 20353*x^8 - 1644*x^7 + 49532*x^6 + 15954*x^5 + 43350*x^4 - 5499*x^3 + 2977*x^2 - 438*x + 963)
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 - 6*x^15 + 54*x^14 - 162*x^13 + 800*x^12 - 1524*x^11 + 5317*x^10 - 5127*x^9 + 20353*x^8 - 1644*x^7 + 49532*x^6 + 15954*x^5 + 43350*x^4 - 5499*x^3 + 2977*x^2 - 438*x + 963, 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 - 6*x^15 + 54*x^14 - 162*x^13 + 800*x^12 - 1524*x^11 + 5317*x^10 - 5127*x^9 + 20353*x^8 - 1644*x^7 + 49532*x^6 + 15954*x^5 + 43350*x^4 - 5499*x^3 + 2977*x^2 - 438*x + 963);
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 - 6*x^15 + 54*x^14 - 162*x^13 + 800*x^12 - 1524*x^11 + 5317*x^10 - 5127*x^9 + 20353*x^8 - 1644*x^7 + 49532*x^6 + 15954*x^5 + 43350*x^4 - 5499*x^3 + 2977*x^2 - 438*x + 963);
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
$\OD_{16}:C_2$ (as 16T36):
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 11 conjugacy class representatives for $\OD_{16}:C_2$ |
| Character table for $\OD_{16}:C_2$ |
Intermediate fields
| \(\Q(\sqrt{37}) \), \(\Q(\sqrt{-259}) \), \(\Q(\sqrt{-7}) \), 4.0.50653.1, 4.4.2481997.1, \(\Q(\sqrt{-7}, \sqrt{37})\), 8.0.4651661979517.1 x2, 8.4.227931436996333.1 x2, 8.0.6160309108009.1 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{8}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{8}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 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 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(37\)
| 37.8.7.2 | $x^{8} + 37$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 37.8.7.2 | $x^{8} + 37$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |