Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - x^13 + 54*x^12 + 210*x^11 + 209*x^10 - 589*x^9 - 630*x^8 + 1721*x^7 + 4011*x^6 + 14085*x^5 + 52958*x^4 + 81624*x^3 + 79362*x^2 + 77319*x + 77291)
gp: K = bnfinit(y^16 - 8*y^14 - y^13 + 54*y^12 + 210*y^11 + 209*y^10 - 589*y^9 - 630*y^8 + 1721*y^7 + 4011*y^6 + 14085*y^5 + 52958*y^4 + 81624*y^3 + 79362*y^2 + 77319*y + 77291, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 - x^13 + 54*x^12 + 210*x^11 + 209*x^10 - 589*x^9 - 630*x^8 + 1721*x^7 + 4011*x^6 + 14085*x^5 + 52958*x^4 + 81624*x^3 + 79362*x^2 + 77319*x + 77291);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 - x^13 + 54*x^12 + 210*x^11 + 209*x^10 - 589*x^9 - 630*x^8 + 1721*x^7 + 4011*x^6 + 14085*x^5 + 52958*x^4 + 81624*x^3 + 79362*x^2 + 77319*x + 77291)
\( x^{16} - 8 x^{14} - x^{13} + 54 x^{12} + 210 x^{11} + 209 x^{10} - 589 x^{9} - 630 x^{8} + 1721 x^{7} + \cdots + 77291 \)
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: |
\(818629961123679547712831809\)
\(\medspace = 17^{10}\cdot 67^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(48.09\) | 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: | $17^{3/4}67^{1/2}\approx 68.52895233095656$ | ||
| Ramified primes: |
\(17\), \(67\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{76}a^{13}-\frac{9}{76}a^{12}+\frac{7}{38}a^{11}-\frac{13}{76}a^{10}+\frac{1}{76}a^{9}-\frac{25}{76}a^{8}-\frac{9}{38}a^{7}+\frac{1}{19}a^{6}-\frac{7}{19}a^{5}+\frac{7}{19}a^{4}-\frac{1}{4}a^{3}+\frac{9}{19}a^{2}-\frac{25}{76}a+\frac{5}{19}$, $\frac{1}{4078692}a^{14}+\frac{6167}{2039346}a^{13}-\frac{27361}{226594}a^{12}-\frac{47281}{679782}a^{11}+\frac{151895}{679782}a^{10}-\frac{179125}{453188}a^{9}-\frac{399925}{1019673}a^{8}-\frac{1038125}{4078692}a^{7}-\frac{273413}{679782}a^{6}+\frac{179011}{679782}a^{5}+\frac{220963}{453188}a^{4}+\frac{96665}{1359564}a^{3}-\frac{45313}{1019673}a^{2}-\frac{258205}{4078692}a+\frac{1259731}{4078692}$, $\frac{1}{13\!\cdots\!36}a^{15}-\frac{45\!\cdots\!79}{45\!\cdots\!12}a^{14}+\frac{52\!\cdots\!67}{13\!\cdots\!36}a^{13}+\frac{48\!\cdots\!91}{11\!\cdots\!03}a^{12}-\frac{18\!\cdots\!99}{15\!\cdots\!04}a^{11}-\frac{19\!\cdots\!64}{11\!\cdots\!03}a^{10}+\frac{66\!\cdots\!01}{13\!\cdots\!36}a^{9}-\frac{29\!\cdots\!77}{13\!\cdots\!36}a^{8}+\frac{60\!\cdots\!33}{33\!\cdots\!09}a^{7}-\frac{12\!\cdots\!14}{37\!\cdots\!01}a^{6}+\frac{16\!\cdots\!13}{45\!\cdots\!12}a^{5}-\frac{84\!\cdots\!33}{22\!\cdots\!06}a^{4}-\frac{57\!\cdots\!82}{33\!\cdots\!09}a^{3}+\frac{13\!\cdots\!07}{11\!\cdots\!03}a^{2}+\frac{51\!\cdots\!09}{13\!\cdots\!36}a-\frac{11\!\cdots\!14}{33\!\cdots\!09}$
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: |
\( -1 \)
(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: |
$\frac{21\!\cdots\!55}{15\!\cdots\!04}a^{15}-\frac{21\!\cdots\!71}{75\!\cdots\!02}a^{14}-\frac{12\!\cdots\!97}{15\!\cdots\!04}a^{13}+\frac{74\!\cdots\!14}{37\!\cdots\!01}a^{12}+\frac{76\!\cdots\!97}{15\!\cdots\!04}a^{11}+\frac{67\!\cdots\!80}{37\!\cdots\!01}a^{10}-\frac{83\!\cdots\!91}{75\!\cdots\!02}a^{9}-\frac{14\!\cdots\!75}{15\!\cdots\!04}a^{8}+\frac{17\!\cdots\!61}{15\!\cdots\!04}a^{7}+\frac{64\!\cdots\!19}{37\!\cdots\!01}a^{6}+\frac{18\!\cdots\!31}{15\!\cdots\!04}a^{5}+\frac{24\!\cdots\!75}{15\!\cdots\!04}a^{4}+\frac{58\!\cdots\!37}{15\!\cdots\!04}a^{3}+\frac{12\!\cdots\!47}{75\!\cdots\!02}a^{2}+\frac{31\!\cdots\!01}{75\!\cdots\!02}a+\frac{44\!\cdots\!15}{15\!\cdots\!04}$, $\frac{35\!\cdots\!17}{13\!\cdots\!36}a^{15}-\frac{36\!\cdots\!85}{13\!\cdots\!36}a^{14}+\frac{71\!\cdots\!35}{67\!\cdots\!18}a^{13}+\frac{24\!\cdots\!62}{11\!\cdots\!03}a^{12}+\frac{92\!\cdots\!19}{22\!\cdots\!06}a^{11}-\frac{52\!\cdots\!33}{45\!\cdots\!12}a^{10}-\frac{49\!\cdots\!89}{13\!\cdots\!36}a^{9}-\frac{10\!\cdots\!11}{15\!\cdots\!04}a^{8}+\frac{42\!\cdots\!55}{13\!\cdots\!36}a^{7}+\frac{27\!\cdots\!98}{11\!\cdots\!03}a^{6}-\frac{31\!\cdots\!99}{45\!\cdots\!12}a^{5}-\frac{13\!\cdots\!76}{11\!\cdots\!03}a^{4}+\frac{11\!\cdots\!25}{13\!\cdots\!36}a^{3}-\frac{74\!\cdots\!21}{13\!\cdots\!36}a^{2}+\frac{22\!\cdots\!29}{22\!\cdots\!06}a+\frac{17\!\cdots\!83}{13\!\cdots\!36}$, $\frac{98\!\cdots\!31}{37\!\cdots\!81}a^{15}-\frac{11\!\cdots\!57}{11\!\cdots\!03}a^{14}-\frac{93\!\cdots\!34}{33\!\cdots\!09}a^{13}+\frac{69\!\cdots\!39}{11\!\cdots\!03}a^{12}+\frac{13\!\cdots\!59}{75\!\cdots\!02}a^{11}+\frac{54\!\cdots\!84}{11\!\cdots\!03}a^{10}+\frac{13\!\cdots\!91}{67\!\cdots\!18}a^{9}-\frac{85\!\cdots\!85}{33\!\cdots\!09}a^{8}-\frac{48\!\cdots\!26}{33\!\cdots\!09}a^{7}+\frac{55\!\cdots\!79}{75\!\cdots\!02}a^{6}+\frac{17\!\cdots\!91}{22\!\cdots\!06}a^{5}+\frac{24\!\cdots\!86}{11\!\cdots\!03}a^{4}+\frac{95\!\cdots\!27}{67\!\cdots\!18}a^{3}+\frac{32\!\cdots\!31}{22\!\cdots\!06}a^{2}-\frac{81\!\cdots\!07}{67\!\cdots\!18}a-\frac{92\!\cdots\!50}{33\!\cdots\!09}$, $\frac{11\!\cdots\!33}{35\!\cdots\!22}a^{15}-\frac{59\!\cdots\!23}{71\!\cdots\!44}a^{14}+\frac{76\!\cdots\!73}{35\!\cdots\!22}a^{13}+\frac{19\!\cdots\!17}{11\!\cdots\!74}a^{12}-\frac{53\!\cdots\!57}{59\!\cdots\!37}a^{11}-\frac{55\!\cdots\!77}{59\!\cdots\!37}a^{10}-\frac{66\!\cdots\!87}{71\!\cdots\!44}a^{9}+\frac{14\!\cdots\!77}{11\!\cdots\!74}a^{8}+\frac{24\!\cdots\!51}{71\!\cdots\!44}a^{7}-\frac{54\!\cdots\!05}{59\!\cdots\!37}a^{6}+\frac{66\!\cdots\!77}{11\!\cdots\!74}a^{5}-\frac{31\!\cdots\!37}{23\!\cdots\!48}a^{4}-\frac{47\!\cdots\!83}{71\!\cdots\!44}a^{3}-\frac{20\!\cdots\!09}{17\!\cdots\!11}a^{2}-\frac{40\!\cdots\!75}{23\!\cdots\!48}a-\frac{62\!\cdots\!59}{37\!\cdots\!76}$, $\frac{18\!\cdots\!03}{50\!\cdots\!27}a^{15}-\frac{25\!\cdots\!27}{13\!\cdots\!36}a^{14}+\frac{82\!\cdots\!35}{75\!\cdots\!02}a^{13}+\frac{19\!\cdots\!51}{22\!\cdots\!06}a^{12}-\frac{42\!\cdots\!72}{11\!\cdots\!03}a^{11}+\frac{59\!\cdots\!08}{37\!\cdots\!01}a^{10}-\frac{21\!\cdots\!67}{13\!\cdots\!36}a^{9}-\frac{39\!\cdots\!67}{67\!\cdots\!18}a^{8}+\frac{36\!\cdots\!49}{45\!\cdots\!12}a^{7}-\frac{76\!\cdots\!45}{11\!\cdots\!03}a^{6}-\frac{28\!\cdots\!95}{75\!\cdots\!02}a^{5}+\frac{18\!\cdots\!11}{45\!\cdots\!12}a^{4}-\frac{27\!\cdots\!63}{13\!\cdots\!36}a^{3}-\frac{63\!\cdots\!40}{33\!\cdots\!09}a^{2}+\frac{18\!\cdots\!07}{13\!\cdots\!36}a-\frac{38\!\cdots\!79}{15\!\cdots\!04}$, $\frac{93\!\cdots\!37}{22\!\cdots\!06}a^{15}-\frac{25\!\cdots\!97}{37\!\cdots\!01}a^{14}-\frac{77\!\cdots\!81}{22\!\cdots\!06}a^{13}+\frac{54\!\cdots\!95}{75\!\cdots\!02}a^{12}+\frac{16\!\cdots\!53}{75\!\cdots\!02}a^{11}+\frac{19\!\cdots\!24}{37\!\cdots\!01}a^{10}-\frac{50\!\cdots\!41}{11\!\cdots\!03}a^{9}-\frac{43\!\cdots\!29}{11\!\cdots\!03}a^{8}+\frac{54\!\cdots\!58}{11\!\cdots\!03}a^{7}+\frac{13\!\cdots\!87}{75\!\cdots\!02}a^{6}+\frac{93\!\cdots\!65}{37\!\cdots\!01}a^{5}+\frac{73\!\cdots\!29}{75\!\cdots\!02}a^{4}+\frac{23\!\cdots\!67}{11\!\cdots\!03}a^{3}+\frac{61\!\cdots\!22}{37\!\cdots\!01}a^{2}+\frac{11\!\cdots\!71}{22\!\cdots\!06}a+\frac{86\!\cdots\!51}{11\!\cdots\!03}$, $\frac{11\!\cdots\!03}{33\!\cdots\!09}a^{15}-\frac{12\!\cdots\!84}{33\!\cdots\!09}a^{14}-\frac{22\!\cdots\!51}{67\!\cdots\!18}a^{13}+\frac{54\!\cdots\!87}{22\!\cdots\!06}a^{12}+\frac{27\!\cdots\!10}{11\!\cdots\!03}a^{11}+\frac{64\!\cdots\!90}{11\!\cdots\!03}a^{10}-\frac{38\!\cdots\!41}{67\!\cdots\!18}a^{9}-\frac{18\!\cdots\!24}{37\!\cdots\!01}a^{8}-\frac{36\!\cdots\!52}{33\!\cdots\!09}a^{7}+\frac{22\!\cdots\!36}{11\!\cdots\!03}a^{6}+\frac{61\!\cdots\!99}{22\!\cdots\!06}a^{5}+\frac{43\!\cdots\!99}{22\!\cdots\!06}a^{4}+\frac{17\!\cdots\!91}{67\!\cdots\!18}a^{3}-\frac{10\!\cdots\!31}{67\!\cdots\!18}a^{2}-\frac{54\!\cdots\!86}{11\!\cdots\!03}a-\frac{36\!\cdots\!39}{67\!\cdots\!18}$
| 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: | \( 2782055.44714 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2782055.44714 \cdot 4}{2\cdot\sqrt{818629961123679547712831809}}\cr\approx \mathstrut & 0.472379352127 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - x^13 + 54*x^12 + 210*x^11 + 209*x^10 - 589*x^9 - 630*x^8 + 1721*x^7 + 4011*x^6 + 14085*x^5 + 52958*x^4 + 81624*x^3 + 79362*x^2 + 77319*x + 77291)
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 - 8*x^14 - x^13 + 54*x^12 + 210*x^11 + 209*x^10 - 589*x^9 - 630*x^8 + 1721*x^7 + 4011*x^6 + 14085*x^5 + 52958*x^4 + 81624*x^3 + 79362*x^2 + 77319*x + 77291, 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 - 8*x^14 - x^13 + 54*x^12 + 210*x^11 + 209*x^10 - 589*x^9 - 630*x^8 + 1721*x^7 + 4011*x^6 + 14085*x^5 + 52958*x^4 + 81624*x^3 + 79362*x^2 + 77319*x + 77291);
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 - 8*x^14 - x^13 + 54*x^12 + 210*x^11 + 209*x^10 - 589*x^9 - 630*x^8 + 1721*x^7 + 4011*x^6 + 14085*x^5 + 52958*x^4 + 81624*x^3 + 79362*x^2 + 77319*x + 77291);
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
$C_2^3:C_4$ (as 16T33):
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 $C_2^3:C_4$ |
| Character table for $C_2^3:C_4$ |
Intermediate fields
| \(\Q(\sqrt{-1139}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-67}) \), 4.0.76313.1 x2, \(\Q(\sqrt{17}, \sqrt{-67})\), 4.2.19363.1 x2, 8.0.1683041777041.1 x2, 8.0.28611710209697.2 x2, 8.0.1683041777041.2 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.4.3.1 | $x^{4} + 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 17.4.3.1 | $x^{4} + 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(67\)
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |