Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 15*x^13 + 6*x^12 - 118*x^11 + 317*x^10 - 460*x^9 + 1614*x^8 - 2361*x^7 + 2820*x^6 - 2374*x^5 + 4393*x^4 + 3912*x^3 + 10208*x^2 + 8568*x + 2448)
gp: K = bnfinit(y^16 - 4*y^15 + 4*y^14 + 15*y^13 + 6*y^12 - 118*y^11 + 317*y^10 - 460*y^9 + 1614*y^8 - 2361*y^7 + 2820*y^6 - 2374*y^5 + 4393*y^4 + 3912*y^3 + 10208*y^2 + 8568*y + 2448, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 4*x^14 + 15*x^13 + 6*x^12 - 118*x^11 + 317*x^10 - 460*x^9 + 1614*x^8 - 2361*x^7 + 2820*x^6 - 2374*x^5 + 4393*x^4 + 3912*x^3 + 10208*x^2 + 8568*x + 2448);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 4*x^14 + 15*x^13 + 6*x^12 - 118*x^11 + 317*x^10 - 460*x^9 + 1614*x^8 - 2361*x^7 + 2820*x^6 - 2374*x^5 + 4393*x^4 + 3912*x^3 + 10208*x^2 + 8568*x + 2448)
\( x^{16} - 4 x^{15} + 4 x^{14} + 15 x^{13} + 6 x^{12} - 118 x^{11} + 317 x^{10} - 460 x^{9} + 1614 x^{8} + \cdots + 2448 \)
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: |
\(166101760110563345913601\)
\(\medspace = 17^{8}\cdot 47^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(28.27\) | 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: | $17^{1/2}47^{1/2}\approx 28.26658805020514$ | ||
| Ramified primes: |
\(17\), \(47\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{21}a^{12}+\frac{1}{7}a^{10}-\frac{1}{7}a^{9}+\frac{1}{7}a^{8}+\frac{1}{7}a^{7}-\frac{1}{7}a^{6}-\frac{2}{7}a^{5}-\frac{1}{21}a^{4}-\frac{1}{7}a^{2}-\frac{2}{7}a+\frac{2}{7}$, $\frac{1}{42}a^{13}-\frac{2}{21}a^{11}-\frac{1}{14}a^{10}-\frac{2}{21}a^{9}-\frac{2}{21}a^{8}-\frac{17}{42}a^{7}+\frac{4}{21}a^{6}+\frac{1}{7}a^{5}-\frac{1}{6}a^{4}-\frac{5}{21}a^{3}+\frac{4}{21}a^{2}-\frac{1}{42}a$, $\frac{1}{12852}a^{14}-\frac{22}{3213}a^{13}-\frac{1}{1071}a^{12}+\frac{1147}{12852}a^{11}+\frac{17}{126}a^{10}+\frac{263}{6426}a^{9}+\frac{605}{12852}a^{8}-\frac{1499}{3213}a^{7}-\frac{505}{6426}a^{6}-\frac{6241}{12852}a^{5}-\frac{1096}{3213}a^{4}-\frac{1}{14}a^{3}-\frac{835}{12852}a^{2}+\frac{1}{21}a-\frac{2}{21}$, $\frac{1}{34\!\cdots\!04}a^{15}+\frac{11\!\cdots\!23}{56\!\cdots\!84}a^{14}-\frac{17\!\cdots\!81}{23\!\cdots\!98}a^{13}-\frac{50\!\cdots\!33}{34\!\cdots\!04}a^{12}-\frac{33\!\cdots\!67}{42\!\cdots\!63}a^{11}-\frac{10\!\cdots\!21}{17\!\cdots\!52}a^{10}-\frac{46\!\cdots\!95}{37\!\cdots\!56}a^{9}+\frac{94\!\cdots\!23}{56\!\cdots\!84}a^{8}-\frac{46\!\cdots\!85}{17\!\cdots\!52}a^{7}-\frac{28\!\cdots\!75}{67\!\cdots\!04}a^{6}-\frac{50\!\cdots\!15}{24\!\cdots\!36}a^{5}+\frac{75\!\cdots\!77}{17\!\cdots\!52}a^{4}-\frac{13\!\cdots\!83}{34\!\cdots\!04}a^{3}+\frac{68\!\cdots\!59}{17\!\cdots\!52}a^{2}-\frac{11\!\cdots\!28}{27\!\cdots\!71}a-\frac{99\!\cdots\!00}{27\!\cdots\!71}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
$C_{10}$, which has order $10$ (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: |
$\frac{53\!\cdots\!17}{17\!\cdots\!52}a^{15}-\frac{34\!\cdots\!49}{17\!\cdots\!52}a^{14}+\frac{27\!\cdots\!25}{77\!\cdots\!66}a^{13}+\frac{12\!\cdots\!83}{17\!\cdots\!52}a^{12}-\frac{33\!\cdots\!09}{18\!\cdots\!28}a^{11}-\frac{53\!\cdots\!93}{85\!\cdots\!26}a^{10}+\frac{38\!\cdots\!35}{17\!\cdots\!52}a^{9}-\frac{31\!\cdots\!09}{17\!\cdots\!52}a^{8}+\frac{16\!\cdots\!28}{42\!\cdots\!63}a^{7}-\frac{26\!\cdots\!99}{17\!\cdots\!52}a^{6}+\frac{22\!\cdots\!41}{17\!\cdots\!52}a^{5}+\frac{22\!\cdots\!78}{14\!\cdots\!21}a^{4}+\frac{32\!\cdots\!95}{17\!\cdots\!52}a^{3}-\frac{12\!\cdots\!99}{56\!\cdots\!84}a^{2}-\frac{22\!\cdots\!91}{55\!\cdots\!42}a+\frac{63\!\cdots\!93}{93\!\cdots\!57}$, $\frac{48\!\cdots\!55}{56\!\cdots\!52}a^{15}-\frac{20\!\cdots\!35}{39\!\cdots\!64}a^{14}+\frac{25\!\cdots\!46}{26\!\cdots\!93}a^{13}+\frac{33\!\cdots\!83}{23\!\cdots\!92}a^{12}-\frac{18\!\cdots\!71}{39\!\cdots\!64}a^{11}-\frac{10\!\cdots\!70}{99\!\cdots\!41}a^{10}+\frac{23\!\cdots\!83}{39\!\cdots\!64}a^{9}-\frac{34\!\cdots\!07}{39\!\cdots\!64}a^{8}+\frac{82\!\cdots\!65}{73\!\cdots\!66}a^{7}-\frac{95\!\cdots\!03}{39\!\cdots\!64}a^{6}+\frac{42\!\cdots\!57}{13\!\cdots\!88}a^{5}+\frac{23\!\cdots\!91}{19\!\cdots\!82}a^{4}-\frac{17\!\cdots\!83}{56\!\cdots\!52}a^{3}-\frac{37\!\cdots\!87}{39\!\cdots\!64}a^{2}+\frac{46\!\cdots\!87}{64\!\cdots\!97}a+\frac{86\!\cdots\!29}{64\!\cdots\!97}$, $\frac{30\!\cdots\!68}{14\!\cdots\!21}a^{15}-\frac{14\!\cdots\!15}{17\!\cdots\!52}a^{14}+\frac{13\!\cdots\!17}{33\!\cdots\!14}a^{13}+\frac{26\!\cdots\!58}{47\!\cdots\!07}a^{12}-\frac{57\!\cdots\!05}{17\!\cdots\!52}a^{11}-\frac{14\!\cdots\!33}{47\!\cdots\!07}a^{10}+\frac{67\!\cdots\!83}{85\!\cdots\!26}a^{9}-\frac{87\!\cdots\!15}{17\!\cdots\!52}a^{8}+\frac{98\!\cdots\!97}{85\!\cdots\!26}a^{7}-\frac{10\!\cdots\!71}{12\!\cdots\!18}a^{6}-\frac{62\!\cdots\!77}{17\!\cdots\!52}a^{5}+\frac{11\!\cdots\!69}{85\!\cdots\!26}a^{4}-\frac{43\!\cdots\!59}{28\!\cdots\!42}a^{3}+\frac{60\!\cdots\!51}{24\!\cdots\!36}a^{2}+\frac{34\!\cdots\!31}{55\!\cdots\!42}a+\frac{31\!\cdots\!53}{27\!\cdots\!71}$, $\frac{70\!\cdots\!85}{56\!\cdots\!84}a^{15}+\frac{16\!\cdots\!08}{15\!\cdots\!69}a^{14}-\frac{29\!\cdots\!35}{77\!\cdots\!66}a^{13}-\frac{19\!\cdots\!69}{56\!\cdots\!84}a^{12}+\frac{13\!\cdots\!63}{28\!\cdots\!42}a^{11}+\frac{29\!\cdots\!05}{14\!\cdots\!21}a^{10}-\frac{25\!\cdots\!01}{11\!\cdots\!84}a^{9}+\frac{80\!\cdots\!36}{47\!\cdots\!07}a^{8}+\frac{10\!\cdots\!36}{14\!\cdots\!21}a^{7}+\frac{29\!\cdots\!27}{63\!\cdots\!76}a^{6}-\frac{33\!\cdots\!53}{20\!\cdots\!03}a^{5}-\frac{68\!\cdots\!96}{14\!\cdots\!21}a^{4}+\frac{47\!\cdots\!89}{56\!\cdots\!84}a^{3}+\frac{58\!\cdots\!20}{14\!\cdots\!21}a^{2}+\frac{57\!\cdots\!21}{55\!\cdots\!42}a+\frac{18\!\cdots\!16}{93\!\cdots\!57}$, $\frac{11\!\cdots\!39}{67\!\cdots\!04}a^{15}-\frac{62\!\cdots\!93}{16\!\cdots\!26}a^{14}-\frac{56\!\cdots\!55}{50\!\cdots\!22}a^{13}+\frac{40\!\cdots\!77}{67\!\cdots\!04}a^{12}+\frac{46\!\cdots\!15}{11\!\cdots\!84}a^{11}-\frac{95\!\cdots\!81}{33\!\cdots\!52}a^{10}+\frac{11\!\cdots\!47}{67\!\cdots\!04}a^{9}+\frac{22\!\cdots\!09}{23\!\cdots\!18}a^{8}-\frac{11\!\cdots\!95}{33\!\cdots\!52}a^{7}+\frac{18\!\cdots\!61}{67\!\cdots\!04}a^{6}-\frac{16\!\cdots\!55}{16\!\cdots\!26}a^{5}+\frac{27\!\cdots\!91}{15\!\cdots\!12}a^{4}-\frac{58\!\cdots\!85}{67\!\cdots\!04}a^{3}+\frac{16\!\cdots\!90}{93\!\cdots\!57}a^{2}+\frac{37\!\cdots\!69}{13\!\cdots\!51}a+\frac{11\!\cdots\!04}{93\!\cdots\!57}$, $\frac{53\!\cdots\!79}{56\!\cdots\!52}a^{15}-\frac{13\!\cdots\!11}{39\!\cdots\!64}a^{14}+\frac{25\!\cdots\!06}{26\!\cdots\!93}a^{13}+\frac{76\!\cdots\!59}{39\!\cdots\!64}a^{12}+\frac{32\!\cdots\!93}{39\!\cdots\!64}a^{11}-\frac{11\!\cdots\!79}{99\!\cdots\!41}a^{10}+\frac{97\!\cdots\!71}{39\!\cdots\!64}a^{9}-\frac{92\!\cdots\!15}{39\!\cdots\!64}a^{8}+\frac{67\!\cdots\!09}{73\!\cdots\!66}a^{7}-\frac{28\!\cdots\!79}{39\!\cdots\!64}a^{6}+\frac{35\!\cdots\!85}{13\!\cdots\!88}a^{5}+\frac{24\!\cdots\!11}{19\!\cdots\!82}a^{4}-\frac{60\!\cdots\!71}{56\!\cdots\!52}a^{3}+\frac{29\!\cdots\!01}{39\!\cdots\!64}a^{2}+\frac{16\!\cdots\!29}{21\!\cdots\!99}a+\frac{69\!\cdots\!90}{64\!\cdots\!97}$, $\frac{29\!\cdots\!05}{13\!\cdots\!02}a^{15}-\frac{20\!\cdots\!09}{24\!\cdots\!36}a^{14}+\frac{22\!\cdots\!53}{23\!\cdots\!98}a^{13}+\frac{49\!\cdots\!01}{16\!\cdots\!26}a^{12}+\frac{25\!\cdots\!19}{17\!\cdots\!52}a^{11}-\frac{33\!\cdots\!72}{14\!\cdots\!21}a^{10}+\frac{30\!\cdots\!10}{42\!\cdots\!63}a^{9}-\frac{18\!\cdots\!91}{17\!\cdots\!52}a^{8}+\frac{31\!\cdots\!61}{85\!\cdots\!26}a^{7}-\frac{22\!\cdots\!55}{42\!\cdots\!63}a^{6}+\frac{12\!\cdots\!67}{17\!\cdots\!52}a^{5}-\frac{56\!\cdots\!27}{85\!\cdots\!26}a^{4}+\frac{58\!\cdots\!70}{47\!\cdots\!07}a^{3}+\frac{61\!\cdots\!57}{17\!\cdots\!52}a^{2}+\frac{41\!\cdots\!23}{18\!\cdots\!14}a+\frac{31\!\cdots\!12}{27\!\cdots\!71}$
| 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: | \( 25496.198852 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 25496.198852 \cdot 10}{2\cdot\sqrt{166101760110563345913601}}\cr\approx \mathstrut & 0.75979647019 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 15*x^13 + 6*x^12 - 118*x^11 + 317*x^10 - 460*x^9 + 1614*x^8 - 2361*x^7 + 2820*x^6 - 2374*x^5 + 4393*x^4 + 3912*x^3 + 10208*x^2 + 8568*x + 2448)
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 - 4*x^15 + 4*x^14 + 15*x^13 + 6*x^12 - 118*x^11 + 317*x^10 - 460*x^9 + 1614*x^8 - 2361*x^7 + 2820*x^6 - 2374*x^5 + 4393*x^4 + 3912*x^3 + 10208*x^2 + 8568*x + 2448, 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 - 4*x^15 + 4*x^14 + 15*x^13 + 6*x^12 - 118*x^11 + 317*x^10 - 460*x^9 + 1614*x^8 - 2361*x^7 + 2820*x^6 - 2374*x^5 + 4393*x^4 + 3912*x^3 + 10208*x^2 + 8568*x + 2448);
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 - 4*x^15 + 4*x^14 + 15*x^13 + 6*x^12 - 118*x^11 + 317*x^10 - 460*x^9 + 1614*x^8 - 2361*x^7 + 2820*x^6 - 2374*x^5 + 4393*x^4 + 3912*x^3 + 10208*x^2 + 8568*x + 2448);
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
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 16 |
| The 7 conjugacy class representatives for $D_{8}$ |
| Character table for $D_{8}$ |
Intermediate fields
| \(\Q(\sqrt{-799}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{17}, \sqrt{-47})\), 4.2.13583.1 x2, 4.0.37553.1 x2, 8.0.407555836801.2, 8.2.8671400783.1 x4, 8.0.23973872753.1 x4 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.2.0.1}{2} }^{8}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 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 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 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 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(47\)
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |