Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + x^18 - 4*x^17 - 3*x^16 + 15*x^15 + 3*x^14 - 28*x^13 + 2*x^12 + 19*x^11 + 29*x^10 - 56*x^9 + 17*x^8 - 3*x^7 + 26*x^6 - 34*x^5 + 37*x^4 - 35*x^3 + 21*x^2 - 7*x + 1)
gp: K = bnfinit(y^20 - y^19 + y^18 - 4*y^17 - 3*y^16 + 15*y^15 + 3*y^14 - 28*y^13 + 2*y^12 + 19*y^11 + 29*y^10 - 56*y^9 + 17*y^8 - 3*y^7 + 26*y^6 - 34*y^5 + 37*y^4 - 35*y^3 + 21*y^2 - 7*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 + x^18 - 4*x^17 - 3*x^16 + 15*x^15 + 3*x^14 - 28*x^13 + 2*x^12 + 19*x^11 + 29*x^10 - 56*x^9 + 17*x^8 - 3*x^7 + 26*x^6 - 34*x^5 + 37*x^4 - 35*x^3 + 21*x^2 - 7*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 + x^18 - 4*x^17 - 3*x^16 + 15*x^15 + 3*x^14 - 28*x^13 + 2*x^12 + 19*x^11 + 29*x^10 - 56*x^9 + 17*x^8 - 3*x^7 + 26*x^6 - 34*x^5 + 37*x^4 - 35*x^3 + 21*x^2 - 7*x + 1)
\( x^{20} - x^{19} + x^{18} - 4 x^{17} - 3 x^{16} + 15 x^{15} + 3 x^{14} - 28 x^{13} + 2 x^{12} + 19 x^{11} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1723568365103679045632\)
\(\medspace = 2^{15}\cdot 47^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.53\) | 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: | $2^{3/2}47^{1/2}\approx 19.390719429665317$ | ||
| Ramified primes: |
\(2\), \(47\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $10$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{47}a^{18}-\frac{16}{47}a^{17}-\frac{15}{47}a^{16}-\frac{7}{47}a^{15}-\frac{6}{47}a^{14}+\frac{17}{47}a^{13}+\frac{15}{47}a^{12}+\frac{1}{47}a^{11}+\frac{1}{47}a^{10}-\frac{17}{47}a^{9}-\frac{19}{47}a^{8}+\frac{22}{47}a^{7}-\frac{8}{47}a^{6}-\frac{16}{47}a^{5}+\frac{11}{47}a^{4}-\frac{4}{47}a^{3}+\frac{7}{47}a^{2}-\frac{9}{47}a+\frac{9}{47}$, $\frac{1}{24657195037}a^{19}+\frac{14423459}{24657195037}a^{18}+\frac{6052526827}{24657195037}a^{17}+\frac{6076674939}{24657195037}a^{16}+\frac{490195653}{24657195037}a^{15}-\frac{3702211118}{24657195037}a^{14}-\frac{3221954579}{24657195037}a^{13}-\frac{1683844526}{24657195037}a^{12}-\frac{5510042910}{24657195037}a^{11}+\frac{6610333898}{24657195037}a^{10}-\frac{215387826}{524621171}a^{9}+\frac{2552398449}{24657195037}a^{8}-\frac{8641400464}{24657195037}a^{7}+\frac{5276938446}{24657195037}a^{6}+\frac{7916454968}{24657195037}a^{5}-\frac{7803064182}{24657195037}a^{4}-\frac{4935834703}{24657195037}a^{3}+\frac{6673552933}{24657195037}a^{2}-\frac{4891694897}{24657195037}a-\frac{6491582894}{24657195037}$
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
Trivial group, which has order $1$
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: | $9$ | 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: |
$\frac{5480998887964}{24657195037}a^{19}-\frac{2705858205680}{24657195037}a^{18}+\frac{4109065027076}{24657195037}a^{17}-\frac{19843517699273}{24657195037}a^{16}-\frac{26491863446166}{24657195037}a^{15}+\frac{68807124820422}{24657195037}a^{14}+\frac{51292712573652}{24657195037}a^{13}-\frac{127516497779395}{24657195037}a^{12}-\frac{53628022468519}{24657195037}a^{11}+\frac{77019068384271}{24657195037}a^{10}+\frac{197977878613986}{24657195037}a^{9}-\frac{206701440792921}{24657195037}a^{8}-\frac{11550650649486}{24657195037}a^{7}-\frac{22268111751795}{24657195037}a^{6}+\frac{131256349241091}{24657195037}a^{5}-\frac{119856650893652}{24657195037}a^{4}+\frac{142071308290184}{24657195037}a^{3}-\frac{119890461913823}{24657195037}a^{2}+\frac{54344702035321}{24657195037}a-\frac{10799261699363}{24657195037}$, $\frac{2583028267356}{24657195037}a^{19}-\frac{1285060972434}{24657195037}a^{18}+\frac{1941863590891}{24657195037}a^{17}-\frac{9357580953053}{24657195037}a^{16}-\frac{12449574917915}{24657195037}a^{15}+\frac{32472166085848}{24657195037}a^{14}+\frac{24039487028713}{24657195037}a^{13}-\frac{60189883168722}{24657195037}a^{12}-\frac{25011563347548}{24657195037}a^{11}+\frac{36410535757503}{24657195037}a^{10}+\frac{93103773144411}{24657195037}a^{9}-\frac{97807299591623}{24657195037}a^{8}-\frac{5009784095596}{24657195037}a^{7}-\frac{10388115733700}{24657195037}a^{6}+\frac{61838894141826}{24657195037}a^{5}-\frac{56793053157113}{24657195037}a^{4}+\frac{67177484198107}{24657195037}a^{3}-\frac{56700382590493}{24657195037}a^{2}+\frac{25819632846469}{24657195037}a-\frac{5174012966636}{24657195037}$, $\frac{5436226585702}{24657195037}a^{19}-\frac{2677155629433}{24657195037}a^{18}+\frac{4081228867994}{24657195037}a^{17}-\frac{19675628703414}{24657195037}a^{16}-\frac{26293148498288}{24657195037}a^{15}+\frac{68184904889870}{24657195037}a^{14}+\frac{50896976895063}{24657195037}a^{13}-\frac{126330380545760}{24657195037}a^{12}-\frac{53202467360930}{24657195037}a^{11}+\frac{76183846204613}{24657195037}a^{10}+\frac{196266236199580}{24657195037}a^{9}-\frac{204733971595442}{24657195037}a^{8}-\frac{11340696967805}{24657195037}a^{7}-\frac{22232533113166}{24657195037}a^{6}+\frac{2766420392906}{524621171}a^{5}-\frac{118824086068186}{24657195037}a^{4}+\frac{140935586048729}{24657195037}a^{3}-\frac{118840635217277}{24657195037}a^{2}+\frac{53932456382740}{24657195037}a-\frac{10757472487966}{24657195037}$, $\frac{2891478538813}{24657195037}a^{19}-\frac{1431792286119}{24657195037}a^{18}+\frac{2166692681531}{24657195037}a^{17}-\frac{10471684009602}{24657195037}a^{16}-\frac{13961336863929}{24657195037}a^{15}+\frac{36332500363225}{24657195037}a^{14}+\frac{27028894982022}{24657195037}a^{13}-\frac{67339901424706}{24657195037}a^{12}-\frac{28245538037761}{24657195037}a^{11}+\frac{40713409229703}{24657195037}a^{10}+\frac{104457666092036}{24657195037}a^{9}-\frac{109194684134552}{24657195037}a^{8}-\frac{6075315320426}{24657195037}a^{7}-\frac{11725614867254}{24657195037}a^{6}+\frac{69304548799725}{24657195037}a^{5}-\frac{63255405449180}{24657195037}a^{4}+\frac{74997506422845}{24657195037}a^{3}-\frac{63360919489344}{24657195037}a^{2}+\frac{28699883408514}{24657195037}a-\frac{5710585904942}{24657195037}$, $\frac{4066295860717}{24657195037}a^{19}-\frac{2016131149983}{24657195037}a^{18}+\frac{3049820185617}{24657195037}a^{17}-\frac{14727012290985}{24657195037}a^{16}-\frac{19624066735353}{24657195037}a^{15}+\frac{51101479050551}{24657195037}a^{14}+\frac{37962757928700}{24657195037}a^{13}-\frac{94718514287515}{24657195037}a^{12}-\frac{39619762805172}{24657195037}a^{11}+\frac{57280375743121}{24657195037}a^{10}+\frac{146795934632763}{24657195037}a^{9}-\frac{153679566070242}{24657195037}a^{8}-\frac{8352033506680}{24657195037}a^{7}-\frac{16431432182804}{24657195037}a^{6}+\frac{97418698299275}{24657195037}a^{5}-\frac{89104488350200}{24657195037}a^{4}+\frac{105546106930088}{24657195037}a^{3}-\frac{89129142439035}{24657195037}a^{2}+\frac{40423000155958}{24657195037}a-\frac{8051872106684}{24657195037}$, $\frac{1493295134129}{24657195037}a^{19}-\frac{734900395412}{24657195037}a^{18}+\frac{1123850845761}{24657195037}a^{17}-\frac{5401173454041}{24657195037}a^{16}-\frac{7220439895540}{24657195037}a^{15}+\frac{18719821541222}{24657195037}a^{14}+\frac{13957268039633}{24657195037}a^{13}-\frac{34697715676024}{24657195037}a^{12}-\frac{14567069894634}{24657195037}a^{11}+\frac{20934787988544}{24657195037}a^{10}+\frac{53843134167435}{24657195037}a^{9}-\frac{56293373014406}{24657195037}a^{8}-\frac{3041510187600}{24657195037}a^{7}-\frac{6026561138656}{24657195037}a^{6}+\frac{35696090048280}{24657195037}a^{5}-\frac{32718077590466}{24657195037}a^{4}+\frac{38701404352874}{24657195037}a^{3}-\frac{32631778279372}{24657195037}a^{2}+\frac{14851066977162}{24657195037}a-\frac{62829391209}{524621171}$, $\frac{4516761286788}{24657195037}a^{19}-\frac{2217128251905}{24657195037}a^{18}+\frac{3383102998920}{24657195037}a^{17}-\frac{16343242652908}{24657195037}a^{16}-\frac{21873120117310}{24657195037}a^{15}+\frac{56631907510053}{24657195037}a^{14}+\frac{42411936554957}{24657195037}a^{13}-\frac{104935298609050}{24657195037}a^{12}-\frac{44458739922892}{24657195037}a^{11}+\frac{63295804832621}{24657195037}a^{10}+\frac{163296706684474}{24657195037}a^{9}-\frac{169870529613723}{24657195037}a^{8}-\frac{9903036374799}{24657195037}a^{7}-\frac{18449706648143}{24657195037}a^{6}+\frac{108123268274122}{24657195037}a^{5}-\frac{98511846110402}{24657195037}a^{4}+\frac{116862389734695}{24657195037}a^{3}-\frac{98518002167963}{24657195037}a^{2}+\frac{44573775790867}{24657195037}a-\frac{8839759954053}{24657195037}$, $a$, $\frac{3859277408905}{24657195037}a^{19}-\frac{1925809512460}{24657195037}a^{18}+\frac{2895671005862}{24657195037}a^{17}-\frac{13985128294338}{24657195037}a^{16}-\frac{18583057361569}{24657195037}a^{15}+\frac{1033528773504}{524621171}a^{14}+\frac{35901438316249}{24657195037}a^{13}-\frac{90070128654302}{24657195037}a^{12}-\frac{37377150910160}{24657195037}a^{11}+\frac{54608577802149}{24657195037}a^{10}+\frac{139240653503031}{24657195037}a^{9}-\frac{146405809422505}{24657195037}a^{8}-\frac{7691561721632}{24657195037}a^{7}-\frac{15359598052206}{24657195037}a^{6}+\frac{92616109379485}{24657195037}a^{5}-\frac{84884784894300}{24657195037}a^{4}+\frac{100266453700579}{24657195037}a^{3}-\frac{84802150286496}{24657195037}a^{2}+\frac{38573598812824}{24657195037}a-\frac{7687157596026}{24657195037}$
| 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: | \( 143.606119616 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 143.606119616 \cdot 1}{2\cdot\sqrt{1723568365103679045632}}\cr\approx \mathstrut & 0.165854545312 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + x^18 - 4*x^17 - 3*x^16 + 15*x^15 + 3*x^14 - 28*x^13 + 2*x^12 + 19*x^11 + 29*x^10 - 56*x^9 + 17*x^8 - 3*x^7 + 26*x^6 - 34*x^5 + 37*x^4 - 35*x^3 + 21*x^2 - 7*x + 1)
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^20 - x^19 + x^18 - 4*x^17 - 3*x^16 + 15*x^15 + 3*x^14 - 28*x^13 + 2*x^12 + 19*x^11 + 29*x^10 - 56*x^9 + 17*x^8 - 3*x^7 + 26*x^6 - 34*x^5 + 37*x^4 - 35*x^3 + 21*x^2 - 7*x + 1, 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^20 - x^19 + x^18 - 4*x^17 - 3*x^16 + 15*x^15 + 3*x^14 - 28*x^13 + 2*x^12 + 19*x^11 + 29*x^10 - 56*x^9 + 17*x^8 - 3*x^7 + 26*x^6 - 34*x^5 + 37*x^4 - 35*x^3 + 21*x^2 - 7*x + 1);
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^20 - x^19 + x^18 - 4*x^17 - 3*x^16 + 15*x^15 + 3*x^14 - 28*x^13 + 2*x^12 + 19*x^11 + 29*x^10 - 56*x^9 + 17*x^8 - 3*x^7 + 26*x^6 - 34*x^5 + 37*x^4 - 35*x^3 + 21*x^2 - 7*x + 1);
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 40 |
| The 13 conjugacy class representatives for $C_5:D_4$ |
| Character table for $C_5:D_4$ |
Intermediate fields
| \(\Q(\sqrt{-47}) \), 4.0.17672.1, 5.1.2209.1 x5, 10.0.229345007.1 |
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 40 |
| Degree 20 sibling: | 20.2.1201657195483347978027008.1 |
| Minimal sibling: | 20.2.1201657195483347978027008.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{5}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{5}$ | ${\href{/padicField/13.4.0.1}{4} }^{5}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{5}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.4.0.1}{4} }^{5}$ | ${\href{/padicField/31.2.0.1}{2} }^{10}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 2.10.15.1 | $x^{10} + 70 x^{8} + 2 x^{7} + 1960 x^{6} - 26 x^{5} + 27441 x^{4} - 2240 x^{3} + 192110 x^{2} - 14504 x + 537993$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ | |
|
\(47\)
| 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 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |