Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 7*x^16 + 21*x^15 - 49*x^14 - 5*x^13 + 87*x^12 - 97*x^11 + 63*x^10 + 116*x^9 - 329*x^8 + 93*x^7 + 293*x^6 - 176*x^5 - 84*x^4 + 59*x^3 + 12*x^2 - 6*x - 1)
gp: K = bnfinit(y^18 - 6*y^17 + 7*y^16 + 21*y^15 - 49*y^14 - 5*y^13 + 87*y^12 - 97*y^11 + 63*y^10 + 116*y^9 - 329*y^8 + 93*y^7 + 293*y^6 - 176*y^5 - 84*y^4 + 59*y^3 + 12*y^2 - 6*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 7*x^16 + 21*x^15 - 49*x^14 - 5*x^13 + 87*x^12 - 97*x^11 + 63*x^10 + 116*x^9 - 329*x^8 + 93*x^7 + 293*x^6 - 176*x^5 - 84*x^4 + 59*x^3 + 12*x^2 - 6*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 7*x^16 + 21*x^15 - 49*x^14 - 5*x^13 + 87*x^12 - 97*x^11 + 63*x^10 + 116*x^9 - 329*x^8 + 93*x^7 + 293*x^6 - 176*x^5 - 84*x^4 + 59*x^3 + 12*x^2 - 6*x - 1)
\( x^{18} - 6 x^{17} + 7 x^{16} + 21 x^{15} - 49 x^{14} - 5 x^{13} + 87 x^{12} - 97 x^{11} + 63 x^{10} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[10, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(10672332302001983204197\)
\(\medspace = 19^{16}\cdot 37\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.74\) | 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: | $19^{8/9}37^{1/2}\approx 83.32411881951172$ | ||
| Ramified primes: |
\(19\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{379}a^{16}-\frac{133}{379}a^{15}-\frac{115}{379}a^{14}-\frac{56}{379}a^{13}-\frac{41}{379}a^{12}-\frac{182}{379}a^{11}-\frac{124}{379}a^{10}+\frac{48}{379}a^{9}+\frac{129}{379}a^{8}+\frac{151}{379}a^{7}-\frac{65}{379}a^{6}-\frac{91}{379}a^{5}+\frac{24}{379}a^{4}+\frac{155}{379}a^{3}+\frac{189}{379}a^{2}-\frac{9}{379}$, $\frac{1}{143641}a^{17}+\frac{120}{143641}a^{16}+\frac{15127}{143641}a^{15}+\frac{38690}{143641}a^{14}-\frac{8903}{143641}a^{13}+\frac{27345}{143641}a^{12}-\frac{1827}{143641}a^{11}+\frac{56983}{143641}a^{10}-\frac{2129}{143641}a^{9}+\frac{19144}{143641}a^{8}-\frac{30082}{143641}a^{7}-\frac{55573}{143641}a^{6}+\frac{36504}{143641}a^{5}+\frac{2816}{143641}a^{4}+\frac{67450}{143641}a^{3}+\frac{23940}{143641}a^{2}-\frac{9}{143641}a-\frac{1140}{143641}$
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: | $13$ | 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: |
$\frac{9320092}{143641}a^{17}-\frac{58028079}{143641}a^{16}+\frac{79803389}{143641}a^{15}+\frac{169281372}{143641}a^{14}-\frac{485974095}{143641}a^{13}+\frac{93008681}{143641}a^{12}+\frac{726577292}{143641}a^{11}-\frac{1079015977}{143641}a^{10}+\frac{939007143}{143641}a^{9}+\frac{744756498}{143641}a^{8}-\frac{3138630634}{143641}a^{7}+\frac{1729673775}{143641}a^{6}+\frac{1915810364}{143641}a^{5}-\frac{1974077479}{143641}a^{4}-\frac{2312378}{143641}a^{3}+\frac{377692496}{143641}a^{2}-\frac{41219451}{143641}a-\frac{15119377}{143641}$, $\frac{8255512}{143641}a^{17}-\frac{52696377}{143641}a^{16}+\frac{78263247}{143641}a^{15}+\frac{141752183}{143641}a^{14}-\frac{457181167}{143641}a^{13}+\frac{139768815}{143641}a^{12}+\frac{652782794}{143641}a^{11}-\frac{1053540780}{143641}a^{10}+\frac{944201285}{143641}a^{9}+\frac{572661265}{143641}a^{8}-\frac{2917444541}{143641}a^{7}+\frac{1916238212}{143641}a^{6}+\frac{1604584211}{143641}a^{5}-\frac{2052337821}{143641}a^{4}+\frac{155943627}{143641}a^{3}+\frac{397226593}{143641}a^{2}-\frac{65681148}{143641}a-\frac{19052353}{143641}$, $\frac{1369339}{143641}a^{17}-\frac{5848804}{143641}a^{16}-\frac{3890452}{143641}a^{15}+\frac{41681667}{143641}a^{14}-\frac{16297716}{143641}a^{13}-\frac{104169740}{143641}a^{12}+\frac{87613109}{143641}a^{11}+\frac{41901941}{143641}a^{10}-\frac{93485053}{143641}a^{9}+\frac{289365038}{143641}a^{8}-\frac{178007380}{143641}a^{7}-\frac{534694173}{143641}a^{6}+\frac{470486734}{143641}a^{5}+\frac{329749011}{143641}a^{4}-\frac{323636582}{143641}a^{3}-\frac{69246334}{143641}a^{2}+\frac{52601681}{143641}a+\frac{8687402}{143641}$, $\frac{3951877}{143641}a^{17}-\frac{23823966}{143641}a^{16}+\frac{29332011}{143641}a^{15}+\frac{76409324}{143641}a^{14}-\frac{189757945}{143641}a^{13}+\frac{6125253}{143641}a^{12}+\frac{300691359}{143641}a^{11}-\frac{399878943}{143641}a^{10}+\frac{333856868}{143641}a^{9}+\frac{364989326}{143641}a^{8}-\frac{1245596905}{143641}a^{7}+\frac{508976251}{143641}a^{6}+\frac{855045956}{143641}a^{5}-\frac{656204419}{143641}a^{4}-\frac{85317976}{143641}a^{3}+\frac{121267612}{143641}a^{2}-\frac{2242181}{143641}a-\frac{3161750}{143641}$, $\frac{1238620}{143641}a^{17}-\frac{10383388}{143641}a^{16}+\frac{26167608}{143641}a^{15}+\frac{5834725}{143641}a^{14}-\frac{119651269}{143641}a^{13}+\frac{129519164}{143641}a^{12}+\frac{116328597}{143641}a^{11}-\frac{343147417}{143641}a^{10}+\frac{354530761}{143641}a^{9}-\frac{79040992}{143641}a^{8}-\frac{700597743}{143641}a^{7}+\frac{1014660593}{143641}a^{6}+\frac{70597163}{143641}a^{5}-\frac{881490842}{143641}a^{4}+\frac{318494727}{143641}a^{3}+\frac{176499534}{143641}a^{2}-\frac{63432904}{143641}a-\frac{13185933}{143641}$, $\frac{583546}{143641}a^{17}-\frac{3920033}{143641}a^{16}+\frac{6684016}{143641}a^{15}+\frac{8712215}{143641}a^{14}-\frac{36231662}{143641}a^{13}+\frac{18740737}{143641}a^{12}+\frac{46870000}{143641}a^{11}-\frac{89103660}{143641}a^{10}+\frac{84712331}{143641}a^{9}+\frac{26090984}{143641}a^{8}-\frac{225800464}{143641}a^{7}+\frac{194144224}{143641}a^{6}+\frac{95219545}{143641}a^{5}-\frac{188740495}{143641}a^{4}+\frac{38008625}{143641}a^{3}+\frac{35463049}{143641}a^{2}-\frac{9561144}{143641}a-\frac{2030719}{143641}$, $\frac{414088}{143641}a^{17}-\frac{6697818}{143641}a^{16}+\frac{27605543}{143641}a^{15}-\frac{18473379}{143641}a^{14}-\frac{106428115}{143641}a^{13}+\frac{186196669}{143641}a^{12}+\frac{61319347}{143641}a^{11}-\frac{357738023}{143641}a^{10}+\frac{399238664}{143641}a^{9}-\frac{244412699}{143641}a^{8}-\frac{575367422}{143641}a^{7}+\frac{1295923897}{143641}a^{6}-\frac{209853188}{143641}a^{5}-\frac{1046940649}{143641}a^{4}+\frac{502199269}{143641}a^{3}+\frac{210214082}{143641}a^{2}-\frac{93215135}{143641}a-\frac{17568447}{143641}$, $\frac{8188872}{143641}a^{17}-\frac{50319947}{143641}a^{16}+\frac{66238876}{143641}a^{15}+\frac{152883945}{143641}a^{14}-\frac{413185107}{143641}a^{13}+\frac{52412584}{143641}a^{12}+\frac{633137133}{143641}a^{11}-\frac{897719304}{143641}a^{10}+\frac{768112005}{143641}a^{9}+\frac{697966533}{143641}a^{8}-\frac{2686219222}{143641}a^{7}+\frac{1323024244}{143641}a^{6}+\frac{1727375763}{143641}a^{5}-\frac{1576634256}{143641}a^{4}-\frac{80897273}{143641}a^{3}+\frac{297870469}{143641}a^{2}-\frac{21558165}{143641}a-\frac{10155520}{143641}$, $\frac{7367793}{143641}a^{17}-\frac{45761806}{143641}a^{16}+\frac{62415663}{143641}a^{15}+\frac{134641185}{143641}a^{14}-\frac{381957790}{143641}a^{13}+\frac{68089231}{143641}a^{12}+\frac{574322168}{143641}a^{11}-\frac{844042634}{143641}a^{10}+\frac{731223129}{143641}a^{9}+\frac{597479057}{143641}a^{8}-\frac{2470273593}{143641}a^{7}+\frac{1331330427}{143641}a^{6}+\frac{1527829139}{143641}a^{5}-\frac{1533480923}{143641}a^{4}-\frac{20022587}{143641}a^{3}+\frac{294008676}{143641}a^{2}-\frac{29250759}{143641}a-\frac{11537996}{143641}$, $\frac{3050601}{143641}a^{17}-\frac{18805791}{143641}a^{16}+\frac{25033476}{143641}a^{15}+\frac{56545107}{143641}a^{14}-\frac{155161737}{143641}a^{13}+\frac{22324810}{143641}a^{12}+\frac{236209242}{143641}a^{11}-\frac{339215412}{143641}a^{10}+\frac{291549721}{143641}a^{9}+\frac{255951721}{143641}a^{8}-\frac{1007023906}{143641}a^{7}+\frac{511426800}{143641}a^{6}+\frac{638693642}{143641}a^{5}-\frac{602766943}{143641}a^{4}-\frac{22044516}{143641}a^{3}+\frac{114938946}{143641}a^{2}-\frac{9931207}{143641}a-\frac{4159994}{143641}$, $\frac{6172062}{143641}a^{17}-\frac{39145060}{143641}a^{16}+\frac{57034159}{143641}a^{15}+\frac{107592024}{143641}a^{14}-\frac{336618350}{143641}a^{13}+\frac{93265862}{143641}a^{12}+\frac{486303781}{143641}a^{11}-\frac{768528054}{143641}a^{10}+\frac{684052545}{143641}a^{9}+\frac{445013484}{143641}a^{8}-\frac{2154776259}{143641}a^{7}+\frac{1357806295}{143641}a^{6}+\frac{1217687160}{143641}a^{5}-\frac{1474399742}{143641}a^{4}+\frac{86280302}{143641}a^{3}+\frac{283435079}{143641}a^{2}-\frac{43339073}{143641}a-\frac{12990745}{143641}$, $\frac{9606644}{143641}a^{17}-\frac{57446744}{143641}a^{16}+\frac{68418821}{143641}a^{15}+\frac{189508740}{143641}a^{14}-\frac{452153174}{143641}a^{13}-\frac{9250366}{143641}a^{12}+\frac{732425271}{143641}a^{11}-\frac{933085504}{143641}a^{10}+\frac{761805205}{143641}a^{9}+\frac{926959358}{143641}a^{8}-\frac{2984491492}{143641}a^{7}+\frac{1078940380}{143641}a^{6}+\frac{2146166892}{143641}a^{5}-\frac{1478019523}{143641}a^{4}-\frac{293897122}{143641}a^{3}+\frac{274918177}{143641}a^{2}+\frac{11072443}{143641}a-\frac{5368928}{143641}$, $\frac{5171407}{143641}a^{17}-\frac{35219408}{143641}a^{16}+\frac{61905170}{143641}a^{15}+\frac{74960874}{143641}a^{14}-\frac{331862801}{143641}a^{13}+\frac{184601721}{143641}a^{12}+\frac{424859526}{143641}a^{11}-\frac{825042388}{143641}a^{10}+\frac{782078869}{143641}a^{9}+\frac{210415278}{143641}a^{8}-\frac{2061225285}{143641}a^{7}+\frac{1850138630}{143641}a^{6}+\frac{850443331}{143641}a^{5}-\frac{1795903114}{143641}a^{4}+\frac{362986689}{143641}a^{3}+\frac{351044210}{143641}a^{2}-\frac{89060399}{143641}a-\frac{20601538}{143641}$
| 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: | \( 18497.6611092 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{4}\cdot 18497.6611092 \cdot 1}{2\cdot\sqrt{10672332302001983204197}}\cr\approx \mathstrut & 0.142881691627 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 7*x^16 + 21*x^15 - 49*x^14 - 5*x^13 + 87*x^12 - 97*x^11 + 63*x^10 + 116*x^9 - 329*x^8 + 93*x^7 + 293*x^6 - 176*x^5 - 84*x^4 + 59*x^3 + 12*x^2 - 6*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^18 - 6*x^17 + 7*x^16 + 21*x^15 - 49*x^14 - 5*x^13 + 87*x^12 - 97*x^11 + 63*x^10 + 116*x^9 - 329*x^8 + 93*x^7 + 293*x^6 - 176*x^5 - 84*x^4 + 59*x^3 + 12*x^2 - 6*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^18 - 6*x^17 + 7*x^16 + 21*x^15 - 49*x^14 - 5*x^13 + 87*x^12 - 97*x^11 + 63*x^10 + 116*x^9 - 329*x^8 + 93*x^7 + 293*x^6 - 176*x^5 - 84*x^4 + 59*x^3 + 12*x^2 - 6*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^18 - 6*x^17 + 7*x^16 + 21*x^15 - 49*x^14 - 5*x^13 + 87*x^12 - 97*x^11 + 63*x^10 + 116*x^9 - 329*x^8 + 93*x^7 + 293*x^6 - 176*x^5 - 84*x^4 + 59*x^3 + 12*x^2 - 6*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
$C_2\wr C_9$ (as 18T460):
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 4608 |
| The 80 conjugacy class representatives for $C_2\wr C_9$ |
| Character table for $C_2\wr C_9$ |
Intermediate fields
| 3.3.361.1, \(\Q(\zeta_{19})^+\) |
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
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| 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 | $18$ | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | $18$ | $18$ | R | $18$ | $18$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| 19.18.16.1 | $x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$ | $9$ | $2$ | $16$ | $C_{18}$ | $[\ ]_{9}^{2}$ |
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |