Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^18 - 2*x^17 + 2*x^16 - 2*x^15 + 8*x^14 - 11*x^13 + 2*x^12 + 2*x^11 - 9*x^10 + 17*x^9 - x^8 - 28*x^7 + 49*x^6 - 46*x^5 + 31*x^4 - 20*x^3 + 12*x^2 - 5*x + 1)
gp: K = bnfinit(y^20 - y^19 - y^18 - 2*y^17 + 2*y^16 - 2*y^15 + 8*y^14 - 11*y^13 + 2*y^12 + 2*y^11 - 9*y^10 + 17*y^9 - y^8 - 28*y^7 + 49*y^6 - 46*y^5 + 31*y^4 - 20*y^3 + 12*y^2 - 5*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 - x^18 - 2*x^17 + 2*x^16 - 2*x^15 + 8*x^14 - 11*x^13 + 2*x^12 + 2*x^11 - 9*x^10 + 17*x^9 - x^8 - 28*x^7 + 49*x^6 - 46*x^5 + 31*x^4 - 20*x^3 + 12*x^2 - 5*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 - x^18 - 2*x^17 + 2*x^16 - 2*x^15 + 8*x^14 - 11*x^13 + 2*x^12 + 2*x^11 - 9*x^10 + 17*x^9 - x^8 - 28*x^7 + 49*x^6 - 46*x^5 + 31*x^4 - 20*x^3 + 12*x^2 - 5*x + 1)
\( x^{20} - x^{19} - x^{18} - 2 x^{17} + 2 x^{16} - 2 x^{15} + 8 x^{14} - 11 x^{13} + 2 x^{12} + 2 x^{11} + \cdots + 1 \)
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: | $[2, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-53112053233392982274207\)
\(\medspace = -\,47^{9}\cdot 83^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.69\) | 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: | $47^{1/2}83^{1/2}\approx 62.457985878508765$ | ||
| Ramified primes: |
\(47\), \(83\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-47}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{371964527509}a^{19}+\frac{146661806959}{371964527509}a^{18}-\frac{106968022877}{371964527509}a^{17}+\frac{171278320129}{371964527509}a^{16}-\frac{34295505397}{371964527509}a^{15}-\frac{58447289071}{371964527509}a^{14}+\frac{163135832866}{371964527509}a^{13}-\frac{181980568727}{371964527509}a^{12}-\frac{67721582707}{371964527509}a^{11}-\frac{107736196998}{371964527509}a^{10}+\frac{44099949217}{371964527509}a^{9}-\frac{179160330562}{371964527509}a^{8}+\frac{133210778749}{371964527509}a^{7}+\frac{46545565534}{371964527509}a^{6}-\frac{66675617602}{371964527509}a^{5}+\frac{173054946221}{371964527509}a^{4}+\frac{49326225560}{371964527509}a^{3}+\frac{55135156143}{371964527509}a^{2}+\frac{159444404372}{371964527509}a-\frac{15620399623}{371964527509}$
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: | $10$ | 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: |
$\frac{1004722217836}{371964527509}a^{19}-\frac{210267268642}{371964527509}a^{18}-\frac{1250946892029}{371964527509}a^{17}-\frac{3048472500443}{371964527509}a^{16}-\frac{287510024892}{371964527509}a^{15}-\frac{1912143376936}{371964527509}a^{14}+\frac{6748907223187}{371964527509}a^{13}-\frac{5549275768303}{371964527509}a^{12}-\frac{2796085253265}{371964527509}a^{11}-\frac{177295858927}{371964527509}a^{10}-\frac{8565494245948}{371964527509}a^{9}+\frac{10515129925040}{371964527509}a^{8}+\frac{7989212540823}{371964527509}a^{7}-\frac{22003210359523}{371964527509}a^{6}+\frac{30440723272178}{371964527509}a^{5}-\frac{20738002836279}{371964527509}a^{4}+\frac{13750765356152}{371964527509}a^{3}-\frac{9597803874042}{371964527509}a^{2}+\frac{4915564640607}{371964527509}a-\frac{1483983344804}{371964527509}$, $\frac{302926176989}{371964527509}a^{19}-\frac{206327497294}{371964527509}a^{18}-\frac{309825065345}{371964527509}a^{17}-\frac{650285531677}{371964527509}a^{16}+\frac{370935937667}{371964527509}a^{15}-\frac{736595344757}{371964527509}a^{14}+\frac{1886959562755}{371964527509}a^{13}-\frac{3069070175086}{371964527509}a^{12}-\frac{184834490770}{371964527509}a^{11}+\frac{554731497311}{371964527509}a^{10}-\frac{2736196824765}{371964527509}a^{9}+\frac{3904676475953}{371964527509}a^{8}+\frac{36772939974}{371964527509}a^{7}-\frac{8560862687130}{371964527509}a^{6}+\frac{12795732591445}{371964527509}a^{5}-\frac{10268794898977}{371964527509}a^{4}+\frac{7014189873413}{371964527509}a^{3}-\frac{4236282451126}{371964527509}a^{2}+\frac{2658520195102}{371964527509}a-\frac{673001089531}{371964527509}$, $a$, $\frac{510256898123}{371964527509}a^{19}-\frac{284757036718}{371964527509}a^{18}-\frac{694060066450}{371964527509}a^{17}-\frac{1310656535012}{371964527509}a^{16}+\frac{482119688392}{371964527509}a^{15}-\frac{657802475226}{371964527509}a^{14}+\frac{3836910007931}{371964527509}a^{13}-\frac{3670669783865}{371964527509}a^{12}-\frac{894986165605}{371964527509}a^{11}+\frac{1023732587438}{371964527509}a^{10}-\frac{4225069116847}{371964527509}a^{9}+\frac{6752056356980}{371964527509}a^{8}+\frac{3184298825435}{371964527509}a^{7}-\frac{13332574800645}{371964527509}a^{6}+\frac{19020884078175}{371964527509}a^{5}-\frac{13877636009859}{371964527509}a^{4}+\frac{7082549463095}{371964527509}a^{3}-\frac{5430433552332}{371964527509}a^{2}+\frac{2551986804705}{371964527509}a-\frac{660955422003}{371964527509}$, $\frac{395032092609}{371964527509}a^{19}-\frac{297349247966}{371964527509}a^{18}-\frac{418406971638}{371964527509}a^{17}-\frac{808460188702}{371964527509}a^{16}+\frac{517820853171}{371964527509}a^{15}-\frac{927541390417}{371964527509}a^{14}+\frac{2632944771143}{371964527509}a^{13}-\frac{3860973181303}{371964527509}a^{12}-\frac{57309087120}{371964527509}a^{11}+\frac{1061791359598}{371964527509}a^{10}-\frac{3905827260874}{371964527509}a^{9}+\frac{5407255981674}{371964527509}a^{8}+\frac{528540440870}{371964527509}a^{7}-\frac{11432305149362}{371964527509}a^{6}+\frac{17797137009398}{371964527509}a^{5}-\frac{14063085819976}{371964527509}a^{4}+\frac{8263272120089}{371964527509}a^{3}-\frac{3980779272870}{371964527509}a^{2}+\frac{2524182401200}{371964527509}a-\frac{924444330686}{371964527509}$, $\frac{360102271408}{371964527509}a^{19}+\frac{165089406414}{371964527509}a^{18}-\frac{371519135075}{371964527509}a^{17}-\frac{1420733252805}{371964527509}a^{16}-\frac{1038304368448}{371964527509}a^{15}-\frac{1159428896398}{371964527509}a^{14}+\frac{1992437235525}{371964527509}a^{13}-\frac{425939545292}{371964527509}a^{12}-\frac{1340030199645}{371964527509}a^{11}-\frac{1343259077244}{371964527509}a^{10}-\frac{3758049303663}{371964527509}a^{9}+\frac{1740717720122}{371964527509}a^{8}+\frac{4597384083605}{371964527509}a^{7}-\frac{4475252386673}{371964527509}a^{6}+\frac{7074013290176}{371964527509}a^{5}-\frac{3254598719976}{371964527509}a^{4}+\frac{3104360457795}{371964527509}a^{3}-\frac{1975192009850}{371964527509}a^{2}+\frac{336529475538}{371964527509}a-\frac{141073352127}{371964527509}$, $\frac{161222096161}{371964527509}a^{19}+\frac{63763497032}{371964527509}a^{18}-\frac{192137330550}{371964527509}a^{17}-\frac{667714612825}{371964527509}a^{16}-\frac{448113396446}{371964527509}a^{15}-\frac{354174450588}{371964527509}a^{14}+\frac{1192378511856}{371964527509}a^{13}+\frac{17107934794}{371964527509}a^{12}-\frac{688304619108}{371964527509}a^{11}-\frac{846585634579}{371964527509}a^{10}-\frac{1881293920305}{371964527509}a^{9}+\frac{1241483906330}{371964527509}a^{8}+\frac{2617893136967}{371964527509}a^{7}-\frac{1912685794503}{371964527509}a^{6}+\frac{2809433518242}{371964527509}a^{5}-\frac{2487912329934}{371964527509}a^{4}+\frac{962473388316}{371964527509}a^{3}-\frac{242770947768}{371964527509}a^{2}-\frac{131558772082}{371964527509}a-\frac{52056722945}{371964527509}$, $\frac{392410708545}{371964527509}a^{19}-\frac{52640277625}{371964527509}a^{18}-\frac{527045998255}{371964527509}a^{17}-\frac{1243113358777}{371964527509}a^{16}-\frac{134123806779}{371964527509}a^{15}-\frac{608276817962}{371964527509}a^{14}+\frac{2630663574881}{371964527509}a^{13}-\frac{2006774898987}{371964527509}a^{12}-\frac{1520655792698}{371964527509}a^{11}-\frac{204526771999}{371964527509}a^{10}-\frac{3030676070065}{371964527509}a^{9}+\frac{3934971082053}{371964527509}a^{8}+\frac{3567544318325}{371964527509}a^{7}-\frac{8574446638591}{371964527509}a^{6}+\frac{10752854441507}{371964527509}a^{5}-\frac{6576645597341}{371964527509}a^{4}+\frac{4859882212940}{371964527509}a^{3}-\frac{3648155775848}{371964527509}a^{2}+\frac{1629896019662}{371964527509}a-\frac{212128284609}{371964527509}$, $\frac{516392125141}{371964527509}a^{19}-\frac{411273741053}{371964527509}a^{18}-\frac{617200037968}{371964527509}a^{17}-\frac{1126199026461}{371964527509}a^{16}+\frac{779888696943}{371964527509}a^{15}-\frac{891659804369}{371964527509}a^{14}+\frac{3904409398098}{371964527509}a^{13}-\frac{4698429033285}{371964527509}a^{12}+\frac{53570394522}{371964527509}a^{11}+\frac{1461815837946}{371964527509}a^{10}-\frac{4592652560069}{371964527509}a^{9}+\frac{7833217925997}{371964527509}a^{8}+\frac{1535960083970}{371964527509}a^{7}-\frac{14260915628244}{371964527509}a^{6}+\frac{23114301314816}{371964527509}a^{5}-\frac{18378540167698}{371964527509}a^{4}+\frac{10674511227248}{371964527509}a^{3}-\frac{6391018070503}{371964527509}a^{2}+\frac{3643418030738}{371964527509}a-\frac{1176024378719}{371964527509}$, $\frac{302879281117}{371964527509}a^{19}-\frac{3095072975}{371964527509}a^{18}-\frac{226586404190}{371964527509}a^{17}-\frac{928586547103}{371964527509}a^{16}-\frac{448096303290}{371964527509}a^{15}-\frac{1195421906662}{371964527509}a^{14}+\frac{1536242801499}{371964527509}a^{13}-\frac{1759514955868}{371964527509}a^{12}-\frac{403603647986}{371964527509}a^{11}-\frac{675196014690}{371964527509}a^{10}-\frac{3391231037421}{371964527509}a^{9}+\frac{2277743857804}{371964527509}a^{8}+\frac{1521269248254}{371964527509}a^{7}-\frac{5495389808114}{371964527509}a^{6}+\frac{9489773806936}{371964527509}a^{5}-\frac{7294700718136}{371964527509}a^{4}+\frac{5975629604833}{371964527509}a^{3}-\frac{3363014624225}{371964527509}a^{2}+\frac{2030923466996}{371964527509}a-\frac{845551685024}{371964527509}$
| 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: | \( 1486.6023737 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 1486.6023737 \cdot 1}{2\cdot\sqrt{53112053233392982274207}}\cr\approx \mathstrut & 0.19690062168 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^18 - 2*x^17 + 2*x^16 - 2*x^15 + 8*x^14 - 11*x^13 + 2*x^12 + 2*x^11 - 9*x^10 + 17*x^9 - x^8 - 28*x^7 + 49*x^6 - 46*x^5 + 31*x^4 - 20*x^3 + 12*x^2 - 5*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 - 2*x^17 + 2*x^16 - 2*x^15 + 8*x^14 - 11*x^13 + 2*x^12 + 2*x^11 - 9*x^10 + 17*x^9 - x^8 - 28*x^7 + 49*x^6 - 46*x^5 + 31*x^4 - 20*x^3 + 12*x^2 - 5*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 - 2*x^17 + 2*x^16 - 2*x^15 + 8*x^14 - 11*x^13 + 2*x^12 + 2*x^11 - 9*x^10 + 17*x^9 - x^8 - 28*x^7 + 49*x^6 - 46*x^5 + 31*x^4 - 20*x^3 + 12*x^2 - 5*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 - 2*x^17 + 2*x^16 - 2*x^15 + 8*x^14 - 11*x^13 + 2*x^12 + 2*x^11 - 9*x^10 + 17*x^9 - x^8 - 28*x^7 + 49*x^6 - 46*x^5 + 31*x^4 - 20*x^3 + 12*x^2 - 5*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
$C_2\wr D_5$ (as 20T87):
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 320 |
| The 20 conjugacy class representatives for $C_2\wr D_5$ |
| Character table for $C_2\wr D_5$ |
Intermediate fields
| 5.1.2209.1, 10.2.33616122409.2 |
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
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.0.405013523.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{5}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{5}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.4.0.1}{4} }^{5}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(47\)
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(83\)
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |