Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 48*x^13 - 32*x^12 + 882*x^11 + 1176*x^10 - 7303*x^9 - 15390*x^8 + 21330*x^7 + 81960*x^6 + 35883*x^5 - 123750*x^4 - 208680*x^3 - 143280*x^2 - 47760*x - 6368)
gp: K = bnfinit(y^15 - 48*y^13 - 32*y^12 + 882*y^11 + 1176*y^10 - 7303*y^9 - 15390*y^8 + 21330*y^7 + 81960*y^6 + 35883*y^5 - 123750*y^4 - 208680*y^3 - 143280*y^2 - 47760*y - 6368, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 48*x^13 - 32*x^12 + 882*x^11 + 1176*x^10 - 7303*x^9 - 15390*x^8 + 21330*x^7 + 81960*x^6 + 35883*x^5 - 123750*x^4 - 208680*x^3 - 143280*x^2 - 47760*x - 6368);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 48*x^13 - 32*x^12 + 882*x^11 + 1176*x^10 - 7303*x^9 - 15390*x^8 + 21330*x^7 + 81960*x^6 + 35883*x^5 - 123750*x^4 - 208680*x^3 - 143280*x^2 - 47760*x - 6368)
\( x^{15} - 48 x^{13} - 32 x^{12} + 882 x^{11} + 1176 x^{10} - 7303 x^{9} - 15390 x^{8} + 21330 x^{7} + \cdots - 6368 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[15, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(19616877526450987759846617\)
\(\medspace = 3^{15}\cdot 11^{13}\cdot 199^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(48.55\) | 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: | $3^{727/486}11^{9/10}199^{2/3}\approx 1525.9508812936347$ | ||
| Ramified primes: |
\(3\), \(11\), \(199\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{33}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{7}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{3}{8}a+\frac{1}{4}$, $\frac{1}{64}a^{12}-\frac{1}{32}a^{11}+\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{1}{32}a^{8}-\frac{3}{16}a^{7}-\frac{7}{64}a^{6}+\frac{1}{4}a^{5}-\frac{13}{32}a^{4}+\frac{1}{16}a^{3}+\frac{27}{64}a^{2}+\frac{1}{16}a+\frac{3}{16}$, $\frac{1}{512}a^{13}-\frac{1}{128}a^{12}-\frac{5}{128}a^{11}-\frac{63}{256}a^{9}-\frac{7}{32}a^{8}+\frac{81}{512}a^{7}+\frac{47}{256}a^{6}+\frac{83}{256}a^{5}-\frac{1}{64}a^{4}-\frac{237}{512}a^{3}+\frac{119}{256}a^{2}+\frac{1}{128}a-\frac{19}{64}$, $\frac{1}{4096}a^{14}+\frac{1}{2048}a^{13}+\frac{5}{1024}a^{12}-\frac{31}{512}a^{11}-\frac{447}{2048}a^{10}-\frac{153}{1024}a^{9}-\frac{207}{4096}a^{8}-\frac{111}{1024}a^{7}-\frac{499}{2048}a^{6}-\frac{9}{1024}a^{5}-\frac{669}{4096}a^{4}-\frac{21}{128}a^{3}-\frac{245}{512}a^{2}+\frac{9}{32}a-\frac{9}{256}$
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 $1$ (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: | $14$ | 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: |
$\frac{4431663}{4096}a^{14}-\frac{1542525}{2048}a^{13}-\frac{52642909}{1024}a^{12}+\frac{596619}{512}a^{11}+\frac{1952688879}{2048}a^{10}+\frac{623243069}{1024}a^{9}-\frac{34099402113}{4096}a^{8}-\frac{11116355607}{1024}a^{7}+\frac{62738552571}{2048}a^{6}+\frac{68966771733}{1024}a^{5}-\frac{33002339859}{4096}a^{4}-\frac{65677690461}{512}a^{3}-\frac{69881716197}{512}a^{2}-\frac{7682589423}{128}a-\frac{2534349699}{256}$, $\frac{767637}{2048}a^{14}-\frac{260253}{1024}a^{13}-\frac{9123435}{512}a^{12}+\frac{22599}{256}a^{11}+\frac{338468949}{1024}a^{10}+\frac{110932929}{512}a^{9}-\frac{5907016827}{2048}a^{8}-\frac{488037945}{128}a^{7}+\frac{10834595141}{1024}a^{6}+\frac{12055742541}{512}a^{5}-\frac{5156226945}{2048}a^{4}-\frac{22875758211}{512}a^{3}-\frac{3066946011}{64}a^{2}-\frac{2714423481}{128}a-\frac{450446351}{128}$, $\frac{767637}{2048}a^{14}-\frac{260253}{1024}a^{13}-\frac{9123435}{512}a^{12}+\frac{22599}{256}a^{11}+\frac{338468949}{1024}a^{10}+\frac{110932929}{512}a^{9}-\frac{5907016827}{2048}a^{8}-\frac{488037945}{128}a^{7}+\frac{10834595141}{1024}a^{6}+\frac{12055742541}{512}a^{5}-\frac{5156226945}{2048}a^{4}-\frac{22875758211}{512}a^{3}-\frac{3066946011}{64}a^{2}-\frac{2714423481}{128}a-\frac{450446479}{128}$, $\frac{1358127}{4096}a^{14}-\frac{461457}{2048}a^{13}-\frac{16140789}{1024}a^{12}+\frac{51759}{512}a^{11}+\frac{598798575}{2048}a^{10}+\frac{195831081}{1024}a^{9}-\frac{10450903041}{4096}a^{8}-\frac{3449918913}{1024}a^{7}+\frac{19174071331}{2048}a^{6}+\frac{21313449369}{1024}a^{5}-\frac{9207591123}{4096}a^{4}-\frac{5056896863}{128}a^{3}-\frac{21680135595}{512}a^{2}-\frac{599288301}{32}a-\frac{795136967}{256}$, $\frac{2533761}{4096}a^{14}-\frac{866727}{2048}a^{13}-\frac{30108699}{1024}a^{12}+\frac{164313}{512}a^{11}+\frac{1116943553}{2048}a^{10}+\frac{362849727}{1024}a^{9}-\frac{19497190351}{4096}a^{8}-\frac{6413929091}{1024}a^{7}+\frac{35799405565}{2048}a^{6}+\frac{39671046911}{1024}a^{5}-\frac{17649931805}{4096}a^{4}-\frac{18842865369}{256}a^{3}-\frac{40309860457}{512}a^{2}-\frac{2224954919}{64}a-\frac{1473778593}{256}$, $\frac{9225377}{4096}a^{14}-\frac{3237095}{2048}a^{13}-\frac{109568763}{1024}a^{12}+\frac{1545177}{512}a^{11}+\frac{4064063457}{2048}a^{10}+\frac{1286217023}{1024}a^{9}-\frac{70983854255}{4096}a^{8}-\frac{23040862099}{1024}a^{7}+\frac{130729502429}{2048}a^{6}+\frac{143156748095}{1024}a^{5}-\frac{70837458109}{4096}a^{4}-\frac{68246545469}{256}a^{3}-\frac{144853901625}{512}a^{2}-\frac{7945748091}{64}a-\frac{5231498145}{256}$, $\frac{3754685}{2048}a^{14}-\frac{1303827}{1024}a^{13}-\frac{44603391}{512}a^{12}+\frac{469885}{256}a^{11}+\frac{1654504189}{1024}a^{10}+\frac{529347899}{512}a^{9}-\frac{28890767187}{2048}a^{8}-\frac{9429949851}{512}a^{7}+\frac{53140972441}{1024}a^{6}+\frac{58479824107}{512}a^{5}-\frac{27720338025}{2048}a^{4}-\frac{869882073}{4}a^{3}-\frac{59277334001}{256}a^{2}-\frac{815033809}{8}a-\frac{2152002725}{128}$, $\frac{105835}{4096}a^{14}-\frac{4805}{2048}a^{13}-\frac{1280025}{1024}a^{12}-\frac{357941}{512}a^{11}+\frac{47753515}{2048}a^{10}+\frac{28590045}{1024}a^{9}-\frac{818291717}{4096}a^{8}-\frac{387997501}{1024}a^{7}+\frac{1355605167}{2048}a^{6}+\frac{2152573741}{1024}a^{5}+\frac{1725108449}{4096}a^{4}-\frac{58040197}{16}a^{3}-\frac{2407747495}{512}a^{2}-\frac{37662275}{16}a-\frac{110741155}{256}$, $\frac{4005681}{4096}a^{14}-\frac{1398887}{2048}a^{13}-\frac{47579787}{1024}a^{12}+\frac{593465}{512}a^{11}+\frac{1764860529}{2048}a^{10}+\frac{561330463}{1024}a^{9}-\frac{30822224159}{4096}a^{8}-\frac{10029891467}{1024}a^{7}+\frac{56733551949}{2048}a^{6}+\frac{62264220415}{1024}a^{5}-\frac{30236277261}{4096}a^{4}-\frac{29662925763}{256}a^{3}-\frac{63049395073}{512}a^{2}-\frac{3462278333}{64}a-\frac{2281923073}{256}$, $\frac{2661991}{2048}a^{14}-\frac{938369}{1024}a^{13}-\frac{31613165}{512}a^{12}+\frac{495919}{256}a^{11}+\frac{1172544551}{1024}a^{10}+\frac{369293961}{512}a^{9}-\frac{20482134089}{2048}a^{8}-\frac{6631965277}{512}a^{7}+\frac{37742208139}{1024}a^{6}+\frac{41240070633}{512}a^{5}-\frac{20785781931}{2048}a^{4}-\frac{19673486989}{128}a^{3}-\frac{41696932255}{256}a^{2}-\frac{2284604715}{32}a-\frac{1502520007}{128}$, $\frac{394061}{2048}a^{14}-\frac{134951}{1024}a^{13}-\frac{4682535}{512}a^{12}+\frac{27361}{256}a^{11}+\frac{173707661}{1024}a^{10}+\frac{56364935}{512}a^{9}-\frac{3032313411}{2048}a^{8}-\frac{996921469}{512}a^{7}+\frac{5568572977}{1024}a^{6}+\frac{6167360111}{512}a^{5}-\frac{2758925433}{2048}a^{4}-\frac{5859772235}{256}a^{3}-\frac{6265244735}{256}a^{2}-\frac{691396585}{64}a-\frac{228903401}{128}$, $\frac{333171}{1024}a^{14}-\frac{29033}{128}a^{13}-\frac{3957567}{256}a^{12}+\frac{23393}{64}a^{11}+\frac{146797427}{512}a^{10}+\frac{23391895}{128}a^{9}-\frac{2563585853}{1024}a^{8}-\frac{1670169697}{512}a^{7}+\frac{4717506509}{512}a^{6}+\frac{161946083}{8}a^{5}-\frac{2494987575}{1024}a^{4}-\frac{19744754053}{512}a^{3}-\frac{10499385971}{256}a^{2}-\frac{2307659727}{128}a-\frac{95120581}{32}$, $\frac{1938895}{4096}a^{14}-\frac{642921}{2048}a^{13}-\frac{23054437}{1024}a^{12}-\frac{110297}{512}a^{11}+\frac{855427471}{2048}a^{10}+\frac{286351889}{1024}a^{9}-\frac{14922391649}{4096}a^{8}-\frac{4985777309}{1024}a^{7}+\frac{27304807731}{2048}a^{6}+\frac{30677872913}{1024}a^{5}-\frac{11918404467}{4096}a^{4}-\frac{14510824059}{256}a^{3}-\frac{31313499887}{512}a^{2}-\frac{1739774521}{64}a-\frac{1159402879}{256}$, $\frac{3317453}{4096}a^{14}-\frac{1212527}{2048}a^{13}-\frac{39367287}{1024}a^{12}+\frac{1119625}{512}a^{11}+\frac{1459818765}{2048}a^{10}+\frac{441733695}{1024}a^{9}-\frac{25522516675}{4096}a^{8}-\frac{8099628961}{1024}a^{7}+\frac{47238055809}{2048}a^{6}+\frac{50713925591}{1024}a^{5}-\frac{29377115257}{4096}a^{4}-\frac{48653154561}{512}a^{3}-\frac{50952054859}{512}a^{2}-\frac{5530382979}{128}a-\frac{1801633617}{256}$
| 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: | \( 63974103.6431 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 63974103.6431 \cdot 1}{2\cdot\sqrt{19616877526450987759846617}}\cr\approx \mathstrut & 0.236651471117 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 48*x^13 - 32*x^12 + 882*x^11 + 1176*x^10 - 7303*x^9 - 15390*x^8 + 21330*x^7 + 81960*x^6 + 35883*x^5 - 123750*x^4 - 208680*x^3 - 143280*x^2 - 47760*x - 6368)
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^15 - 48*x^13 - 32*x^12 + 882*x^11 + 1176*x^10 - 7303*x^9 - 15390*x^8 + 21330*x^7 + 81960*x^6 + 35883*x^5 - 123750*x^4 - 208680*x^3 - 143280*x^2 - 47760*x - 6368, 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^15 - 48*x^13 - 32*x^12 + 882*x^11 + 1176*x^10 - 7303*x^9 - 15390*x^8 + 21330*x^7 + 81960*x^6 + 35883*x^5 - 123750*x^4 - 208680*x^3 - 143280*x^2 - 47760*x - 6368);
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^15 - 48*x^13 - 32*x^12 + 882*x^11 + 1176*x^10 - 7303*x^9 - 15390*x^8 + 21330*x^7 + 81960*x^6 + 35883*x^5 - 123750*x^4 - 208680*x^3 - 143280*x^2 - 47760*x - 6368);
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
$C_7^3:C_6$ (as 15T44):
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 2430 |
| The 39 conjugacy class representatives for $C_7^3:C_6$ |
| Character table for $C_7^3:C_6$ is not computed |
Intermediate fields
| \(\Q(\zeta_{11})^+\) |
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 15 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 45 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 | ${\href{/padicField/2.5.0.1}{5} }^{3}$ | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $15$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{5}$ | $15$ | $15$ | $15$ | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.15.15.41 | $x^{15} - 33 x^{14} + 741 x^{13} - 3054 x^{12} - 22212 x^{11} + 52065 x^{10} + 398718 x^{9} + 775089 x^{8} + 832599 x^{7} + 626886 x^{6} + 184761 x^{5} - 104409 x^{4} - 34101 x^{3} + 16281 x^{2} - 2673 x + 243$ | $3$ | $5$ | $15$ | 15T44 | $[3/2, 3/2, 3/2, 3/2, 3/2]_{2}^{5}$ |
|
\(11\)
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
|
\(199\)
| $\Q_{199}$ | $x + 196$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{199}$ | $x + 196$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{199}$ | $x + 196$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 199.3.2.2 | $x^{3} + 398$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 199.3.0.1 | $x^{3} + x + 196$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 199.3.0.1 | $x^{3} + x + 196$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 199.3.0.1 | $x^{3} + x + 196$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |