Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 8*x^20 + 8*x^19 + 94*x^18 - 269*x^17 - 195*x^16 + 1577*x^15 - 1066*x^14 - 3297*x^13 + 4973*x^12 + 1604*x^11 - 7061*x^10 + 2541*x^9 + 3722*x^8 - 3133*x^7 - 183*x^6 + 1045*x^5 - 344*x^4 - 56*x^3 + 58*x^2 - 13*x + 1)
gp: K = bnfinit(y^21 - 8*y^20 + 8*y^19 + 94*y^18 - 269*y^17 - 195*y^16 + 1577*y^15 - 1066*y^14 - 3297*y^13 + 4973*y^12 + 1604*y^11 - 7061*y^10 + 2541*y^9 + 3722*y^8 - 3133*y^7 - 183*y^6 + 1045*y^5 - 344*y^4 - 56*y^3 + 58*y^2 - 13*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 8*x^20 + 8*x^19 + 94*x^18 - 269*x^17 - 195*x^16 + 1577*x^15 - 1066*x^14 - 3297*x^13 + 4973*x^12 + 1604*x^11 - 7061*x^10 + 2541*x^9 + 3722*x^8 - 3133*x^7 - 183*x^6 + 1045*x^5 - 344*x^4 - 56*x^3 + 58*x^2 - 13*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 8*x^20 + 8*x^19 + 94*x^18 - 269*x^17 - 195*x^16 + 1577*x^15 - 1066*x^14 - 3297*x^13 + 4973*x^12 + 1604*x^11 - 7061*x^10 + 2541*x^9 + 3722*x^8 - 3133*x^7 - 183*x^6 + 1045*x^5 - 344*x^4 - 56*x^3 + 58*x^2 - 13*x + 1)
\( x^{21} - 8 x^{20} + 8 x^{19} + 94 x^{18} - 269 x^{17} - 195 x^{16} + 1577 x^{15} - 1066 x^{14} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[21, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(84034131400742875647626644438054873\)
\(\medspace = 71^{3}\cdot 109^{2}\cdot 29437^{2}\cdot 283583^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(46.03\) | 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: | $71^{1/2}109^{2/3}29437^{2/3}283583^{1/2}\approx 97614867.21334988$ | ||
| Ramified primes: |
\(71\), \(109\), \(29437\), \(283583\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{20134393}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| 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: | $20$ | 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: |
$a^{19}-7a^{18}+a^{17}+95a^{16}-174a^{15}-369a^{14}+1208a^{13}+142a^{12}-3155a^{11}+1818a^{10}+3422a^{9}-3639a^{8}-1098a^{7}+2624a^{6}-509a^{5}-692a^{4}+353a^{3}+9a^{2}-48a+11$, $1919a^{20}-14287a^{19}+7423a^{18}+184506a^{17}-413815a^{16}-603867a^{15}+2691145a^{14}-552127a^{13}-6633433a^{12}+5861858a^{11}+6331383a^{10}-10036528a^{9}-693831a^{8}+6757690a^{7}-2262034a^{6}-1606591a^{5}+1113822a^{4}-42020a^{3}-130789a^{2}+38722a-3458$, $3838a^{20}-28784a^{19}+16316a^{18}+368854a^{17}-847892a^{16}-1171467a^{15}+5464525a^{14}-1362169a^{13}-13321294a^{12}+12425535a^{11}+12333772a^{10}-20912763a^{9}-663121a^{8}+13914441a^{7}-5086155a^{6}-3216292a^{5}+2402345a^{4}-128056a^{3}-276691a^{2}+85040a-7759$, $1919a^{20}-14497a^{19}+8893a^{18}+184348a^{17}-434077a^{16}-567600a^{15}+2773380a^{14}-810042a^{13}-6687861a^{12}+6563677a^{11}+6002389a^{10}-10876235a^{9}+30710a^{8}+7156751a^{7}-2824121a^{6}-1609701a^{5}+1288523a^{4}-86036a^{3}-145902a^{2}+46318a-4301$, $a^{20}-8a^{19}+8a^{18}+94a^{17}-269a^{16}-195a^{15}+1577a^{14}-1066a^{13}-3297a^{12}+4973a^{11}+1604a^{10}-7061a^{9}+2541a^{8}+3722a^{7}-3133a^{6}-183a^{5}+1045a^{4}-344a^{3}-56a^{2}+58a-12$, $a-1$, $1919a^{20}-14497a^{19}+8893a^{18}+184348a^{17}-434077a^{16}-567600a^{15}+2773380a^{14}-810042a^{13}-6687861a^{12}+6563677a^{11}+6002389a^{10}-10876235a^{9}+30710a^{8}+7156751a^{7}-2824121a^{6}-1609701a^{5}+1288523a^{4}-86036a^{3}-145902a^{2}+46318a-4300$, $3838a^{20}-28784a^{19}+16316a^{18}+368854a^{17}-847892a^{16}-1171467a^{15}+5464525a^{14}-1362169a^{13}-13321294a^{12}+12425535a^{11}+12333772a^{10}-20912763a^{9}-663121a^{8}+13914441a^{7}-5086155a^{6}-3216292a^{5}+2402345a^{4}-128056a^{3}-276691a^{2}+85040a-7758$, $3458a^{20}-25744a^{19}+13370a^{18}+332476a^{17}-745601a^{16}-1088299a^{15}+4849030a^{14}-993875a^{13}-11953011a^{12}+10560046a^{11}+11410308a^{10}-18082133a^{9}-1253389a^{8}+12175747a^{7}-4073600a^{6}-2895357a^{5}+2006327a^{4}-75377a^{3}-235659a^{2}+69728a-6221$, $853a^{20}-6350a^{19}+3295a^{18}+82016a^{17}-183884a^{16}-268554a^{15}+1196041a^{14}-244571a^{13}-2948807a^{12}+2603578a^{11}+2816248a^{10}-4459457a^{9}-311544a^{8}+3003775a^{7}-1003487a^{6}-714875a^{5}+494661a^{4}-18351a^{3}-58153a^{2}+17174a-1526$, $a^{20}-7a^{19}+a^{18}+95a^{17}-174a^{16}-369a^{15}+1208a^{14}+142a^{13}-3155a^{12}+1818a^{11}+3422a^{10}-3639a^{9}-1098a^{8}+2624a^{7}-509a^{6}-692a^{5}+353a^{4}+9a^{3}-47a^{2}+10a-2$, $a^{19}-7a^{18}+a^{17}+95a^{16}-174a^{15}-369a^{14}+1208a^{13}+142a^{12}-3155a^{11}+1818a^{10}+3422a^{9}-3639a^{8}-1098a^{7}+2624a^{6}-509a^{5}-692a^{4}+353a^{3}+9a^{2}-47a+10$, $1920a^{20}-14504a^{19}+8894a^{18}+184443a^{17}-434251a^{16}-567969a^{15}+2774588a^{14}-809900a^{13}-6691016a^{12}+6565495a^{11}+6005811a^{10}-10879874a^{9}+29612a^{8}+7159375a^{7}-2824630a^{6}-1610393a^{5}+1288876a^{4}-86027a^{3}-145949a^{2}+46329a-4303$, $745a^{20}-5440a^{19}+2108a^{18}+71895a^{17}-150325a^{16}-255512a^{15}+1006877a^{14}-71176a^{13}-2577342a^{12}+1901664a^{11}+2711196a^{10}-3482881a^{9}-751935a^{8}+2475572a^{7}-521744a^{6}-670721a^{5}+326824a^{4}+22811a^{3}-43199a^{2}+9237a-576$, $28a^{20}-415a^{19}+1578a^{18}+2380a^{17}-26018a^{16}+29142a^{15}+117036a^{14}-272466a^{13}-125757a^{12}+792904a^{11}-303384a^{10}-966973a^{9}+800204a^{8}+452106a^{7}-648207a^{6}+9653a^{5}+204171a^{4}-56524a^{3}-17259a^{2}+9488a-1101$, $6649a^{20}-50475a^{19}+32556a^{18}+638379a^{17}-1527712a^{16}-1921810a^{15}+9701722a^{14}-3119133a^{13}-23207925a^{12}+23577287a^{11}+20331953a^{10}-38650396a^{9}+1060461a^{8}+25208281a^{7}-10508344a^{6}-5533220a^{5}+4684442a^{4}-365794a^{3}-523074a^{2}+171207a-16206$, $2602a^{20}-19784a^{19}+12963a^{18}+249771a^{17}-600863a^{16}-746304a^{15}+3808210a^{14}-1260567a^{13}-9085375a^{12}+9332326a^{11}+7894136a^{10}-15244691a^{9}+540043a^{8}+9911283a^{7}-4205701a^{6}-2155331a^{5}+1860887a^{4}-153173a^{3}-206658a^{2}+68522a-6549$, $997a^{20}-7493a^{19}+4333a^{18}+95909a^{17}-221746a^{16}-303092a^{15}+1427767a^{14}-366204a^{13}-3480797a^{12}+3269605a^{11}+3226606a^{10}-5500019a^{9}-181271a^{8}+3671094a^{7}-1323642a^{6}-859876a^{5}+627238a^{4}-28197a^{3}-72807a^{2}+21691a-1926$, $1639a^{20}-12012a^{19}+4971a^{18}+157967a^{17}-334999a^{16}-552098a^{15}+2228811a^{14}-221695a^{13}-5654267a^{12}+4346157a^{11}+5816827a^{10}-7824291a^{9}-1398174a^{8}+5476476a^{7}-1329171a^{6}-1429269a^{5}+769929a^{4}+24727a^{3}-97732a^{2}+23797a-1798$, $4301a^{20}-32489a^{19}+19911a^{18}+413187a^{17}-972621a^{16}-1272772a^{15}+6215077a^{14}-1811486a^{13}-14990439a^{12}+14701012a^{11}+13462481a^{10}-24366972a^{9}+52606a^{8}+16039032a^{7}-6318282a^{6}-3611204a^{5}+2884844a^{4}-191021a^{3}-326892a^{2}+103556a-9596$
| 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: | \( 54641615186.6 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 54641615186.6 \cdot 1}{2\cdot\sqrt{84034131400742875647626644438054873}}\cr\approx \mathstrut & 0.197649438095 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 8*x^20 + 8*x^19 + 94*x^18 - 269*x^17 - 195*x^16 + 1577*x^15 - 1066*x^14 - 3297*x^13 + 4973*x^12 + 1604*x^11 - 7061*x^10 + 2541*x^9 + 3722*x^8 - 3133*x^7 - 183*x^6 + 1045*x^5 - 344*x^4 - 56*x^3 + 58*x^2 - 13*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^21 - 8*x^20 + 8*x^19 + 94*x^18 - 269*x^17 - 195*x^16 + 1577*x^15 - 1066*x^14 - 3297*x^13 + 4973*x^12 + 1604*x^11 - 7061*x^10 + 2541*x^9 + 3722*x^8 - 3133*x^7 - 183*x^6 + 1045*x^5 - 344*x^4 - 56*x^3 + 58*x^2 - 13*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^21 - 8*x^20 + 8*x^19 + 94*x^18 - 269*x^17 - 195*x^16 + 1577*x^15 - 1066*x^14 - 3297*x^13 + 4973*x^12 + 1604*x^11 - 7061*x^10 + 2541*x^9 + 3722*x^8 - 3133*x^7 - 183*x^6 + 1045*x^5 - 344*x^4 - 56*x^3 + 58*x^2 - 13*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^21 - 8*x^20 + 8*x^19 + 94*x^18 - 269*x^17 - 195*x^16 + 1577*x^15 - 1066*x^14 - 3297*x^13 + 4973*x^12 + 1604*x^11 - 7061*x^10 + 2541*x^9 + 3722*x^8 - 3133*x^7 - 183*x^6 + 1045*x^5 - 344*x^4 - 56*x^3 + 58*x^2 - 13*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
$C_3^7.S_7$ (as 21T139):
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 non-solvable group of order 11022480 |
| The 429 conjugacy class representatives for $C_3^7.S_7$ |
| Character table for $C_3^7.S_7$ |
Intermediate fields
| 7.7.20134393.1 |
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
| Degree 21 sibling: | data not computed |
| Degree 42 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 | $21$ | $21$ | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{3}$ | $18{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.7.0.1}{7} }^{3}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | $21$ | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | $15{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.7.0.1}{7} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{5}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ | $15{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(71\)
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.3.0.1 | $x^{3} + 4 x + 64$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 71.6.3.1 | $x^{6} - 3550 x^{5} + 161634624 x^{4} + 10165888006904 x^{3} + 6625668596 x^{2} - 569794312 x - 22906304$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(109\)
| 109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.3.2.2 | $x^{3} + 327$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 109.12.0.1 | $x^{12} + x^{8} + 50 x^{7} + 53 x^{6} + 37 x^{5} + 8 x^{4} + 65 x^{3} + 103 x^{2} + 28 x + 6$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(29437\)
| $\Q_{29437}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{29437}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{29437}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $3$ | $1$ | $2$ | ||||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
|
\(283583\)
| $\Q_{283583}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{283583}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{283583}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $6$ | $2$ | $3$ | $3$ |