Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 9*x^20 + 23*x^19 + 24*x^18 - 196*x^17 + 152*x^16 + 587*x^15 - 1008*x^14 - 562*x^13 + 2180*x^12 - 404*x^11 - 2259*x^10 + 1317*x^9 + 1079*x^8 - 1081*x^7 - 177*x^6 + 477*x^5 - 106*x^4 - 81*x^3 + 53*x^2 - 12*x + 1)
gp: K = bnfinit(y^21 - 9*y^20 + 23*y^19 + 24*y^18 - 196*y^17 + 152*y^16 + 587*y^15 - 1008*y^14 - 562*y^13 + 2180*y^12 - 404*y^11 - 2259*y^10 + 1317*y^9 + 1079*y^8 - 1081*y^7 - 177*y^6 + 477*y^5 - 106*y^4 - 81*y^3 + 53*y^2 - 12*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 9*x^20 + 23*x^19 + 24*x^18 - 196*x^17 + 152*x^16 + 587*x^15 - 1008*x^14 - 562*x^13 + 2180*x^12 - 404*x^11 - 2259*x^10 + 1317*x^9 + 1079*x^8 - 1081*x^7 - 177*x^6 + 477*x^5 - 106*x^4 - 81*x^3 + 53*x^2 - 12*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 9*x^20 + 23*x^19 + 24*x^18 - 196*x^17 + 152*x^16 + 587*x^15 - 1008*x^14 - 562*x^13 + 2180*x^12 - 404*x^11 - 2259*x^10 + 1317*x^9 + 1079*x^8 - 1081*x^7 - 177*x^6 + 477*x^5 - 106*x^4 - 81*x^3 + 53*x^2 - 12*x + 1)
\( x^{21} - 9 x^{20} + 23 x^{19} + 24 x^{18} - 196 x^{17} + 152 x^{16} + 587 x^{15} - 1008 x^{14} + \cdots + 1 \)
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: | $[3, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-19019216778782670169504453271\)
\(\medspace = -\,67^{2}\cdot 71^{9}\cdot 9613^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.21\) | 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: | $67^{2/3}71^{1/2}9613^{2/3}\approx 62842.242032253074$ | ||
| Ramified primes: |
\(67\), \(71\), \(9613\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-71}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11}a^{18}-\frac{1}{11}a^{17}-\frac{1}{11}a^{16}-\frac{2}{11}a^{15}+\frac{3}{11}a^{14}+\frac{2}{11}a^{12}-\frac{5}{11}a^{11}+\frac{4}{11}a^{10}+\frac{2}{11}a^{9}+\frac{4}{11}a^{8}+\frac{3}{11}a^{7}-\frac{1}{11}a^{6}-\frac{4}{11}a^{5}-\frac{2}{11}a^{3}+\frac{3}{11}a^{2}+\frac{5}{11}a+\frac{1}{11}$, $\frac{1}{11}a^{19}-\frac{2}{11}a^{17}-\frac{3}{11}a^{16}+\frac{1}{11}a^{15}+\frac{3}{11}a^{14}+\frac{2}{11}a^{13}-\frac{3}{11}a^{12}-\frac{1}{11}a^{11}-\frac{5}{11}a^{10}-\frac{5}{11}a^{9}-\frac{4}{11}a^{8}+\frac{2}{11}a^{7}-\frac{5}{11}a^{6}-\frac{4}{11}a^{5}-\frac{2}{11}a^{4}+\frac{1}{11}a^{3}-\frac{3}{11}a^{2}-\frac{5}{11}a+\frac{1}{11}$, $\frac{1}{11}a^{20}-\frac{5}{11}a^{17}-\frac{1}{11}a^{16}-\frac{1}{11}a^{15}-\frac{3}{11}a^{14}-\frac{3}{11}a^{13}+\frac{3}{11}a^{12}-\frac{4}{11}a^{11}+\frac{3}{11}a^{10}-\frac{1}{11}a^{8}+\frac{1}{11}a^{7}+\frac{5}{11}a^{6}+\frac{1}{11}a^{5}+\frac{1}{11}a^{4}+\frac{4}{11}a^{3}+\frac{1}{11}a^{2}+\frac{2}{11}$
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: | $11$ | 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: |
$\frac{12141}{11}a^{20}-\frac{102794}{11}a^{19}+\frac{224363}{11}a^{18}+\frac{411516}{11}a^{17}-\frac{2161100}{11}a^{16}+\frac{690625}{11}a^{15}+682252a^{14}-748735a^{13}-\frac{11251799}{11}a^{12}+\frac{20495154}{11}a^{11}+\frac{6088085}{11}a^{10}-\frac{24258918}{11}a^{9}+\frac{2998715}{11}a^{8}+\frac{14799585}{11}a^{7}-\frac{5215501}{11}a^{6}-\frac{4996664}{11}a^{5}+\frac{3131398}{11}a^{4}+\frac{408669}{11}a^{3}-\frac{772768}{11}a^{2}+\frac{227125}{11}a-\frac{22130}{11}$, $\frac{2185}{11}a^{20}-\frac{17903}{11}a^{19}+\frac{35521}{11}a^{18}+\frac{83509}{11}a^{17}-\frac{365675}{11}a^{16}+\frac{25734}{11}a^{15}+122978a^{14}-101235a^{13}-\frac{2310298}{11}a^{12}+\frac{3046165}{11}a^{11}+\frac{1893555}{11}a^{10}-\frac{3814194}{11}a^{9}-\frac{470566}{11}a^{8}+\frac{2490456}{11}a^{7}-\frac{270988}{11}a^{6}-\frac{943810}{11}a^{5}+\frac{305577}{11}a^{4}+\frac{145493}{11}a^{3}-\frac{96428}{11}a^{2}+\frac{16485}{11}a-\frac{573}{11}$, $\frac{2394}{11}a^{20}-\frac{20070}{11}a^{19}+\frac{42594}{11}a^{18}+\frac{84466}{11}a^{17}-\frac{418508}{11}a^{16}+\frac{101975}{11}a^{15}+134816a^{14}-136061a^{13}-\frac{2325588}{11}a^{12}+\frac{3820081}{11}a^{11}+\frac{1494134}{11}a^{10}-\frac{4593203}{11}a^{9}+\frac{217880}{11}a^{8}+\frac{2855953}{11}a^{7}-\frac{778099}{11}a^{6}-\frac{996390}{11}a^{5}+\frac{519780}{11}a^{4}+\frac{103264}{11}a^{3}-\frac{135610}{11}a^{2}+\frac{36736}{11}a-\frac{3411}{11}$, $a^{20}-8a^{19}+15a^{18}+39a^{17}-157a^{16}-5a^{15}+582a^{14}-426a^{13}-988a^{12}+1192a^{11}+788a^{10}-1471a^{9}-154a^{8}+925a^{7}-156a^{6}-333a^{5}+144a^{4}+38a^{3}-43a^{2}+10a-1$, $a^{20}-8a^{19}+15a^{18}+39a^{17}-157a^{16}-5a^{15}+582a^{14}-426a^{13}-988a^{12}+1192a^{11}+788a^{10}-1471a^{9}-154a^{8}+925a^{7}-156a^{6}-333a^{5}+144a^{4}+38a^{3}-43a^{2}+10a-2$, $\frac{11054}{11}a^{20}-\frac{94893}{11}a^{19}+\frac{214737}{11}a^{18}+\frac{355137}{11}a^{17}-\frac{2020189}{11}a^{16}+\frac{837753}{11}a^{15}+622652a^{14}-754179a^{13}-\frac{9706359}{11}a^{12}+\frac{20097740}{11}a^{11}+\frac{3968889}{11}a^{10}-\frac{23417248}{11}a^{9}+\frac{4751767}{11}a^{8}+\frac{14023584}{11}a^{7}-\frac{6092634}{11}a^{6}-\frac{4569012}{11}a^{5}+\frac{3374659}{11}a^{4}+\frac{261931}{11}a^{3}-\frac{793524}{11}a^{2}+\frac{250788}{11}a-\frac{25787}{11}$, $\frac{18382}{11}a^{20}-\frac{158011}{11}a^{19}+\frac{358777}{11}a^{18}+\frac{587474}{11}a^{17}-\frac{3368071}{11}a^{16}+\frac{1426613}{11}a^{15}+\frac{11393713}{11}a^{14}-\frac{13924675}{11}a^{13}-\frac{16059127}{11}a^{12}+\frac{33657952}{11}a^{11}+\frac{6354390}{11}a^{10}-\frac{39168565}{11}a^{9}+\frac{8221024}{11}a^{8}+\frac{23425150}{11}a^{7}-\frac{10346374}{11}a^{6}-\frac{7612679}{11}a^{5}+\frac{5696849}{11}a^{4}+\frac{422637}{11}a^{3}-\frac{1335067}{11}a^{2}+\frac{423111}{11}a-\frac{43443}{11}$, $\frac{8869}{11}a^{20}-\frac{75365}{11}a^{19}+\frac{166084}{11}a^{18}+\frac{296624}{11}a^{17}-\frac{1590014}{11}a^{16}+\frac{547957}{11}a^{15}+\frac{5488359}{11}a^{14}-\frac{6184976}{11}a^{13}-737832a^{12}+\frac{15284440}{11}a^{11}+\frac{4133292}{11}a^{10}-\frac{18028462}{11}a^{9}+\frac{2600976}{11}a^{8}+\frac{10962382}{11}a^{7}-\frac{4086988}{11}a^{6}-\frac{3679461}{11}a^{5}+217932a^{4}+\frac{285600}{11}a^{3}-53147a^{2}+\frac{173190}{11}a-\frac{16811}{11}$, $\frac{15354}{11}a^{20}-11908a^{19}+\frac{291663}{11}a^{18}+\frac{505814}{11}a^{17}-\frac{2773415}{11}a^{16}+\frac{1030860}{11}a^{15}+864371a^{14}-\frac{11020058}{11}a^{13}-\frac{13840056}{11}a^{12}+\frac{27019494}{11}a^{11}+\frac{6542038}{11}a^{10}-\frac{31717756}{11}a^{9}+\frac{5284163}{11}a^{8}+\frac{19171210}{11}a^{7}-\frac{7587709}{11}a^{6}-\frac{6359391}{11}a^{5}+\frac{4346450}{11}a^{4}+\frac{442889}{11}a^{3}-\frac{1044128}{11}a^{2}+\frac{318759}{11}a-\frac{31961}{11}$, $\frac{8257}{11}a^{20}-\frac{68724}{11}a^{19}+13003a^{18}+\frac{297914}{11}a^{17}-\frac{1422407}{11}a^{16}+\frac{278187}{11}a^{15}+\frac{5094586}{11}a^{14}-\frac{4875499}{11}a^{13}-\frac{8163278}{11}a^{12}+\frac{12645304}{11}a^{11}+\frac{5613059}{11}a^{10}-\frac{15331743}{11}a^{9}+\frac{159004}{11}a^{8}+\frac{9623486}{11}a^{7}-\frac{2301383}{11}a^{6}-\frac{3420217}{11}a^{5}+\frac{1646292}{11}a^{4}+\frac{394019}{11}a^{3}-\frac{443987}{11}a^{2}+\frac{111410}{11}a-\frac{9244}{11}$, $\frac{14015}{11}a^{20}-\frac{116420}{11}a^{19}+\frac{241388}{11}a^{18}+\frac{505675}{11}a^{17}-\frac{2399722}{11}a^{16}+\frac{455659}{11}a^{15}+\frac{8580571}{11}a^{14}-\frac{8166087}{11}a^{13}-\frac{13688676}{11}a^{12}+\frac{21128981}{11}a^{11}+\frac{9278202}{11}a^{10}-\frac{25455596}{11}a^{9}+\frac{501896}{11}a^{8}+\frac{15774868}{11}a^{7}-\frac{4039197}{11}a^{6}-\frac{5485940}{11}a^{5}+\frac{2832185}{11}a^{4}+50876a^{3}-\frac{750089}{11}a^{2}+18850a-\frac{19636}{11}$
| 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: | \( 520928.798224 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 520928.798224 \cdot 1}{2\cdot\sqrt{19019216778782670169504453271}}\cr\approx \mathstrut & 0.230600826363 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 9*x^20 + 23*x^19 + 24*x^18 - 196*x^17 + 152*x^16 + 587*x^15 - 1008*x^14 - 562*x^13 + 2180*x^12 - 404*x^11 - 2259*x^10 + 1317*x^9 + 1079*x^8 - 1081*x^7 - 177*x^6 + 477*x^5 - 106*x^4 - 81*x^3 + 53*x^2 - 12*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 - 9*x^20 + 23*x^19 + 24*x^18 - 196*x^17 + 152*x^16 + 587*x^15 - 1008*x^14 - 562*x^13 + 2180*x^12 - 404*x^11 - 2259*x^10 + 1317*x^9 + 1079*x^8 - 1081*x^7 - 177*x^6 + 477*x^5 - 106*x^4 - 81*x^3 + 53*x^2 - 12*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 - 9*x^20 + 23*x^19 + 24*x^18 - 196*x^17 + 152*x^16 + 587*x^15 - 1008*x^14 - 562*x^13 + 2180*x^12 - 404*x^11 - 2259*x^10 + 1317*x^9 + 1079*x^8 - 1081*x^7 - 177*x^6 + 477*x^5 - 106*x^4 - 81*x^3 + 53*x^2 - 12*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 - 9*x^20 + 23*x^19 + 24*x^18 - 196*x^17 + 152*x^16 + 587*x^15 - 1008*x^14 - 562*x^13 + 2180*x^12 - 404*x^11 - 2259*x^10 + 1317*x^9 + 1079*x^8 - 1081*x^7 - 177*x^6 + 477*x^5 - 106*x^4 - 81*x^3 + 53*x^2 - 12*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
$C_3^7:D_7$ (as 21T76):
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 30618 |
| The 288 conjugacy class representatives for $C_3^7:D_7$ are not computed |
| Character table for $C_3^7:D_7$ is not computed |
Intermediate fields
| 7.1.357911.1 |
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 siblings: | 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 | $21$ | ${\href{/padicField/3.7.0.1}{7} }^{3}$ | $21$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | $21$ | ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | $21$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | $21$ | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.7.0.1}{7} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(67\)
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.3.2.1 | $x^{3} + 67$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 67.6.0.1 | $x^{6} + 63 x^{3} + 49 x^{2} + 55 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(71\)
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.3.0.1 | $x^{3} + 4 x + 64$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 71.6.3.2 | $x^{6} + 221 x^{4} + 128 x^{3} + 15139 x^{2} - 26752 x + 322815$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 71.6.3.2 | $x^{6} + 221 x^{4} + 128 x^{3} + 15139 x^{2} - 26752 x + 322815$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(9613\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $3$ | $3$ | $1$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |