Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^21 + 15*x^20 - 35*x^19 + 68*x^18 - 114*x^17 + 170*x^16 - 230*x^15 + 285*x^14 - 325*x^13 + 343*x^12 - 335*x^11 + 306*x^10 - 258*x^9 + 202*x^8 - 146*x^7 + 97*x^6 - 59*x^5 + 32*x^4 - 16*x^3 + 7*x^2 - 2*x + 1)
gp: K = bnfinit(y^22 - 5*y^21 + 15*y^20 - 35*y^19 + 68*y^18 - 114*y^17 + 170*y^16 - 230*y^15 + 285*y^14 - 325*y^13 + 343*y^12 - 335*y^11 + 306*y^10 - 258*y^9 + 202*y^8 - 146*y^7 + 97*y^6 - 59*y^5 + 32*y^4 - 16*y^3 + 7*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 5*x^21 + 15*x^20 - 35*x^19 + 68*x^18 - 114*x^17 + 170*x^16 - 230*x^15 + 285*x^14 - 325*x^13 + 343*x^12 - 335*x^11 + 306*x^10 - 258*x^9 + 202*x^8 - 146*x^7 + 97*x^6 - 59*x^5 + 32*x^4 - 16*x^3 + 7*x^2 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 5*x^21 + 15*x^20 - 35*x^19 + 68*x^18 - 114*x^17 + 170*x^16 - 230*x^15 + 285*x^14 - 325*x^13 + 343*x^12 - 335*x^11 + 306*x^10 - 258*x^9 + 202*x^8 - 146*x^7 + 97*x^6 - 59*x^5 + 32*x^4 - 16*x^3 + 7*x^2 - 2*x + 1)
\( x^{22} - 5 x^{21} + 15 x^{20} - 35 x^{19} + 68 x^{18} - 114 x^{17} + 170 x^{16} - 230 x^{15} + 285 x^{14} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-103946711888571010237690363\)
\(\medspace = -\,79\cdot 22109\cdot 59513372976908430233\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.23\) | 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: | $79^{1/2}22109^{1/2}59513372976908430233^{1/2}\approx 10195426027811.246$ | ||
| Ramified primes: |
\(79\), \(22109\), \(59513372976908430233\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-10394\!\cdots\!90363}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{43}a^{21}+\frac{7}{43}a^{20}+\frac{13}{43}a^{19}-\frac{8}{43}a^{18}+\frac{15}{43}a^{17}-\frac{20}{43}a^{16}+\frac{16}{43}a^{15}+\frac{5}{43}a^{14}+\frac{1}{43}a^{13}-\frac{12}{43}a^{12}-\frac{16}{43}a^{11}-\frac{11}{43}a^{10}+\frac{2}{43}a^{9}-\frac{19}{43}a^{8}+\frac{17}{43}a^{7}+\frac{15}{43}a^{6}+\frac{19}{43}a^{5}-\frac{3}{43}a^{4}-\frac{4}{43}a^{3}-\frac{21}{43}a^{2}+\frac{13}{43}a-\frac{18}{43}$
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: | $10$ | 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: |
$a$, $\frac{131}{43}a^{21}-\frac{717}{43}a^{20}+\frac{2219}{43}a^{19}-\frac{5176}{43}a^{18}+\frac{9963}{43}a^{17}-\frac{16423}{43}a^{16}+\frac{23854}{43}a^{15}-\frac{31251}{43}a^{14}+\frac{37369}{43}a^{13}-\frac{40917}{43}a^{12}+\frac{41162}{43}a^{11}-\frac{38034}{43}a^{10}+\frac{32598}{43}a^{9}-\frac{25666}{43}a^{8}+\frac{18395}{43}a^{7}-\frac{11924}{43}a^{6}+\frac{7004}{43}a^{5}-\frac{3661}{43}a^{4}+\frac{1583}{43}a^{3}-\frac{644}{43}a^{2}+\frac{241}{43}a+\frac{7}{43}$, $\frac{30}{43}a^{21}-\frac{177}{43}a^{20}+\frac{605}{43}a^{19}-\frac{1530}{43}a^{18}+\frac{3159}{43}a^{17}-\frac{5545}{43}a^{16}+\frac{8521}{43}a^{15}-\frac{11718}{43}a^{14}+\frac{14607}{43}a^{13}-\frac{16614}{43}a^{12}+\frac{17322}{43}a^{11}-\frac{16541}{43}a^{10}+\frac{14508}{43}a^{9}-\frac{11621}{43}a^{8}+\frac{8508}{43}a^{7}-\frac{5527}{43}a^{6}+\frac{3193}{43}a^{5}-\frac{1509}{43}a^{4}+\frac{611}{43}a^{3}-\frac{200}{43}a^{2}+\frac{46}{43}a+\frac{19}{43}$, $\frac{118}{43}a^{21}-\frac{464}{43}a^{20}+\frac{1018}{43}a^{19}-\frac{1632}{43}a^{18}+\frac{1856}{43}a^{17}-\frac{984}{43}a^{16}-\frac{1466}{43}a^{15}+\frac{5535}{43}a^{14}-\frac{10933}{43}a^{13}+\frac{16988}{43}a^{12}-\frac{22485}{43}a^{11}+\frac{26308}{43}a^{10}-\frac{27198}{43}a^{9}+\frac{25837}{43}a^{8}-\frac{22289}{43}a^{7}+\frac{17336}{43}a^{6}-\frac{12077}{43}a^{5}+\frac{7472}{43}a^{4}-\frac{4127}{43}a^{3}+\frac{1865}{43}a^{2}-\frac{659}{43}a+\frac{284}{43}$, $\frac{137}{43}a^{21}-\frac{632}{43}a^{20}+\frac{1695}{43}a^{19}-\frac{3504}{43}a^{18}+\frac{5968}{43}a^{17}-\frac{8588}{43}a^{16}+\frac{10749}{43}a^{15}-\frac{11957}{43}a^{14}+\frac{11790}{43}a^{13}-\frac{10029}{43}a^{12}+\frac{7053}{43}a^{11}-\frac{3442}{43}a^{10}+\frac{403}{43}a^{9}+\frac{2127}{43}a^{8}-\frac{3562}{43}a^{7}+\frac{3861}{43}a^{6}-\frac{3331}{43}a^{5}+\frac{2384}{43}a^{4}-\frac{1537}{43}a^{3}+\frac{735}{43}a^{2}-\frac{283}{43}a+\frac{200}{43}$, $\frac{244}{43}a^{21}-\frac{1388}{43}a^{20}+\frac{4462}{43}a^{19}-\frac{10767}{43}a^{18}+\frac{21333}{43}a^{17}-\frac{36141}{43}a^{16}+\frac{53913}{43}a^{15}-\frac{72396}{43}a^{14}+\frac{88652}{43}a^{13}-\frac{99549}{43}a^{12}+\frac{102865}{43}a^{11}-\frac{97800}{43}a^{10}+\frac{86144}{43}a^{9}-\frac{69910}{43}a^{8}+\frac{51964}{43}a^{7}-\frac{35169}{43}a^{6}+\frac{21449}{43}a^{5}-\frac{11740}{43}a^{4}+\frac{5560}{43}a^{3}-\frac{2243}{43}a^{2}+\frac{807}{43}a-\frac{135}{43}$, $\frac{6}{43}a^{21}-\frac{130}{43}a^{20}+\frac{680}{43}a^{19}-\frac{2112}{43}a^{18}+\frac{4949}{43}a^{17}-\frac{9580}{43}a^{16}+\frac{15920}{43}a^{15}-\frac{23319}{43}a^{14}+\frac{30794}{43}a^{13}-\frac{37138}{43}a^{12}+\frac{41055}{43}a^{11}-\frac{41690}{43}a^{10}+\frac{38927}{43}a^{9}-\frac{33611}{43}a^{8}+\frac{26762}{43}a^{7}-\frac{19389}{43}a^{6}+\frac{12670}{43}a^{5}-\frac{7457}{43}a^{4}+\frac{3932}{43}a^{3}-\frac{1717}{43}a^{2}+\frac{637}{43}a-\frac{194}{43}$, $\frac{190}{43}a^{21}-\frac{1035}{43}a^{20}+\frac{3244}{43}a^{19}-\frac{7712}{43}a^{18}+\frac{15148}{43}a^{17}-\frac{25558}{43}a^{16}+\frac{38128}{43}a^{15}-\frac{51381}{43}a^{14}+\frac{63271}{43}a^{13}-\frac{71510}{43}a^{12}+\frac{74446}{43}a^{11}-\frac{71406}{43}a^{10}+\frac{63633}{43}a^{9}-\frac{52243}{43}a^{8}+\frac{39350}{43}a^{7}-\frac{26992}{43}a^{6}+\frac{16768}{43}a^{5}-\frac{9342}{43}a^{4}+\frac{4443}{43}a^{3}-\frac{1840}{43}a^{2}+\frac{621}{43}a-\frac{66}{43}$, $\frac{138}{43}a^{21}-\frac{668}{43}a^{20}+\frac{1923}{43}a^{19}-\frac{4286}{43}a^{18}+\frac{7918}{43}a^{17}-\frac{12564}{43}a^{16}+\frac{17645}{43}a^{15}-\frac{22401}{43}a^{14}+\frac{25938}{43}a^{13}-\frac{27413}{43}a^{12}+\frac{26516}{43}a^{11}-\frac{23362}{43}a^{10}+\frac{19067}{43}a^{9}-\frac{13888}{43}a^{8}+\frac{9054}{43}a^{7}-\frac{5111}{43}a^{6}+\frac{2364}{43}a^{5}-\frac{887}{43}a^{4}+\frac{50}{43}a^{3}+\frac{69}{43}a^{2}-\frac{55}{43}a+\frac{96}{43}$, $\frac{94}{43}a^{21}-\frac{417}{43}a^{20}+\frac{1050}{43}a^{19}-\frac{1999}{43}a^{18}+\frac{3044}{43}a^{17}-\frac{3686}{43}a^{16}+\frac{3439}{43}a^{15}-\frac{2067}{43}a^{14}-\frac{465}{43}a^{13}+\frac{3946}{43}a^{12}-\frac{7696}{43}a^{11}+\frac{10963}{43}a^{10}-\frac{12755}{43}a^{9}+\frac{13221}{43}a^{8}-\frac{12248}{43}a^{7}+\frac{10139}{43}a^{6}-\frac{7502}{43}a^{5}+\frac{4921}{43}a^{4}-\frac{2913}{43}a^{3}+\frac{1423}{43}a^{2}-\frac{584}{43}a+\frac{243}{43}$
| 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: | \( 14889.9809115 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 14889.9809115 \cdot 1}{2\cdot\sqrt{103946711888571010237690363}}\cr\approx \mathstrut & 0.439984441313 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^21 + 15*x^20 - 35*x^19 + 68*x^18 - 114*x^17 + 170*x^16 - 230*x^15 + 285*x^14 - 325*x^13 + 343*x^12 - 335*x^11 + 306*x^10 - 258*x^9 + 202*x^8 - 146*x^7 + 97*x^6 - 59*x^5 + 32*x^4 - 16*x^3 + 7*x^2 - 2*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^22 - 5*x^21 + 15*x^20 - 35*x^19 + 68*x^18 - 114*x^17 + 170*x^16 - 230*x^15 + 285*x^14 - 325*x^13 + 343*x^12 - 335*x^11 + 306*x^10 - 258*x^9 + 202*x^8 - 146*x^7 + 97*x^6 - 59*x^5 + 32*x^4 - 16*x^3 + 7*x^2 - 2*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^22 - 5*x^21 + 15*x^20 - 35*x^19 + 68*x^18 - 114*x^17 + 170*x^16 - 230*x^15 + 285*x^14 - 325*x^13 + 343*x^12 - 335*x^11 + 306*x^10 - 258*x^9 + 202*x^8 - 146*x^7 + 97*x^6 - 59*x^5 + 32*x^4 - 16*x^3 + 7*x^2 - 2*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^22 - 5*x^21 + 15*x^20 - 35*x^19 + 68*x^18 - 114*x^17 + 170*x^16 - 230*x^15 + 285*x^14 - 325*x^13 + 343*x^12 - 335*x^11 + 306*x^10 - 258*x^9 + 202*x^8 - 146*x^7 + 97*x^6 - 59*x^5 + 32*x^4 - 16*x^3 + 7*x^2 - 2*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
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 1124000727777607680000 |
| The 1002 conjugacy class representatives for $S_{22}$ |
| Character table for $S_{22}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 44 sibling: | 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 | ${\href{/padicField/2.14.0.1}{14} }{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/13.7.0.1}{7} }$ | $22$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | $22$ | ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(79\)
| $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.3.0.1 | $x^{3} + 9 x + 76$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 79.6.0.1 | $x^{6} + 19 x^{3} + 28 x^{2} + 68 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 79.8.0.1 | $x^{8} + 60 x^{3} + 59 x^{2} + 48 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(22109\)
| $\Q_{22109}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(59513372976908430233\)
| $\Q_{59513372976908430233}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |