Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 13*x^18 + 14*x^17 + 70*x^16 - 82*x^15 - 203*x^14 + 261*x^13 + 351*x^12 - 492*x^11 - 397*x^10 + 566*x^9 + 346*x^8 - 403*x^7 - 262*x^6 + 180*x^5 + 142*x^4 - 44*x^3 - 36*x^2 + 1)
gp: K = bnfinit(y^20 - y^19 - 13*y^18 + 14*y^17 + 70*y^16 - 82*y^15 - 203*y^14 + 261*y^13 + 351*y^12 - 492*y^11 - 397*y^10 + 566*y^9 + 346*y^8 - 403*y^7 - 262*y^6 + 180*y^5 + 142*y^4 - 44*y^3 - 36*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 - 13*x^18 + 14*x^17 + 70*x^16 - 82*x^15 - 203*x^14 + 261*x^13 + 351*x^12 - 492*x^11 - 397*x^10 + 566*x^9 + 346*x^8 - 403*x^7 - 262*x^6 + 180*x^5 + 142*x^4 - 44*x^3 - 36*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 - 13*x^18 + 14*x^17 + 70*x^16 - 82*x^15 - 203*x^14 + 261*x^13 + 351*x^12 - 492*x^11 - 397*x^10 + 566*x^9 + 346*x^8 - 403*x^7 - 262*x^6 + 180*x^5 + 142*x^4 - 44*x^3 - 36*x^2 + 1)
\( x^{20} - x^{19} - 13 x^{18} + 14 x^{17} + 70 x^{16} - 82 x^{15} - 203 x^{14} + 261 x^{13} + 351 x^{12} + \cdots + 1 \)
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: | $[10, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-195532563918689056742899679\)
\(\medspace = -\,16493^{2}\cdot 520151\cdot 1175561^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.63\) | 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: | $16493^{1/2}520151^{1/2}1175561^{1/2}\approx 100423911.52322002$ | ||
| Ramified primes: |
\(16493\), \(520151\), \(1175561\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-520151}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5}a^{17}-\frac{1}{5}a^{16}-\frac{1}{5}a^{15}+\frac{2}{5}a^{11}+\frac{1}{5}a^{10}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{18}-\frac{2}{5}a^{16}-\frac{1}{5}a^{15}+\frac{2}{5}a^{12}-\frac{2}{5}a^{11}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}+\frac{2}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{19}+\frac{2}{5}a^{16}-\frac{2}{5}a^{15}+\frac{2}{5}a^{13}-\frac{2}{5}a^{12}-\frac{2}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$
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: |
$\frac{4}{5}a^{18}-\frac{4}{5}a^{17}-\frac{44}{5}a^{16}+10a^{15}+38a^{14}-50a^{13}-\frac{407}{5}a^{12}+\frac{639}{5}a^{11}+93a^{10}-\frac{891}{5}a^{9}-\frac{341}{5}a^{8}+\frac{697}{5}a^{7}+\frac{274}{5}a^{6}-\frac{346}{5}a^{5}-\frac{194}{5}a^{4}+24a^{3}+\frac{52}{5}a^{2}-\frac{3}{5}a-1$, $\frac{1}{5}a^{18}+\frac{6}{5}a^{17}-\frac{18}{5}a^{16}-\frac{62}{5}a^{15}+26a^{14}+50a^{13}-\frac{493}{5}a^{12}-100a^{11}+\frac{1072}{5}a^{10}+\frac{551}{5}a^{9}-\frac{1372}{5}a^{8}-88a^{7}+\frac{1047}{5}a^{6}+\frac{388}{5}a^{5}-\frac{504}{5}a^{4}-\frac{262}{5}a^{3}+\frac{148}{5}a^{2}+\frac{64}{5}a-\frac{4}{5}$, $\frac{3}{5}a^{18}+\frac{2}{5}a^{17}-\frac{38}{5}a^{16}-3a^{15}+40a^{14}+6a^{13}-\frac{569}{5}a^{12}+\frac{18}{5}a^{11}+194a^{10}-\frac{92}{5}a^{9}-\frac{1072}{5}a^{8}-\frac{6}{5}a^{7}+\frac{833}{5}a^{6}+\frac{168}{5}a^{5}-\frac{438}{5}a^{4}-32a^{3}+\frac{104}{5}a^{2}+\frac{64}{5}a$, $\frac{4}{5}a^{18}-\frac{3}{5}a^{17}-9a^{16}+\frac{34}{5}a^{15}+42a^{14}-32a^{13}-\frac{542}{5}a^{12}+\frac{411}{5}a^{11}+\frac{891}{5}a^{10}-\frac{631}{5}a^{9}-206a^{8}+\frac{573}{5}a^{7}+\frac{867}{5}a^{6}-52a^{5}-\frac{513}{5}a^{4}+\frac{19}{5}a^{3}+\frac{182}{5}a^{2}+4a-\frac{12}{5}$, $\frac{4}{5}a^{18}-\frac{7}{5}a^{17}-\frac{41}{5}a^{16}+\frac{78}{5}a^{15}+32a^{14}-70a^{13}-\frac{292}{5}a^{12}+\frac{818}{5}a^{11}+\frac{252}{5}a^{10}-\frac{1096}{5}a^{9}-\frac{139}{5}a^{8}+\frac{914}{5}a^{7}+34a^{6}-\frac{534}{5}a^{5}-\frac{167}{5}a^{4}+\frac{213}{5}a^{3}+\frac{62}{5}a^{2}-\frac{27}{5}a-\frac{9}{5}$, $\frac{4}{5}a^{17}-\frac{4}{5}a^{16}-\frac{44}{5}a^{15}+10a^{14}+38a^{13}-50a^{12}-\frac{407}{5}a^{11}+\frac{639}{5}a^{10}+93a^{9}-\frac{891}{5}a^{8}-\frac{341}{5}a^{7}+\frac{697}{5}a^{6}+\frac{274}{5}a^{5}-\frac{346}{5}a^{4}-\frac{194}{5}a^{3}+24a^{2}+\frac{47}{5}a+\frac{2}{5}$, $\frac{3}{5}a^{18}-\frac{4}{5}a^{17}-\frac{32}{5}a^{16}+\frac{46}{5}a^{15}+27a^{14}-43a^{13}-\frac{289}{5}a^{12}+\frac{531}{5}a^{11}+\frac{349}{5}a^{10}-\frac{762}{5}a^{9}-\frac{288}{5}a^{8}+\frac{668}{5}a^{7}+46a^{6}-\frac{368}{5}a^{5}-\frac{169}{5}a^{4}+\frac{126}{5}a^{3}+\frac{74}{5}a^{2}-\frac{24}{5}a-\frac{3}{5}$, $\frac{3}{5}a^{19}+\frac{1}{5}a^{18}-\frac{43}{5}a^{17}+\frac{2}{5}a^{16}+\frac{251}{5}a^{15}-18a^{14}-\frac{769}{5}a^{13}+\frac{471}{5}a^{12}+\frac{1342}{5}a^{11}-\frac{1078}{5}a^{10}-\frac{1403}{5}a^{9}+\frac{1247}{5}a^{8}+\frac{999}{5}a^{7}-\frac{771}{5}a^{6}-119a^{5}+62a^{4}+47a^{3}-\frac{99}{5}a^{2}-\frac{16}{5}a+\frac{7}{5}$, $\frac{1}{5}a^{19}+\frac{6}{5}a^{18}-\frac{18}{5}a^{17}-\frac{62}{5}a^{16}+26a^{15}+50a^{14}-\frac{493}{5}a^{13}-100a^{12}+\frac{1072}{5}a^{11}+\frac{551}{5}a^{10}-\frac{1372}{5}a^{9}-88a^{8}+\frac{1047}{5}a^{7}+\frac{388}{5}a^{6}-\frac{504}{5}a^{5}-\frac{262}{5}a^{4}+\frac{148}{5}a^{3}+\frac{64}{5}a^{2}-\frac{4}{5}a$, $\frac{3}{5}a^{19}-\frac{3}{5}a^{18}-6a^{17}+\frac{32}{5}a^{16}+\frac{117}{5}a^{15}-27a^{14}-\frac{229}{5}a^{13}+\frac{283}{5}a^{12}+\frac{261}{5}a^{11}-\frac{299}{5}a^{10}-\frac{237}{5}a^{9}+\frac{132}{5}a^{8}+\frac{211}{5}a^{7}+\frac{17}{5}a^{6}-24a^{5}-\frac{62}{5}a^{4}+\frac{21}{5}a^{3}+\frac{39}{5}a^{2}+\frac{4}{5}a-\frac{6}{5}$, $\frac{3}{5}a^{18}-\frac{3}{5}a^{17}-\frac{33}{5}a^{16}+8a^{15}+28a^{14}-42a^{13}-\frac{279}{5}a^{12}+\frac{553}{5}a^{11}+49a^{10}-\frac{777}{5}a^{9}-\frac{62}{5}a^{8}+\frac{609}{5}a^{7}+\frac{23}{5}a^{6}-\frac{332}{5}a^{5}-\frac{43}{5}a^{4}+29a^{3}-\frac{6}{5}a^{2}-\frac{21}{5}a$, $\frac{7}{5}a^{17}-\frac{7}{5}a^{16}-\frac{77}{5}a^{15}+17a^{14}+67a^{13}-83a^{12}-\frac{736}{5}a^{11}+\frac{1042}{5}a^{10}+180a^{9}-\frac{1438}{5}a^{8}-\frac{743}{5}a^{7}+\frac{1116}{5}a^{6}+\frac{587}{5}a^{5}-\frac{523}{5}a^{4}-\frac{357}{5}a^{3}+29a^{2}+\frac{81}{5}a+\frac{6}{5}$, $\frac{2}{5}a^{17}+\frac{3}{5}a^{16}-\frac{32}{5}a^{15}-3a^{14}+38a^{13}-5a^{12}-\frac{546}{5}a^{11}+\frac{282}{5}a^{10}+164a^{9}-\frac{618}{5}a^{8}-\frac{688}{5}a^{7}+\frac{551}{5}a^{6}+\frac{432}{5}a^{5}-\frac{248}{5}a^{4}-\frac{252}{5}a^{3}+20a^{2}+\frac{56}{5}a-\frac{14}{5}$, $a^{19}-\frac{61}{5}a^{17}+\frac{6}{5}a^{16}+\frac{316}{5}a^{15}-13a^{14}-184a^{13}+57a^{12}+\frac{1683}{5}a^{11}-\frac{636}{5}a^{10}-408a^{9}+\frac{739}{5}a^{8}+\frac{1689}{5}a^{7}-\frac{398}{5}a^{6}-\frac{996}{5}a^{5}+\frac{69}{5}a^{4}+\frac{391}{5}a^{3}-2a^{2}-\frac{58}{5}a+\frac{7}{5}$
| 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: | \( 434433.516854 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{5}\cdot 434433.516854 \cdot 1}{2\cdot\sqrt{195532563918689056742899679}}\cr\approx \mathstrut & 0.155769734355 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 13*x^18 + 14*x^17 + 70*x^16 - 82*x^15 - 203*x^14 + 261*x^13 + 351*x^12 - 492*x^11 - 397*x^10 + 566*x^9 + 346*x^8 - 403*x^7 - 262*x^6 + 180*x^5 + 142*x^4 - 44*x^3 - 36*x^2 + 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 - 13*x^18 + 14*x^17 + 70*x^16 - 82*x^15 - 203*x^14 + 261*x^13 + 351*x^12 - 492*x^11 - 397*x^10 + 566*x^9 + 346*x^8 - 403*x^7 - 262*x^6 + 180*x^5 + 142*x^4 - 44*x^3 - 36*x^2 + 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 - 13*x^18 + 14*x^17 + 70*x^16 - 82*x^15 - 203*x^14 + 261*x^13 + 351*x^12 - 492*x^11 - 397*x^10 + 566*x^9 + 346*x^8 - 403*x^7 - 262*x^6 + 180*x^5 + 142*x^4 - 44*x^3 - 36*x^2 + 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 - 13*x^18 + 14*x^17 + 70*x^16 - 82*x^15 - 203*x^14 + 261*x^13 + 351*x^12 - 492*x^11 - 397*x^10 + 566*x^9 + 346*x^8 - 403*x^7 - 262*x^6 + 180*x^5 + 142*x^4 - 44*x^3 - 36*x^2 + 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
$C_2^{10}.S_{10}$ (as 20T1110):
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 3715891200 |
| The 481 conjugacy class representatives for $C_2^{10}.S_{10}$ |
| Character table for $C_2^{10}.S_{10}$ |
Intermediate fields
| 10.6.19388527573.1 |
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 20 siblings: | data not computed |
| Degree 40 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 | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }$ | $20$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ | $18{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(16493\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(520151\)
| $\Q_{520151}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{520151}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(1175561\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |