Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^31 - x - 4)
gp: K = bnfinit(y^31 - y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^31 - x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^31 - x - 4)
\( x^{31} - x - 4 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $31$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-18327886165296381817189229292602149543677096810012151519\)
\(\medspace = -\,23\cdot 1693\cdot 3697\cdot 19843\cdot 14132537\cdot 75771043\cdot 5991663366536301415229073661\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(60.63\) | 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: | $23^{1/2}1693^{1/2}3697^{1/2}19843^{1/2}14132537^{1/2}75771043^{1/2}5991663366536301415229073661^{1/2}\approx 4.28110805344789e+27$ | ||
| Ramified primes: |
\(23\), \(1693\), \(3697\), \(19843\), \(14132537\), \(75771043\), \(5991663366536301415229073661\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-18327\!\cdots\!51519}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2}a^{16}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{11}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{12}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{13}$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{14}$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{15}$
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: | $15$ | 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: |
$\frac{1}{2}a^{16}-\frac{1}{2}a-1$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{21}+\frac{1}{2}a^{16}-\frac{1}{2}a^{11}+\frac{1}{2}a^{6}-\frac{1}{2}a-1$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{25}+\frac{1}{2}a^{22}-\frac{1}{2}a^{19}+\frac{1}{2}a^{16}-\frac{1}{2}a^{13}+\frac{1}{2}a^{10}-\frac{1}{2}a^{7}+\frac{1}{2}a^{4}-\frac{1}{2}a-1$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{29}+\frac{1}{2}a^{28}-\frac{1}{2}a^{27}+\frac{1}{2}a^{26}-\frac{1}{2}a^{25}+\frac{1}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{1}{2}a-1$, $\frac{1}{2}a^{30}+\frac{1}{2}a^{29}-\frac{1}{2}a^{23}-a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-a^{19}-a^{18}-\frac{3}{2}a^{17}-\frac{3}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-a^{13}-a^{12}-a^{11}+a^{9}+\frac{1}{2}a^{8}+\frac{1}{2}a^{6}+\frac{3}{2}a^{5}+2a^{4}+2a^{3}+\frac{3}{2}a^{2}+\frac{3}{2}a+1$, $a^{30}-a^{29}+\frac{1}{2}a^{28}-\frac{1}{2}a^{27}+a^{26}-\frac{1}{2}a^{23}-\frac{1}{2}a^{22}+a^{21}-\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-\frac{3}{2}a^{18}+\frac{1}{2}a^{17}+a^{15}-\frac{3}{2}a^{13}+\frac{1}{2}a^{12}-a^{11}+2a^{10}-2a^{9}+\frac{1}{2}a^{8}-\frac{5}{2}a^{7}+2a^{6}+\frac{1}{2}a^{5}+\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{3}{2}a^{2}+2a-1$, $\frac{1}{2}a^{29}+a^{28}-2a^{26}+\frac{1}{2}a^{25}+\frac{1}{2}a^{24}+a^{23}-\frac{1}{2}a^{22}-a^{21}-\frac{3}{2}a^{20}+\frac{3}{2}a^{19}+2a^{18}-\frac{1}{2}a^{17}-2a^{16}-\frac{1}{2}a^{14}+3a^{13}-2a^{11}-\frac{5}{2}a^{10}+\frac{5}{2}a^{9}+2a^{8}+\frac{1}{2}a^{7}-a^{6}-\frac{7}{2}a^{5}+\frac{1}{2}a^{4}+5a^{3}+\frac{1}{2}a^{2}-3a-3$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-a^{14}+a^{12}-\frac{1}{2}a^{10}+\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-a^{7}+\frac{1}{2}a^{4}+\frac{3}{2}a^{3}+\frac{1}{2}a^{2}-\frac{1}{2}a-1$, $a^{30}+\frac{1}{2}a^{29}-a^{27}-\frac{3}{2}a^{26}-a^{25}-\frac{1}{2}a^{24}+\frac{1}{2}a^{23}+\frac{3}{2}a^{22}+2a^{21}+2a^{20}+a^{19}-a^{17}-2a^{16}-2a^{15}-\frac{3}{2}a^{14}+2a^{12}+\frac{7}{2}a^{11}+4a^{10}+\frac{5}{2}a^{9}+\frac{1}{2}a^{8}-\frac{3}{2}a^{7}-4a^{6}-4a^{5}-2a^{4}+3a^{2}+5a+5$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{28}-\frac{1}{2}a^{27}+\frac{1}{2}a^{26}-\frac{1}{2}a^{24}+a^{23}-\frac{1}{2}a^{21}+a^{20}-\frac{1}{2}a^{19}-a^{18}-a^{15}+\frac{1}{2}a^{14}+\frac{1}{2}a^{13}-\frac{3}{2}a^{12}+\frac{3}{2}a^{11}+a^{10}-\frac{3}{2}a^{9}-\frac{3}{2}a^{6}+\frac{3}{2}a^{4}-a^{3}+a^{2}+3a-1$, $\frac{1}{2}a^{30}+\frac{1}{2}a^{29}+\frac{1}{2}a^{28}-\frac{1}{2}a^{26}-a^{25}+a^{22}+a^{21}+a^{20}-\frac{3}{2}a^{18}-a^{17}+\frac{1}{2}a^{15}+\frac{3}{2}a^{14}+\frac{3}{2}a^{13}+a^{12}-\frac{1}{2}a^{11}-2a^{10}-a^{9}-a^{8}+a^{7}+2a^{6}+2a^{5}-\frac{3}{2}a^{3}-3a^{2}-a-1$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{28}-a^{27}+\frac{3}{2}a^{26}-a^{24}-\frac{1}{2}a^{23}+a^{22}+\frac{1}{2}a^{21}-\frac{3}{2}a^{20}+a^{19}+\frac{3}{2}a^{18}+\frac{1}{2}a^{17}-\frac{3}{2}a^{16}+\frac{1}{2}a^{15}+2a^{14}-\frac{1}{2}a^{13}-a^{12}+\frac{3}{2}a^{11}+2a^{10}-a^{9}-\frac{5}{2}a^{8}+a^{7}+\frac{3}{2}a^{6}-\frac{5}{2}a^{5}-a^{4}+\frac{3}{2}a^{3}+\frac{5}{2}a^{2}-\frac{9}{2}a-3$, $a^{29}-\frac{1}{2}a^{28}-a^{27}-\frac{3}{2}a^{25}-\frac{1}{2}a^{24}-\frac{1}{2}a^{22}+\frac{3}{2}a^{21}+a^{20}+a^{19}+a^{18}-\frac{1}{2}a^{17}-a^{15}-a^{14}-\frac{1}{2}a^{13}-a^{12}+a^{11}+\frac{1}{2}a^{10}+\frac{1}{2}a^{9}+a^{8}-\frac{3}{2}a^{7}-\frac{1}{2}a^{6}-a^{5}-a^{4}+a^{3}-\frac{1}{2}a^{2}+3a+3$, $\frac{3}{2}a^{29}+\frac{1}{2}a^{28}-a^{27}-a^{26}+\frac{1}{2}a^{25}+\frac{1}{2}a^{24}+a^{23}-a^{21}-\frac{3}{2}a^{20}+a^{19}+\frac{5}{2}a^{18}-\frac{1}{2}a^{17}-3a^{16}-a^{15}+\frac{7}{2}a^{14}+\frac{3}{2}a^{13}-2a^{12}-2a^{11}+\frac{3}{2}a^{10}+\frac{1}{2}a^{9}-a^{8}+a^{7}+a^{6}-\frac{5}{2}a^{5}-3a^{4}+\frac{9}{2}a^{3}+\frac{9}{2}a^{2}-3a-7$, $\frac{1}{2}a^{30}+\frac{1}{2}a^{29}+\frac{1}{2}a^{28}+\frac{3}{2}a^{27}+\frac{3}{2}a^{26}+2a^{25}+\frac{3}{2}a^{24}+a^{23}+2a^{22}+\frac{3}{2}a^{21}+3a^{20}+\frac{3}{2}a^{19}+2a^{18}+\frac{1}{2}a^{17}+a^{16}+\frac{3}{2}a^{15}+\frac{5}{2}a^{14}+\frac{5}{2}a^{13}+\frac{5}{2}a^{12}+\frac{3}{2}a^{11}+3a^{10}+\frac{5}{2}a^{9}+6a^{8}+4a^{7}+\frac{11}{2}a^{6}+2a^{5}+\frac{7}{2}a^{4}+3a^{3}+\frac{9}{2}a^{2}+5a+3$
| 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: | \( 4169052629874545.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 4169052629874545.0 \cdot 1}{2\cdot\sqrt{18327886165296381817189229292602149543677096810012151519}}\cr\approx \mathstrut & 0.914490532415702 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^31 - x - 4)
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^31 - x - 4, 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^31 - x - 4);
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^31 - x - 4);
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 8222838654177922817725562880000000 |
| The 6842 conjugacy class representatives for $S_{31}$ are not computed |
| Character table for $S_{31}$ is not computed |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}{,}\,{\href{/padicField/2.4.0.1}{4} }^{3}{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | $23{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $30{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | R | $28{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | $31$ | $30{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | $25{,}\,{\href{/padicField/53.6.0.1}{6} }$ | $30{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.7.0.1 | $x^{7} + 21 x + 18$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 23.18.0.1 | $x^{18} + x^{12} + 18 x^{11} + 2 x^{10} + x^{9} + 18 x^{8} + 3 x^{7} + 16 x^{6} + 21 x^{5} + 11 x^{3} + 3 x^{2} + 19 x + 5$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
|
\(1693\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
| Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
|
\(3697\)
| $\Q_{3697}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{3697}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
|
\(19843\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(14132537\)
| $\Q_{14132537}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $26$ | $1$ | $26$ | $0$ | $C_{26}$ | $[\ ]^{26}$ | ||
|
\(75771043\)
| $\Q_{75771043}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(599\!\cdots\!661\)
| $\Q_{59\!\cdots\!61}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |