Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 3*x + 5)
gp: K = bnfinit(y^26 - 3*y + 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 3*x + 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 3*x + 5)
\( x^{26} - 3x + 5 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-73386655124093395346715550870539255750179290771484375\)
\(\medspace = -\,5^{23}\cdot 61\cdot 331\cdot 449\cdot 63806909\cdot 3529639661\cdot 3015121007861\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(107.97\) | 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: | $5^{23/24}61^{1/2}331^{1/2}449^{1/2}63806909^{1/2}3529639661^{1/2}3015121007861^{1/2}\approx 1.160111551152547e+19$ | ||
| Ramified primes: |
\(5\), \(61\), \(331\), \(449\), \(63806909\), \(3529639661\), \(3015121007861\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-30780\!\cdots\!28255}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{5}a^{25}-\frac{2}{5}a^{24}-\frac{1}{5}a^{23}+\frac{2}{5}a^{22}+\frac{1}{5}a^{21}-\frac{2}{5}a^{20}-\frac{1}{5}a^{19}+\frac{2}{5}a^{18}+\frac{1}{5}a^{17}-\frac{2}{5}a^{16}-\frac{1}{5}a^{15}+\frac{2}{5}a^{14}+\frac{1}{5}a^{13}-\frac{2}{5}a^{12}-\frac{1}{5}a^{11}+\frac{2}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{8}-\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a$
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
$C_{2}$, which has order $2$ (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: | $12$ | 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: |
$2a^{25}-3a^{24}+3a^{23}-2a^{22}+a^{21}-a^{20}+2a^{19}-3a^{18}+3a^{17}-2a^{16}+a^{15}-a^{14}+2a^{13}-3a^{12}+3a^{11}-2a^{10}+a^{9}-a^{8}+2a^{7}-3a^{6}+3a^{5}-2a^{4}+a^{3}-3a+4$, $\frac{52}{5}a^{25}+\frac{26}{5}a^{24}-\frac{47}{5}a^{23}-\frac{56}{5}a^{22}+\frac{27}{5}a^{21}+\frac{81}{5}a^{20}+\frac{23}{5}a^{19}-\frac{76}{5}a^{18}-\frac{68}{5}a^{17}+\frac{56}{5}a^{16}+\frac{108}{5}a^{15}+\frac{4}{5}a^{14}-\frac{118}{5}a^{13}-\frac{64}{5}a^{12}+\frac{108}{5}a^{11}+\frac{129}{5}a^{10}-\frac{48}{5}a^{9}-\frac{184}{5}a^{8}-\frac{62}{5}a^{7}+\frac{164}{5}a^{6}+\frac{132}{5}a^{5}-\frac{114}{5}a^{4}-\frac{222}{5}a^{3}+\frac{4}{5}a^{2}+\frac{272}{5}a$, $\frac{38}{5}a^{25}-\frac{11}{5}a^{24}-\frac{63}{5}a^{23}-\frac{44}{5}a^{22}+\frac{38}{5}a^{21}+\frac{89}{5}a^{20}+\frac{47}{5}a^{19}-\frac{49}{5}a^{18}-\frac{92}{5}a^{17}-\frac{31}{5}a^{16}+\frac{57}{5}a^{15}+\frac{66}{5}a^{14}-\frac{2}{5}a^{13}-\frac{61}{5}a^{12}-\frac{53}{5}a^{11}+\frac{6}{5}a^{10}+\frac{63}{5}a^{9}+\frac{69}{5}a^{8}+\frac{37}{5}a^{7}-\frac{24}{5}a^{6}-\frac{102}{5}a^{5}-\frac{111}{5}a^{4}+\frac{7}{5}a^{3}+\frac{146}{5}a^{2}+\frac{138}{5}a-33$, $\frac{23}{5}a^{25}+\frac{19}{5}a^{24}+\frac{37}{5}a^{23}+\frac{41}{5}a^{22}+\frac{23}{5}a^{21}+\frac{19}{5}a^{20}+\frac{7}{5}a^{19}-\frac{34}{5}a^{18}-\frac{37}{5}a^{17}-\frac{31}{5}a^{16}-\frac{63}{5}a^{15}-\frac{34}{5}a^{14}-\frac{7}{5}a^{13}-\frac{26}{5}a^{12}-\frac{13}{5}a^{11}+\frac{1}{5}a^{10}-\frac{52}{5}a^{9}-\frac{71}{5}a^{8}-\frac{63}{5}a^{7}-\frac{129}{5}a^{6}-\frac{117}{5}a^{5}-\frac{71}{5}a^{4}-\frac{83}{5}a^{3}-\frac{69}{5}a^{2}+\frac{18}{5}a-14$, $\frac{17}{5}a^{25}+\frac{91}{5}a^{24}-\frac{162}{5}a^{23}+\frac{109}{5}a^{22}+\frac{7}{5}a^{21}-\frac{154}{5}a^{20}+\frac{193}{5}a^{19}-\frac{121}{5}a^{18}-\frac{63}{5}a^{17}+\frac{221}{5}a^{16}-\frac{267}{5}a^{15}+\frac{109}{5}a^{14}+\frac{157}{5}a^{13}-\frac{369}{5}a^{12}+\frac{333}{5}a^{11}-\frac{26}{5}a^{10}-\frac{368}{5}a^{9}+\frac{551}{5}a^{8}-\frac{317}{5}a^{7}-\frac{211}{5}a^{6}+\frac{677}{5}a^{5}-\frac{659}{5}a^{4}+\frac{143}{5}a^{3}+\frac{579}{5}a^{2}-\frac{933}{5}a+120$, $\frac{3}{5}a^{25}+\frac{4}{5}a^{24}+\frac{2}{5}a^{23}+\frac{6}{5}a^{22}+\frac{8}{5}a^{21}+\frac{9}{5}a^{20}+\frac{12}{5}a^{19}+\frac{1}{5}a^{18}-\frac{2}{5}a^{17}-\frac{1}{5}a^{16}-\frac{3}{5}a^{15}+\frac{1}{5}a^{14}-\frac{2}{5}a^{13}-\frac{1}{5}a^{12}-\frac{13}{5}a^{11}-\frac{24}{5}a^{10}-\frac{22}{5}a^{9}-\frac{31}{5}a^{8}-\frac{8}{5}a^{7}+\frac{16}{5}a^{6}+\frac{28}{5}a^{5}+\frac{44}{5}a^{4}+\frac{27}{5}a^{3}+\frac{21}{5}a^{2}+\frac{3}{5}a-4$, $\frac{14}{5}a^{25}-\frac{38}{5}a^{24}+\frac{36}{5}a^{23}-\frac{2}{5}a^{22}-\frac{16}{5}a^{21}-\frac{13}{5}a^{20}+\frac{1}{5}a^{19}+\frac{23}{5}a^{18}-\frac{76}{5}a^{17}+\frac{32}{5}a^{16}+\frac{26}{5}a^{15}-\frac{57}{5}a^{14}+\frac{14}{5}a^{13}+\frac{22}{5}a^{12}+\frac{31}{5}a^{11}-\frac{47}{5}a^{10}+\frac{9}{5}a^{9}+\frac{112}{5}a^{8}-\frac{89}{5}a^{7}-\frac{12}{5}a^{6}+\frac{109}{5}a^{5}-\frac{68}{5}a^{4}-\frac{9}{5}a^{3}-\frac{52}{5}a^{2}+\frac{124}{5}a-26$, $\frac{3}{5}a^{25}+\frac{9}{5}a^{24}+\frac{7}{5}a^{23}+\frac{16}{5}a^{22}+\frac{18}{5}a^{21}+\frac{14}{5}a^{20}+\frac{17}{5}a^{19}+\frac{1}{5}a^{18}+\frac{8}{5}a^{17}+\frac{4}{5}a^{16}+\frac{7}{5}a^{15}+\frac{31}{5}a^{14}+\frac{8}{5}a^{13}+\frac{24}{5}a^{12}+\frac{7}{5}a^{11}-\frac{19}{5}a^{10}+\frac{8}{5}a^{9}-\frac{26}{5}a^{8}+\frac{12}{5}a^{7}+\frac{16}{5}a^{6}-\frac{7}{5}a^{5}+\frac{29}{5}a^{4}-\frac{33}{5}a^{3}-\frac{19}{5}a^{2}-\frac{22}{5}a-9$, $5a^{25}+4a^{24}-10a^{23}+6a^{22}+8a^{21}-16a^{20}+4a^{19}+14a^{18}-14a^{17}+3a^{16}+15a^{15}-17a^{14}+27a^{12}-24a^{11}-10a^{10}+32a^{9}-20a^{8}-10a^{7}+36a^{6}-22a^{5}-24a^{4}+57a^{3}-18a^{2}-46a+42$, $\frac{49}{5}a^{25}+\frac{7}{5}a^{24}-\frac{64}{5}a^{23}+\frac{48}{5}a^{22}+\frac{14}{5}a^{21}-\frac{78}{5}a^{20}+\frac{66}{5}a^{19}+\frac{43}{5}a^{18}-\frac{111}{5}a^{17}+\frac{57}{5}a^{16}+\frac{76}{5}a^{15}-\frac{152}{5}a^{14}+\frac{39}{5}a^{13}+\frac{132}{5}a^{12}-\frac{154}{5}a^{11}+\frac{8}{5}a^{10}+\frac{174}{5}a^{9}-\frac{163}{5}a^{8}-\frac{44}{5}a^{7}+\frac{198}{5}a^{6}-\frac{176}{5}a^{5}-\frac{38}{5}a^{4}+\frac{271}{5}a^{3}-\frac{207}{5}a^{2}-\frac{111}{5}a+41$, $\frac{27}{5}a^{25}+\frac{26}{5}a^{24}+\frac{28}{5}a^{23}+\frac{9}{5}a^{22}-\frac{13}{5}a^{21}-\frac{34}{5}a^{20}-\frac{37}{5}a^{19}-\frac{16}{5}a^{18}-\frac{33}{5}a^{17}-\frac{9}{5}a^{16}+\frac{58}{5}a^{15}+\frac{44}{5}a^{14}+\frac{17}{5}a^{13}+\frac{46}{5}a^{12}+\frac{18}{5}a^{11}-\frac{26}{5}a^{10}-\frac{68}{5}a^{9}-\frac{59}{5}a^{8}-\frac{17}{5}a^{7}-\frac{41}{5}a^{6}-\frac{13}{5}a^{5}+\frac{96}{5}a^{4}+\frac{53}{5}a^{3}+\frac{49}{5}a^{2}+\frac{82}{5}a-22$, $\frac{484}{5}a^{25}+\frac{537}{5}a^{24}+\frac{401}{5}a^{23}+\frac{108}{5}a^{22}-\frac{166}{5}a^{21}-\frac{418}{5}a^{20}-\frac{589}{5}a^{19}-\frac{532}{5}a^{18}-\frac{361}{5}a^{17}-\frac{108}{5}a^{16}+\frac{296}{5}a^{15}+\frac{663}{5}a^{14}+\frac{799}{5}a^{13}+\frac{747}{5}a^{12}+\frac{446}{5}a^{11}-\frac{197}{5}a^{10}-\frac{791}{5}a^{9}-\frac{1128}{5}a^{8}-\frac{1224}{5}a^{7}-\frac{712}{5}a^{6}+\frac{164}{5}a^{5}+\frac{872}{5}a^{4}+\frac{1431}{5}a^{3}+\frac{1553}{5}a^{2}+\frac{904}{5}a-302$
| 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: | \( 16503473635912132 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 16503473635912132 \cdot 2}{2\cdot\sqrt{73386655124093395346715550870539255750179290771484375}}\cr\approx \mathstrut & 1.44912251999500 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^26 - 3*x + 5)
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^26 - 3*x + 5, 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^26 - 3*x + 5);
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^26 - 3*x + 5);
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 403291461126605635584000000 |
| The 2436 conjugacy class representatives for $S_{26}$ |
| Character table for $S_{26}$ |
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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.3.0.1}{3} }^{8}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | $21{,}\,{\href{/padicField/7.5.0.1}{5} }$ | $26$ | $24{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/19.8.0.1}{8} }$ | $26$ | ${\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $21{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $24$ | $24$ | $1$ | $23$ | ||||
|
\(61\)
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.3.0.1 | $x^{3} + 7 x + 59$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 61.6.0.1 | $x^{6} + 49 x^{3} + 3 x^{2} + 29 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 61.11.0.1 | $x^{11} + 18 x + 59$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
|
\(331\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
|
\(449\)
| $\Q_{449}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
|
\(63806909\)
| $\Q_{63806909}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{63806909}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{63806909}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
|
\(3529639661\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(3015121007861\)
| $\Q_{3015121007861}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |