Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 11*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 44*x^7 + 15*x^6 - 17*x^5 - 16*x^4 + 6*x^2 + x - 1)
gp: K = bnfinit(y^23 - 3*y^21 - 6*y^20 + y^19 + 17*y^18 + 18*y^17 - 10*y^16 - 43*y^15 - 25*y^14 + 33*y^13 + 60*y^12 + 11*y^11 - 54*y^10 - 47*y^9 + 12*y^8 + 44*y^7 + 15*y^6 - 17*y^5 - 16*y^4 + 6*y^2 + y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 11*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 44*x^7 + 15*x^6 - 17*x^5 - 16*x^4 + 6*x^2 + x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 11*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 44*x^7 + 15*x^6 - 17*x^5 - 16*x^4 + 6*x^2 + x - 1)
\( x^{23} - 3 x^{21} - 6 x^{20} + x^{19} + 17 x^{18} + 18 x^{17} - 10 x^{16} - 43 x^{15} - 25 x^{14} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[3, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1215307246476151828207513456186973\)
\(\medspace = 7\cdot 41\cdot 599\cdot 7069315563547560848845133621\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.45\) | 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: | $7^{1/2}41^{1/2}599^{1/2}7069315563547560848845133621^{1/2}\approx 3.4861257098334132e+16$ | ||
| Ramified primes: |
\(7\), \(41\), \(599\), \(7069315563547560848845133621\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{12153\!\cdots\!86973}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3}a^{21}+\frac{1}{3}a^{20}-\frac{1}{3}a^{19}-\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{33}a^{22}+\frac{5}{33}a^{21}+\frac{5}{33}a^{19}+\frac{5}{11}a^{18}-\frac{7}{33}a^{17}+\frac{16}{33}a^{16}-\frac{7}{33}a^{15}+\frac{10}{33}a^{14}-\frac{8}{33}a^{13}+\frac{4}{33}a^{12}+\frac{14}{33}a^{11}-\frac{7}{33}a^{10}-\frac{4}{11}a^{9}+\frac{1}{11}a^{8}+\frac{16}{33}a^{7}+\frac{1}{11}a^{6}-\frac{1}{11}a^{5}+\frac{1}{33}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{16}{33}a-\frac{2}{33}$
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: | $12$ | 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: |
$a$, $\frac{46}{33}a^{22}-\frac{34}{33}a^{21}-4a^{20}-\frac{199}{33}a^{19}+\frac{76}{11}a^{18}+\frac{767}{33}a^{17}+\frac{406}{33}a^{16}-\frac{916}{33}a^{15}-\frac{1817}{33}a^{14}-\frac{203}{33}a^{13}+\frac{1999}{33}a^{12}+\frac{2129}{33}a^{11}-\frac{619}{33}a^{10}-\frac{844}{11}a^{9}-\frac{372}{11}a^{8}+\frac{1198}{33}a^{7}+\frac{563}{11}a^{6}-\frac{2}{11}a^{5}-\frac{746}{33}a^{4}-\frac{40}{3}a^{3}+\frac{14}{3}a^{2}+\frac{188}{33}a-\frac{26}{33}$, $a^{22}-3a^{20}-6a^{19}+a^{18}+17a^{17}+18a^{16}-10a^{15}-43a^{14}-25a^{13}+33a^{12}+60a^{11}+11a^{10}-54a^{9}-47a^{8}+12a^{7}+44a^{6}+15a^{5}-17a^{4}-16a^{3}+6a+2$, $\frac{58}{33}a^{22}+\frac{37}{33}a^{21}-\frac{14}{3}a^{20}-\frac{149}{11}a^{19}-\frac{73}{11}a^{18}+\frac{881}{33}a^{17}+\frac{1610}{33}a^{16}+\frac{125}{11}a^{15}-\frac{2390}{33}a^{14}-\frac{987}{11}a^{13}+\frac{265}{33}a^{12}+\frac{3925}{33}a^{11}+\frac{3037}{33}a^{10}-\frac{1642}{33}a^{9}-\frac{4028}{33}a^{8}-\frac{545}{11}a^{7}+\frac{652}{11}a^{6}+\frac{734}{11}a^{5}+\frac{157}{33}a^{4}-32a^{3}-\frac{58}{3}a^{2}+\frac{73}{33}a+\frac{42}{11}$, $\frac{68}{33}a^{22}+\frac{76}{33}a^{21}-6a^{20}-\frac{617}{33}a^{19}-\frac{133}{11}a^{18}+\frac{1141}{33}a^{17}+\frac{2375}{33}a^{16}+\frac{778}{33}a^{15}-\frac{3247}{33}a^{14}-\frac{4537}{33}a^{13}+\frac{8}{33}a^{12}+\frac{5506}{33}a^{11}+\frac{4738}{33}a^{10}-\frac{668}{11}a^{9}-\frac{1934}{11}a^{8}-\frac{2542}{33}a^{7}+\frac{882}{11}a^{6}+\frac{1065}{11}a^{5}+\frac{233}{33}a^{4}-\frac{125}{3}a^{3}-\frac{80}{3}a^{2}+\frac{100}{33}a+\frac{161}{33}$, $\frac{67}{33}a^{22}+\frac{49}{33}a^{21}-\frac{17}{3}a^{20}-\frac{178}{11}a^{19}-\frac{83}{11}a^{18}+\frac{1049}{33}a^{17}+\frac{1919}{33}a^{16}+\frac{126}{11}a^{15}-\frac{2828}{33}a^{14}-\frac{1143}{11}a^{13}+\frac{433}{33}a^{12}+\frac{4513}{33}a^{11}+\frac{3304}{33}a^{10}-\frac{1948}{33}a^{9}-\frac{4463}{33}a^{8}-\frac{530}{11}a^{7}+\frac{716}{11}a^{6}+\frac{758}{11}a^{5}+\frac{100}{33}a^{4}-31a^{3}-\frac{55}{3}a^{2}-\frac{5}{33}a+\frac{47}{11}$, $\frac{12}{11}a^{22}+\frac{16}{11}a^{21}-4a^{20}-\frac{116}{11}a^{19}-\frac{62}{11}a^{18}+\frac{257}{11}a^{17}+\frac{456}{11}a^{16}+\frac{48}{11}a^{15}-\frac{749}{11}a^{14}-\frac{844}{11}a^{13}+\frac{268}{11}a^{12}+\frac{1257}{11}a^{11}+\frac{763}{11}a^{10}-\frac{760}{11}a^{9}-\frac{1251}{11}a^{8}-\frac{215}{11}a^{7}+\frac{806}{11}a^{6}+\frac{613}{11}a^{5}-\frac{164}{11}a^{4}-33a^{3}-11a^{2}+\frac{72}{11}a+\frac{53}{11}$, $\frac{17}{11}a^{22}+\frac{46}{33}a^{21}-\frac{10}{3}a^{20}-\frac{460}{33}a^{19}-\frac{108}{11}a^{18}+\frac{233}{11}a^{17}+\frac{1751}{33}a^{16}+\frac{875}{33}a^{15}-\frac{699}{11}a^{14}-\frac{3521}{33}a^{13}-\frac{240}{11}a^{12}+\frac{3838}{33}a^{11}+\frac{4087}{33}a^{10}-\frac{656}{33}a^{9}-\frac{4456}{33}a^{8}-\frac{2693}{33}a^{7}+\frac{546}{11}a^{6}+\frac{917}{11}a^{5}+\frac{215}{11}a^{4}-\frac{106}{3}a^{3}-25a^{2}+\frac{31}{33}a+\frac{184}{33}$, $\frac{2}{11}a^{22}-\frac{58}{33}a^{21}+\frac{1}{3}a^{20}+\frac{118}{33}a^{19}+\frac{96}{11}a^{18}-\frac{25}{11}a^{17}-\frac{773}{33}a^{16}-\frac{746}{33}a^{15}+\frac{174}{11}a^{14}+\frac{1943}{33}a^{13}+\frac{294}{11}a^{12}-\frac{1522}{33}a^{11}-\frac{2539}{33}a^{10}-\frac{193}{33}a^{9}+\frac{2317}{33}a^{8}+\frac{1724}{33}a^{7}-\frac{203}{11}a^{6}-\frac{578}{11}a^{5}-\frac{119}{11}a^{4}+\frac{52}{3}a^{3}+17a^{2}-\frac{52}{33}a-\frac{166}{33}$, $\frac{17}{33}a^{22}+\frac{19}{33}a^{21}-2a^{20}-\frac{146}{33}a^{19}-\frac{14}{11}a^{18}+\frac{343}{33}a^{17}+\frac{503}{33}a^{16}-\frac{86}{33}a^{15}-\frac{886}{33}a^{14}-\frac{697}{33}a^{13}+\frac{596}{33}a^{12}+\frac{1228}{33}a^{11}+\frac{244}{33}a^{10}-\frac{365}{11}a^{9}-\frac{313}{11}a^{8}+\frac{371}{33}a^{7}+\frac{281}{11}a^{6}+\frac{71}{11}a^{5}-\frac{379}{33}a^{4}-\frac{20}{3}a^{3}+\frac{1}{3}a^{2}+\frac{25}{33}a+\frac{32}{33}$, $\frac{27}{11}a^{22}-\frac{35}{33}a^{21}-\frac{19}{3}a^{20}-\frac{376}{33}a^{19}+\frac{64}{11}a^{18}+\frac{383}{11}a^{17}+\frac{845}{33}a^{16}-\frac{952}{33}a^{15}-\frac{852}{11}a^{14}-\frac{626}{33}a^{13}+\frac{834}{11}a^{12}+\frac{2905}{33}a^{11}-\frac{710}{33}a^{10}-\frac{3524}{33}a^{9}-\frac{1429}{33}a^{8}+\frac{1747}{33}a^{7}+\frac{752}{11}a^{6}-\frac{114}{11}a^{5}-\frac{424}{11}a^{4}-\frac{40}{3}a^{3}+11a^{2}+\frac{343}{33}a-\frac{140}{33}$, $\frac{60}{11}a^{22}+\frac{97}{33}a^{21}-\frac{46}{3}a^{20}-\frac{1366}{33}a^{19}-\frac{167}{11}a^{18}+\frac{977}{11}a^{17}+\frac{4871}{33}a^{16}+\frac{533}{33}a^{15}-\frac{2645}{11}a^{14}-\frac{8810}{33}a^{13}+\frac{647}{11}a^{12}+\frac{12607}{33}a^{11}+\frac{8530}{33}a^{10}-\frac{6197}{33}a^{9}-\frac{12352}{33}a^{8}-\frac{3863}{33}a^{7}+\frac{2248}{11}a^{6}+\frac{2130}{11}a^{5}-\frac{61}{11}a^{4}-\frac{298}{3}a^{3}-49a^{2}+\frac{376}{33}a+\frac{388}{33}$
| 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: | \( 53774282.4892 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{10}\cdot 53774282.4892 \cdot 1}{2\cdot\sqrt{1215307246476151828207513456186973}}\cr\approx \mathstrut & 0.591684585834 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^23 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 11*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 44*x^7 + 15*x^6 - 17*x^5 - 16*x^4 + 6*x^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^23 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 11*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 44*x^7 + 15*x^6 - 17*x^5 - 16*x^4 + 6*x^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^23 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 11*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 44*x^7 + 15*x^6 - 17*x^5 - 16*x^4 + 6*x^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^23 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 11*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 44*x^7 + 15*x^6 - 17*x^5 - 16*x^4 + 6*x^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 25852016738884976640000 |
| The 1255 conjugacy class representatives for $S_{23}$ |
| Character table for $S_{23}$ |
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 46 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 | ${\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.10.0.1}{10} }$ | $18{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.9.0.1}{9} }$ | R | ${\href{/padicField/11.7.0.1}{7} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | $18{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/37.5.0.1}{5} }$ | R | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 7.13.0.1 | $x^{13} + 6 x^{2} + 4$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
|
\(41\)
| 41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 41.3.0.1 | $x^{3} + x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 41.18.0.1 | $x^{18} + x^{11} + 7 x^{10} + 20 x^{9} + 23 x^{8} + 35 x^{7} + 38 x^{6} + 24 x^{5} + 12 x^{4} + 29 x^{3} + 10 x^{2} + 6 x + 6$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
|
\(599\)
| $\Q_{599}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(706\!\cdots\!621\)
| $\Q_{70\!\cdots\!21}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{70\!\cdots\!21}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |