Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^20 - 4*x^19 + 3*x^18 + 3*x^17 + 6*x^16 - 9*x^15 + x^14 - 3*x^13 + 9*x^12 - 10*x^11 + x^10 - 2*x^9 + 8*x^8 - 5*x^7 - x^5 + 4*x^4 - 2*x^3 + x^2 + 1)
gp: K = bnfinit(y^22 - y^20 - 4*y^19 + 3*y^18 + 3*y^17 + 6*y^16 - 9*y^15 + y^14 - 3*y^13 + 9*y^12 - 10*y^11 + y^10 - 2*y^9 + 8*y^8 - 5*y^7 - y^5 + 4*y^4 - 2*y^3 + y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^20 - 4*x^19 + 3*x^18 + 3*x^17 + 6*x^16 - 9*x^15 + x^14 - 3*x^13 + 9*x^12 - 10*x^11 + x^10 - 2*x^9 + 8*x^8 - 5*x^7 - x^5 + 4*x^4 - 2*x^3 + x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^20 - 4*x^19 + 3*x^18 + 3*x^17 + 6*x^16 - 9*x^15 + x^14 - 3*x^13 + 9*x^12 - 10*x^11 + x^10 - 2*x^9 + 8*x^8 - 5*x^7 - x^5 + 4*x^4 - 2*x^3 + x^2 + 1)
\( x^{22} - x^{20} - 4 x^{19} + 3 x^{18} + 3 x^{17} + 6 x^{16} - 9 x^{15} + x^{14} - 3 x^{13} + 9 x^{12} + \cdots + 1 \)
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: |
\(-6250054957079127310759347\)
\(\medspace = -\,3^{11}\cdot 12917^{2}\cdot 459847^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.40\) | 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: | $3^{1/2}12917^{1/2}459847^{1/2}\approx 133489.8164542899$ | ||
| Ramified primes: |
\(3\), \(12917\), \(459847\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{10919}a^{21}-\frac{1899}{10919}a^{20}+\frac{2930}{10919}a^{19}+\frac{4616}{10919}a^{18}+\frac{2176}{10919}a^{17}-\frac{4839}{10919}a^{16}-\frac{4531}{10919}a^{15}+\frac{188}{10919}a^{14}+\frac{3316}{10919}a^{13}+\frac{3176}{10919}a^{12}-\frac{3927}{10919}a^{11}-\frac{314}{10919}a^{10}-\frac{4258}{10919}a^{9}-\frac{5039}{10919}a^{8}+\frac{4025}{10919}a^{7}-\frac{180}{10919}a^{6}+\frac{3331}{10919}a^{5}-\frac{3469}{10919}a^{4}+\frac{3478}{10919}a^{3}+\frac{1271}{10919}a^{2}-\frac{529}{10919}a+\frac{23}{10919}$
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: |
\( \frac{4653}{10919} a^{21} - \frac{2576}{10919} a^{20} - \frac{4541}{10919} a^{19} - \frac{10344}{10919} a^{18} + \frac{24853}{10919} a^{17} - \frac{889}{10919} a^{16} + \frac{1846}{10919} a^{15} - \frac{42432}{10919} a^{14} + \frac{44477}{10919} a^{13} + \frac{4521}{10919} a^{12} + \frac{16994}{10919} a^{11} - \frac{63410}{10919} a^{10} + \frac{27349}{10919} a^{9} + \frac{7545}{10919} a^{8} + \frac{2240}{10919} a^{7} - \frac{40453}{10919} a^{6} + \frac{16001}{10919} a^{5} + \frac{18863}{10919} a^{4} + \frac{1176}{10919} a^{3} - \frac{15054}{10919} a^{2} + \frac{6257}{10919} a + \frac{8748}{10919} \)
(order $6$)
| 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: |
$\frac{2576}{10919}a^{21}-\frac{112}{10919}a^{20}-\frac{8268}{10919}a^{19}-\frac{10894}{10919}a^{18}+\frac{14848}{10919}a^{17}+\frac{26072}{10919}a^{16}+\frac{555}{10919}a^{15}-\frac{39824}{10919}a^{14}-\frac{18480}{10919}a^{13}+\frac{24883}{10919}a^{12}+\frac{16880}{10919}a^{11}-\frac{22696}{10919}a^{10}-\frac{16851}{10919}a^{9}+\frac{34984}{10919}a^{8}+\frac{17188}{10919}a^{7}-\frac{16001}{10919}a^{6}-\frac{23516}{10919}a^{5}+\frac{17436}{10919}a^{4}+\frac{5748}{10919}a^{3}-\frac{1604}{10919}a^{2}-\frac{8748}{10919}a+\frac{4653}{10919}$, $\frac{2977}{10919}a^{21}-\frac{8200}{10919}a^{20}-\frac{12590}{10919}a^{19}-\frac{5189}{10919}a^{18}+\frac{57580}{10919}a^{17}+\frac{29215}{10919}a^{16}-\frac{36579}{10919}a^{15}-\frac{139140}{10919}a^{14}+\frac{11875}{10919}a^{13}+\frac{97369}{10919}a^{12}+\frac{112760}{10919}a^{11}-\frac{94015}{10919}a^{10}-\frac{53702}{10919}a^{9}+\frac{34360}{10919}a^{8}+\frac{80715}{10919}a^{7}-\frac{33586}{10919}a^{6}-\frac{52660}{10919}a^{5}+\frac{13080}{10919}a^{4}+\frac{35551}{10919}a^{3}-\frac{16045}{10919}a^{2}-\frac{24335}{10919}a-\frac{7962}{10919}$, $\frac{3958}{10919}a^{21}-\frac{3970}{10919}a^{20}-\frac{9957}{10919}a^{19}-\frac{8278}{10919}a^{18}+\frac{30274}{10919}a^{17}+\frac{21002}{10919}a^{16}-\frac{15619}{10919}a^{15}-\frac{63902}{10919}a^{14}+\frac{90}{10919}a^{13}+\frac{57434}{10919}a^{12}+\frac{27428}{10919}a^{11}-\frac{41722}{10919}a^{10}-\frac{48823}{10919}a^{9}+\frac{37408}{10919}a^{8}+\frac{21967}{10919}a^{7}-\frac{2705}{10919}a^{6}-\frac{16973}{10919}a^{5}+\frac{16719}{10919}a^{4}+\frac{18903}{10919}a^{3}-\frac{3041}{10919}a^{2}-\frac{8253}{10919}a+\frac{3682}{10919}$, $\frac{3628}{10919}a^{21}-\frac{21521}{10919}a^{20}-\frac{26904}{10919}a^{19}+\frac{8021}{10919}a^{18}+\frac{131119}{10919}a^{17}+\frac{45536}{10919}a^{16}-\frac{125482}{10919}a^{15}-\frac{289727}{10919}a^{14}+\frac{52305}{10919}a^{13}+\frac{243201}{10919}a^{12}+\frac{209600}{10919}a^{11}-\frac{200158}{10919}a^{10}-\frac{106829}{10919}a^{9}+\frac{73347}{10919}a^{8}+\frac{145944}{10919}a^{7}-\frac{85252}{10919}a^{6}-\frac{78898}{10919}a^{5}+\frac{25913}{10919}a^{4}+\frac{72253}{10919}a^{3}-\frac{29387}{10919}a^{2}-\frac{41144}{10919}a-\frac{25746}{10919}$, $\frac{1870}{10919}a^{21}-\frac{13374}{10919}a^{20}-\frac{13157}{10919}a^{19}+\frac{5910}{10919}a^{18}+\frac{72766}{10919}a^{17}+\frac{13840}{10919}a^{16}-\frac{65340}{10919}a^{15}-\frac{150714}{10919}a^{14}+\frac{42604}{10919}a^{13}+\frac{108374}{10919}a^{12}+\frac{103268}{10919}a^{11}-\frac{106744}{10919}a^{10}-\frac{35266}{10919}a^{9}+\frac{32924}{10919}a^{8}+\frac{69073}{10919}a^{7}-\frac{41787}{10919}a^{6}-\frac{27617}{10919}a^{5}+\frac{9775}{10919}a^{4}+\frac{39812}{10919}a^{3}-\frac{14491}{10919}a^{2}-\frac{17439}{10919}a-\frac{22504}{10919}$, $\frac{12891}{10919}a^{21}+\frac{389}{10919}a^{20}-\frac{20029}{10919}a^{19}-\frac{58289}{10919}a^{18}+\frac{43581}{10919}a^{17}+\frac{77131}{10919}a^{16}+\frac{83962}{10919}a^{15}-\frac{153376}{10919}a^{14}-\frac{66843}{10919}a^{13}-\frac{15353}{10919}a^{12}+\frac{172231}{10919}a^{11}-\frac{73258}{10919}a^{10}-\frac{43741}{10919}a^{9}-\frac{33375}{10919}a^{8}+\frac{119296}{10919}a^{7}-\frac{27390}{10919}a^{6}-\frac{26344}{10919}a^{5}-\frac{27412}{10919}a^{4}+\frac{56079}{10919}a^{3}-\frac{4958}{10919}a^{2}+\frac{5036}{10919}a+\frac{1680}{10919}$, $\frac{15208}{10919}a^{21}+\frac{763}{10919}a^{20}-\frac{22837}{10919}a^{19}-\frac{74556}{10919}a^{18}+\frac{51714}{10919}a^{17}+\frac{100819}{10919}a^{16}+\frac{122470}{10919}a^{15}-\frac{209135}{10919}a^{14}-\frac{114323}{10919}a^{13}-\frac{37805}{10919}a^{12}+\frac{289008}{10919}a^{11}-\frac{58304}{10919}a^{10}-\frac{71508}{10919}a^{9}-\frac{123679}{10919}a^{8}+\frac{164071}{10919}a^{7}-\frac{7690}{10919}a^{6}-\frac{28150}{10919}a^{5}-\frac{61458}{10919}a^{4}+\frac{78221}{10919}a^{3}+\frac{2738}{10919}a^{2}+\frac{13190}{10919}a-\frac{21462}{10919}$, $\frac{16238}{10919}a^{21}-\frac{706}{10919}a^{20}-\frac{29500}{10919}a^{19}-\frac{69841}{10919}a^{18}+\frac{65518}{10919}a^{17}+\frac{106632}{10919}a^{16}+\frac{85196}{10919}a^{15}-\frac{212037}{10919}a^{14}-\frac{94652}{10919}a^{13}+\frac{12370}{10919}a^{12}+\frac{229633}{10919}a^{11}-\frac{75992}{10919}a^{10}-\frac{67810}{10919}a^{9}-\frac{50891}{10919}a^{8}+\frac{127844}{10919}a^{7}-\frac{29305}{10919}a^{6}-\frac{25786}{10919}a^{5}-\frac{20339}{10919}a^{4}+\frac{57291}{10919}a^{3}+\frac{1588}{10919}a^{2}+\frac{3351}{10919}a-\frac{19610}{10919}$, $\frac{18716}{10919}a^{21}-\frac{339}{10919}a^{20}-\frac{30095}{10919}a^{19}-\frac{74505}{10919}a^{18}+\frac{63660}{10919}a^{17}+\frac{104652}{10919}a^{16}+\frac{93029}{10919}a^{15}-\frac{193852}{10919}a^{14}-\frac{88692}{10919}a^{13}+\frac{9899}{10919}a^{12}+\frac{183680}{10919}a^{11}-\frac{67916}{10919}a^{10}-\frac{71380}{10919}a^{9}-\frac{2521}{10919}a^{8}+\frac{99990}{10919}a^{7}-\frac{27666}{10919}a^{6}-\frac{48170}{10919}a^{5}-\frac{12349}{10919}a^{4}+\frac{60684}{10919}a^{3}-\frac{4465}{10919}a^{2}+\frac{2769}{10919}a-\frac{6292}{10919}$, $\frac{16868}{10919}a^{21}+\frac{25852}{10919}a^{20}-\frac{17992}{10919}a^{19}-\frac{109891}{10919}a^{18}-\frac{59505}{10919}a^{17}+\frac{148139}{10919}a^{16}+\frac{255229}{10919}a^{15}-\frac{6245}{10919}a^{14}-\frac{298562}{10919}a^{13}-\frac{181469}{10919}a^{12}+\frac{146884}{10919}a^{11}+\frac{173867}{10919}a^{10}-\frac{107952}{10919}a^{9}-\frac{124465}{10919}a^{8}+\frac{64872}{10919}a^{7}+\frac{119351}{10919}a^{6}-\frac{45542}{10919}a^{5}-\frac{76604}{10919}a^{4}+\frac{20955}{10919}a^{3}+\frac{59826}{10919}a^{2}+\frac{19489}{10919}a-\frac{5120}{10919}$
| 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: | \( 4357.91042138 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 4357.91042138 \cdot 1}{6\cdot\sqrt{6250054957079127310759347}}\cr\approx \mathstrut & 0.175050631724 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - x^20 - 4*x^19 + 3*x^18 + 3*x^17 + 6*x^16 - 9*x^15 + x^14 - 3*x^13 + 9*x^12 - 10*x^11 + x^10 - 2*x^9 + 8*x^8 - 5*x^7 - x^5 + 4*x^4 - 2*x^3 + 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^22 - x^20 - 4*x^19 + 3*x^18 + 3*x^17 + 6*x^16 - 9*x^15 + x^14 - 3*x^13 + 9*x^12 - 10*x^11 + x^10 - 2*x^9 + 8*x^8 - 5*x^7 - x^5 + 4*x^4 - 2*x^3 + 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^22 - x^20 - 4*x^19 + 3*x^18 + 3*x^17 + 6*x^16 - 9*x^15 + x^14 - 3*x^13 + 9*x^12 - 10*x^11 + x^10 - 2*x^9 + 8*x^8 - 5*x^7 - x^5 + 4*x^4 - 2*x^3 + 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^22 - x^20 - 4*x^19 + 3*x^18 + 3*x^17 + 6*x^16 - 9*x^15 + x^14 - 3*x^13 + 9*x^12 - 10*x^11 + x^10 - 2*x^9 + 8*x^8 - 5*x^7 - x^5 + 4*x^4 - 2*x^3 + 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
$C_2\times S_{11}$ (as 22T47):
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 79833600 |
| The 112 conjugacy class representatives for $C_2\times S_{11}$ are not computed |
| Character table for $C_2\times S_{11}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 11.1.5939843699.1 |
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 22 sibling: | data not computed |
| 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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.6.0.1}{6} }^{2}$ | R | $22$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | $18{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.9.0.1}{9} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.7.0.1}{7} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.7.0.1}{7} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.22.11.2 | $x^{22} + 33 x^{20} + 495 x^{18} + 4455 x^{16} + 26730 x^{14} + 4 x^{13} + 112266 x^{12} - 406 x^{11} + 336798 x^{10} + 1650 x^{9} + 721710 x^{8} + 20196 x^{7} + 1082565 x^{6} - 35640 x^{5} + 1082569 x^{4} - 37422 x^{3} + 649567 x^{2} + 20898 x + 177172$ | $2$ | $11$ | $11$ | 22T1 | $[\ ]_{2}^{11}$ |
|
\(12917\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
|
\(459847\)
| $\Q_{459847}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{459847}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{459847}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{459847}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |