Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 3*x^20 - 36*x^18 - 60*x^16 + 213*x^14 + 9*x^12 - 114*x^10 + 180*x^8 - 108*x^6 - 6*x^4 + 36*x^2 - 54)
gp: K = bnfinit(y^22 + 3*y^20 - 36*y^18 - 60*y^16 + 213*y^14 + 9*y^12 - 114*y^10 + 180*y^8 - 108*y^6 - 6*y^4 + 36*y^2 - 54, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 3*x^20 - 36*x^18 - 60*x^16 + 213*x^14 + 9*x^12 - 114*x^10 + 180*x^8 - 108*x^6 - 6*x^4 + 36*x^2 - 54);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 3*x^20 - 36*x^18 - 60*x^16 + 213*x^14 + 9*x^12 - 114*x^10 + 180*x^8 - 108*x^6 - 6*x^4 + 36*x^2 - 54)
\( x^{22} + 3 x^{20} - 36 x^{18} - 60 x^{16} + 213 x^{14} + 9 x^{12} - 114 x^{10} + 180 x^{8} - 108 x^{6} + \cdots - 54 \)
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: | $[6, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(6246529265931622277982705180820920018628868505654981941197288960425984\)
\(\medspace = 2^{43}\cdot 3^{29}\cdot 7^{4}\cdot 23^{4}\cdot 137^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(1487.75\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(23\), \(137\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{6}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{12}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{13}$, $\frac{1}{12}a^{18}-\frac{1}{4}a^{16}-\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}$, $\frac{1}{12}a^{19}-\frac{1}{4}a^{17}-\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a$, $\frac{1}{27000}a^{20}-\frac{41}{1125}a^{18}-\frac{23}{750}a^{16}+\frac{299}{1125}a^{14}-\frac{2833}{9000}a^{12}-\frac{157}{500}a^{10}-\frac{97}{1125}a^{8}+\frac{27}{250}a^{6}+\frac{2}{5}a^{4}+\frac{899}{4500}a^{2}-\frac{269}{1500}$, $\frac{1}{162000}a^{21}-\frac{1}{54000}a^{20}+\frac{293}{13500}a^{19}+\frac{41}{2250}a^{18}+\frac{551}{2250}a^{17}+\frac{23}{1500}a^{16}-\frac{2777}{13500}a^{15}+\frac{413}{1125}a^{14}+\frac{19667}{54000}a^{13}-\frac{6167}{18000}a^{12}-\frac{157}{3000}a^{11}-\frac{343}{1000}a^{10}+\frac{514}{3375}a^{9}-\frac{514}{1125}a^{8}+\frac{9}{500}a^{7}-\frac{27}{500}a^{6}+\frac{7}{30}a^{5}+\frac{3}{10}a^{4}-\frac{8101}{27000}a^{3}-\frac{899}{9000}a^{2}-\frac{269}{9000}a-\frac{1231}{3000}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $13$ | 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: |
$\frac{6142481}{13500}a^{20}+\frac{2525857}{4500}a^{18}-\frac{29714377}{1500}a^{16}-\frac{8283823}{4500}a^{14}+\frac{203415988}{1125}a^{12}-\frac{11202446}{125}a^{10}-\frac{291006814}{1125}a^{8}+\frac{5244362}{125}a^{6}-\frac{332646}{5}a^{4}+\frac{94521169}{2250}a^{2}+\frac{39062368}{375}$, $\frac{508699009}{2250}a^{20}+\frac{1647333721}{1500}a^{18}-\frac{3051757281}{500}a^{16}-\frac{37377460819}{1500}a^{14}+\frac{3007227631}{1500}a^{12}+\frac{1518511897}{250}a^{10}-\frac{5585356117}{375}a^{8}+\frac{1673395208}{125}a^{6}+\frac{1572946}{5}a^{4}-\frac{325829284}{375}a^{2}+\frac{1657693783}{250}$, $\frac{363317363}{3375}a^{20}+\frac{1880220619}{4500}a^{18}-\frac{5264387959}{1500}a^{16}-\frac{43036151941}{4500}a^{14}+\frac{65570640859}{4500}a^{12}+\frac{3515856661}{250}a^{10}-\frac{152874313}{1125}a^{8}+\frac{2330677204}{125}a^{6}+\frac{24095048}{5}a^{4}+\frac{4017895049}{1125}a^{2}+\frac{5041575287}{750}$, $\frac{263579161}{13500}a^{20}+\frac{168680971}{2250}a^{18}-\frac{244136453}{375}a^{16}-\frac{3963580069}{2250}a^{14}+\frac{13711625387}{4500}a^{12}+\frac{940577473}{250}a^{10}-\frac{825803984}{1125}a^{8}+\frac{78565347}{125}a^{6}-\frac{6649071}{5}a^{4}-\frac{3539402761}{2250}a^{2}+\frac{90641941}{750}$, $\frac{15007510961}{13500}a^{20}+\frac{9184635667}{4500}a^{18}-\frac{63649347037}{1500}a^{16}-\frac{79030164913}{4500}a^{14}+\frac{289553895778}{1125}a^{12}-\frac{36023709251}{125}a^{10}+\frac{232986135191}{1125}a^{8}-\frac{4958192428}{125}a^{6}-\frac{371152016}{5}a^{4}+\frac{178478908939}{2250}a^{2}-\frac{19449889442}{375}$, $\frac{7774529}{2700}a^{20}+\frac{113563}{900}a^{18}-\frac{30371593}{300}a^{16}+\frac{108117143}{900}a^{14}+\frac{40791742}{225}a^{12}-\frac{6868139}{25}a^{10}+\frac{56920349}{225}a^{8}-\frac{2235417}{25}a^{6}-54806a^{4}+\frac{35069971}{450}a^{2}-\frac{4824338}{75}$, $\frac{43\!\cdots\!73}{3375}a^{20}+\frac{66\!\cdots\!37}{2250}a^{18}-\frac{36\!\cdots\!07}{750}a^{16}-\frac{96\!\cdots\!43}{2250}a^{14}+\frac{68\!\cdots\!57}{2250}a^{12}-\frac{25\!\cdots\!22}{125}a^{10}-\frac{46\!\cdots\!48}{1125}a^{8}+\frac{29\!\cdots\!84}{125}a^{6}-\frac{15\!\cdots\!92}{5}a^{4}+\frac{23\!\cdots\!79}{1125}a^{2}-\frac{36\!\cdots\!24}{375}$, $\frac{28\!\cdots\!41}{13500}a^{20}+\frac{88\!\cdots\!01}{2250}a^{18}-\frac{30\!\cdots\!93}{375}a^{16}-\frac{76\!\cdots\!89}{2250}a^{14}+\frac{22\!\cdots\!47}{4500}a^{12}-\frac{13\!\cdots\!87}{250}a^{10}+\frac{44\!\cdots\!46}{1125}a^{8}-\frac{95\!\cdots\!18}{125}a^{6}-\frac{71\!\cdots\!56}{5}a^{4}+\frac{34\!\cdots\!09}{2250}a^{2}-\frac{74\!\cdots\!79}{750}$, $\frac{83\!\cdots\!57}{40500}a^{21}-\frac{24\!\cdots\!09}{13500}a^{20}+\frac{10\!\cdots\!79}{13500}a^{19}-\frac{13\!\cdots\!99}{2250}a^{18}-\frac{32\!\cdots\!19}{4500}a^{17}+\frac{24\!\cdots\!82}{375}a^{16}-\frac{22\!\cdots\!81}{13500}a^{15}+\frac{29\!\cdots\!11}{2250}a^{14}+\frac{25\!\cdots\!47}{6750}a^{13}-\frac{17\!\cdots\!03}{4500}a^{12}+\frac{12\!\cdots\!63}{375}a^{11}-\frac{49\!\cdots\!87}{250}a^{10}-\frac{61\!\cdots\!33}{3375}a^{9}+\frac{31\!\cdots\!46}{1125}a^{8}+\frac{97\!\cdots\!88}{125}a^{7}-\frac{10\!\cdots\!68}{125}a^{6}-\frac{23\!\cdots\!62}{15}a^{5}+\frac{77\!\cdots\!54}{5}a^{4}-\frac{89\!\cdots\!57}{6750}a^{3}+\frac{17\!\cdots\!59}{2250}a^{2}+\frac{63\!\cdots\!21}{1125}a-\frac{74\!\cdots\!29}{750}$, $\frac{13\!\cdots\!63}{27000}a^{21}+\frac{59\!\cdots\!27}{5400}a^{20}+\frac{88\!\cdots\!09}{2250}a^{19}+\frac{19\!\cdots\!93}{225}a^{18}+\frac{64\!\cdots\!63}{375}a^{17}+\frac{57\!\cdots\!79}{150}a^{16}-\frac{23\!\cdots\!63}{1125}a^{15}-\frac{21\!\cdots\!79}{450}a^{14}+\frac{67\!\cdots\!21}{9000}a^{13}+\frac{30\!\cdots\!09}{1800}a^{12}+\frac{40\!\cdots\!09}{500}a^{11}+\frac{18\!\cdots\!61}{100}a^{10}-\frac{17\!\cdots\!61}{1125}a^{9}-\frac{78\!\cdots\!19}{225}a^{8}+\frac{26\!\cdots\!51}{250}a^{7}+\frac{11\!\cdots\!79}{50}a^{6}+\frac{46\!\cdots\!56}{5}a^{5}+20\!\cdots\!19a^{4}-\frac{11\!\cdots\!63}{4500}a^{3}-\frac{50\!\cdots\!27}{900}a^{2}+\frac{80\!\cdots\!53}{1500}a+\frac{35\!\cdots\!37}{300}$, $\frac{66\!\cdots\!63}{2700}a^{21}-\frac{37\!\cdots\!33}{6750}a^{20}+\frac{17\!\cdots\!61}{900}a^{19}-\frac{49\!\cdots\!38}{1125}a^{18}+\frac{25\!\cdots\!79}{300}a^{17}-\frac{72\!\cdots\!57}{375}a^{16}-\frac{94\!\cdots\!79}{900}a^{15}+\frac{26\!\cdots\!32}{1125}a^{14}+\frac{16\!\cdots\!23}{450}a^{13}-\frac{18\!\cdots\!11}{2250}a^{12}+\frac{10\!\cdots\!42}{25}a^{11}-\frac{11\!\cdots\!19}{125}a^{10}-\frac{17\!\cdots\!97}{225}a^{9}+\frac{19\!\cdots\!54}{1125}a^{8}+\frac{13\!\cdots\!01}{25}a^{7}-\frac{14\!\cdots\!57}{125}a^{6}+46\!\cdots\!32a^{5}-\frac{52\!\cdots\!19}{5}a^{4}-\frac{56\!\cdots\!13}{450}a^{3}+\frac{31\!\cdots\!08}{1125}a^{2}+\frac{20\!\cdots\!64}{75}a-\frac{22\!\cdots\!98}{375}$, $\frac{43\!\cdots\!97}{40500}a^{21}-\frac{65\!\cdots\!99}{4500}a^{20}+\frac{17\!\cdots\!71}{3375}a^{19}-\frac{10\!\cdots\!03}{1500}a^{18}-\frac{32\!\cdots\!31}{1125}a^{17}+\frac{19\!\cdots\!33}{500}a^{16}-\frac{79\!\cdots\!13}{6750}a^{15}+\frac{24\!\cdots\!67}{1500}a^{14}+\frac{13\!\cdots\!49}{13500}a^{13}-\frac{51\!\cdots\!02}{375}a^{12}+\frac{21\!\cdots\!21}{750}a^{11}-\frac{47\!\cdots\!73}{125}a^{10}-\frac{23\!\cdots\!93}{3375}a^{9}+\frac{35\!\cdots\!06}{375}a^{8}+\frac{79\!\cdots\!98}{125}a^{7}-\frac{10\!\cdots\!69}{125}a^{6}+\frac{20\!\cdots\!18}{15}a^{5}-\frac{91\!\cdots\!68}{5}a^{4}-\frac{26\!\cdots\!47}{6750}a^{3}+\frac{40\!\cdots\!49}{750}a^{2}+\frac{70\!\cdots\!57}{2250}a-\frac{53\!\cdots\!72}{125}$, $\frac{31\!\cdots\!61}{2250}a^{21}-\frac{11\!\cdots\!47}{2700}a^{20}+\frac{13\!\cdots\!21}{375}a^{19}-\frac{18\!\cdots\!09}{900}a^{18}-\frac{67\!\cdots\!81}{125}a^{17}+\frac{55\!\cdots\!49}{300}a^{16}-\frac{24\!\cdots\!69}{375}a^{15}-\frac{11\!\cdots\!49}{900}a^{14}+\frac{27\!\cdots\!87}{750}a^{13}-\frac{74\!\cdots\!37}{450}a^{12}-\frac{83\!\cdots\!56}{125}a^{11}+\frac{56\!\cdots\!27}{25}a^{10}-\frac{14\!\cdots\!18}{375}a^{9}+\frac{37\!\cdots\!93}{225}a^{8}+\frac{44\!\cdots\!57}{125}a^{7}-\frac{83\!\cdots\!69}{25}a^{6}-\frac{48\!\cdots\!76}{5}a^{5}+12\!\cdots\!59a^{4}-\frac{59\!\cdots\!86}{375}a^{3}+\frac{42\!\cdots\!97}{450}a^{2}+\frac{68\!\cdots\!91}{125}a-\frac{70\!\cdots\!91}{75}$
| 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: | \( 18112506929300000000000000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 18112506929300000000000000000 \cdot 1}{2\cdot\sqrt{6246529265931622277982705180820920018628868505654981941197288960425984}}\cr\approx \mathstrut & 17.8134525093832 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 3*x^20 - 36*x^18 - 60*x^16 + 213*x^14 + 9*x^12 - 114*x^10 + 180*x^8 - 108*x^6 - 6*x^4 + 36*x^2 - 54)
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 + 3*x^20 - 36*x^18 - 60*x^16 + 213*x^14 + 9*x^12 - 114*x^10 + 180*x^8 - 108*x^6 - 6*x^4 + 36*x^2 - 54, 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 + 3*x^20 - 36*x^18 - 60*x^16 + 213*x^14 + 9*x^12 - 114*x^10 + 180*x^8 - 108*x^6 - 6*x^4 + 36*x^2 - 54);
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 + 3*x^20 - 36*x^18 - 60*x^16 + 213*x^14 + 9*x^12 - 114*x^10 + 180*x^8 - 108*x^6 - 6*x^4 + 36*x^2 - 54);
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
$C_2^{11}.A_{11}$ (as 22T52):
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 40874803200 |
| The 400 conjugacy class representatives for $C_2^{11}.A_{11}$ are not computed |
| Character table for $C_2^{11}.A_{11}$ is not computed |
Intermediate fields
| 11.7.63019333158425674204677255696384.1 |
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 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 | R | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | $22$ | $18{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | $22$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | R | ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.4.0.1}{4} }$ | $22$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.7.0.1}{7} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 2.8.18.61 | $x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{3} + 2$ | $8$ | $1$ | $18$ | $C_2^3 : C_4 $ | $[2, 2, 3]^{4}$ | |
| 2.12.25.86 | $x^{12} + 4 x^{11} + 4 x^{9} + 4 x^{8} + 2 x^{6} + 4 x^{3} + 6 x^{2} + 14$ | $12$ | $1$ | $25$ | 12T224 | $[4/3, 4/3, 8/3, 8/3, 3, 3, 19/6, 19/6]_{3}^{2}$ | |
|
\(3\)
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $18$ | $18$ | $1$ | $27$ | ||||
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.14.0.1 | $x^{14} + 5 x^{7} + 6 x^{5} + 2 x^{4} + 3 x^{2} + 6 x + 3$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(23\)
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.7.0.1 | $x^{7} + 21 x + 18$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 23.7.0.1 | $x^{7} + 21 x + 18$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
|
\(137\)
| $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 137.5.4.1 | $x^{5} + 137$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 137.5.4.1 | $x^{5} + 137$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 137.10.8.1 | $x^{10} + 655 x^{9} + 171625 x^{8} + 22488770 x^{7} + 1474044185 x^{6} + 38714410755 x^{5} + 4422222290 x^{4} + 225901280 x^{3} + 3082903315 x^{2} + 201591447850 x + 5280771081809$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |