Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 14*x^20 + 49*x^18 - 84*x^16 - 873*x^14 - 2112*x^12 - 2265*x^10 - 576*x^8 + 1248*x^6 + 1432*x^4 + 632*x^2 + 64)
gp: K = bnfinit(y^22 + 14*y^20 + 49*y^18 - 84*y^16 - 873*y^14 - 2112*y^12 - 2265*y^10 - 576*y^8 + 1248*y^6 + 1432*y^4 + 632*y^2 + 64, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 14*x^20 + 49*x^18 - 84*x^16 - 873*x^14 - 2112*x^12 - 2265*x^10 - 576*x^8 + 1248*x^6 + 1432*x^4 + 632*x^2 + 64);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 14*x^20 + 49*x^18 - 84*x^16 - 873*x^14 - 2112*x^12 - 2265*x^10 - 576*x^8 + 1248*x^6 + 1432*x^4 + 632*x^2 + 64)
\( x^{22} + 14 x^{20} + 49 x^{18} - 84 x^{16} - 873 x^{14} - 2112 x^{12} - 2265 x^{10} - 576 x^{8} + \cdots + 64 \)
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: | $[4, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-66629645503270637631815521928756480198707930726986474039437748911210496\)
\(\medspace = -\,2^{48}\cdot 3^{28}\cdot 7^{4}\cdot 23^{4}\cdot 137^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(1656.75\) | 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: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{12}a^{18}-\frac{1}{4}a^{14}-\frac{1}{2}a^{12}+\frac{1}{4}a^{10}-\frac{1}{2}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{4}+\frac{1}{3}$, $\frac{1}{12}a^{19}-\frac{1}{4}a^{15}-\frac{1}{2}a^{13}+\frac{1}{4}a^{11}-\frac{1}{2}a^{9}+\frac{1}{4}a^{7}-\frac{1}{2}a^{5}+\frac{1}{3}a$, $\frac{1}{27000}a^{20}-\frac{487}{13500}a^{18}+\frac{143}{1000}a^{16}-\frac{323}{1125}a^{14}+\frac{1201}{9000}a^{12}+\frac{59}{750}a^{10}-\frac{2759}{9000}a^{8}-\frac{13}{90}a^{6}+\frac{193}{750}a^{4}-\frac{649}{3375}a^{2}+\frac{41}{3375}$, $\frac{1}{54000}a^{21}-\frac{487}{27000}a^{19}-\frac{357}{2000}a^{17}-\frac{323}{2250}a^{15}-\frac{3299}{18000}a^{13}-\frac{691}{1500}a^{11}+\frac{1741}{18000}a^{9}-\frac{13}{180}a^{7}+\frac{142}{375}a^{5}+\frac{1363}{3375}a^{3}+\frac{41}{6750}a$
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: | $12$ | 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: |
$\frac{6142481}{13500}a^{20}+\frac{69002381}{13500}a^{18}+\frac{2858533}{500}a^{16}-\frac{384798779}{4500}a^{14}-\frac{1098184819}{4500}a^{12}+\frac{48982141}{1500}a^{10}+\frac{3242770421}{4500}a^{8}+\frac{117960343}{180}a^{6}-\frac{85997917}{375}a^{4}-\frac{1771364338}{3375}a^{2}-\frac{224473933}{3375}$, $\frac{508699009}{2250}a^{20}+\frac{15115981343}{4500}a^{18}+\frac{1744308493}{125}a^{16}-\frac{10619700887}{1500}a^{14}-\frac{152694357941}{750}a^{12}-\frac{325813993627}{500}a^{10}-\frac{800987950331}{750}a^{8}-\frac{62394563501}{60}a^{6}-\frac{75318717651}{125}a^{4}-\frac{211913525614}{1125}a^{2}-\frac{19180439899}{1125}$, $\frac{100471139344}{3375}a^{20}+\frac{13794480250651}{13500}a^{18}+\frac{2312322014259}{250}a^{16}+\frac{80685055516541}{4500}a^{14}-\frac{118935715626356}{1125}a^{12}-\frac{775837123615639}{1500}a^{10}-\frac{871309591640096}{1125}a^{8}-\frac{44087068627237}{180}a^{6}+\frac{185688992261443}{375}a^{4}+\frac{16\!\cdots\!27}{3375}a^{2}+\frac{190904878304707}{3375}$, $\frac{363317363}{3375}a^{20}+\frac{20173356377}{13500}a^{18}+\frac{1273770193}{250}a^{16}-\frac{43562742593}{4500}a^{14}-\frac{104555904412}{1125}a^{12}-\frac{324161556953}{1500}a^{10}-\frac{243443728642}{1125}a^{8}-\frac{5955511139}{180}a^{6}+\frac{52471892861}{375}a^{4}+\frac{459546452054}{3375}a^{2}+\frac{167768134439}{3375}$, $\frac{15007510961}{13500}a^{20}+\frac{177629016611}{13500}a^{18}+\frac{12980704923}{500}a^{16}-\frac{675667171349}{4500}a^{14}-\frac{2919505559089}{4500}a^{12}-\frac{1427462928329}{1500}a^{10}-\frac{2103718166149}{4500}a^{8}+\frac{66308933113}{180}a^{6}+\frac{446070833071}{750}a^{4}+\frac{1048571008247}{3375}a^{2}+\frac{111407195177}{3375}$, $\frac{7774529}{2700}a^{20}+\frac{78085979}{2700}a^{18}+\frac{2947247}{100}a^{16}-\frac{305731661}{900}a^{14}-\frac{1077470521}{900}a^{12}-\frac{477616481}{300}a^{10}-\frac{596487661}{900}a^{8}+\frac{24930109}{36}a^{6}+\frac{148377469}{150}a^{4}+\frac{328774058}{675}a^{2}+\frac{34312703}{675}$, $\frac{43\!\cdots\!73}{3375}a^{20}+\frac{10\!\cdots\!71}{6750}a^{18}+\frac{45\!\cdots\!39}{125}a^{16}-\frac{38\!\cdots\!39}{2250}a^{14}-\frac{93\!\cdots\!52}{1125}a^{12}-\frac{96\!\cdots\!69}{750}a^{10}-\frac{79\!\cdots\!57}{1125}a^{8}+\frac{41\!\cdots\!83}{90}a^{6}+\frac{30\!\cdots\!81}{375}a^{4}+\frac{15\!\cdots\!84}{3375}a^{2}+\frac{16\!\cdots\!69}{3375}$, $\frac{22\!\cdots\!81}{13500}a^{20}+\frac{52\!\cdots\!64}{3375}a^{18}+\frac{555577533694383}{500}a^{16}-\frac{41\!\cdots\!27}{2250}a^{14}-\frac{27\!\cdots\!19}{4500}a^{12}-\frac{58\!\cdots\!17}{750}a^{10}-\frac{13\!\cdots\!79}{4500}a^{8}+\frac{31\!\cdots\!49}{90}a^{6}+\frac{17\!\cdots\!33}{375}a^{4}+\frac{77\!\cdots\!37}{3375}a^{2}+\frac{80\!\cdots\!17}{3375}$, $\frac{98\!\cdots\!01}{54000}a^{21}+\frac{22\!\cdots\!49}{13500}a^{20}+\frac{73\!\cdots\!13}{27000}a^{19}+\frac{33\!\cdots\!99}{13500}a^{18}+\frac{22\!\cdots\!43}{2000}a^{17}+\frac{51\!\cdots\!07}{500}a^{16}-\frac{12\!\cdots\!73}{2250}a^{15}-\frac{23\!\cdots\!41}{4500}a^{14}-\frac{29\!\cdots\!99}{18000}a^{13}-\frac{68\!\cdots\!01}{4500}a^{12}-\frac{78\!\cdots\!91}{1500}a^{11}-\frac{72\!\cdots\!61}{1500}a^{10}-\frac{15\!\cdots\!59}{18000}a^{9}-\frac{35\!\cdots\!41}{4500}a^{8}-\frac{15\!\cdots\!53}{180}a^{7}-\frac{13\!\cdots\!03}{180}a^{6}-\frac{36\!\cdots\!91}{750}a^{5}-\frac{33\!\cdots\!61}{750}a^{4}-\frac{10\!\cdots\!49}{6750}a^{3}-\frac{47\!\cdots\!27}{3375}a^{2}-\frac{92\!\cdots\!59}{6750}a-\frac{42\!\cdots\!57}{3375}$, $\frac{32\!\cdots\!61}{54000}a^{21}-\frac{29\!\cdots\!87}{500}a^{20}+\frac{19\!\cdots\!93}{27000}a^{19}-\frac{12\!\cdots\!61}{1500}a^{18}+\frac{32\!\cdots\!23}{2000}a^{17}-\frac{15\!\cdots\!57}{500}a^{16}-\frac{19\!\cdots\!03}{2250}a^{15}+\frac{22\!\cdots\!99}{500}a^{14}-\frac{73\!\cdots\!39}{18000}a^{13}+\frac{27\!\cdots\!39}{500}a^{12}-\frac{70\!\cdots\!51}{1500}a^{11}+\frac{68\!\cdots\!37}{500}a^{10}+\frac{23\!\cdots\!01}{18000}a^{9}+\frac{65\!\cdots\!99}{500}a^{8}+\frac{10\!\cdots\!07}{180}a^{7}-\frac{12\!\cdots\!83}{20}a^{6}+\frac{15\!\cdots\!49}{750}a^{5}-\frac{13\!\cdots\!69}{125}a^{4}-\frac{12\!\cdots\!89}{6750}a^{3}-\frac{89\!\cdots\!24}{125}a^{2}-\frac{18\!\cdots\!49}{6750}a-\frac{29\!\cdots\!27}{375}$, $\frac{24\!\cdots\!61}{27000}a^{21}-\frac{44\!\cdots\!99}{250}a^{20}+\frac{10\!\cdots\!59}{6750}a^{19}-\frac{79\!\cdots\!49}{250}a^{18}+\frac{10\!\cdots\!23}{1000}a^{17}-\frac{26\!\cdots\!32}{125}a^{16}+\frac{15\!\cdots\!63}{4500}a^{15}-\frac{17\!\cdots\!77}{250}a^{14}+\frac{54\!\cdots\!61}{9000}a^{13}-\frac{15\!\cdots\!61}{125}a^{12}+\frac{77\!\cdots\!73}{1500}a^{11}-\frac{25\!\cdots\!51}{250}a^{10}+\frac{32\!\cdots\!01}{9000}a^{9}-\frac{89\!\cdots\!51}{125}a^{8}-\frac{66\!\cdots\!41}{180}a^{7}+\frac{73\!\cdots\!69}{10}a^{6}-\frac{26\!\cdots\!77}{750}a^{5}+\frac{17\!\cdots\!23}{250}a^{4}-\frac{49\!\cdots\!39}{3375}a^{3}+\frac{36\!\cdots\!04}{125}a^{2}-\frac{48\!\cdots\!74}{3375}a+\frac{35\!\cdots\!89}{125}$, $\frac{36\!\cdots\!21}{54000}a^{21}+\frac{45\!\cdots\!92}{3375}a^{20}+\frac{33\!\cdots\!73}{27000}a^{19}+\frac{33\!\cdots\!93}{13500}a^{18}+\frac{16\!\cdots\!03}{2000}a^{17}+\frac{40\!\cdots\!87}{250}a^{16}+\frac{60\!\cdots\!67}{2250}a^{15}+\frac{24\!\cdots\!63}{4500}a^{14}+\frac{83\!\cdots\!21}{18000}a^{13}+\frac{10\!\cdots\!67}{1125}a^{12}+\frac{59\!\cdots\!89}{1500}a^{11}+\frac{11\!\cdots\!23}{1500}a^{10}+\frac{49\!\cdots\!61}{18000}a^{9}+\frac{61\!\cdots\!47}{1125}a^{8}-\frac{51\!\cdots\!93}{180}a^{7}-\frac{10\!\cdots\!71}{180}a^{6}-\frac{10\!\cdots\!68}{375}a^{5}-\frac{20\!\cdots\!51}{375}a^{4}-\frac{75\!\cdots\!29}{6750}a^{3}-\frac{75\!\cdots\!39}{3375}a^{2}-\frac{74\!\cdots\!39}{6750}a-\frac{74\!\cdots\!49}{3375}$
| 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: | \( 13265885053200000000000000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{9}\cdot 13265885053200000000000000000 \cdot 1}{2\cdot\sqrt{66629645503270637631815521928756480198707930726986474039437748911210496}}\cr\approx \mathstrut & 6.27496460971433 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 14*x^20 + 49*x^18 - 84*x^16 - 873*x^14 - 2112*x^12 - 2265*x^10 - 576*x^8 + 1248*x^6 + 1432*x^4 + 632*x^2 + 64)
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 + 14*x^20 + 49*x^18 - 84*x^16 - 873*x^14 - 2112*x^12 - 2265*x^10 - 576*x^8 + 1248*x^6 + 1432*x^4 + 632*x^2 + 64, 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 + 14*x^20 + 49*x^18 - 84*x^16 - 873*x^14 - 2112*x^12 - 2265*x^10 - 576*x^8 + 1248*x^6 + 1432*x^4 + 632*x^2 + 64);
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 + 14*x^20 + 49*x^18 - 84*x^16 - 873*x^14 - 2112*x^12 - 2265*x^10 - 576*x^8 + 1248*x^6 + 1432*x^4 + 632*x^2 + 64);
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 | $22$ | ${\href{/padicField/13.9.0.1}{9} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | $22$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.8.21.1 | $x^{8} + 14 x^{6} + 8 x^{4} + 2$ | $8$ | $1$ | $21$ | $C_2 \wr C_2\wr C_2$ | $[2, 2, 3, 7/2, 7/2, 15/4]^{2}$ | |
| 2.12.24.424 | $x^{12} + 2 x^{10} + 4 x^{9} + 2 x^{8} + 4 x^{6} + 4 x^{5} + 6 x^{2} + 4 x + 10$ | $12$ | $1$ | $24$ | 12T149 | $[4/3, 4/3, 2, 7/3, 7/3, 3]_{3}^{2}$ | |
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| Deg $18$ | $9$ | $2$ | $26$ | ||||
|
\(7\)
| 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 | $x^{14} + 5 x^{7} + 6 x^{5} + 2 x^{4} + 3 x^{2} + 6 x + 3$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(23\)
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.14.0.1 | $x^{14} + x^{8} + 5 x^{7} + 16 x^{6} + x^{5} + 18 x^{4} + 19 x^{3} + x^{2} + 22 x + 5$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(137\)
| $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | 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 | $x^{5} + 137$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 137.5.4.1 | $x^{5} + 137$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 137.5.4.1 | $x^{5} + 137$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |