Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 - 12*x^17 + 27*x^16 - 39*x^15 + 68*x^14 - 127*x^13 + 178*x^12 - 215*x^11 + 277*x^10 - 328*x^9 + 297*x^8 - 211*x^7 + 139*x^6 - 82*x^5 + 29*x^4 + x^3 - 3*x^2 - x + 1)
gp: K = bnfinit(y^20 - 2*y^19 + 3*y^18 - 12*y^17 + 27*y^16 - 39*y^15 + 68*y^14 - 127*y^13 + 178*y^12 - 215*y^11 + 277*y^10 - 328*y^9 + 297*y^8 - 211*y^7 + 139*y^6 - 82*y^5 + 29*y^4 + y^3 - 3*y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 3*x^18 - 12*x^17 + 27*x^16 - 39*x^15 + 68*x^14 - 127*x^13 + 178*x^12 - 215*x^11 + 277*x^10 - 328*x^9 + 297*x^8 - 211*x^7 + 139*x^6 - 82*x^5 + 29*x^4 + x^3 - 3*x^2 - x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 + 3*x^18 - 12*x^17 + 27*x^16 - 39*x^15 + 68*x^14 - 127*x^13 + 178*x^12 - 215*x^11 + 277*x^10 - 328*x^9 + 297*x^8 - 211*x^7 + 139*x^6 - 82*x^5 + 29*x^4 + x^3 - 3*x^2 - x + 1)
\( x^{20} - 2 x^{19} + 3 x^{18} - 12 x^{17} + 27 x^{16} - 39 x^{15} + 68 x^{14} - 127 x^{13} + 178 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2788613721199293001609\)
\(\medspace = 7^{2}\cdot 53^{4}\cdot 139^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.81\) | 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: | $7^{1/2}53^{1/2}139^{1/2}\approx 227.08808863522543$ | ||
| Ramified primes: |
\(7\), \(53\), \(139\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{74019941}a^{19}-\frac{25336160}{74019941}a^{18}-\frac{34882728}{74019941}a^{17}-\frac{12443997}{74019941}a^{16}+\frac{20770159}{74019941}a^{15}+\frac{6077301}{74019941}a^{14}+\frac{8699359}{74019941}a^{13}+\frac{29973500}{74019941}a^{12}+\frac{26432076}{74019941}a^{11}+\frac{20199767}{74019941}a^{10}-\frac{31349346}{74019941}a^{9}-\frac{29727334}{74019941}a^{8}-\frac{6873169}{74019941}a^{7}+\frac{12448186}{74019941}a^{6}-\frac{9340461}{74019941}a^{5}+\frac{21719426}{74019941}a^{4}+\frac{8670726}{74019941}a^{3}+\frac{10743901}{74019941}a^{2}-\frac{26025510}{74019941}a-\frac{3564028}{74019941}$
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 $1$
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: | $9$ | 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: |
$\frac{1193179739}{74019941}a^{19}-\frac{1343337790}{74019941}a^{18}+\frac{2049129060}{74019941}a^{17}-\frac{12275268327}{74019941}a^{16}+\frac{20964555193}{74019941}a^{15}-\frac{24747961565}{74019941}a^{14}+\frac{54669139544}{74019941}a^{13}-\frac{98237447496}{74019941}a^{12}+\frac{112261460128}{74019941}a^{11}-\frac{134754526927}{74019941}a^{10}+\frac{188468735575}{74019941}a^{9}-\frac{195420388461}{74019941}a^{8}+\frac{139054491636}{74019941}a^{7}-\frac{88950600315}{74019941}a^{6}+\frac{61748987087}{74019941}a^{5}-\frac{26140133516}{74019941}a^{4}-\frac{813394797}{74019941}a^{3}+\frac{3919542159}{74019941}a^{2}+\frac{964985651}{74019941}a-\frac{816942117}{74019941}$, $\frac{1156032250}{74019941}a^{19}-\frac{1419281582}{74019941}a^{18}+\frac{2059243141}{74019941}a^{17}-\frac{12077433009}{74019941}a^{16}+\frac{21446057676}{74019941}a^{15}-\frac{25507334683}{74019941}a^{14}+\frac{54841675197}{74019941}a^{13}-\frac{99902211422}{74019941}a^{12}+\frac{116404376105}{74019941}a^{11}-\frac{138661886171}{74019941}a^{10}+\frac{193177626037}{74019941}a^{9}-\frac{203995642427}{74019941}a^{8}+\frac{148593318552}{74019941}a^{7}-\frac{95609085939}{74019941}a^{6}+\frac{66245343373}{74019941}a^{5}-\frac{29959395597}{74019941}a^{4}+\frac{817040043}{74019941}a^{3}+\frac{3835404880}{74019941}a^{2}+\frac{867875413}{74019941}a-\frac{931226842}{74019941}$, $\frac{64176579}{74019941}a^{19}-\frac{98023209}{74019941}a^{18}+\frac{135080323}{74019941}a^{17}-\frac{688046290}{74019941}a^{16}+\frac{1396962250}{74019941}a^{15}-\frac{1710366930}{74019941}a^{14}+\frac{3322097470}{74019941}a^{13}-\frac{6370733153}{74019941}a^{12}+\frac{7816348939}{74019941}a^{11}-\frac{9025476259}{74019941}a^{10}+\frac{12406117443}{74019941}a^{9}-\frac{13746952315}{74019941}a^{8}+\frac{10413052567}{74019941}a^{7}-\frac{6496624268}{74019941}a^{6}+\frac{4330951422}{74019941}a^{5}-\frac{2005398614}{74019941}a^{4}-\frac{31751057}{74019941}a^{3}+\frac{476028116}{74019941}a^{2}+\frac{15580021}{74019941}a-\frac{29334578}{74019941}$, $\frac{686555619}{74019941}a^{19}-\frac{577880090}{74019941}a^{18}+\frac{1007375123}{74019941}a^{17}-\frac{6773934988}{74019941}a^{16}+\frac{10119517754}{74019941}a^{15}-\frac{11284655537}{74019941}a^{14}+\frac{28169715798}{74019941}a^{13}-\frac{48331802694}{74019941}a^{12}+\frac{50458263021}{74019941}a^{11}-\frac{62719978993}{74019941}a^{10}+\frac{89873010917}{74019941}a^{9}-\frac{85764271422}{74019941}a^{8}+\frac{54445326866}{74019941}a^{7}-\frac{34444471772}{74019941}a^{6}+\frac{24307132633}{74019941}a^{5}-\frac{7093188982}{74019941}a^{4}-\frac{2997807351}{74019941}a^{3}+\frac{1735876541}{74019941}a^{2}+\frac{849580213}{74019941}a-\frac{205017100}{74019941}$, $\frac{1043021688}{74019941}a^{19}-\frac{1530410157}{74019941}a^{18}+\frac{2042888541}{74019941}a^{17}-\frac{11251297760}{74019941}a^{16}+\frac{21786048256}{74019941}a^{15}-\frac{26467082708}{74019941}a^{14}+\frac{53296379357}{74019941}a^{13}-\frac{100124533414}{74019941}a^{12}+\frac{121779116958}{74019941}a^{11}-\frac{142042052128}{74019941}a^{10}+\frac{195942565931}{74019941}a^{9}-\frac{215319890847}{74019941}a^{8}+\frac{162810324614}{74019941}a^{7}-\frac{104102996634}{74019941}a^{6}+\frac{71700605082}{74019941}a^{5}-\frac{35415607228}{74019941}a^{4}+\frac{2726362420}{74019941}a^{3}+\frac{4544524868}{74019941}a^{2}+\frac{376237622}{74019941}a-\frac{1193179739}{74019941}$, $\frac{564030254}{74019941}a^{19}-\frac{557760749}{74019941}a^{18}+\frac{895416688}{74019941}a^{17}-\frac{5695336043}{74019941}a^{16}+\frac{9145963094}{74019941}a^{15}-\frac{10485822422}{74019941}a^{14}+\frac{24589753642}{74019941}a^{13}-\frac{43297610888}{74019941}a^{12}+\frac{47397826511}{74019941}a^{11}-\frac{57987719027}{74019941}a^{10}+\frac{82233003985}{74019941}a^{9}-\frac{82071572041}{74019941}a^{8}+\frac{56014010401}{74019941}a^{7}-\frac{36415793100}{74019941}a^{6}+\frac{25629300651}{74019941}a^{5}-\frac{9745087856}{74019941}a^{4}-\frac{858581641}{74019941}a^{3}+\frac{1360988253}{74019941}a^{2}+\frac{611666378}{74019941}a-\frac{272253784}{74019941}$, $\frac{645247522}{74019941}a^{19}-\frac{620355109}{74019941}a^{18}+\frac{1033691075}{74019941}a^{17}-\frac{6488700941}{74019941}a^{16}+\frac{10319882717}{74019941}a^{15}-\frac{11951302181}{74019941}a^{14}+\frac{27990351359}{74019941}a^{13}-\frac{49035817938}{74019941}a^{12}+\frac{53778999487}{74019941}a^{11}-\frac{65922026589}{74019941}a^{10}+\frac{93274790125}{74019941}a^{9}-\frac{92991961210}{74019941}a^{8}+\frac{63460014771}{74019941}a^{7}-\frac{41298476843}{74019941}a^{6}+\frac{29072885210}{74019941}a^{5}-\frac{10721331395}{74019941}a^{4}-\frac{1316569103}{74019941}a^{3}+\frac{1651796941}{74019941}a^{2}+\frac{628349798}{74019941}a-\frac{210673331}{74019941}$, $\frac{984181412}{74019941}a^{19}-\frac{1513710055}{74019941}a^{18}+\frac{1931220847}{74019941}a^{17}-\frac{10732304641}{74019941}a^{16}+\frac{21159852895}{74019941}a^{15}-\frac{25534107315}{74019941}a^{14}+\frac{51153913219}{74019941}a^{13}-\frac{96802615835}{74019941}a^{12}+\frac{118087527901}{74019941}a^{11}-\frac{137215691942}{74019941}a^{10}+\frac{189504832272}{74019941}a^{9}-\frac{209124503150}{74019941}a^{8}+\frac{158553458089}{74019941}a^{7}-\frac{101212585101}{74019941}a^{6}+\frac{69386006964}{74019941}a^{5}-\frac{34690322507}{74019941}a^{4}+\frac{2514160355}{74019941}a^{3}+\frac{4708353146}{74019941}a^{2}+\frac{193849380}{74019941}a-\frac{1287533046}{74019941}$, $\frac{802810485}{74019941}a^{19}-\frac{1378002180}{74019941}a^{18}+\frac{1714021895}{74019941}a^{17}-\frac{8937412161}{74019941}a^{16}+\frac{18726395552}{74019941}a^{15}-\frac{23062058089}{74019941}a^{14}+\frac{44006677733}{74019941}a^{13}-\frac{85078836424}{74019941}a^{12}+\frac{106924000746}{74019941}a^{11}-\frac{122588909301}{74019941}a^{10}+\frac{168155132520}{74019941}a^{9}-\frac{190581730194}{74019941}a^{8}+\frac{148006941656}{74019941}a^{7}-\frac{94591343086}{74019941}a^{6}+\frac{65139075659}{74019941}a^{5}-\frac{34400295775}{74019941}a^{4}+\frac{4204010020}{74019941}a^{3}+\frac{4034207551}{74019941}a^{2}+\frac{111422549}{74019941}a-\frac{1188676220}{74019941}$
| 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: | \( 192.366272752 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 192.366272752 \cdot 1}{2\cdot\sqrt{2788613721199293001609}}\cr\approx \mathstrut & 0.174664014658 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 - 12*x^17 + 27*x^16 - 39*x^15 + 68*x^14 - 127*x^13 + 178*x^12 - 215*x^11 + 277*x^10 - 328*x^9 + 297*x^8 - 211*x^7 + 139*x^6 - 82*x^5 + 29*x^4 + x^3 - 3*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^20 - 2*x^19 + 3*x^18 - 12*x^17 + 27*x^16 - 39*x^15 + 68*x^14 - 127*x^13 + 178*x^12 - 215*x^11 + 277*x^10 - 328*x^9 + 297*x^8 - 211*x^7 + 139*x^6 - 82*x^5 + 29*x^4 + x^3 - 3*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^20 - 2*x^19 + 3*x^18 - 12*x^17 + 27*x^16 - 39*x^15 + 68*x^14 - 127*x^13 + 178*x^12 - 215*x^11 + 277*x^10 - 328*x^9 + 297*x^8 - 211*x^7 + 139*x^6 - 82*x^5 + 29*x^4 + x^3 - 3*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^20 - 2*x^19 + 3*x^18 - 12*x^17 + 27*x^16 - 39*x^15 + 68*x^14 - 127*x^13 + 178*x^12 - 215*x^11 + 277*x^10 - 328*x^9 + 297*x^8 - 211*x^7 + 139*x^6 - 82*x^5 + 29*x^4 + x^3 - 3*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
$C_2^{10}.S_5$ (as 20T799):
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 122880 |
| The 252 conjugacy class representatives for $C_2^{10}.S_5$ |
| Character table for $C_2^{10}.S_5$ |
Intermediate fields
| 5.3.7367.1, 10.4.7543903771.1, 10.2.52807326397.1, 10.0.379908823.1 |
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 20 siblings: | data not computed |
| Degree 40 siblings: | 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 | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(53\)
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.4.0.1 | $x^{4} + 9 x^{2} + 38 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 53.4.0.1 | $x^{4} + 9 x^{2} + 38 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(139\)
| 139.2.1.2 | $x^{2} + 139$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 139.2.1.2 | $x^{2} + 139$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 139.4.0.1 | $x^{4} + 7 x^{2} + 96 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 139.4.2.1 | $x^{4} + 276 x^{3} + 19326 x^{2} + 38916 x + 2665885$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 139.4.2.1 | $x^{4} + 276 x^{3} + 19326 x^{2} + 38916 x + 2665885$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 139.4.0.1 | $x^{4} + 7 x^{2} + 96 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |