Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 2*x^18 - 2*x^17 + 4*x^16 - 4*x^15 - 14*x^13 + 9*x^12 - 2*x^11 + 6*x^10 - 2*x^9 + 7*x^8 - 12*x^7 + 26*x^6 - 28*x^5 + 19*x^4 - 20*x^3 + 24*x^2 - 16*x + 4)
gp: K = bnfinit(y^20 + 2*y^18 - 2*y^17 + 4*y^16 - 4*y^15 - 14*y^13 + 9*y^12 - 2*y^11 + 6*y^10 - 2*y^9 + 7*y^8 - 12*y^7 + 26*y^6 - 28*y^5 + 19*y^4 - 20*y^3 + 24*y^2 - 16*y + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 2*x^18 - 2*x^17 + 4*x^16 - 4*x^15 - 14*x^13 + 9*x^12 - 2*x^11 + 6*x^10 - 2*x^9 + 7*x^8 - 12*x^7 + 26*x^6 - 28*x^5 + 19*x^4 - 20*x^3 + 24*x^2 - 16*x + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 2*x^18 - 2*x^17 + 4*x^16 - 4*x^15 - 14*x^13 + 9*x^12 - 2*x^11 + 6*x^10 - 2*x^9 + 7*x^8 - 12*x^7 + 26*x^6 - 28*x^5 + 19*x^4 - 20*x^3 + 24*x^2 - 16*x + 4)
\( x^{20} + 2 x^{18} - 2 x^{17} + 4 x^{16} - 4 x^{15} - 14 x^{13} + 9 x^{12} - 2 x^{11} + 6 x^{10} + \cdots + 4 \)
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: |
\(865929930053558147743744\)
\(\medspace = 2^{30}\cdot 73^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.74\) | 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: | $2^{3/2}73^{1/2}\approx 24.166091947189145$ | ||
| Ramified primes: |
\(2\), \(73\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{366754196932}a^{19}-\frac{5930761099}{183377098466}a^{18}-\frac{18196035718}{91688549233}a^{17}-\frac{20387056869}{183377098466}a^{16}-\frac{19768327097}{91688549233}a^{15}+\frac{10280964097}{91688549233}a^{14}-\frac{44145253789}{91688549233}a^{13}-\frac{75447815195}{183377098466}a^{12}-\frac{120271512395}{366754196932}a^{11}-\frac{21527910992}{91688549233}a^{10}+\frac{15747933907}{91688549233}a^{9}-\frac{47198905435}{183377098466}a^{8}-\frac{67877354561}{366754196932}a^{7}-\frac{41308070853}{183377098466}a^{6}-\frac{28803188068}{91688549233}a^{5}-\frac{25015085867}{91688549233}a^{4}+\frac{11746510495}{366754196932}a^{3}+\frac{2507175921}{183377098466}a^{2}-\frac{775135241}{183377098466}a+\frac{21014649745}{91688549233}$
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: |
$\frac{1628799941283}{183377098466}a^{19}+\frac{1047854593529}{183377098466}a^{18}+\frac{1967352762375}{91688549233}a^{17}-\frac{365842983515}{91688549233}a^{16}+\frac{3024770239821}{91688549233}a^{15}-\frac{1317060113974}{91688549233}a^{14}-\frac{835610291294}{91688549233}a^{13}-\frac{11948778878788}{91688549233}a^{12}-\frac{708098794763}{183377098466}a^{11}-\frac{3734740712451}{183377098466}a^{10}+\frac{3729559123654}{91688549233}a^{9}+\frac{752112077014}{91688549233}a^{8}+\frac{12324799792343}{183377098466}a^{7}-\frac{11614136832045}{183377098466}a^{6}+\frac{17481040156749}{91688549233}a^{5}-\frac{11541489491187}{91688549233}a^{4}+\frac{16296289338833}{183377098466}a^{3}-\frac{22393359267045}{183377098466}a^{2}+\frac{12392921345562}{91688549233}a-\frac{5110129873196}{91688549233}$, $\frac{2862816654080}{91688549233}a^{19}+\frac{1799974299652}{91688549233}a^{18}+\frac{6857447042890}{91688549233}a^{17}-\frac{1416276500098}{91688549233}a^{16}+\frac{10560985427228}{91688549233}a^{15}-\frac{4822389783635}{91688549233}a^{14}-\frac{3035263318304}{91688549233}a^{13}-\frac{42009416851556}{91688549233}a^{12}-\frac{644537667714}{91688549233}a^{11}-\frac{6140028522861}{91688549233}a^{10}+\frac{13333644393820}{91688549233}a^{9}+\frac{2673192698727}{91688549233}a^{8}+\frac{21800745433144}{91688549233}a^{7}-\frac{20614728886275}{91688549233}a^{6}+\frac{61419742991548}{91688549233}a^{5}-\frac{41578160572616}{91688549233}a^{4}+\frac{28256495240844}{91688549233}a^{3}-\frac{39575809228517}{91688549233}a^{2}+\frac{43959122685335}{91688549233}a-\frac{18270177126247}{91688549233}$, $\frac{11808325711605}{366754196932}a^{19}+\frac{1861122372517}{91688549233}a^{18}+\frac{7079561927861}{91688549233}a^{17}-\frac{2880497593931}{183377098466}a^{16}+\frac{10905500000648}{91688549233}a^{15}-\frac{4937471896239}{91688549233}a^{14}-\frac{3105210570819}{91688549233}a^{13}-\frac{86598131135505}{183377098466}a^{12}-\frac{2912895924699}{366754196932}a^{11}-\frac{12795100142363}{183377098466}a^{10}+\frac{13691997076887}{91688549233}a^{9}+\frac{5465283567703}{183377098466}a^{8}+\frac{89703478288851}{366754196932}a^{7}-\frac{21254903567762}{91688549233}a^{6}+\frac{63391647612736}{91688549233}a^{5}-\frac{42713224718912}{91688549233}a^{4}+\frac{116663471016303}{366754196932}a^{3}-\frac{40739131354929}{91688549233}a^{2}+\frac{90422519964193}{183377098466}a-\frac{18766276607035}{91688549233}$, $\frac{7696257738789}{366754196932}a^{19}+\frac{2402824292877}{183377098466}a^{18}+\frac{4598568004181}{91688549233}a^{17}-\frac{1959401552701}{183377098466}a^{16}+\frac{7088683778371}{91688549233}a^{15}-\frac{3270093344790}{91688549233}a^{14}-\frac{2027362745342}{91688549233}a^{13}-\frac{56415473846419}{183377098466}a^{12}-\frac{1112749579063}{366754196932}a^{11}-\frac{4012147051263}{91688549233}a^{10}+\frac{9065697812303}{91688549233}a^{9}+\frac{3481075229941}{183377098466}a^{8}+\frac{58097615157755}{366754196932}a^{7}-\frac{28013600092743}{183377098466}a^{6}+\frac{41269253285529}{91688549233}a^{5}-\frac{28106803016259}{91688549233}a^{4}+\frac{76508681410967}{366754196932}a^{3}-\frac{53328897894303}{183377098466}a^{2}+\frac{59172354155451}{183377098466}a-\frac{12272652309418}{91688549233}$, $\frac{2087066405455}{183377098466}a^{19}+\frac{663531292849}{91688549233}a^{18}+\frac{2511295136430}{91688549233}a^{17}-\frac{486468100428}{91688549233}a^{16}+\frac{3869591139112}{91688549233}a^{15}-\frac{1713429667021}{91688549233}a^{14}-\frac{1087511571186}{91688549233}a^{13}-\frac{15303315786082}{91688549233}a^{12}-\frac{699461058197}{183377098466}a^{11}-\frac{2356372433372}{91688549233}a^{10}+\frac{4734834939648}{91688549233}a^{9}+\frac{951699104546}{91688549233}a^{8}+\frac{15869667203029}{183377098466}a^{7}-\frac{7451282284400}{91688549233}a^{6}+\frac{22397977282418}{91688549233}a^{5}-\frac{14951668409821}{91688549233}a^{4}+\frac{20554971374501}{183377098466}a^{3}-\frac{14254991724161}{91688549233}a^{2}+\frac{15917554882625}{91688549233}a-\frac{6598981695147}{91688549233}$, $\frac{10100437985889}{366754196932}a^{19}+\frac{3189185917259}{183377098466}a^{18}+\frac{6054594685466}{91688549233}a^{17}-\frac{2464303751479}{183377098466}a^{16}+\frac{9309132499086}{91688549233}a^{15}-\frac{4237241319674}{91688549233}a^{14}-\frac{2695646733215}{91688549233}a^{13}-\frac{74138035434051}{183377098466}a^{12}-\frac{2739033194487}{366754196932}a^{11}-\frac{5453104338565}{91688549233}a^{10}+\frac{11770032447474}{91688549233}a^{9}+\frac{4907383692947}{183377098466}a^{8}+\frac{77105386879151}{366754196932}a^{7}-\frac{36360767498135}{183377098466}a^{6}+\frac{54089841277185}{91688549233}a^{5}-\frac{36543446531567}{91688549233}a^{4}+\frac{99677184037071}{366754196932}a^{3}-\frac{69563669039391}{183377098466}a^{2}+\frac{77167969677201}{183377098466}a-\frac{16065297008728}{91688549233}$, $\frac{3530854287850}{91688549233}a^{19}+\frac{2213947819587}{91688549233}a^{18}+\frac{8449753030159}{91688549233}a^{17}-\frac{1758114925200}{91688549233}a^{16}+\frac{13028774005520}{91688549233}a^{15}-\frac{5944207696942}{91688549233}a^{14}-\frac{3723516749397}{91688549233}a^{13}-\frac{51762684663084}{91688549233}a^{12}-\frac{671677454061}{91688549233}a^{11}-\frac{7516989605516}{91688549233}a^{10}+\frac{16391268923635}{91688549233}a^{9}+\frac{3162054782426}{91688549233}a^{8}+\frac{26766692130764}{91688549233}a^{7}-\frac{25559762968717}{91688549233}a^{6}+\frac{75816404134355}{91688549233}a^{5}-\frac{51248213700214}{91688549233}a^{4}+\frac{34932982211998}{91688549233}a^{3}-\frac{48700605943830}{91688549233}a^{2}+\frac{54290290475593}{91688549233}a-\frac{22585596979825}{91688549233}$, $\frac{8743122917703}{366754196932}a^{19}+\frac{1376808530505}{91688549233}a^{18}+\frac{5239283705539}{91688549233}a^{17}-\frac{2145847569443}{183377098466}a^{16}+\frac{8063636614378}{91688549233}a^{15}-\frac{3670090592250}{91688549233}a^{14}-\frac{2322899714024}{91688549233}a^{13}-\frac{64140784902103}{183377098466}a^{12}-\frac{2194479483669}{366754196932}a^{11}-\frac{9428578258357}{183377098466}a^{10}+\frac{10161207960646}{91688549233}a^{9}+\frac{4133407775673}{183377098466}a^{8}+\frac{66532610998889}{366754196932}a^{7}-\frac{15738540384646}{91688549233}a^{6}+\frac{46920810516385}{91688549233}a^{5}-\frac{31761313192093}{91688549233}a^{4}+\frac{86634986435477}{366754196932}a^{3}-\frac{30173328469548}{91688549233}a^{2}+\frac{66983283092739}{183377098466}a-\frac{13906254124137}{91688549233}$, $\frac{1594399267457}{183377098466}a^{19}+\frac{515786651255}{91688549233}a^{18}+\frac{1913629330462}{91688549233}a^{17}-\frac{373422212679}{91688549233}a^{16}+\frac{2902225963789}{91688549233}a^{15}-\frac{1322459141814}{91688549233}a^{14}-\frac{913331727421}{91688549233}a^{13}-\frac{11742767150422}{91688549233}a^{12}-\frac{821781474013}{183377098466}a^{11}-\frac{1615820977146}{91688549233}a^{10}+\frac{3861514969855}{91688549233}a^{9}+\frac{1005503664038}{91688549233}a^{8}+\frac{12240325695423}{183377098466}a^{7}-\frac{5692525839563}{91688549233}a^{6}+\frac{16880471309326}{91688549233}a^{5}-\frac{11356364829837}{91688549233}a^{4}+\frac{15107177322601}{183377098466}a^{3}-\frac{10973372235726}{91688549233}a^{2}+\frac{11910723090826}{91688549233}a-\frac{4842356208183}{91688549233}$
| 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: | \( 15341.0740173 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 15341.0740173 \cdot 1}{2\cdot\sqrt{865929930053558147743744}}\cr\approx \mathstrut & 0.790465787513 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 + 2*x^18 - 2*x^17 + 4*x^16 - 4*x^15 - 14*x^13 + 9*x^12 - 2*x^11 + 6*x^10 - 2*x^9 + 7*x^8 - 12*x^7 + 26*x^6 - 28*x^5 + 19*x^4 - 20*x^3 + 24*x^2 - 16*x + 4)
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^18 - 2*x^17 + 4*x^16 - 4*x^15 - 14*x^13 + 9*x^12 - 2*x^11 + 6*x^10 - 2*x^9 + 7*x^8 - 12*x^7 + 26*x^6 - 28*x^5 + 19*x^4 - 20*x^3 + 24*x^2 - 16*x + 4, 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^18 - 2*x^17 + 4*x^16 - 4*x^15 - 14*x^13 + 9*x^12 - 2*x^11 + 6*x^10 - 2*x^9 + 7*x^8 - 12*x^7 + 26*x^6 - 28*x^5 + 19*x^4 - 20*x^3 + 24*x^2 - 16*x + 4);
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^18 - 2*x^17 + 4*x^16 - 4*x^15 - 14*x^13 + 9*x^12 - 2*x^11 + 6*x^10 - 2*x^9 + 7*x^8 - 12*x^7 + 26*x^6 - 28*x^5 + 19*x^4 - 20*x^3 + 24*x^2 - 16*x + 4);
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
$C_2\times A_5$ (as 20T36):
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 120 |
| The 10 conjugacy class representatives for $C_2\times A_5$ |
| Character table for $C_2\times A_5$ |
Intermediate fields
| 10.2.116319195136.1 |
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 10 sibling: | data not computed |
| Degree 12 siblings: | data not computed |
| Degree 20 sibling: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 40 sibling: | data not computed |
| Minimal sibling: | 10.2.8491301244928.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.3.0.1}{3} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.7 | $x^{4} + 2 x^{3} + 2 x^{2} + 2$ | $4$ | $1$ | $6$ | $A_4$ | $[2, 2]^{3}$ |
| 2.4.6.7 | $x^{4} + 2 x^{3} + 2 x^{2} + 2$ | $4$ | $1$ | $6$ | $A_4$ | $[2, 2]^{3}$ | |
| 2.12.18.59 | $x^{12} + 6 x^{11} + 22 x^{10} + 56 x^{9} + 126 x^{8} + 240 x^{7} + 332 x^{6} - 18 x^{5} - 459 x^{4} - 394 x^{3} - 344 x^{2} + 138 x + 423$ | $4$ | $3$ | $18$ | $A_4$ | $[2, 2]^{3}$ | |
|
\(73\)
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.4.2.1 | $x^{4} + 8024 x^{3} + 16372240 x^{2} + 1107697152 x + 99582464$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 73.4.2.1 | $x^{4} + 8024 x^{3} + 16372240 x^{2} + 1107697152 x + 99582464$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 73.4.2.1 | $x^{4} + 8024 x^{3} + 16372240 x^{2} + 1107697152 x + 99582464$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 73.4.2.1 | $x^{4} + 8024 x^{3} + 16372240 x^{2} + 1107697152 x + 99582464$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |