Properties

Label 20.4.721...184.1
Degree $20$
Signature $[4, 8]$
Discriminant $7.210\times 10^{25}$
Root discriminant \(19.63\)
Ramified primes $2,73$
Class number $1$
Class group trivial
Galois group $C_2\times A_5$ (as 20T31)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - x^18 + 16*x^17 - 16*x^15 - 34*x^14 + 2*x^13 + 52*x^12 + 60*x^11 - 16*x^10 - 54*x^9 - 60*x^8 - 10*x^7 + 41*x^6 + 4*x^5 + 7*x^4 + 10*x^3 - 16*x^2 + 2)
 
gp: K = bnfinit(y^20 - 4*y^19 - y^18 + 16*y^17 - 16*y^15 - 34*y^14 + 2*y^13 + 52*y^12 + 60*y^11 - 16*y^10 - 54*y^9 - 60*y^8 - 10*y^7 + 41*y^6 + 4*y^5 + 7*y^4 + 10*y^3 - 16*y^2 + 2, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 - x^18 + 16*x^17 - 16*x^15 - 34*x^14 + 2*x^13 + 52*x^12 + 60*x^11 - 16*x^10 - 54*x^9 - 60*x^8 - 10*x^7 + 41*x^6 + 4*x^5 + 7*x^4 + 10*x^3 - 16*x^2 + 2);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 - x^18 + 16*x^17 - 16*x^15 - 34*x^14 + 2*x^13 + 52*x^12 + 60*x^11 - 16*x^10 - 54*x^9 - 60*x^8 - 10*x^7 + 41*x^6 + 4*x^5 + 7*x^4 + 10*x^3 - 16*x^2 + 2)
 

\( x^{20} - 4 x^{19} - x^{18} + 16 x^{17} - 16 x^{15} - 34 x^{14} + 2 x^{13} + 52 x^{12} + 60 x^{11} - 16 x^{10} - 54 x^{9} - 60 x^{8} - 10 x^{7} + 41 x^{6} + 4 x^{5} + 7 x^{4} + 10 x^{3} - 16 x^{2} + \cdots + 2 \) Copy content Toggle raw display

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:  $[4, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(72102196832115802645725184\) \(\medspace = 2^{24}\cdot 73^{10}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(19.63\)
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))
 
Ramified primes:   \(2\), \(73\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}$, $\frac{1}{164}a^{18}-\frac{9}{41}a^{17}-\frac{8}{41}a^{16}+\frac{1}{41}a^{15}+\frac{2}{41}a^{14}+\frac{20}{41}a^{13}+\frac{39}{82}a^{12}-\frac{23}{82}a^{11}-\frac{29}{82}a^{10}-\frac{1}{2}a^{9}+\frac{23}{82}a^{8}+\frac{8}{41}a^{7}-\frac{35}{82}a^{6}-\frac{19}{82}a^{5}-\frac{65}{164}a^{4}-\frac{15}{82}a^{3}-\frac{19}{82}a^{2}-\frac{5}{41}a-\frac{7}{82}$, $\frac{1}{166369144}a^{19}-\frac{411491}{166369144}a^{18}+\frac{4054375}{41592286}a^{17}-\frac{2899542}{20796143}a^{16}-\frac{2707289}{41592286}a^{15}+\frac{430531}{20796143}a^{14}-\frac{4587349}{83184572}a^{13}+\frac{9136434}{20796143}a^{12}+\frac{5703915}{20796143}a^{11}-\frac{14579597}{41592286}a^{10}-\frac{5487383}{20796143}a^{9}-\frac{25792273}{83184572}a^{8}+\frac{15482311}{83184572}a^{7}-\frac{1080973}{41592286}a^{6}-\frac{76416499}{166369144}a^{5}+\frac{49473797}{166369144}a^{4}+\frac{17476575}{41592286}a^{3}-\frac{8255021}{83184572}a^{2}-\frac{6142509}{83184572}a+\frac{13582907}{83184572}$ Copy content Toggle raw display

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:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{7064798}{20796143}a^{19}-\frac{29267214}{20796143}a^{18}-\frac{13166189}{41592286}a^{17}+\frac{125362035}{20796143}a^{16}-\frac{9172947}{41592286}a^{15}-\frac{155492748}{20796143}a^{14}-\frac{259710160}{20796143}a^{13}+\frac{62861206}{20796143}a^{12}+\frac{1002528893}{41592286}a^{11}+\frac{956352407}{41592286}a^{10}-\frac{231100324}{20796143}a^{9}-\frac{1275230045}{41592286}a^{8}-\frac{565596241}{20796143}a^{7}-\frac{37576221}{20796143}a^{6}+\frac{533494699}{20796143}a^{5}+\frac{471853697}{41592286}a^{4}+\frac{174564471}{20796143}a^{3}+\frac{108255064}{20796143}a^{2}-\frac{142128987}{20796143}a-\frac{36649118}{20796143}$, $\frac{15998117}{83184572}a^{19}-\frac{58038013}{83184572}a^{18}-\frac{9079461}{20796143}a^{17}+\frac{117987983}{41592286}a^{16}+\frac{22617067}{20796143}a^{15}-\frac{48775807}{20796143}a^{14}-\frac{318055195}{41592286}a^{13}-\frac{123980601}{41592286}a^{12}+\frac{361593663}{41592286}a^{11}+\frac{325398262}{20796143}a^{10}+\frac{157553957}{41592286}a^{9}-\frac{371009833}{41592286}a^{8}-\frac{705385971}{41592286}a^{7}-\frac{196496837}{20796143}a^{6}+\frac{443924387}{83184572}a^{5}+\frac{351066101}{83184572}a^{4}+\frac{76323120}{20796143}a^{3}+\frac{80605569}{41592286}a^{2}-\frac{104608061}{41592286}a-\frac{29247383}{41592286}$, $\frac{17812425}{41592286}a^{19}-\frac{3562199}{2028892}a^{18}-\frac{6797436}{20796143}a^{17}+\frac{7144943}{1014446}a^{16}-\frac{7876825}{41592286}a^{15}-\frac{315107935}{41592286}a^{14}-\frac{322149047}{20796143}a^{13}+\frac{91278089}{41592286}a^{12}+\frac{528548612}{20796143}a^{11}+\frac{1190813601}{41592286}a^{10}-\frac{207777054}{20796143}a^{9}-\frac{625940917}{20796143}a^{8}-\frac{1366336511}{41592286}a^{7}-\frac{135866507}{41592286}a^{6}+\frac{509444204}{20796143}a^{5}+\frac{884571361}{83184572}a^{4}+\frac{310677201}{41592286}a^{3}+\frac{205848427}{41592286}a^{2}-\frac{165213609}{20796143}a-\frac{36116805}{41592286}$, $\frac{19626733}{83184572}a^{19}-\frac{44006073}{41592286}a^{18}+\frac{2282025}{20796143}a^{17}+\frac{87477340}{20796143}a^{16}-\frac{53110959}{41592286}a^{15}-\frac{217556321}{41592286}a^{14}-\frac{326242899}{41592286}a^{13}+\frac{107629345}{20796143}a^{12}+\frac{695503561}{41592286}a^{11}+\frac{540017077}{41592286}a^{10}-\frac{573108065}{41592286}a^{9}-\frac{880872001}{41592286}a^{8}-\frac{330475270}{20796143}a^{7}+\frac{257127167}{41592286}a^{6}+\frac{1593852429}{83184572}a^{5}+\frac{133376315}{20796143}a^{4}+\frac{158030961}{41592286}a^{3}+\frac{62621429}{20796143}a^{2}-\frac{225819157}{41592286}a-\frac{24230854}{20796143}$, $\frac{85043085}{166369144}a^{19}-\frac{280953959}{166369144}a^{18}-\frac{32436434}{20796143}a^{17}+\frac{136002426}{20796143}a^{16}+\frac{192384409}{41592286}a^{15}-\frac{127043843}{41592286}a^{14}-\frac{1655676869}{83184572}a^{13}-\frac{605637823}{41592286}a^{12}+\frac{497107489}{41592286}a^{11}+\frac{846035675}{20796143}a^{10}+\frac{534197010}{20796143}a^{9}-\frac{171141725}{83184572}a^{8}-\frac{2941668807}{83184572}a^{7}-\frac{1466490341}{41592286}a^{6}-\frac{2074037955}{166369144}a^{5}-\frac{1045368483}{166369144}a^{4}+\frac{87333025}{41592286}a^{3}+\frac{644696131}{83184572}a^{2}-\frac{73743993}{83184572}a-\frac{29058685}{83184572}$, $\frac{51154205}{166369144}a^{19}-\frac{170594759}{166369144}a^{18}-\frac{21758477}{20796143}a^{17}+\frac{91605731}{20796143}a^{16}+\frac{128216537}{41592286}a^{15}-\frac{71993134}{20796143}a^{14}-\frac{1101392291}{83184572}a^{13}-\frac{340662715}{41592286}a^{12}+\frac{262500861}{20796143}a^{11}+\frac{586219624}{20796143}a^{10}+\frac{275139462}{20796143}a^{9}-\frac{957209701}{83184572}a^{8}-\frac{2375478221}{83184572}a^{7}-\frac{451262574}{20796143}a^{6}+\frac{275800429}{166369144}a^{5}+\frac{754917585}{166369144}a^{4}+\frac{184954521}{41592286}a^{3}+\frac{490231811}{83184572}a^{2}-\frac{175655157}{83184572}a-\frac{58376897}{83184572}$, $\frac{5723347}{20796143}a^{19}-\frac{85327991}{83184572}a^{18}-\frac{18152845}{41592286}a^{17}+\frac{161486913}{41592286}a^{16}+\frac{15580432}{20796143}a^{15}-\frac{107952783}{41592286}a^{14}-\frac{195988826}{20796143}a^{13}-\frac{126999513}{41592286}a^{12}+\frac{398997775}{41592286}a^{11}+\frac{354136041}{20796143}a^{10}+\frac{86010585}{20796143}a^{9}-\frac{270765647}{41592286}a^{8}-\frac{655894411}{41592286}a^{7}-\frac{402359511}{41592286}a^{6}+\frac{101333183}{41592286}a^{5}-\frac{240093957}{83184572}a^{4}+\frac{62842399}{41592286}a^{3}+\frac{86946063}{41592286}a^{2}-\frac{62138416}{20796143}a+\frac{65228609}{41592286}$, $\frac{101542051}{166369144}a^{19}-\frac{372475189}{166369144}a^{18}-\frac{28394185}{20796143}a^{17}+\frac{194280584}{20796143}a^{16}+\frac{65668042}{20796143}a^{15}-\frac{182888690}{20796143}a^{14}-\frac{1998200523}{83184572}a^{13}-\frac{136720749}{20796143}a^{12}+\frac{1270039141}{41592286}a^{11}+\frac{962068568}{20796143}a^{10}+\frac{106055530}{20796143}a^{9}-\frac{2703584973}{83184572}a^{8}-\frac{3851423527}{83184572}a^{7}-\frac{789037279}{41592286}a^{6}+\frac{2995886263}{166369144}a^{5}+\frac{1033435631}{166369144}a^{4}+\frac{176455707}{41592286}a^{3}+\frac{644871969}{83184572}a^{2}-\frac{364693899}{83184572}a-\frac{82715343}{83184572}$, $\frac{25282924}{20796143}a^{19}-\frac{357956455}{83184572}a^{18}-\frac{65939223}{20796143}a^{17}+\frac{742630003}{41592286}a^{16}+\frac{172765208}{20796143}a^{15}-\frac{315987499}{20796143}a^{14}-\frac{1012240217}{20796143}a^{13}-\frac{421476290}{20796143}a^{12}+\frac{1106101947}{20796143}a^{11}+\frac{4067092013}{41592286}a^{10}+\frac{544611627}{20796143}a^{9}-\frac{2161180645}{41592286}a^{8}-\frac{2011802720}{20796143}a^{7}-\frac{2339554723}{41592286}a^{6}+\frac{440865689}{20796143}a^{5}+\frac{1028598025}{83184572}a^{4}+\frac{591188005}{41592286}a^{3}+\frac{776579681}{41592286}a^{2}-\frac{191433061}{20796143}a-\frac{142167733}{41592286}$, $\frac{62341381}{83184572}a^{19}-\frac{219486455}{83184572}a^{18}-\frac{41488122}{20796143}a^{17}+\frac{454069249}{41592286}a^{16}+\frac{110789395}{20796143}a^{15}-\frac{187016476}{20796143}a^{14}-\frac{1255196565}{41592286}a^{13}-\frac{13694715}{1014446}a^{12}+\frac{1318063567}{41592286}a^{11}+\frac{1270985336}{20796143}a^{10}+\frac{797762019}{41592286}a^{9}-\frac{1274009819}{41592286}a^{8}-\frac{2518338729}{41592286}a^{7}-\frac{809638745}{20796143}a^{6}+\frac{994820739}{83184572}a^{5}+\frac{17260451}{2028892}a^{4}+\frac{221657785}{20796143}a^{3}+\frac{498148225}{41592286}a^{2}-\frac{242529527}{41592286}a-\frac{99288213}{41592286}$, $\frac{16134147}{83184572}a^{19}-\frac{64144031}{83184572}a^{18}-\frac{3117233}{20796143}a^{17}+\frac{117061969}{41592286}a^{16}+\frac{8780937}{41592286}a^{15}-\frac{49997668}{20796143}a^{14}-\frac{145537210}{20796143}a^{13}+\frac{3444266}{20796143}a^{12}+\frac{166279662}{20796143}a^{11}+\frac{264170512}{20796143}a^{10}-\frac{101402637}{41592286}a^{9}-\frac{154072694}{20796143}a^{8}-\frac{524927581}{41592286}a^{7}-\frac{31948209}{41592286}a^{6}+\frac{333852663}{83184572}a^{5}+\frac{107213061}{83184572}a^{4}+\frac{2288725}{20796143}a^{3}+\frac{1925007}{1014446}a^{2}-\frac{105428295}{41592286}a+\frac{15701985}{41592286}$ Copy content Toggle raw display
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:  \( 363553.530459 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 363553.530459 \cdot 1}{2\cdot\sqrt{72102196832115802645725184}}\cr\approx \mathstrut & 0.831999473971 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - x^18 + 16*x^17 - 16*x^15 - 34*x^14 + 2*x^13 + 52*x^12 + 60*x^11 - 16*x^10 - 54*x^9 - 60*x^8 - 10*x^7 + 41*x^6 + 4*x^5 + 7*x^4 + 10*x^3 - 16*x^2 + 2)
 
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 - 4*x^19 - x^18 + 16*x^17 - 16*x^15 - 34*x^14 + 2*x^13 + 52*x^12 + 60*x^11 - 16*x^10 - 54*x^9 - 60*x^8 - 10*x^7 + 41*x^6 + 4*x^5 + 7*x^4 + 10*x^3 - 16*x^2 + 2, 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 - 4*x^19 - x^18 + 16*x^17 - 16*x^15 - 34*x^14 + 2*x^13 + 52*x^12 + 60*x^11 - 16*x^10 - 54*x^9 - 60*x^8 - 10*x^7 + 41*x^6 + 4*x^5 + 7*x^4 + 10*x^3 - 16*x^2 + 2);
 
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 - 4*x^19 - x^18 + 16*x^17 - 16*x^15 - 34*x^14 + 2*x^13 + 52*x^12 + 60*x^11 - 16*x^10 - 54*x^9 - 60*x^8 - 10*x^7 + 41*x^6 + 4*x^5 + 7*x^4 + 10*x^3 - 16*x^2 + 2);
 
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 20T31):

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

\(\Q(\sqrt{73}) \), 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 ${\href{/padicField/3.5.0.1}{5} }^{4}$ ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.3.0.1}{3} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.5.0.1}{5} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.2.0.1}{2} }^{10}$ ${\href{/padicField/37.3.0.1}{3} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 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}$
2.6.6.1$x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$$2$$3$$6$$A_4$$[2, 2]^{3}$
\(73\) Copy content Toggle raw display 73.2.1.1$x^{2} + 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} + 73$$2$$1$$1$$C_2$$[\ ]_{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}$