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Properties

Label 20.4.223...625.1
Degree $20$
Signature $[4, 8]$
Discriminant $2.239\times 10^{23}$
Root discriminant \(14.71\)
Ramified primes $5,13,41$
Class number $1$
Class group trivial
Galois group $C_2\times S_5$ (as 20T62)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^18 - 4*x^17 - x^16 + 15*x^15 + 33*x^14 - 21*x^13 - 80*x^12 + 16*x^11 + 108*x^10 - 15*x^9 - 97*x^8 + 34*x^7 + 55*x^6 - 35*x^5 - 15*x^4 + 13*x^3 + 2*x^2 - 5*x + 1)
 
Copy content gp:K = bnfinit(y^20 - 4*y^18 - 4*y^17 - y^16 + 15*y^15 + 33*y^14 - 21*y^13 - 80*y^12 + 16*y^11 + 108*y^10 - 15*y^9 - 97*y^8 + 34*y^7 + 55*y^6 - 35*y^5 - 15*y^4 + 13*y^3 + 2*y^2 - 5*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^18 - 4*x^17 - x^16 + 15*x^15 + 33*x^14 - 21*x^13 - 80*x^12 + 16*x^11 + 108*x^10 - 15*x^9 - 97*x^8 + 34*x^7 + 55*x^6 - 35*x^5 - 15*x^4 + 13*x^3 + 2*x^2 - 5*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 4*x^18 - 4*x^17 - x^16 + 15*x^15 + 33*x^14 - 21*x^13 - 80*x^12 + 16*x^11 + 108*x^10 - 15*x^9 - 97*x^8 + 34*x^7 + 55*x^6 - 35*x^5 - 15*x^4 + 13*x^3 + 2*x^2 - 5*x + 1)
 

\( x^{20} - 4 x^{18} - 4 x^{17} - x^{16} + 15 x^{15} + 33 x^{14} - 21 x^{13} - 80 x^{12} + 16 x^{11} + \cdots + 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $20$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[4, 8]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(223904745130829775390625\) \(\medspace = 5^{10}\cdot 13^{6}\cdot 41^{6}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(14.71\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{1/2}13^{1/2}41^{1/2}\approx 51.62363799656123$
Ramified primes:   \(5\), \(13\), \(41\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{293778824857}a^{19}-\frac{122731789841}{293778824857}a^{18}+\frac{29668497640}{293778824857}a^{17}-\frac{98870289971}{293778824857}a^{16}+\frac{2433481172}{41968403551}a^{15}-\frac{46223756263}{293778824857}a^{14}+\frac{119542485757}{293778824857}a^{13}+\frac{107084334337}{293778824857}a^{12}-\frac{145660048092}{293778824857}a^{11}-\frac{8696669444}{293778824857}a^{10}+\frac{127801650794}{293778824857}a^{9}+\frac{104836246363}{293778824857}a^{8}-\frac{45954012795}{293778824857}a^{7}+\frac{2941182564}{41968403551}a^{6}+\frac{112182798419}{293778824857}a^{5}+\frac{70817478413}{293778824857}a^{4}-\frac{74692251805}{293778824857}a^{3}-\frac{111589157014}{293778824857}a^{2}-\frac{128671736649}{293778824857}a+\frac{28596241987}{293778824857}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $11$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{127630150707}{293778824857}a^{19}+\frac{261868052185}{293778824857}a^{18}-\frac{318824545836}{293778824857}a^{17}-\frac{1512789020107}{293778824857}a^{16}-\frac{281857408787}{41968403551}a^{15}+\frac{621988581084}{293778824857}a^{14}+\frac{7865184182309}{293778824857}a^{13}+\frac{9215814292863}{293778824857}a^{12}-\frac{8187254461131}{293778824857}a^{11}-\frac{21610421096764}{293778824857}a^{10}-\frac{541339985336}{293778824857}a^{9}+\frac{23189942192087}{293778824857}a^{8}+\frac{7801352766122}{293778824857}a^{7}-\frac{1835212684251}{41968403551}a^{6}-\frac{4033753117446}{293778824857}a^{5}+\frac{5172318808414}{293778824857}a^{4}+\frac{53728532849}{293778824857}a^{3}-\frac{2357884077595}{293778824857}a^{2}-\frac{730459974534}{293778824857}a+\frac{89404902357}{293778824857}$, $\frac{50234512289}{293778824857}a^{19}+\frac{24543585988}{293778824857}a^{18}-\frac{154845699644}{293778824857}a^{17}-\frac{283711114282}{293778824857}a^{16}-\frac{43318390061}{41968403551}a^{15}+\frac{490610348840}{293778824857}a^{14}+\frac{1806545809606}{293778824857}a^{13}+\frac{283366890768}{293778824857}a^{12}-\frac{2875496381132}{293778824857}a^{11}-\frac{1219569148906}{293778824857}a^{10}+\frac{2907207935636}{293778824857}a^{9}+\frac{1173407006526}{293778824857}a^{8}-\frac{2261611253392}{293778824857}a^{7}-\frac{185809275}{41968403551}a^{6}+\frac{1387352499670}{293778824857}a^{5}-\frac{35483738211}{293778824857}a^{4}-\frac{378088540972}{293778824857}a^{3}+\frac{144870782276}{293778824857}a^{2}+\frac{103721862304}{293778824857}a-\frac{211721691880}{293778824857}$, $a$, $\frac{144161935495}{293778824857}a^{19}-\frac{87605732387}{293778824857}a^{18}-\frac{704018235768}{293778824857}a^{17}-\frac{276081793168}{293778824857}a^{16}+\frac{100333590855}{41968403551}a^{15}+\frac{2990420256331}{293778824857}a^{14}+\frac{3829431613638}{293778824857}a^{13}-\frac{7825875035653}{293778824857}a^{12}-\frac{14801264720553}{293778824857}a^{11}+\frac{10044261888097}{293778824857}a^{10}+\frac{25375275687009}{293778824857}a^{9}-\frac{8294438886565}{293778824857}a^{8}-\frac{26349940393937}{293778824857}a^{7}+\frac{1053735167159}{41968403551}a^{6}+\frac{15907272857969}{293778824857}a^{5}-\frac{6468163763210}{293778824857}a^{4}-\frac{5318572842231}{293778824857}a^{3}+\frac{2700129053535}{293778824857}a^{2}+\frac{1427324389046}{293778824857}a-\frac{627930060843}{293778824857}$, $\frac{351945174325}{293778824857}a^{19}+\frac{533497637664}{293778824857}a^{18}-\frac{1026836956921}{293778824857}a^{17}-\frac{3453839462592}{293778824857}a^{16}-\frac{576798715847}{41968403551}a^{15}+\frac{2759699222283}{293778824857}a^{14}+\frac{18999245103866}{293778824857}a^{13}+\frac{16366051102565}{293778824857}a^{12}-\frac{24929480354811}{293778824857}a^{11}-\frac{42386725269136}{293778824857}a^{10}+\frac{11498361387678}{293778824857}a^{9}+\frac{48102619409403}{293778824857}a^{8}+\frac{4466441394092}{293778824857}a^{7}-\frac{3916209563825}{41968403551}a^{6}-\frac{2843657353386}{293778824857}a^{5}+\frac{10843053070404}{293778824857}a^{4}-\frac{442278685669}{293778824857}a^{3}-\frac{4791938044464}{293778824857}a^{2}-\frac{1180179891894}{293778824857}a+\frac{335592097926}{293778824857}$, $\frac{13756737283}{293778824857}a^{19}-\frac{20280876181}{293778824857}a^{18}-\frac{43199940977}{293778824857}a^{17}-\frac{9641225128}{293778824857}a^{16}+\frac{7826398203}{41968403551}a^{15}+\frac{282799194435}{293778824857}a^{14}+\frac{196322667685}{293778824857}a^{13}-\frac{778231141889}{293778824857}a^{12}-\frac{818943161090}{293778824857}a^{11}+\frac{888766935684}{293778824857}a^{10}+\frac{1447773215397}{293778824857}a^{9}-\frac{713779787514}{293778824857}a^{8}-\frac{1419142695372}{293778824857}a^{7}+\frac{64303654188}{41968403551}a^{6}+\frac{1078824264033}{293778824857}a^{5}-\frac{76673117333}{293778824857}a^{4}-\frac{983469842522}{293778824857}a^{3}-\frac{1007146796}{293778824857}a^{2}+\frac{678670148154}{293778824857}a+\frac{19599067316}{293778824857}$, $\frac{208090067481}{293778824857}a^{19}+\frac{245986529902}{293778824857}a^{18}-\frac{749280358600}{293778824857}a^{17}-\frac{1918754668855}{293778824857}a^{16}-\frac{226395209464}{41968403551}a^{15}+\frac{2888848981160}{293778824857}a^{14}+\frac{11189653492413}{293778824857}a^{13}+\frac{5597240661483}{293778824857}a^{12}-\frac{20435496228481}{293778824857}a^{11}-\frac{22628642828003}{293778824857}a^{10}+\frac{18556261934215}{293778824857}a^{9}+\frac{32705785221883}{293778824857}a^{8}-\frac{8477008137685}{293778824857}a^{7}-\frac{3533775086457}{41968403551}a^{6}+\frac{3436680121978}{293778824857}a^{5}+\frac{11718515914370}{293778824857}a^{4}-\frac{2480209252793}{293778824857}a^{3}-\frac{4326203012402}{293778824857}a^{2}+\frac{925377537215}{293778824857}a+\frac{868356038378}{293778824857}$, $\frac{307527105583}{293778824857}a^{19}+\frac{83672514526}{293778824857}a^{18}-\frac{1273929065888}{293778824857}a^{17}-\frac{1703360352705}{293778824857}a^{16}-\frac{77415849261}{41968403551}a^{15}+\frac{5228083755291}{293778824857}a^{14}+\frac{12304061778018}{293778824857}a^{13}-\frac{3785111506521}{293778824857}a^{12}-\frac{29700688749419}{293778824857}a^{11}-\frac{6578297393099}{293778824857}a^{10}+\frac{37992183793860}{293778824857}a^{9}+\frac{15443742462893}{293778824857}a^{8}-\frac{31048196972713}{293778824857}a^{7}-\frac{1485941474054}{41968403551}a^{6}+\frac{16989560467609}{293778824857}a^{5}+\frac{2478136219980}{293778824857}a^{4}-\frac{6736582797127}{293778824857}a^{3}-\frac{1292621893691}{293778824857}a^{2}+\frac{2271687133914}{293778824857}a-\frac{222174627608}{293778824857}$, $\frac{420622221158}{293778824857}a^{19}+\frac{217351350471}{293778824857}a^{18}-\frac{1651935679350}{293778824857}a^{17}-\frac{2639483076225}{293778824857}a^{16}-\frac{204872563576}{41968403551}a^{15}+\frac{6277455951507}{293778824857}a^{14}+\frac{17623115577996}{293778824857}a^{13}-\frac{876308482127}{293778824857}a^{12}-\frac{38523496186778}{293778824857}a^{11}-\frac{14760178768201}{293778824857}a^{10}+\frac{46642650174674}{293778824857}a^{9}+\frac{24608992730320}{293778824857}a^{8}-\frac{37518682134922}{293778824857}a^{7}-\frac{2063814984101}{41968403551}a^{6}+\frac{22551556605691}{293778824857}a^{5}+\frac{2566085453201}{293778824857}a^{4}-\frac{9326155003232}{293778824857}a^{3}-\frac{274300261719}{293778824857}a^{2}+\frac{2173469102295}{293778824857}a-\frac{556425842103}{293778824857}$, $\frac{131160160619}{293778824857}a^{19}-\frac{57931060944}{293778824857}a^{18}-\frac{659876540872}{293778824857}a^{17}-\frac{341300941965}{293778824857}a^{16}+\frac{95072781931}{41968403551}a^{15}+\frac{2814901195524}{293778824857}a^{14}+\frac{3721197405649}{293778824857}a^{13}-\frac{6929079999060}{293778824857}a^{12}-\frac{14842201174327}{293778824857}a^{11}+\frac{8034100781422}{293778824857}a^{10}+\frac{26577608847176}{293778824857}a^{9}-\frac{4761332465082}{293778824857}a^{8}-\frac{28771044053574}{293778824857}a^{7}+\frac{454233110528}{41968403551}a^{6}+\frac{19150003241469}{293778824857}a^{5}-\frac{4027214040492}{293778824857}a^{4}-\frac{7890787529283}{293778824857}a^{3}+\frac{2800007567425}{293778824857}a^{2}+\frac{2501752415773}{293778824857}a-\frac{1102940949799}{293778824857}$, $\frac{383699805936}{293778824857}a^{19}+\frac{390698617823}{293778824857}a^{18}-\frac{1404088214905}{293778824857}a^{17}-\frac{3179068329593}{293778824857}a^{16}-\frac{368807614257}{41968403551}a^{15}+\frac{5065802310281}{293778824857}a^{14}+\frac{19168311793993}{293778824857}a^{13}+\frac{7713399231803}{293778824857}a^{12}-\frac{35257462699516}{293778824857}a^{11}-\frac{32108343564471}{293778824857}a^{10}+\frac{33658808232309}{293778824857}a^{9}+\frac{44063006825892}{293778824857}a^{8}-\frac{19235272438286}{293778824857}a^{7}-\frac{4168136179987}{41968403551}a^{6}+\frac{10183489802564}{293778824857}a^{5}+\frac{10638387320054}{293778824857}a^{4}-\frac{5354325618870}{293778824857}a^{3}-\frac{3730896143129}{293778824857}a^{2}+\frac{937280827085}{293778824857}a+\frac{310748660616}{293778824857}$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 4695.57166134 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 4695.57166134 \cdot 1}{2\cdot\sqrt{223904745130829775390625}}\cr\approx \mathstrut & 0.192834953745 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^18 - 4*x^17 - x^16 + 15*x^15 + 33*x^14 - 21*x^13 - 80*x^12 + 16*x^11 + 108*x^10 - 15*x^9 - 97*x^8 + 34*x^7 + 55*x^6 - 35*x^5 - 15*x^4 + 13*x^3 + 2*x^2 - 5*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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^20 - 4*x^18 - 4*x^17 - x^16 + 15*x^15 + 33*x^14 - 21*x^13 - 80*x^12 + 16*x^11 + 108*x^10 - 15*x^9 - 97*x^8 + 34*x^7 + 55*x^6 - 35*x^5 - 15*x^4 + 13*x^3 + 2*x^2 - 5*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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^18 - 4*x^17 - x^16 + 15*x^15 + 33*x^14 - 21*x^13 - 80*x^12 + 16*x^11 + 108*x^10 - 15*x^9 - 97*x^8 + 34*x^7 + 55*x^6 - 35*x^5 - 15*x^4 + 13*x^3 + 2*x^2 - 5*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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^18 - 4*x^17 - x^16 + 15*x^15 + 33*x^14 - 21*x^13 - 80*x^12 + 16*x^11 + 108*x^10 - 15*x^9 - 97*x^8 + 34*x^7 + 55*x^6 - 35*x^5 - 15*x^4 + 13*x^3 + 2*x^2 - 5*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

$C_2\times S_5$ (as 20T62):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 240
The 14 conjugacy class representatives for $C_2\times S_5$
Character table for $C_2\times S_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.18927429625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 10 siblings: data not computed
Degree 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 10.2.887778125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }^{2}$ ${\href{/padicField/3.10.0.1}{10} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.5.0.1}{5} }^{4}$ R ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{10}$ ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }$ R ${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{10}$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
5.6.2.6a1.2$x^{12} + 2 x^{10} + 8 x^{9} + 3 x^{8} + 8 x^{7} + 22 x^{6} + 8 x^{5} + 5 x^{4} + 16 x^{3} + 4 x^{2} + 9$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(13\) Copy content Toggle raw display 13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.6.1.0a1.1$x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
13.6.2.6a1.2$x^{12} + 20 x^{9} + 22 x^{8} + 22 x^{7} + 104 x^{6} + 220 x^{5} + 341 x^{4} + 282 x^{3} + 165 x^{2} + 44 x + 17$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(41\) Copy content Toggle raw display $\Q_{41}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{41}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{41}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{41}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
41.1.2.1a1.1$x^{2} + 41$$2$$1$$1$$C_2$$$[\ ]_{2}$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.1.2.1a1.1$x^{2} + 41$$2$$1$$1$$C_2$$$[\ ]_{2}$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.2.2a1.2$x^{4} + 76 x^{3} + 1456 x^{2} + 456 x + 77$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
41.2.2.2a1.2$x^{4} + 76 x^{3} + 1456 x^{2} + 456 x + 77$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)