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Properties

Label 20.0.866...552.2
Degree $20$
Signature $[0, 10]$
Discriminant $8.667\times 10^{34}$
Root discriminant \(55.83\)
Ramified primes $2,59$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $C_2^9.(C_2\times F_5)$ (as 20T514)

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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 + 32*x^16 - 8*x^14 + 498*x^12 - 112*x^10 + 2540*x^8 - 248*x^6 + 6706*x^4 - 10688*x^2 + 12800)
 
Copy content gp:K = bnfinit(y^20 + 32*y^16 - 8*y^14 + 498*y^12 - 112*y^10 + 2540*y^8 - 248*y^6 + 6706*y^4 - 10688*y^2 + 12800, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 32*x^16 - 8*x^14 + 498*x^12 - 112*x^10 + 2540*x^8 - 248*x^6 + 6706*x^4 - 10688*x^2 + 12800);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 + 32*x^16 - 8*x^14 + 498*x^12 - 112*x^10 + 2540*x^8 - 248*x^6 + 6706*x^4 - 10688*x^2 + 12800)
 

\( x^{20} + 32x^{16} - 8x^{14} + 498x^{12} - 112x^{10} + 2540x^{8} - 248x^{6} + 6706x^{4} - 10688x^{2} + 12800 \) 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:  $[0, 10]$
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:   \(86673392150966032007026116292247552\) \(\medspace = 2^{69}\cdot 59^{8}\) 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:  \(55.83\)
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:  not computed
Ramified primes:   \(2\), \(59\) 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:  \(\Q(\sqrt{2}) \)
$\Aut(K/\Q)$:   $C_2$
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.
Maximal CM subfield:  10.0.813182331387904.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5}a^{15}-\frac{2}{5}a^{13}-\frac{2}{5}a^{11}+\frac{2}{5}a^{9}+\frac{2}{5}a^{5}+\frac{1}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{10}a^{16}-\frac{1}{5}a^{14}-\frac{1}{5}a^{12}+\frac{1}{5}a^{10}+\frac{1}{5}a^{6}-\frac{2}{5}a^{4}+\frac{2}{5}a^{2}$, $\frac{1}{40}a^{17}-\frac{2}{5}a^{13}+\frac{1}{5}a^{11}-\frac{3}{20}a^{9}-\frac{1}{5}a^{7}-\frac{1}{2}a^{5}+\frac{2}{5}a^{3}+\frac{9}{20}a$, $\frac{1}{45\cdots 60}a^{18}-\frac{1}{80}a^{17}+\frac{700026280075151}{28\cdots 60}a^{16}-\frac{22\cdots 13}{47\cdots 60}a^{14}+\frac{1}{5}a^{13}-\frac{23\cdots 73}{56\cdots 20}a^{12}-\frac{1}{10}a^{11}+\frac{68\cdots 53}{22\cdots 80}a^{10}-\frac{17}{40}a^{9}+\frac{45\cdots 49}{94\cdots 20}a^{8}-\frac{2}{5}a^{7}-\frac{47\cdots 57}{11\cdots 40}a^{6}+\frac{1}{4}a^{5}+\frac{29\cdots 49}{56\cdots 20}a^{4}+\frac{3}{10}a^{3}+\frac{23\cdots 81}{22\cdots 80}a^{2}-\frac{9}{40}a-\frac{539838157322683}{28\cdots 96}$, $\frac{1}{18\cdots 40}a^{19}+\frac{700026280075151}{11\cdots 40}a^{17}-\frac{1}{20}a^{16}+\frac{307971594056423}{37\cdots 28}a^{15}-\frac{2}{5}a^{14}+\frac{11\cdots 63}{45\cdots 36}a^{13}-\frac{2}{5}a^{12}+\frac{77\cdots 81}{18\cdots 44}a^{11}+\frac{2}{5}a^{10}+\frac{786130375153421}{37\cdots 80}a^{9}-\frac{1}{2}a^{8}-\frac{16\cdots 97}{45\cdots 60}a^{7}+\frac{2}{5}a^{6}-\frac{76\cdots 39}{22\cdots 80}a^{5}+\frac{1}{5}a^{4}+\frac{36\cdots 45}{18\cdots 44}a^{3}-\frac{1}{5}a^{2}+\frac{140368721968681}{56\cdots 20}a-\frac{1}{2}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  $C_{4}$, which has order $4$ (assuming GRH)
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:  $C_{4}$, which has order $4$ (assuming GRH)
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:  $9$
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:   \( -\frac{693560855}{14557177871616} a^{18} - \frac{145025713}{909823616976} a^{16} - \frac{275965573}{151637269496} a^{14} - \frac{7730329705}{1819647233952} a^{12} - \frac{231005311007}{7278588935808} a^{10} - \frac{17334870123}{303274538992} a^{8} - \frac{880449604813}{3639294467904} a^{6} - \frac{274172346959}{1819647233952} a^{4} - \frac{4884517123967}{7278588935808} a^{2} + \frac{89143772641}{454911808488} \)  (order $4$) 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:   $\frac{109435235129}{295787448810635}a^{18}-\frac{206213836441}{295787448810635}a^{16}+\frac{3653725641393}{295787448810635}a^{14}-\frac{7212650465086}{295787448810635}a^{12}+\frac{59251582458044}{295787448810635}a^{10}-\frac{104275644652537}{295787448810635}a^{8}+\frac{311601899090748}{295787448810635}a^{6}-\frac{321940368870102}{295787448810635}a^{4}+\frac{176867616995648}{295787448810635}a^{2}+\frac{16568929007201}{59157489762127}$, $\frac{16175483725381}{22\cdots 80}a^{18}-\frac{3622675768163}{14\cdots 80}a^{16}-\frac{5223779909703}{23\cdots 80}a^{14}-\frac{197299201326299}{28\cdots 60}a^{12}-\frac{778434961458041}{22\cdots 68}a^{10}-\frac{322082934787409}{47\cdots 60}a^{8}-\frac{10\cdots 59}{56\cdots 20}a^{6}-\frac{92\cdots 33}{28\cdots 60}a^{4}-\frac{79\cdots 53}{22\cdots 68}a^{2}-\frac{36351416769401}{14\cdots 48}$, $\frac{261069948529561}{18\cdots 40}a^{19}+\frac{256992760117259}{45\cdots 60}a^{18}-\frac{115061481834527}{11\cdots 40}a^{17}+\frac{14456880645997}{28\cdots 60}a^{16}-\frac{121001849037963}{18\cdots 40}a^{15}+\frac{81486951613537}{47\cdots 60}a^{14}-\frac{67\cdots 51}{22\cdots 80}a^{13}+\frac{695182395547381}{56\cdots 20}a^{12}-\frac{99\cdots 61}{90\cdots 20}a^{11}+\frac{11\cdots 31}{45\cdots 36}a^{10}-\frac{16\cdots 21}{37\cdots 80}a^{9}+\frac{20\cdots 31}{94\cdots 20}a^{8}-\frac{39\cdots 71}{45\cdots 60}a^{7}+\frac{13\cdots 61}{11\cdots 40}a^{6}-\frac{41\cdots 93}{22\cdots 80}a^{5}+\frac{76\cdots 47}{56\cdots 20}a^{4}-\frac{19\cdots 17}{90\cdots 20}a^{3}+\frac{21\cdots 87}{45\cdots 36}a^{2}-\frac{18\cdots 61}{56\cdots 20}a-\frac{692394330773249}{28\cdots 96}$, $\frac{261069948529561}{18\cdots 40}a^{19}-\frac{256992760117259}{45\cdots 60}a^{18}-\frac{115061481834527}{11\cdots 40}a^{17}-\frac{14456880645997}{28\cdots 60}a^{16}-\frac{121001849037963}{18\cdots 40}a^{15}-\frac{81486951613537}{47\cdots 60}a^{14}-\frac{67\cdots 51}{22\cdots 80}a^{13}-\frac{695182395547381}{56\cdots 20}a^{12}-\frac{99\cdots 61}{90\cdots 20}a^{11}-\frac{11\cdots 31}{45\cdots 36}a^{10}-\frac{16\cdots 21}{37\cdots 80}a^{9}-\frac{20\cdots 31}{94\cdots 20}a^{8}-\frac{39\cdots 71}{45\cdots 60}a^{7}-\frac{13\cdots 61}{11\cdots 40}a^{6}-\frac{41\cdots 93}{22\cdots 80}a^{5}-\frac{76\cdots 47}{56\cdots 20}a^{4}-\frac{19\cdots 17}{90\cdots 20}a^{3}-\frac{21\cdots 87}{45\cdots 36}a^{2}-\frac{18\cdots 61}{56\cdots 20}a+\frac{692394330773249}{28\cdots 96}$, $\frac{195570034785817}{18\cdots 40}a^{19}-\frac{29226231258077}{45\cdots 60}a^{18}+\frac{32381783820577}{11\cdots 40}a^{17}-\frac{31041722873899}{28\cdots 60}a^{16}-\frac{9011971177423}{37\cdots 28}a^{15}-\frac{2787518475803}{946519836194032}a^{14}+\frac{19\cdots 17}{22\cdots 80}a^{13}-\frac{14\cdots 71}{56\cdots 20}a^{12}-\frac{26\cdots 69}{90\cdots 20}a^{11}-\frac{86\cdots 33}{22\cdots 80}a^{10}+\frac{904596772865679}{75\cdots 56}a^{9}-\frac{28\cdots 73}{94\cdots 20}a^{8}+\frac{65\cdots 01}{45\cdots 60}a^{7}-\frac{20\cdots 07}{11\cdots 40}a^{6}+\frac{44\cdots 23}{22\cdots 80}a^{5}-\frac{24\cdots 57}{56\cdots 20}a^{4}-\frac{62\cdots 93}{90\cdots 20}a^{3}+\frac{16\cdots 59}{22\cdots 80}a^{2}+\frac{40\cdots 59}{56\cdots 20}a-\frac{37\cdots 45}{28\cdots 96}$, $\frac{58404997420829}{36\cdots 88}a^{19}-\frac{143519770570771}{45\cdots 60}a^{18}-\frac{34923117069431}{11\cdots 40}a^{17}-\frac{2971385386997}{28\cdots 60}a^{16}-\frac{86344835349427}{18\cdots 40}a^{15}-\frac{6771368653269}{946519836194032}a^{14}-\frac{252445350972067}{45\cdots 36}a^{13}+\frac{193182339345427}{56\cdots 20}a^{12}-\frac{57\cdots 97}{90\cdots 20}a^{11}-\frac{19\cdots 79}{22\cdots 80}a^{10}-\frac{21\cdots 01}{37\cdots 80}a^{9}+\frac{547311747951001}{94\cdots 20}a^{8}-\frac{65\cdots 03}{45\cdots 60}a^{7}-\frac{46\cdots 21}{11\cdots 40}a^{6}+\frac{41\cdots 27}{22\cdots 80}a^{5}+\frac{31\cdots 49}{56\cdots 20}a^{4}-\frac{22\cdots 69}{90\cdots 20}a^{3}-\frac{24\cdots 03}{22\cdots 80}a^{2}+\frac{27\cdots 87}{56\cdots 20}a+\frac{16\cdots 05}{28\cdots 96}$, $\frac{24\cdots 99}{18\cdots 40}a^{19}-\frac{78243964604549}{90\cdots 72}a^{18}-\frac{88888235599373}{11\cdots 40}a^{17}-\frac{1424473050527}{28\cdots 60}a^{16}-\frac{767573269759233}{18\cdots 40}a^{15}-\frac{134735566556971}{47\cdots 60}a^{14}-\frac{511409324788945}{45\cdots 36}a^{13}+\frac{14811220399513}{56\cdots 20}a^{12}-\frac{54\cdots 43}{90\cdots 20}a^{11}-\frac{96\cdots 37}{22\cdots 80}a^{10}-\frac{50\cdots 43}{37\cdots 80}a^{9}+\frac{39199578400847}{18\cdots 64}a^{8}-\frac{11\cdots 29}{45\cdots 60}a^{7}-\frac{24\cdots 51}{11\cdots 40}a^{6}-\frac{14\cdots 11}{22\cdots 80}a^{5}+\frac{31\cdots 71}{56\cdots 20}a^{4}-\frac{56\cdots 51}{90\cdots 20}a^{3}-\frac{43\cdots 49}{22\cdots 80}a^{2}+\frac{75\cdots 69}{56\cdots 20}a+\frac{38\cdots 51}{28\cdots 96}$, $\frac{698172539711701}{30\cdots 40}a^{19}+\frac{91934805730939}{37\cdots 80}a^{18}+\frac{33840452631819}{18\cdots 40}a^{17}+\frac{266932414401}{23\cdots 80}a^{16}+\frac{137948496730361}{18\cdots 64}a^{15}+\frac{88913358411987}{11\cdots 40}a^{14}+\frac{12\cdots 79}{37\cdots 80}a^{13}-\frac{19093192663071}{946519836194032}a^{12}+\frac{16\cdots 57}{15\cdots 20}a^{11}+\frac{21\cdots 99}{18\cdots 40}a^{10}+\frac{18\cdots 43}{37\cdots 28}a^{9}-\frac{640010572441907}{23\cdots 80}a^{8}+\frac{40\cdots 47}{75\cdots 60}a^{7}+\frac{48\cdots 73}{94\cdots 20}a^{6}+\frac{66\cdots 41}{37\cdots 80}a^{5}-\frac{11\cdots 13}{946519836194032}a^{4}+\frac{20\cdots 49}{15\cdots 20}a^{3}+\frac{18\cdots 63}{18\cdots 40}a^{2}-\frac{21\cdots 27}{94\cdots 20}a-\frac{67\cdots 21}{236629959048508}$, $\frac{753959087870047}{15\cdots 20}a^{19}-\frac{9785258535589}{37\cdots 28}a^{18}+\frac{77293943935751}{94\cdots 20}a^{17}+\frac{11266052481333}{11\cdots 40}a^{16}-\frac{723007434982023}{47\cdots 60}a^{15}-\frac{65461523265473}{591574897621270}a^{14}+\frac{50\cdots 03}{18\cdots 40}a^{13}+\frac{846788114660273}{23\cdots 80}a^{12}-\frac{17\cdots 87}{75\cdots 60}a^{11}-\frac{20\cdots 77}{94\cdots 20}a^{10}+\frac{31\cdots 19}{94\cdots 20}a^{9}+\frac{14\cdots 09}{236629959048508}a^{8}-\frac{32\cdots 17}{37\cdots 80}a^{7}-\frac{85\cdots 31}{47\cdots 60}a^{6}+\frac{41\cdots 29}{18\cdots 40}a^{5}+\frac{77\cdots 11}{23\cdots 80}a^{4}+\frac{70\cdots 61}{75\cdots 60}a^{3}-\frac{29\cdots 89}{94\cdots 20}a^{2}-\frac{63\cdots 67}{47\cdots 60}a+\frac{15\cdots 61}{118314979524254}$ Copy content Toggle raw display (assuming GRH)
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:  \( 9426852324.645239 \) (assuming GRH)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 9426852324.645239 \cdot 4}{4\cdot\sqrt{86673392150966032007026116292247552}}\cr\approx \mathstrut & 3.07059466022830 \end{aligned}\] (assuming GRH)

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 + 32*x^16 - 8*x^14 + 498*x^12 - 112*x^10 + 2540*x^8 - 248*x^6 + 6706*x^4 - 10688*x^2 + 12800) 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 + 32*x^16 - 8*x^14 + 498*x^12 - 112*x^10 + 2540*x^8 - 248*x^6 + 6706*x^4 - 10688*x^2 + 12800, 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 + 32*x^16 - 8*x^14 + 498*x^12 - 112*x^10 + 2540*x^8 - 248*x^6 + 6706*x^4 - 10688*x^2 + 12800); 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 + 32*x^16 - 8*x^14 + 498*x^12 - 112*x^10 + 2540*x^8 - 248*x^6 + 6706*x^4 - 10688*x^2 + 12800); 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

$C_2^9.(C_2\times F_5)$ (as 20T514):

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 solvable group of order 20480
The 74 conjugacy class representatives for $C_2^9.(C_2\times F_5)$
Character table for $C_2^9.(C_2\times F_5)$

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.7129088.1, 10.0.813182331387904.1

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 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 R ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{5}$ ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.10.0.1}{10} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ R

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
\(2\) Copy content Toggle raw display 2.1.2.2a1.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$$[2]$$
2.1.2.2a1.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$$[2]$$
2.1.16.65c1.80$x^{16} + 16 x^{13} + 16 x^{11} + 16 x^{9} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 18$$16$$1$$65$$C_2^5:C_8$$$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}]$$
\(59\) Copy content Toggle raw display 59.4.1.0a1.1$x^{4} + 2 x^{2} + 40 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
59.2.2.2a1.1$x^{4} + 116 x^{3} + 3368 x^{2} + 291 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
59.2.2.2a1.1$x^{4} + 116 x^{3} + 3368 x^{2} + 291 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
59.2.2.2a1.1$x^{4} + 116 x^{3} + 3368 x^{2} + 291 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
59.2.2.2a1.1$x^{4} + 116 x^{3} + 3368 x^{2} + 291 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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