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Properties

Label 20.4.846...984.1
Degree $20$
Signature $[4, 8]$
Discriminant $8.466\times 10^{22}$
Root discriminant \(14.01\)
Ramified primes $2,137,313$
Class number $1$
Class group trivial
Galois group $S_5\wr C_2$ (as 20T540)

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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^19 + 8*x^18 - 6*x^17 - 10*x^16 + 36*x^15 - 57*x^14 + 54*x^13 - 23*x^12 - 18*x^11 + 26*x^10 + 20*x^9 - 90*x^8 + 128*x^7 - 98*x^6 + 24*x^5 + 32*x^4 - 36*x^3 + 19*x^2 - 6*x + 1)
 
Copy content gp:K = bnfinit(y^20 - 4*y^19 + 8*y^18 - 6*y^17 - 10*y^16 + 36*y^15 - 57*y^14 + 54*y^13 - 23*y^12 - 18*y^11 + 26*y^10 + 20*y^9 - 90*y^8 + 128*y^7 - 98*y^6 + 24*y^5 + 32*y^4 - 36*y^3 + 19*y^2 - 6*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 + 8*x^18 - 6*x^17 - 10*x^16 + 36*x^15 - 57*x^14 + 54*x^13 - 23*x^12 - 18*x^11 + 26*x^10 + 20*x^9 - 90*x^8 + 128*x^7 - 98*x^6 + 24*x^5 + 32*x^4 - 36*x^3 + 19*x^2 - 6*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 4*x^19 + 8*x^18 - 6*x^17 - 10*x^16 + 36*x^15 - 57*x^14 + 54*x^13 - 23*x^12 - 18*x^11 + 26*x^10 + 20*x^9 - 90*x^8 + 128*x^7 - 98*x^6 + 24*x^5 + 32*x^4 - 36*x^3 + 19*x^2 - 6*x + 1)
 

\( x^{20} - 4 x^{19} + 8 x^{18} - 6 x^{17} - 10 x^{16} + 36 x^{15} - 57 x^{14} + 54 x^{13} - 23 x^{12} + \cdots + 1 \) 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:   \(84663181407790756265984\) \(\medspace = 2^{30}\cdot 137^{3}\cdot 313^{3}\) 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:  \(14.01\)
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:  $2^{3/2}137^{1/2}313^{1/2}\approx 585.702996406882$
Ramified primes:   \(2\), \(137\), \(313\) 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:  \(\Q(\sqrt{42881}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2290837327499}a^{19}-\frac{691020715792}{2290837327499}a^{18}+\frac{543393049253}{2290837327499}a^{17}+\frac{224302176205}{2290837327499}a^{16}-\frac{978775788729}{2290837327499}a^{15}-\frac{1037800030453}{2290837327499}a^{14}+\frac{204465148448}{2290837327499}a^{13}+\frac{288103871028}{2290837327499}a^{12}-\frac{1033338056862}{2290837327499}a^{11}-\frac{172381397868}{2290837327499}a^{10}-\frac{1002220843699}{2290837327499}a^{9}-\frac{669986951846}{2290837327499}a^{8}+\frac{1066759142472}{2290837327499}a^{7}+\frac{561636375525}{2290837327499}a^{6}+\frac{866242039442}{2290837327499}a^{5}+\frac{598128707524}{2290837327499}a^{4}+\frac{74298095009}{2290837327499}a^{3}-\frac{7644756979}{327262475357}a^{2}-\frac{591037659319}{2290837327499}a-\frac{521374863133}{2290837327499}$ 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:   $a^{19}-4a^{18}+8a^{17}-6a^{16}-10a^{15}+36a^{14}-57a^{13}+54a^{12}-23a^{11}-18a^{10}+26a^{9}+20a^{8}-90a^{7}+128a^{6}-98a^{5}+24a^{4}+32a^{3}-36a^{2}+19a-6$, $\frac{1574763040527}{2290837327499}a^{19}-\frac{6291102354122}{2290837327499}a^{18}+\frac{12587842643191}{2290837327499}a^{17}-\frac{9420431206918}{2290837327499}a^{16}-\frac{15968063194600}{2290837327499}a^{15}+\frac{57077499739786}{2290837327499}a^{14}-\frac{90219493039230}{2290837327499}a^{13}+\frac{84382014024523}{2290837327499}a^{12}-\frac{34084459878814}{2290837327499}a^{11}-\frac{31552400109228}{2290837327499}a^{10}+\frac{44141630486068}{2290837327499}a^{9}+\frac{30777582455428}{2290837327499}a^{8}-\frac{142904460620506}{2290837327499}a^{7}+\frac{203322173311115}{2290837327499}a^{6}-\frac{152203304440571}{2290837327499}a^{5}+\frac{33062846496649}{2290837327499}a^{4}+\frac{57170371267440}{2290837327499}a^{3}-\frac{8724990869381}{327262475357}a^{2}+\frac{29229127574735}{2290837327499}a-\frac{6783267485810}{2290837327499}$, $\frac{1002739013629}{2290837327499}a^{19}-\frac{3604354348605}{2290837327499}a^{18}+\frac{6446240867452}{2290837327499}a^{17}-\frac{3194540985469}{2290837327499}a^{16}-\frac{11455569265978}{2290837327499}a^{15}+\frac{30834746646377}{2290837327499}a^{14}-\frac{43166724002329}{2290837327499}a^{13}+\frac{35249773527041}{2290837327499}a^{12}-\frac{8603626059060}{2290837327499}a^{11}-\frac{19376927104272}{2290837327499}a^{10}+\frac{14684337175409}{2290837327499}a^{9}+\frac{29057724130212}{2290837327499}a^{8}-\frac{77423695926550}{2290837327499}a^{7}+\frac{93955578957096}{2290837327499}a^{6}-\frac{58192883847208}{2290837327499}a^{5}+\frac{1908962234691}{2290837327499}a^{4}+\frac{27361436397926}{2290837327499}a^{3}-\frac{2583364320215}{327262475357}a^{2}+\frac{9722691600364}{2290837327499}a-\frac{3172363303110}{2290837327499}$, $\frac{140858965875}{2290837327499}a^{19}+\frac{491369812208}{2290837327499}a^{18}-\frac{1864803996233}{2290837327499}a^{17}+\frac{3594859092695}{2290837327499}a^{16}-\frac{1084396479102}{2290837327499}a^{15}-\frac{7430472631651}{2290837327499}a^{14}+\frac{15529101421496}{2290837327499}a^{13}-\frac{19105937867165}{2290837327499}a^{12}+\frac{10443220790721}{2290837327499}a^{11}+\frac{3711368498595}{2290837327499}a^{10}-\frac{15992829124479}{2290837327499}a^{9}+\frac{4685460126169}{2290837327499}a^{8}+\frac{20567773290648}{2290837327499}a^{7}-\frac{39635746097887}{2290837327499}a^{6}+\frac{37594341852236}{2290837327499}a^{5}-\frac{9270914183898}{2290837327499}a^{4}-\frac{14335895502141}{2290837327499}a^{3}+\frac{2269170086618}{327262475357}a^{2}-\frac{1987840982473}{2290837327499}a+\frac{2776399940079}{2290837327499}$, $\frac{52499695003}{327262475357}a^{19}-\frac{365377527858}{327262475357}a^{18}+\frac{942882606104}{327262475357}a^{17}-\frac{1185417250232}{327262475357}a^{16}-\frac{286002705238}{327262475357}a^{15}+\frac{3838213039772}{327262475357}a^{14}-\frac{7441305409010}{327262475357}a^{13}+\frac{8374026310500}{327262475357}a^{12}-\frac{4871066449352}{327262475357}a^{11}-\frac{1265533998939}{327262475357}a^{10}+\frac{5087692437916}{327262475357}a^{9}-\frac{684052922106}{327262475357}a^{8}-\frac{9781126569747}{327262475357}a^{7}+\frac{17876593317879}{327262475357}a^{6}-\frac{16637847675413}{327262475357}a^{5}+\frac{6086728741400}{327262475357}a^{4}+\frac{4217620243863}{327262475357}a^{3}-\frac{6852410531066}{327262475357}a^{2}+\frac{3062093909117}{327262475357}a-\frac{844602069294}{327262475357}$, $\frac{2624076468791}{2290837327499}a^{19}-\frac{7367846637614}{2290837327499}a^{18}+\frac{10295110367785}{2290837327499}a^{17}+\frac{2955733269169}{2290837327499}a^{16}-\frac{33756768516594}{2290837327499}a^{15}+\frac{58147648350494}{2290837327499}a^{14}-\frac{57801562030148}{2290837327499}a^{13}+\frac{18016720223349}{2290837327499}a^{12}+\frac{33476877815914}{2290837327499}a^{11}-\frac{60283074035297}{2290837327499}a^{10}+\frac{845587786752}{2290837327499}a^{9}+\frac{95135821682403}{2290837327499}a^{8}-\frac{146023007088208}{2290837327499}a^{7}+\frac{104528281676449}{2290837327499}a^{6}+\frac{5842016058672}{2290837327499}a^{5}-\frac{82488676054031}{2290837327499}a^{4}+\frac{64243133959730}{2290837327499}a^{3}+\frac{188669299179}{327262475357}a^{2}-\frac{6933744538037}{2290837327499}a+\frac{5867116088619}{2290837327499}$, $\frac{208626075410}{2290837327499}a^{19}+\frac{595464203484}{2290837327499}a^{18}-\frac{2618144835132}{2290837327499}a^{17}+\frac{5379841856236}{2290837327499}a^{16}-\frac{2324865252383}{2290837327499}a^{15}-\frac{9875138702046}{2290837327499}a^{14}+\frac{22964367994397}{2290837327499}a^{13}-\frac{29007611669340}{2290837327499}a^{12}+\frac{17621757576252}{2290837327499}a^{11}+\frac{3897226951334}{2290837327499}a^{10}-\frac{24344841041317}{2290837327499}a^{9}+\frac{10447983659315}{2290837327499}a^{8}+\frac{27490201515092}{2290837327499}a^{7}-\frac{59848755012686}{2290837327499}a^{6}+\frac{57152409974721}{2290837327499}a^{5}-\frac{19044309721585}{2290837327499}a^{4}-\frac{21558917994356}{2290837327499}a^{3}+\frac{3679531426835}{327262475357}a^{2}-\frac{5644357696580}{2290837327499}a+\frac{1285748318429}{2290837327499}$, $\frac{55085033367}{327262475357}a^{19}-\frac{247200274910}{327262475357}a^{18}+\frac{530035941290}{327262475357}a^{17}-\frac{443331887602}{327262475357}a^{16}-\frac{563128422234}{327262475357}a^{15}+\frac{2382592950003}{327262475357}a^{14}-\frac{3805251779913}{327262475357}a^{13}+\frac{3667897411137}{327262475357}a^{12}-\frac{1561038396626}{327262475357}a^{11}-\frac{1452499218249}{327262475357}a^{10}+\frac{2140135412425}{327262475357}a^{9}+\frac{861716664753}{327262475357}a^{8}-\frac{6065037952510}{327262475357}a^{7}+\frac{8562237464622}{327262475357}a^{6}-\frac{6655049781249}{327262475357}a^{5}+\frac{1250619104634}{327262475357}a^{4}+\frac{2947326427634}{327262475357}a^{3}-\frac{2769842595675}{327262475357}a^{2}+\frac{1451803214437}{327262475357}a-\frac{209366591614}{327262475357}$, $\frac{178050377996}{327262475357}a^{19}-\frac{609829431331}{327262475357}a^{18}+\frac{1114535912165}{327262475357}a^{17}-\frac{549296628521}{327262475357}a^{16}-\frac{1902329453938}{327262475357}a^{15}+\frac{5266612829390}{327262475357}a^{14}-\frac{7550972535009}{327262475357}a^{13}+\frac{6242174574604}{327262475357}a^{12}-\frac{1839657276507}{327262475357}a^{11}-\frac{3266549303593}{327262475357}a^{10}+\frac{2685104305450}{327262475357}a^{9}+\frac{4369313527206}{327262475357}a^{8}-\frac{13166164109311}{327262475357}a^{7}+\frac{16307730466516}{327262475357}a^{6}-\frac{10483290387964}{327262475357}a^{5}+\frac{1046792755627}{327262475357}a^{4}+\frac{4777985920748}{327262475357}a^{3}-\frac{4082786583318}{327262475357}a^{2}+\frac{2395205083298}{327262475357}a-\frac{524237375380}{327262475357}$, $\frac{4061349237986}{2290837327499}a^{19}-\frac{15251477508468}{2290837327499}a^{18}+\frac{28341411604895}{2290837327499}a^{17}-\frac{16239590092989}{2290837327499}a^{16}-\frac{46365125718036}{2290837327499}a^{15}+\frac{134671671933233}{2290837327499}a^{14}-\frac{193495237386467}{2290837327499}a^{13}+\frac{162506226124261}{2290837327499}a^{12}-\frac{43207506509519}{2290837327499}a^{11}-\frac{88820532860666}{2290837327499}a^{10}+\frac{81264362876497}{2290837327499}a^{9}+\frac{109761262129138}{2290837327499}a^{8}-\frac{339920260371234}{2290837327499}a^{7}+\frac{423566494506258}{2290837327499}a^{6}-\frac{270927577075770}{2290837327499}a^{5}+\frac{10652399901056}{2290837327499}a^{4}+\frac{137358523471138}{2290837327499}a^{3}-\frac{15173172653734}{327262475357}a^{2}+\frac{41938593859859}{2290837327499}a-\frac{11579780013876}{2290837327499}$, $\frac{2375761232688}{2290837327499}a^{19}-\frac{7773498731478}{2290837327499}a^{18}+\frac{13282502502681}{2290837327499}a^{17}-\frac{4535911326890}{2290837327499}a^{16}-\frac{26985066214367}{2290837327499}a^{15}+\frac{65413889562516}{2290837327499}a^{14}-\frac{87018538693761}{2290837327499}a^{13}+\frac{64672598117404}{2290837327499}a^{12}-\frac{8156371888395}{2290837327499}a^{11}-\frac{46099595870997}{2290837327499}a^{10}+\frac{24920471026837}{2290837327499}a^{9}+\frac{67206430249468}{2290837327499}a^{8}-\frac{163414964239169}{2290837327499}a^{7}+\frac{183274241651050}{2290837327499}a^{6}-\frac{99362156875431}{2290837327499}a^{5}-\frac{13354550444439}{2290837327499}a^{4}+\frac{59092136187762}{2290837327499}a^{3}-\frac{5188261238374}{327262475357}a^{2}+\frac{18094571013280}{2290837327499}a-\frac{3767742744688}{2290837327499}$ 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:  \( 2571.87835428 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 2571.87835428 \cdot 1}{2\cdot\sqrt{84663181407790756265984}}\cr\approx \mathstrut & 0.171763968302 \end{aligned}\]

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^19 + 8*x^18 - 6*x^17 - 10*x^16 + 36*x^15 - 57*x^14 + 54*x^13 - 23*x^12 - 18*x^11 + 26*x^10 + 20*x^9 - 90*x^8 + 128*x^7 - 98*x^6 + 24*x^5 + 32*x^4 - 36*x^3 + 19*x^2 - 6*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^19 + 8*x^18 - 6*x^17 - 10*x^16 + 36*x^15 - 57*x^14 + 54*x^13 - 23*x^12 - 18*x^11 + 26*x^10 + 20*x^9 - 90*x^8 + 128*x^7 - 98*x^6 + 24*x^5 + 32*x^4 - 36*x^3 + 19*x^2 - 6*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^19 + 8*x^18 - 6*x^17 - 10*x^16 + 36*x^15 - 57*x^14 + 54*x^13 - 23*x^12 - 18*x^11 + 26*x^10 + 20*x^9 - 90*x^8 + 128*x^7 - 98*x^6 + 24*x^5 + 32*x^4 - 36*x^3 + 19*x^2 - 6*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^19 + 8*x^18 - 6*x^17 - 10*x^16 + 36*x^15 - 57*x^14 + 54*x^13 - 23*x^12 - 18*x^11 + 26*x^10 + 20*x^9 - 90*x^8 + 128*x^7 - 98*x^6 + 24*x^5 + 32*x^4 - 36*x^3 + 19*x^2 - 6*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

$S_5\wr C_2$ (as 20T540):

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 28800
The 35 conjugacy class representatives for $S_5\wr C_2$
Character table for $S_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{2}) \)

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 sibling: data not computed
Degree 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 25 sibling: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 10.2.1405124608.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.5.0.1}{5} }^{4}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(2\) Copy content Toggle raw display 2.5.2.15a1.9$x^{10} + 2 x^{7} + 6 x^{5} + x^{4} + 6 x^{2} + 7$$2$$5$$15$$C_{10}$$$[3]^{5}$$
2.5.2.15a1.9$x^{10} + 2 x^{7} + 6 x^{5} + x^{4} + 6 x^{2} + 7$$2$$5$$15$$C_{10}$$$[3]^{5}$$
\(137\) Copy content Toggle raw display $\Q_{137}$$x + 134$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{137}$$x + 134$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{137}$$x + 134$$1$$1$$0$Trivial$$[\ ]$$
137.2.1.0a1.1$x^{2} + 131 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
137.2.1.0a1.1$x^{2} + 131 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
137.2.1.0a1.1$x^{2} + 131 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
137.2.1.0a1.1$x^{2} + 131 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
137.3.1.0a1.1$x^{3} + 6 x + 134$$1$$3$$0$$C_3$$$[\ ]^{3}$$
137.3.2.3a1.2$x^{6} + 12 x^{4} + 268 x^{3} + 36 x^{2} + 1608 x + 18093$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(313\) Copy content Toggle raw display $\Q_{313}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{313}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{313}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{313}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{313}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{313}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_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)