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Properties

Label 18.6.644...304.2
Degree $18$
Signature $[6, 6]$
Discriminant $6.449\times 10^{24}$
Root discriminant \(23.89\)
Ramified primes $2,101,479$
Class number $2$
Class group [2]
Galois group $S_4^3.S_4$ (as 18T884)

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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^18 - 10*x^16 - 14*x^15 + 31*x^14 + 68*x^13 + 7*x^12 - 40*x^11 - 23*x^10 - 86*x^9 - 160*x^8 - 94*x^7 + 6*x^6 + 40*x^5 + 39*x^4 + 30*x^3 + 15*x^2 + 2*x - 1)
 
Copy content gp:K = bnfinit(y^18 - 10*y^16 - 14*y^15 + 31*y^14 + 68*y^13 + 7*y^12 - 40*y^11 - 23*y^10 - 86*y^9 - 160*y^8 - 94*y^7 + 6*y^6 + 40*y^5 + 39*y^4 + 30*y^3 + 15*y^2 + 2*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 10*x^16 - 14*x^15 + 31*x^14 + 68*x^13 + 7*x^12 - 40*x^11 - 23*x^10 - 86*x^9 - 160*x^8 - 94*x^7 + 6*x^6 + 40*x^5 + 39*x^4 + 30*x^3 + 15*x^2 + 2*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 10*x^16 - 14*x^15 + 31*x^14 + 68*x^13 + 7*x^12 - 40*x^11 - 23*x^10 - 86*x^9 - 160*x^8 - 94*x^7 + 6*x^6 + 40*x^5 + 39*x^4 + 30*x^3 + 15*x^2 + 2*x - 1)
 

\( x^{18} - 10 x^{16} - 14 x^{15} + 31 x^{14} + 68 x^{13} + 7 x^{12} - 40 x^{11} - 23 x^{10} - 86 x^{9} + \cdots - 1 \) 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:  $18$
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:  $[6, 6]$
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:   \(6448527633882401509474304\) \(\medspace = 2^{18}\cdot 101^{7}\cdot 479^{2}\) 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:  \(23.89\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(101\), \(479\) 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:  \(\Q(\sqrt{101}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3333}a^{16}-\frac{469}{3333}a^{15}-\frac{1462}{3333}a^{14}-\frac{395}{1111}a^{13}+\frac{698}{3333}a^{12}-\frac{5}{1111}a^{11}-\frac{1354}{3333}a^{10}-\frac{30}{1111}a^{9}-\frac{1291}{3333}a^{8}+\frac{1283}{3333}a^{7}+\frac{823}{3333}a^{6}-\frac{742}{3333}a^{5}+\frac{83}{1111}a^{4}+\frac{1457}{3333}a^{3}-\frac{713}{3333}a^{2}+\frac{41}{1111}a-\frac{1087}{3333}$, $\frac{1}{6052184721}a^{17}-\frac{144308}{6052184721}a^{16}-\frac{58468321}{183399537}a^{15}+\frac{60575984}{550198611}a^{14}-\frac{2620114336}{6052184721}a^{13}+\frac{578742442}{6052184721}a^{12}+\frac{131693779}{550198611}a^{11}+\frac{1779446098}{6052184721}a^{10}-\frac{13482881}{31686831}a^{9}+\frac{24857070}{672464969}a^{8}-\frac{2876652043}{6052184721}a^{7}+\frac{532744642}{6052184721}a^{6}+\frac{27059366}{550198611}a^{5}+\frac{2754959774}{6052184721}a^{4}-\frac{1737138031}{6052184721}a^{3}+\frac{2344463117}{6052184721}a^{2}+\frac{2322256031}{6052184721}a-\frac{1214783213}{6052184721}$ 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:  $C_{2}$, which has order $2$
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:  $C_{2}$, which has order $2$
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:   $\frac{4332994574}{6052184721}a^{17}-\frac{2506145188}{6052184721}a^{16}-\frac{4574138462}{672464969}a^{15}-\frac{37554825499}{6052184721}a^{14}+\frac{149335684150}{6052184721}a^{13}+\frac{205043243978}{6052184721}a^{12}-\frac{60506555291}{6052184721}a^{11}-\frac{10424669900}{550198611}a^{10}-\frac{28507016}{2880621}a^{9}-\frac{117452557055}{2017394907}a^{8}-\frac{491225309852}{6052184721}a^{7}-\frac{180603809503}{6052184721}a^{6}+\frac{724494715}{59922621}a^{5}+\frac{140227017109}{6052184721}a^{4}+\frac{117541760863}{6052184721}a^{3}+\frac{87182688304}{6052184721}a^{2}+\frac{32557954162}{6052184721}a+\frac{415052258}{6052184721}$, $\frac{552179264}{6052184721}a^{17}+\frac{90752368}{550198611}a^{16}-\frac{2212359199}{2017394907}a^{15}-\frac{16975219546}{6052184721}a^{14}+\frac{13714992562}{6052184721}a^{13}+\frac{77474972753}{6052184721}a^{12}+\frac{31803765814}{6052184721}a^{11}-\frac{66656429941}{6052184721}a^{10}-\frac{162488173}{31686831}a^{9}-\frac{3569498223}{672464969}a^{8}-\frac{161430875180}{6052184721}a^{7}-\frac{134093098318}{6052184721}a^{6}+\frac{27845408927}{6052184721}a^{5}+\frac{62814607054}{6052184721}a^{4}+\frac{22578666139}{6052184721}a^{3}+\frac{18812274718}{6052184721}a^{2}+\frac{23536766281}{6052184721}a+\frac{4230157808}{6052184721}$, $a$, $\frac{4730209718}{6052184721}a^{17}-\frac{4849659664}{6052184721}a^{16}-\frac{14334769259}{2017394907}a^{15}-\frac{21335588005}{6052184721}a^{14}+\frac{174347509504}{6052184721}a^{13}+\frac{145187234204}{6052184721}a^{12}-\frac{139280894432}{6052184721}a^{11}-\frac{62691552376}{6052184721}a^{10}-\frac{152484025}{31686831}a^{9}-\frac{125487389506}{2017394907}a^{8}-\frac{358902819749}{6052184721}a^{7}-\frac{20541076156}{6052184721}a^{6}+\frac{94079573585}{6052184721}a^{5}+\frac{102420212695}{6052184721}a^{4}+\frac{69725046454}{6052184721}a^{3}+\frac{49677257023}{6052184721}a^{2}+\frac{6992935222}{6052184721}a-\frac{6410363275}{6052184721}$, $\frac{3442692410}{6052184721}a^{17}-\frac{3260685520}{6052184721}a^{16}-\frac{3610343107}{672464969}a^{15}-\frac{17071734973}{6052184721}a^{14}+\frac{12226420598}{550198611}a^{13}+\frac{119329974419}{6052184721}a^{12}-\frac{130009434428}{6052184721}a^{11}-\frac{816876143}{59922621}a^{10}+\frac{81056123}{31686831}a^{9}-\frac{87139895294}{2017394907}a^{8}-\frac{287487508976}{6052184721}a^{7}+\frac{45946768973}{6052184721}a^{6}+\frac{125714784329}{6052184721}a^{5}+\frac{60023021170}{6052184721}a^{4}+\frac{37342236385}{6052184721}a^{3}+\frac{13433426629}{6052184721}a^{2}-\frac{6337105163}{6052184721}a-\frac{1112852971}{6052184721}$, $\frac{227292503}{2017394907}a^{17}+\frac{137192767}{672464969}a^{16}-\frac{2334985153}{2017394907}a^{15}-\frac{7334208482}{2017394907}a^{14}+\frac{1927519075}{2017394907}a^{13}+\frac{29477149342}{2017394907}a^{12}+\frac{27845154178}{2017394907}a^{11}-\frac{12035655407}{2017394907}a^{10}-\frac{123881827}{10562277}a^{9}-\frac{24207477139}{2017394907}a^{8}-\frac{23122765359}{672464969}a^{7}-\frac{27518065032}{672464969}a^{6}-\frac{25267918091}{2017394907}a^{5}+\frac{19623689959}{2017394907}a^{4}+\frac{8401334642}{672464969}a^{3}+\frac{19218172484}{2017394907}a^{2}+\frac{14137214107}{2017394907}a+\frac{5912720020}{2017394907}$, $\frac{3994871674}{6052184721}a^{17}-\frac{2262409472}{6052184721}a^{16}-\frac{13043388520}{2017394907}a^{15}-\frac{34046954519}{6052184721}a^{14}+\frac{148205619140}{6052184721}a^{13}+\frac{196804947172}{6052184721}a^{12}-\frac{98205668614}{6052184721}a^{11}-\frac{149160920384}{6052184721}a^{10}-\frac{81432050}{31686831}a^{9}-\frac{97848389963}{2017394907}a^{8}-\frac{40810762196}{550198611}a^{7}-\frac{88146329345}{6052184721}a^{6}+\frac{153560193256}{6052184721}a^{5}+\frac{122837628224}{6052184721}a^{4}+\frac{59920902524}{6052184721}a^{3}+\frac{32245701347}{6052184721}a^{2}+\frac{17199661118}{6052184721}a+\frac{3117304837}{6052184721}$, $\frac{1988463424}{6052184721}a^{17}-\frac{1896348944}{6052184721}a^{16}-\frac{6085014877}{2017394907}a^{15}-\frac{10370626340}{6052184721}a^{14}+\frac{73293334511}{6052184721}a^{13}+\frac{67013176306}{6052184721}a^{12}-\frac{55775975608}{6052184721}a^{11}-\frac{34370125571}{6052184721}a^{10}-\frac{54362138}{31686831}a^{9}-\frac{54739967915}{2017394907}a^{8}-\frac{166123206241}{6052184721}a^{7}-\frac{5340302675}{6052184721}a^{6}+\frac{50228034787}{6052184721}a^{5}+\frac{3808595881}{550198611}a^{4}+\frac{4203646174}{550198611}a^{3}+\frac{26219874008}{6052184721}a^{2}-\frac{5458771309}{6052184721}a-\frac{8904532544}{6052184721}$, $\frac{1701310282}{6052184721}a^{17}-\frac{1414826114}{6052184721}a^{16}-\frac{5472134498}{2017394907}a^{15}-\frac{9192064397}{6052184721}a^{14}+\frac{64908771212}{6052184721}a^{13}+\frac{61132677097}{6052184721}a^{12}-\frac{58460236861}{6052184721}a^{11}-\frac{19166711}{5447511}a^{10}-\frac{2876626}{2880621}a^{9}-\frac{17500796942}{672464969}a^{8}-\frac{130541783377}{6052184721}a^{7}-\frac{8977033946}{6052184721}a^{6}+\frac{34996903453}{6052184721}a^{5}+\frac{48461529041}{6052184721}a^{4}+\frac{34363383728}{6052184721}a^{3}+\frac{26849789030}{6052184721}a^{2}+\frac{4802537342}{6052184721}a-\frac{4132744334}{6052184721}$, $\frac{3716747864}{6052184721}a^{17}-\frac{1889996332}{6052184721}a^{16}-\frac{12289685750}{2017394907}a^{15}-\frac{32306767480}{6052184721}a^{14}+\frac{137774206864}{6052184721}a^{13}+\frac{182249779517}{6052184721}a^{12}-\frac{95676023891}{6052184721}a^{11}-\frac{108889065931}{6052184721}a^{10}+\frac{68345108}{31686831}a^{9}-\frac{108280243471}{2017394907}a^{8}-\frac{442239966401}{6052184721}a^{7}-\frac{64699883071}{6052184721}a^{6}+\frac{77119194395}{6052184721}a^{5}+\frac{5036294180}{550198611}a^{4}+\frac{86702716}{5447511}a^{3}+\frac{57936457057}{6052184721}a^{2}+\frac{13032264394}{6052184721}a+\frac{3806516765}{6052184721}$, $\frac{828475484}{6052184721}a^{17}-\frac{476289985}{6052184721}a^{16}-\frac{877404976}{672464969}a^{15}-\frac{6349179778}{6052184721}a^{14}+\frac{27358348828}{6052184721}a^{13}+\frac{32618009612}{6052184721}a^{12}-\frac{112137781}{59922621}a^{11}+\frac{7985168540}{6052184721}a^{10}-\frac{39311158}{31686831}a^{9}-\frac{33106281878}{2017394907}a^{8}-\frac{75377302754}{6052184721}a^{7}-\frac{34413509203}{6052184721}a^{6}-\frac{56878920679}{6052184721}a^{5}+\frac{5729111917}{6052184721}a^{4}+\frac{40513841815}{6052184721}a^{3}+\frac{6015610891}{6052184721}a^{2}+\frac{6279486151}{6052184721}a+\frac{6971085365}{6052184721}$ 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:  \( 179188.839877 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 179188.839877 \cdot 2}{2\cdot\sqrt{6448527633882401509474304}}\cr\approx \mathstrut & 0.277868847146 \end{aligned}\]

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^18 - 10*x^16 - 14*x^15 + 31*x^14 + 68*x^13 + 7*x^12 - 40*x^11 - 23*x^10 - 86*x^9 - 160*x^8 - 94*x^7 + 6*x^6 + 40*x^5 + 39*x^4 + 30*x^3 + 15*x^2 + 2*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^18 - 10*x^16 - 14*x^15 + 31*x^14 + 68*x^13 + 7*x^12 - 40*x^11 - 23*x^10 - 86*x^9 - 160*x^8 - 94*x^7 + 6*x^6 + 40*x^5 + 39*x^4 + 30*x^3 + 15*x^2 + 2*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^18 - 10*x^16 - 14*x^15 + 31*x^14 + 68*x^13 + 7*x^12 - 40*x^11 - 23*x^10 - 86*x^9 - 160*x^8 - 94*x^7 + 6*x^6 + 40*x^5 + 39*x^4 + 30*x^3 + 15*x^2 + 2*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 = polynomial_ring(QQ); K, a = number_field(x^18 - 10*x^16 - 14*x^15 + 31*x^14 + 68*x^13 + 7*x^12 - 40*x^11 - 23*x^10 - 86*x^9 - 160*x^8 - 94*x^7 + 6*x^6 + 40*x^5 + 39*x^4 + 30*x^3 + 15*x^2 + 2*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

$S_4^3.S_4$ (as 18T884):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 331776
The 165 conjugacy class representatives for $S_4^3.S_4$
Character table for $S_4^3.S_4$

Intermediate fields

3.3.404.1, 9.7.31584907456.1

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 18.4.3088844736629670323038191616.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ ${\href{/padicField/5.9.0.1}{9} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{5}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.9.0.1}{9} }^{2}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.3.6.18a2.2$x^{18} + 6 x^{16} + 6 x^{15} + 15 x^{14} + 30 x^{13} + 35 x^{12} + 60 x^{11} + 75 x^{10} + 80 x^{9} + 96 x^{8} + 90 x^{7} + 78 x^{6} + 66 x^{5} + 51 x^{4} + 30 x^{3} + 19 x^{2} + 12 x + 5$$6$$3$$18$18T270$$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}]_{3}^{6}$$
\(101\) Copy content Toggle raw display 101.1.2.1a1.1$x^{2} + 101$$2$$1$$1$$C_2$$$[\ ]_{2}$$
101.1.2.1a1.1$x^{2} + 101$$2$$1$$1$$C_2$$$[\ ]_{2}$$
101.1.2.1a1.1$x^{2} + 101$$2$$1$$1$$C_2$$$[\ ]_{2}$$
101.1.2.1a1.1$x^{2} + 101$$2$$1$$1$$C_2$$$[\ ]_{2}$$
101.1.2.1a1.2$x^{2} + 202$$2$$1$$1$$C_2$$$[\ ]_{2}$$
101.2.2.2a1.2$x^{4} + 194 x^{3} + 9413 x^{2} + 388 x + 105$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
101.4.1.0a1.1$x^{4} + x^{2} + 78 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
\(479\) Copy content Toggle raw display 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 $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$

Spectrum of ring of integers

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