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Properties

Label 16.16.202...184.1
Degree $16$
Signature $[16, 0]$
Discriminant $2.022\times 10^{27}$
Root discriminant \(50.89\)
Ramified primes $2,7,199$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^7:F_8:C_3$ (as 16T1800)

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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^16 - 6*x^15 - 22*x^14 + 116*x^13 + 240*x^12 - 674*x^11 - 1326*x^10 + 896*x^9 + 2305*x^8 - 158*x^7 - 1638*x^6 - 308*x^5 + 486*x^4 + 160*x^3 - 44*x^2 - 20*x - 1)
 
Copy content gp:K = bnfinit(y^16 - 6*y^15 - 22*y^14 + 116*y^13 + 240*y^12 - 674*y^11 - 1326*y^10 + 896*y^9 + 2305*y^8 - 158*y^7 - 1638*y^6 - 308*y^5 + 486*y^4 + 160*y^3 - 44*y^2 - 20*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 - 22*x^14 + 116*x^13 + 240*x^12 - 674*x^11 - 1326*x^10 + 896*x^9 + 2305*x^8 - 158*x^7 - 1638*x^6 - 308*x^5 + 486*x^4 + 160*x^3 - 44*x^2 - 20*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 6*x^15 - 22*x^14 + 116*x^13 + 240*x^12 - 674*x^11 - 1326*x^10 + 896*x^9 + 2305*x^8 - 158*x^7 - 1638*x^6 - 308*x^5 + 486*x^4 + 160*x^3 - 44*x^2 - 20*x - 1)
 

\( x^{16} - 6 x^{15} - 22 x^{14} + 116 x^{13} + 240 x^{12} - 674 x^{11} - 1326 x^{10} + 896 x^{9} + \cdots - 1 \) 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:  $16$
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:  $[16, 0]$
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:   \(2021811149789090101470429184\) \(\medspace = 2^{24}\cdot 7^{2}\cdot 199^{8}\) 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:  \(50.89\)
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:   \(2\), \(7\), \(199\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{14059249477}a^{15}-\frac{5328434724}{14059249477}a^{14}+\frac{5948563682}{14059249477}a^{13}+\frac{877100151}{2008464211}a^{12}-\frac{6329962532}{14059249477}a^{11}-\frac{6535737038}{14059249477}a^{10}+\frac{5355806954}{14059249477}a^{9}-\frac{441514253}{14059249477}a^{8}-\frac{5251673337}{14059249477}a^{7}+\frac{7021577284}{14059249477}a^{6}+\frac{4644209572}{14059249477}a^{5}+\frac{3680089438}{14059249477}a^{4}+\frac{5069255790}{14059249477}a^{3}+\frac{998283783}{2008464211}a^{2}-\frac{1538097885}{14059249477}a+\frac{2466577747}{14059249477}$ 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 $1$ (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:  $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  $15$
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:   $a$, $\frac{1784073457}{154497247}a^{15}-\frac{11807422731}{154497247}a^{14}-\frac{31902674284}{154497247}a^{13}+\frac{226343347819}{154497247}a^{12}+\frac{287553710281}{154497247}a^{11}-\frac{1374018650439}{154497247}a^{10}-\frac{1511420975335}{154497247}a^{9}+\frac{2495662548724}{154497247}a^{8}+\frac{2546047295149}{154497247}a^{7}-\frac{1786860871905}{154497247}a^{6}-\frac{1781829858168}{154497247}a^{5}+\frac{505125826347}{154497247}a^{4}+\frac{532064865294}{154497247}a^{3}-\frac{32927728837}{154497247}a^{2}-\frac{53091746611}{154497247}a-\frac{3016407566}{154497247}$, $\frac{207938749008}{14059249477}a^{15}-\frac{1398797605536}{14059249477}a^{14}-\frac{3554813473821}{14059249477}a^{13}+\frac{3812452589844}{2008464211}a^{12}+\frac{30443312690540}{14059249477}a^{11}-\frac{161949689126469}{14059249477}a^{10}-\frac{157355719394758}{14059249477}a^{9}+\frac{298949113486913}{14059249477}a^{8}+\frac{258616185047776}{14059249477}a^{7}-\frac{219549118294810}{14059249477}a^{6}-\frac{176493371885669}{14059249477}a^{5}+\frac{65452991057932}{14059249477}a^{4}+\frac{51603285011476}{14059249477}a^{3}-\frac{771592723802}{2008464211}a^{2}-\frac{5119517962595}{14059249477}a-\frac{247979691839}{14059249477}$, $\frac{152292104702}{14059249477}a^{15}-\frac{1013446558248}{14059249477}a^{14}-\frac{2687632619699}{14059249477}a^{13}+\frac{2775628632086}{2008464211}a^{12}+\frac{23842946248463}{14059249477}a^{11}-\frac{118332868567579}{14059249477}a^{10}-\frac{124588415531787}{14059249477}a^{9}+\frac{218502056753364}{14059249477}a^{8}+\frac{208625152440513}{14059249477}a^{7}-\frac{161525000439408}{14059249477}a^{6}-\frac{144968964074187}{14059249477}a^{5}+\frac{48494979024742}{14059249477}a^{4}+\frac{43262925765771}{14059249477}a^{3}-\frac{569465809259}{2008464211}a^{2}-\frac{4382510462749}{14059249477}a-\frac{184688228391}{14059249477}$, $\frac{356148291659}{14059249477}a^{15}-\frac{2393935066596}{14059249477}a^{14}-\frac{6107734730620}{14059249477}a^{13}+\frac{6531954981884}{2008464211}a^{12}+\frac{52477010270151}{14059249477}a^{11}-\frac{277967627644898}{14059249477}a^{10}-\frac{271622557676051}{14059249477}a^{9}+\frac{515547935548056}{14059249477}a^{8}+\frac{448805514685914}{14059249477}a^{7}-\frac{381613658072060}{14059249477}a^{6}-\frac{308469239276060}{14059249477}a^{5}+\frac{114592656454363}{14059249477}a^{4}+\frac{91226301885099}{14059249477}a^{3}-\frac{1355420139893}{2008464211}a^{2}-\frac{9147152710848}{14059249477}a-\frac{467456664791}{14059249477}$, $\frac{186908726246}{14059249477}a^{15}-\frac{1267937444699}{14059249477}a^{14}-\frac{3138693292897}{14059249477}a^{13}+\frac{3465933630563}{2008464211}a^{12}+\frac{26297780689776}{14059249477}a^{11}-\frac{148855986106274}{14059249477}a^{10}-\frac{136143863638387}{14059249477}a^{9}+\frac{286458512971660}{14059249477}a^{8}+\frac{232731559727789}{14059249477}a^{7}-\frac{222298703173088}{14059249477}a^{6}-\frac{169439934216888}{14059249477}a^{5}+\frac{70015850347851}{14059249477}a^{4}+\frac{53657153879506}{14059249477}a^{3}-\frac{837516818010}{2008464211}a^{2}-\frac{5716412882783}{14059249477}a-\frac{320277636313}{14059249477}$, $\frac{170925379907}{14059249477}a^{15}-\frac{1137401746538}{14059249477}a^{14}-\frac{3010788134533}{14059249477}a^{13}+\frac{3108815169102}{2008464211}a^{12}+\frac{26695348365466}{14059249477}a^{11}-\frac{132002161108471}{14059249477}a^{10}-\frac{139565898346852}{14059249477}a^{9}+\frac{240554976009002}{14059249477}a^{8}+\frac{233095539688548}{14059249477}a^{7}-\frac{173051316963943}{14059249477}a^{6}-\frac{162054920528301}{14059249477}a^{5}+\frac{49354684962353}{14059249477}a^{4}+\frac{48449888476466}{14059249477}a^{3}-\frac{486330053728}{2008464211}a^{2}-\frac{4896219670892}{14059249477}a-\frac{274317003988}{14059249477}$, $\frac{68975354547}{14059249477}a^{15}-\frac{466868983662}{14059249477}a^{14}-\frac{1161762229868}{14059249477}a^{13}+\frac{1272881520067}{2008464211}a^{12}+\frac{9790901325697}{14059249477}a^{11}-\frac{54312271794695}{14059249477}a^{10}-\frac{50769382183722}{14059249477}a^{9}+\frac{102162222180489}{14059249477}a^{8}+\frac{86107547238250}{14059249477}a^{7}-\frac{75759212074503}{14059249477}a^{6}-\frac{62049060274276}{14059249477}a^{5}+\frac{21807739363883}{14059249477}a^{4}+\frac{19399034585354}{14059249477}a^{3}-\frac{188132989891}{2008464211}a^{2}-\frac{2038930221918}{14059249477}a-\frac{170874781556}{14059249477}$, $\frac{443054979944}{14059249477}a^{15}-\frac{2955004251638}{14059249477}a^{14}-\frac{7763980227387}{14059249477}a^{13}+\frac{8079352609678}{2008464211}a^{12}+\frac{68436137304849}{14059249477}a^{11}-\frac{343749646568882}{14059249477}a^{10}-\frac{357597843165959}{14059249477}a^{9}+\frac{632032267579411}{14059249477}a^{8}+\frac{600258048510180}{14059249477}a^{7}-\frac{461127162854180}{14059249477}a^{6}-\frac{421474708804451}{14059249477}a^{5}+\frac{134638293113098}{14059249477}a^{4}+\frac{128164251875028}{14059249477}a^{3}-\frac{1413151767766}{2008464211}a^{2}-\frac{13401815739346}{14059249477}a-\frac{731286281273}{14059249477}$, $\frac{72365750154}{1081480729}a^{15}-\frac{474963221564}{1081480729}a^{14}-\frac{1324253213370}{1081480729}a^{13}+\frac{1305624582438}{154497247}a^{12}+\frac{12213139324046}{1081480729}a^{11}-\frac{55633916539429}{1081480729}a^{10}-\frac{64542939329542}{1081480729}a^{9}+\frac{101094418918027}{1081480729}a^{8}+\frac{109457596072078}{1081480729}a^{7}-\frac{73101478474271}{1081480729}a^{6}-\frac{76768373242026}{1081480729}a^{5}+\frac{21191540941549}{1081480729}a^{4}+\frac{22959307898134}{1081480729}a^{3}-\frac{216296266170}{154497247}a^{2}-\frac{2307338484762}{1081480729}a-\frac{123716736179}{1081480729}$, $\frac{453241347528}{14059249477}a^{15}-\frac{3066823165422}{14059249477}a^{14}-\frac{7628033752046}{14059249477}a^{13}+\frac{8352312814119}{2008464211}a^{12}+\frac{64118616171842}{14059249477}a^{11}-\frac{355450006557365}{14059249477}a^{10}-\frac{330142408070170}{14059249477}a^{9}+\frac{663637482269427}{14059249477}a^{8}+\frac{544324262288019}{14059249477}a^{7}-\frac{493272303844589}{14059249477}a^{6}-\frac{375632805745077}{14059249477}a^{5}+\frac{147707335842383}{14059249477}a^{4}+\frac{112127013711981}{14059249477}a^{3}-\frac{1697624968093}{2008464211}a^{2}-\frac{11399107328575}{14059249477}a-\frac{636419629025}{14059249477}$, $\frac{501359065328}{14059249477}a^{15}-\frac{3322797181257}{14059249477}a^{14}-\frac{8941977410447}{14059249477}a^{13}+\frac{9106918222513}{2008464211}a^{12}+\frac{80299530602589}{14059249477}a^{11}-\frac{387926343915106}{14059249477}a^{10}-\frac{421433554867400}{14059249477}a^{9}+\frac{711345446821160}{14059249477}a^{8}+\frac{710025296893792}{14059249477}a^{7}-\frac{519530188620896}{14059249477}a^{6}-\frac{497096809224501}{14059249477}a^{5}+\frac{152696889046451}{14059249477}a^{4}+\frac{149199076527089}{14059249477}a^{3}-\frac{1637625436435}{2008464211}a^{2}-\frac{15077114885452}{14059249477}a-\frac{799262652733}{14059249477}$, $\frac{2914853373}{1081480729}a^{15}-\frac{20369148246}{1081480729}a^{14}-\frac{44479975662}{1081480729}a^{13}+\frac{55060584287}{154497247}a^{12}+\frac{326013187888}{1081480729}a^{11}-\frac{2350108239154}{1081480729}a^{10}-\frac{1595884627038}{1081480729}a^{9}+\frac{4574360698288}{1081480729}a^{8}+\frac{2456856908676}{1081480729}a^{7}-\frac{3633745401099}{1081480729}a^{6}-\frac{1576704587630}{1081480729}a^{5}+\frac{1227656069135}{1081480729}a^{4}+\frac{429542322516}{1081480729}a^{3}-\frac{19140262963}{154497247}a^{2}-\frac{38921943819}{1081480729}a-\frac{3236496110}{1081480729}$, $\frac{115418284340}{14059249477}a^{15}-\frac{742904036904}{14059249477}a^{14}-\frac{2190227960953}{14059249477}a^{13}+\frac{2028678365091}{2008464211}a^{12}+\frac{20949610866468}{14059249477}a^{11}-\frac{84232457838105}{14059249477}a^{10}-\frac{110235508306775}{14059249477}a^{9}+\frac{137112564727219}{14059249477}a^{8}+\frac{174384554290952}{14059249477}a^{7}-\frac{83697618809593}{14059249477}a^{6}-\frac{109672730019590}{14059249477}a^{5}+\frac{19241463266628}{14059249477}a^{4}+\frac{29106757970571}{14059249477}a^{3}-\frac{87091226328}{2008464211}a^{2}-\frac{2665859825192}{14059249477}a-\frac{179409049896}{14059249477}$, $\frac{268546009434}{14059249477}a^{15}-\frac{1793685405384}{14059249477}a^{14}-\frac{4711843656680}{14059249477}a^{13}+\frac{4926178886231}{2008464211}a^{12}+\frac{41518453486656}{14059249477}a^{11}-\frac{211688315109517}{14059249477}a^{10}-\frac{217636886790852}{14059249477}a^{9}+\frac{402064972657298}{14059249477}a^{8}+\frac{374348785010862}{14059249477}a^{7}-\frac{309368678158754}{14059249477}a^{6}-\frac{271053896947884}{14059249477}a^{5}+\frac{97404264034271}{14059249477}a^{4}+\frac{84905699574827}{14059249477}a^{3}-\frac{1181149861122}{2008464211}a^{2}-\frac{9008121306547}{14059249477}a-\frac{480693992249}{14059249477}$ 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:  \( 612600928.379 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 612600928.379 \cdot 1}{2\cdot\sqrt{2021811149789090101470429184}}\cr\approx \mathstrut & 0.446434030135 \end{aligned}\] (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^16 - 6*x^15 - 22*x^14 + 116*x^13 + 240*x^12 - 674*x^11 - 1326*x^10 + 896*x^9 + 2305*x^8 - 158*x^7 - 1638*x^6 - 308*x^5 + 486*x^4 + 160*x^3 - 44*x^2 - 20*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^16 - 6*x^15 - 22*x^14 + 116*x^13 + 240*x^12 - 674*x^11 - 1326*x^10 + 896*x^9 + 2305*x^8 - 158*x^7 - 1638*x^6 - 308*x^5 + 486*x^4 + 160*x^3 - 44*x^2 - 20*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^16 - 6*x^15 - 22*x^14 + 116*x^13 + 240*x^12 - 674*x^11 - 1326*x^10 + 896*x^9 + 2305*x^8 - 158*x^7 - 1638*x^6 - 308*x^5 + 486*x^4 + 160*x^3 - 44*x^2 - 20*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^16 - 6*x^15 - 22*x^14 + 116*x^13 + 240*x^12 - 674*x^11 - 1326*x^10 + 896*x^9 + 2305*x^8 - 158*x^7 - 1638*x^6 - 308*x^5 + 486*x^4 + 160*x^3 - 44*x^2 - 20*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

$C_2^7:F_8:C_3$ (as 16T1800):

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 21504
The 36 conjugacy class representatives for $C_2^7:F_8:C_3$
Character table for $C_2^7:F_8:C_3$

Intermediate fields

8.8.6423507767296.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ ${\href{/padicField/5.7.0.1}{7} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ R ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.3.0.1}{3} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.7.0.1}{7} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.7.0.1}{7} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.2.8.24a1.1$x^{16} + 8 x^{15} + 36 x^{14} + 112 x^{13} + 266 x^{12} + 504 x^{11} + 786 x^{10} + 1026 x^{9} + 1137 x^{8} + 1076 x^{7} + 874 x^{6} + 606 x^{5} + 356 x^{4} + 172 x^{3} + 66 x^{2} + 18 x + 5$$8$$2$$24$16T712$$[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}]_{7}^{6}$$
\(7\) Copy content Toggle raw display 7.1.2.1a1.2$x^{2} + 21$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.3.1.0a1.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$$[\ ]^{3}$$
7.3.1.0a1.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$$[\ ]^{3}$$
7.6.1.0a1.1$x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(199\) Copy content Toggle raw display 199.4.1.0a1.1$x^{4} + 7 x^{2} + 162 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
199.4.3.8a1.3$x^{12} + 21 x^{10} + 486 x^{9} + 156 x^{8} + 6804 x^{7} + 79201 x^{6} + 26730 x^{5} + 551592 x^{4} + 4271940 x^{3} + 236385 x^{2} + 4374 x + 226$$3$$4$$8$$C_{12}$$$[\ ]_{3}^{4}$$

Spectrum of ring of integers

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