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Properties

Label 20.2.442...775.1
Degree $20$
Signature $[2, 9]$
Discriminant $-4.428\times 10^{22}$
Root discriminant \(13.56\)
Ramified primes $5,37,2239,4903$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^9.C_2^6:S_5$ (as 20T1015)

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

\( x^{20} - 3 x^{19} + 2 x^{18} - 2 x^{17} + 9 x^{16} - 4 x^{15} - 7 x^{14} - 8 x^{13} + 12 x^{12} + \cdots + 1 \) 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:  $[2, 9]$
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:   \(-44283824745892362441775\) \(\medspace = -\,5^{2}\cdot 37^{2}\cdot 2239\cdot 4903^{4}\) 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:  \(13.56\)
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:  $5^{1/2}37^{1/2}2239^{1/2}4903^{1/2}\approx 45065.46510355796$
Ramified primes:   \(5\), \(37\), \(2239\), \(4903\) 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:  \(\Q(\sqrt{-2239}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}+\frac{1}{5}a^{16}+\frac{1}{5}a^{15}+\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{12}+\frac{2}{5}a^{11}+\frac{2}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{710512465}a^{19}-\frac{756981}{54654805}a^{18}+\frac{98950126}{710512465}a^{17}-\frac{329401302}{710512465}a^{16}+\frac{190656253}{710512465}a^{15}-\frac{55092358}{142102493}a^{14}+\frac{327219647}{710512465}a^{13}-\frac{340164921}{710512465}a^{12}-\frac{213439229}{710512465}a^{11}+\frac{15893309}{54654805}a^{10}-\frac{55534489}{142102493}a^{9}-\frac{16050188}{54654805}a^{8}+\frac{349631304}{710512465}a^{7}+\frac{188935584}{710512465}a^{6}+\frac{63509811}{142102493}a^{5}+\frac{209009472}{710512465}a^{4}+\frac{279627077}{710512465}a^{3}+\frac{299353401}{710512465}a^{2}-\frac{281449548}{710512465}a+\frac{279113267}{710512465}$ 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:  Trivial group, which has order $1$ (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:  $10$
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:   $\frac{245608224}{710512465}a^{19}-\frac{55025431}{54654805}a^{18}+\frac{238618089}{710512465}a^{17}+\frac{147492361}{710512465}a^{16}+\frac{1622465266}{710512465}a^{15}-\frac{165162151}{710512465}a^{14}-\frac{712162627}{142102493}a^{13}-\frac{209977239}{142102493}a^{12}+\frac{3327130132}{710512465}a^{11}+\frac{291037972}{54654805}a^{10}-\frac{1340082398}{710512465}a^{9}-\frac{20761090}{10930961}a^{8}-\frac{5562118617}{710512465}a^{7}-\frac{1562531817}{710512465}a^{6}+\frac{5132023592}{710512465}a^{5}+\frac{6587573444}{710512465}a^{4}-\frac{1044022881}{710512465}a^{3}-\frac{806914197}{142102493}a^{2}-\frac{238945114}{142102493}a+\frac{97464682}{710512465}$, $\frac{6541133}{710512465}a^{19}+\frac{4245268}{54654805}a^{18}-\frac{369046247}{710512465}a^{17}+\frac{596470892}{710512465}a^{16}-\frac{480610018}{710512465}a^{15}+\frac{1259573218}{710512465}a^{14}-\frac{402450781}{142102493}a^{13}+\frac{86842469}{142102493}a^{12}-\frac{553287816}{710512465}a^{11}+\frac{217499614}{54654805}a^{10}-\frac{1136956091}{710512465}a^{9}-\frac{2077893}{10930961}a^{8}-\frac{1577920114}{710512465}a^{7}-\frac{803737409}{710512465}a^{6}+\frac{1126094169}{710512465}a^{5}+\frac{2606068188}{710512465}a^{4}+\frac{1257231458}{710512465}a^{3}-\frac{480530420}{142102493}a^{2}-\frac{204017669}{142102493}a-\frac{55175766}{710512465}$, $a$, $\frac{7730552}{54654805}a^{19}-\frac{25260599}{54654805}a^{18}+\frac{8260427}{54654805}a^{17}+\frac{11389533}{54654805}a^{16}+\frac{60450403}{54654805}a^{15}-\frac{7755938}{54654805}a^{14}-\frac{32051545}{10930961}a^{13}-\frac{8734791}{10930961}a^{12}+\frac{133565111}{54654805}a^{11}+\frac{202305983}{54654805}a^{10}-\frac{28381449}{54654805}a^{9}-\frac{26649649}{10930961}a^{8}-\frac{264512156}{54654805}a^{7}-\frac{58492971}{54654805}a^{6}+\frac{236470326}{54654805}a^{5}+\frac{373320187}{54654805}a^{4}+\frac{42483807}{54654805}a^{3}-\frac{46956829}{10930961}a^{2}-\frac{34508938}{10930961}a-\frac{20071704}{54654805}$, $\frac{15398891}{142102493}a^{19}-\frac{407281}{10930961}a^{18}-\frac{54878341}{142102493}a^{17}-\frac{37359344}{142102493}a^{16}+\frac{85730744}{142102493}a^{15}+\frac{230997800}{142102493}a^{14}-\frac{11405159}{142102493}a^{13}-\frac{337885631}{142102493}a^{12}-\frac{352122436}{142102493}a^{11}+\frac{24131808}{10930961}a^{10}+\frac{384632289}{142102493}a^{9}+\frac{21213769}{10930961}a^{8}-\frac{152014841}{142102493}a^{7}-\frac{588315020}{142102493}a^{6}-\frac{664389253}{142102493}a^{5}+\frac{182432753}{142102493}a^{4}+\frac{619848465}{142102493}a^{3}+\frac{301921122}{142102493}a^{2}+\frac{21829402}{142102493}a+\frac{71961530}{142102493}$, $\frac{167538451}{710512465}a^{19}-\frac{42635894}{54654805}a^{18}+\frac{455615801}{710512465}a^{17}-\frac{437725826}{710512465}a^{16}+\frac{1901796419}{710512465}a^{15}-\frac{1535250599}{710512465}a^{14}-\frac{119480137}{142102493}a^{13}-\frac{439761870}{142102493}a^{12}+\frac{3717330663}{710512465}a^{11}+\frac{71519223}{54654805}a^{10}+\frac{282931913}{710512465}a^{9}-\frac{29303212}{10930961}a^{8}-\frac{3102564278}{710512465}a^{7}-\frac{947337808}{710512465}a^{6}+\frac{3445929913}{710512465}a^{5}+\frac{4183541341}{710512465}a^{4}-\frac{1023622804}{710512465}a^{3}-\frac{429232530}{142102493}a^{2}-\frac{202533067}{142102493}a-\frac{538511602}{710512465}$, $\frac{17456731}{54654805}a^{19}-\frac{45823417}{54654805}a^{18}+\frac{22428881}{54654805}a^{17}-\frac{53957741}{54654805}a^{16}+\frac{194473324}{54654805}a^{15}-\frac{76075404}{54654805}a^{14}-\frac{2975546}{10930961}a^{13}-\frac{67711234}{10930961}a^{12}+\frac{230935723}{54654805}a^{11}+\frac{118877429}{54654805}a^{10}+\frac{238070553}{54654805}a^{9}-\frac{26748209}{10930961}a^{8}-\frac{254209473}{54654805}a^{7}-\frac{343699138}{54654805}a^{6}+\frac{58484248}{54654805}a^{5}+\frac{420875246}{54654805}a^{4}+\frac{219403056}{54654805}a^{3}-\frac{11487088}{10930961}a^{2}-\frac{26172509}{10930961}a-\frac{20260802}{54654805}$, $\frac{2623352}{142102493}a^{19}-\frac{21539132}{54654805}a^{18}+\frac{113339364}{142102493}a^{17}+\frac{60848439}{710512465}a^{16}-\frac{115190136}{710512465}a^{15}-\frac{987103211}{710512465}a^{14}-\frac{1553178308}{710512465}a^{13}+\frac{3606639079}{710512465}a^{12}+\frac{556051488}{710512465}a^{11}+\frac{1074401}{54654805}a^{10}-\frac{4304431728}{710512465}a^{9}+\frac{40934692}{54654805}a^{8}-\frac{1364888553}{710512465}a^{7}+\frac{4826637897}{710512465}a^{6}+\frac{2955438547}{710512465}a^{5}-\frac{1321833489}{710512465}a^{4}-\frac{4483082564}{710512465}a^{3}-\frac{1376284119}{710512465}a^{2}+\frac{1349793697}{710512465}a+\frac{320882109}{710512465}$, $\frac{13098043}{142102493}a^{19}+\frac{122594}{54654805}a^{18}-\frac{29430225}{142102493}a^{17}-\frac{531844048}{710512465}a^{16}+\frac{526555832}{710512465}a^{15}+\frac{884843192}{710512465}a^{14}+\frac{1281040031}{710512465}a^{13}-\frac{2180118658}{710512465}a^{12}-\frac{2404217381}{710512465}a^{11}-\frac{15949412}{54654805}a^{10}+\frac{4033365501}{710512465}a^{9}+\frac{194442586}{54654805}a^{8}-\frac{581684719}{710512465}a^{7}-\frac{3726528779}{710512465}a^{6}-\frac{5477414849}{710512465}a^{5}+\frac{184271563}{710512465}a^{4}+\frac{4710760713}{710512465}a^{3}+\frac{4436831238}{710512465}a^{2}-\frac{734104079}{710512465}a-\frac{914570743}{710512465}$, $\frac{298466143}{710512465}a^{19}-\frac{62458222}{54654805}a^{18}+\frac{443623988}{710512465}a^{17}-\frac{712703338}{710512465}a^{16}+\frac{2729432077}{710512465}a^{15}-\frac{878465022}{710512465}a^{14}-\frac{228913807}{142102493}a^{13}-\frac{737303880}{142102493}a^{12}+\frac{3165251684}{710512465}a^{11}+\frac{120896564}{54654805}a^{10}+\frac{2608858969}{710512465}a^{9}-\frac{16128194}{10930961}a^{8}-\frac{4186357374}{710512465}a^{7}-\frac{4279545394}{710512465}a^{6}+\frac{1320294734}{710512465}a^{5}+\frac{5269342113}{710512465}a^{4}+\frac{2285120658}{710512465}a^{3}-\frac{188585884}{142102493}a^{2}-\frac{30534502}{142102493}a-\frac{282750841}{710512465}$ 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:  \( 1195.16140002 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 1195.16140002 \cdot 1}{2\cdot\sqrt{44283824745892362441775}}\cr\approx \mathstrut & 0.173361560480 \end{aligned}\] (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 - 3*x^19 + 2*x^18 - 2*x^17 + 9*x^16 - 4*x^15 - 7*x^14 - 8*x^13 + 12*x^12 + 9*x^11 + 2*x^10 - 6*x^9 - 17*x^8 - 8*x^7 + 13*x^6 + 20*x^5 + x^4 - 10*x^3 - 4*x^2 + 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 - 3*x^19 + 2*x^18 - 2*x^17 + 9*x^16 - 4*x^15 - 7*x^14 - 8*x^13 + 12*x^12 + 9*x^11 + 2*x^10 - 6*x^9 - 17*x^8 - 8*x^7 + 13*x^6 + 20*x^5 + x^4 - 10*x^3 - 4*x^2 + 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 - 3*x^19 + 2*x^18 - 2*x^17 + 9*x^16 - 4*x^15 - 7*x^14 - 8*x^13 + 12*x^12 + 9*x^11 + 2*x^10 - 6*x^9 - 17*x^8 - 8*x^7 + 13*x^6 + 20*x^5 + x^4 - 10*x^3 - 4*x^2 + 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^20 - 3*x^19 + 2*x^18 - 2*x^17 + 9*x^16 - 4*x^15 - 7*x^14 - 8*x^13 + 12*x^12 + 9*x^11 + 2*x^10 - 6*x^9 - 17*x^8 - 8*x^7 + 13*x^6 + 20*x^5 + x^4 - 10*x^3 - 4*x^2 + 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

$C_2^9.C_2^6:S_5$ (as 20T1015):

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 non-solvable group of order 3932160
The 506 conjugacy class representatives for $C_2^9.C_2^6:S_5$
Character table for $C_2^9.C_2^6:S_5$

Intermediate fields

5.3.4903.1, 10.2.889458133.2

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(L)]
 
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 ${\href{/padicField/2.10.0.1}{10} }^{2}$ ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.6.0.1}{6} }^{2}$ R ${\href{/padicField/7.5.0.1}{5} }^{4}$ ${\href{/padicField/11.5.0.1}{5} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ R ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.10.0.1}{10} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

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 5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.2.2.2a1.1$x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
5.12.1.0a1.1$x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$$1$$12$$0$$C_{12}$$$[\ ]^{12}$$
\(37\) Copy content Toggle raw display $\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
37.2.1.0a1.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
37.2.1.0a1.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
37.2.1.0a1.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
37.4.1.0a1.1$x^{4} + 6 x^{2} + 24 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
37.2.2.2a1.2$x^{4} + 66 x^{3} + 1093 x^{2} + 132 x + 41$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
37.4.1.0a1.1$x^{4} + 6 x^{2} + 24 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
\(2239\) Copy content Toggle raw display 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$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $8$$1$$8$$0$$C_8$$$[\ ]^{8}$$
\(4903\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$2$$2$$2$
Deg $12$$1$$12$$0$$C_{12}$$$[\ ]^{12}$$

Spectrum of ring of integers

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