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Properties

Label 16.8.220...125.4
Degree $16$
Signature $[8, 4]$
Discriminant $2.201\times 10^{30}$
Root discriminant \(78.78\)
Ramified primes $5,101$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^6.\SD_{16}$ (as 16T1250)

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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 - 8*x^15 + x^14 + 57*x^13 + 113*x^12 - 40*x^11 - 1432*x^10 - 543*x^9 + 2294*x^8 - 682*x^7 - 234*x^6 + 3073*x^5 - 3074*x^4 - 4287*x^3 + 6343*x^2 - 628*x - 829)
 
Copy content gp:K = bnfinit(y^16 - 8*y^15 + y^14 + 57*y^13 + 113*y^12 - 40*y^11 - 1432*y^10 - 543*y^9 + 2294*y^8 - 682*y^7 - 234*y^6 + 3073*y^5 - 3074*y^4 - 4287*y^3 + 6343*y^2 - 628*y - 829, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + x^14 + 57*x^13 + 113*x^12 - 40*x^11 - 1432*x^10 - 543*x^9 + 2294*x^8 - 682*x^7 - 234*x^6 + 3073*x^5 - 3074*x^4 - 4287*x^3 + 6343*x^2 - 628*x - 829);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^15 + x^14 + 57*x^13 + 113*x^12 - 40*x^11 - 1432*x^10 - 543*x^9 + 2294*x^8 - 682*x^7 - 234*x^6 + 3073*x^5 - 3074*x^4 - 4287*x^3 + 6343*x^2 - 628*x - 829)
 

\( x^{16} - 8 x^{15} + x^{14} + 57 x^{13} + 113 x^{12} - 40 x^{11} - 1432 x^{10} - 543 x^{9} + 2294 x^{8} + \cdots - 829 \) 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:  $[8, 4]$
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:   \(2200830136976503360666408203125\) \(\medspace = 5^{9}\cdot 101^{12}\) 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:  \(78.78\)
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:  $5^{3/4}101^{3/4}\approx 106.52916761335055$
Ramified primes:   \(5\), \(101\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{327455}a^{14}-\frac{39736}{327455}a^{13}-\frac{106867}{327455}a^{12}+\frac{100814}{327455}a^{11}+\frac{55308}{327455}a^{10}-\frac{123503}{327455}a^{9}-\frac{7511}{327455}a^{8}+\frac{128703}{327455}a^{7}+\frac{66806}{327455}a^{6}+\frac{52037}{327455}a^{5}-\frac{76491}{327455}a^{4}-\frac{100986}{327455}a^{3}+\frac{28929}{65491}a^{2}+\frac{128394}{327455}a-\frac{71}{395}$, $\frac{1}{32\cdots 25}a^{15}-\frac{39\cdots 97}{32\cdots 25}a^{14}+\frac{64\cdots 59}{32\cdots 25}a^{13}+\frac{10\cdots 56}{32\cdots 25}a^{12}+\frac{28\cdots 04}{32\cdots 25}a^{11}-\frac{25\cdots 96}{32\cdots 25}a^{10}+\frac{51\cdots 62}{32\cdots 25}a^{9}+\frac{14\cdots 89}{32\cdots 25}a^{8}-\frac{17\cdots 77}{32\cdots 25}a^{7}+\frac{14\cdots 71}{32\cdots 25}a^{6}+\frac{10\cdots 22}{32\cdots 25}a^{5}-\frac{10\cdots 19}{28\cdots 35}a^{4}-\frac{13\cdots 34}{32\cdots 25}a^{3}-\frac{11\cdots 86}{32\cdots 25}a^{2}+\frac{69\cdots 72}{32\cdots 25}a-\frac{49\cdots 09}{39\cdots 25}$ 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 $2$ (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:  $11$
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:   $\frac{92\cdots 83}{56\cdots 47}a^{15}-\frac{68\cdots 98}{56\cdots 47}a^{14}-\frac{32\cdots 41}{56\cdots 47}a^{13}+\frac{51\cdots 50}{56\cdots 47}a^{12}+\frac{13\cdots 12}{56\cdots 47}a^{11}+\frac{37\cdots 50}{56\cdots 47}a^{10}-\frac{13\cdots 27}{56\cdots 47}a^{9}-\frac{13\cdots 07}{56\cdots 47}a^{8}+\frac{14\cdots 96}{56\cdots 47}a^{7}+\frac{93\cdots 17}{56\cdots 47}a^{6}+\frac{98\cdots 91}{56\cdots 47}a^{5}+\frac{33\cdots 83}{56\cdots 47}a^{4}-\frac{87\cdots 19}{56\cdots 47}a^{3}-\frac{51\cdots 28}{56\cdots 47}a^{2}+\frac{12\cdots 04}{56\cdots 47}a+\frac{82\cdots 58}{67\cdots 43}$, $\frac{64\cdots 12}{32\cdots 25}a^{15}-\frac{35\cdots 89}{32\cdots 25}a^{14}-\frac{11\cdots 42}{32\cdots 25}a^{13}+\frac{30\cdots 97}{32\cdots 25}a^{12}+\frac{15\cdots 98}{32\cdots 25}a^{11}+\frac{22\cdots 48}{32\cdots 25}a^{10}-\frac{77\cdots 06}{32\cdots 25}a^{9}-\frac{25\cdots 82}{32\cdots 25}a^{8}-\frac{10\cdots 49}{32\cdots 25}a^{7}+\frac{73\cdots 27}{32\cdots 25}a^{6}-\frac{72\cdots 11}{32\cdots 25}a^{5}+\frac{13\cdots 22}{28\cdots 35}a^{4}+\frac{22\cdots 67}{32\cdots 25}a^{3}-\frac{48\cdots 82}{32\cdots 25}a^{2}-\frac{34\cdots 36}{32\cdots 25}a+\frac{39\cdots 92}{39\cdots 25}$, $\frac{59\cdots 03}{32\cdots 25}a^{15}-\frac{47\cdots 26}{32\cdots 25}a^{14}+\frac{42\cdots 62}{32\cdots 25}a^{13}+\frac{32\cdots 38}{32\cdots 25}a^{12}+\frac{69\cdots 97}{32\cdots 25}a^{11}-\frac{81\cdots 18}{32\cdots 25}a^{10}-\frac{82\cdots 09}{32\cdots 25}a^{9}-\frac{37\cdots 23}{32\cdots 25}a^{8}+\frac{10\cdots 89}{32\cdots 25}a^{7}-\frac{57\cdots 72}{32\cdots 25}a^{6}+\frac{11\cdots 96}{32\cdots 25}a^{5}+\frac{15\cdots 37}{28\cdots 35}a^{4}-\frac{18\cdots 92}{32\cdots 25}a^{3}-\frac{21\cdots 33}{32\cdots 25}a^{2}+\frac{32\cdots 26}{32\cdots 25}a-\frac{11\cdots 17}{39\cdots 25}$, $\frac{51\cdots 94}{56\cdots 47}a^{15}-\frac{39\cdots 10}{56\cdots 47}a^{14}-\frac{45\cdots 13}{56\cdots 47}a^{13}+\frac{27\cdots 28}{56\cdots 47}a^{12}+\frac{66\cdots 99}{56\cdots 47}a^{11}+\frac{13\cdots 86}{56\cdots 47}a^{10}-\frac{71\cdots 60}{56\cdots 47}a^{9}-\frac{51\cdots 15}{56\cdots 47}a^{8}+\frac{73\cdots 74}{56\cdots 47}a^{7}-\frac{24\cdots 97}{56\cdots 47}a^{6}+\frac{70\cdots 76}{56\cdots 47}a^{5}+\frac{13\cdots 19}{56\cdots 47}a^{4}-\frac{11\cdots 53}{56\cdots 47}a^{3}-\frac{13\cdots 13}{56\cdots 47}a^{2}+\frac{49\cdots 67}{56\cdots 47}a+\frac{80\cdots 80}{67\cdots 43}$, $\frac{14\cdots 05}{56\cdots 47}a^{15}-\frac{73\cdots 49}{56\cdots 47}a^{14}-\frac{30\cdots 70}{56\cdots 47}a^{13}+\frac{70\cdots 28}{56\cdots 47}a^{12}+\frac{38\cdots 13}{56\cdots 47}a^{11}+\frac{54\cdots 76}{56\cdots 47}a^{10}-\frac{17\cdots 48}{56\cdots 47}a^{9}-\frac{62\cdots 76}{56\cdots 47}a^{8}-\frac{21\cdots 42}{56\cdots 47}a^{7}+\frac{16\cdots 23}{56\cdots 47}a^{6}-\frac{67\cdots 90}{56\cdots 47}a^{5}+\frac{49\cdots 01}{56\cdots 47}a^{4}+\frac{59\cdots 40}{56\cdots 47}a^{3}-\frac{17\cdots 92}{56\cdots 47}a^{2}+\frac{38\cdots 38}{56\cdots 47}a+\frac{27\cdots 91}{67\cdots 43}$, $\frac{40\cdots 04}{14\cdots 75}a^{15}-\frac{36\cdots 18}{14\cdots 75}a^{14}+\frac{41\cdots 16}{14\cdots 75}a^{13}+\frac{21\cdots 09}{14\cdots 75}a^{12}+\frac{19\cdots 71}{14\cdots 75}a^{11}-\frac{58\cdots 49}{14\cdots 75}a^{10}-\frac{53\cdots 62}{14\cdots 75}a^{9}+\frac{43\cdots 61}{14\cdots 75}a^{8}+\frac{92\cdots 77}{14\cdots 75}a^{7}-\frac{15\cdots 71}{14\cdots 75}a^{6}+\frac{11\cdots 78}{14\cdots 75}a^{5}+\frac{21\cdots 13}{28\cdots 35}a^{4}-\frac{26\cdots 56}{14\cdots 75}a^{3}-\frac{68\cdots 44}{14\cdots 75}a^{2}+\frac{52\cdots 93}{14\cdots 75}a-\frac{38\cdots 31}{16\cdots 75}$, $\frac{43\cdots 91}{64\cdots 05}a^{15}-\frac{36\cdots 69}{64\cdots 05}a^{14}+\frac{15\cdots 66}{64\cdots 05}a^{13}+\frac{46\cdots 70}{12\cdots 81}a^{12}+\frac{42\cdots 51}{64\cdots 05}a^{11}-\frac{20\cdots 72}{64\cdots 05}a^{10}-\frac{58\cdots 77}{64\cdots 05}a^{9}-\frac{46\cdots 89}{64\cdots 05}a^{8}+\frac{80\cdots 32}{64\cdots 05}a^{7}-\frac{87\cdots 96}{64\cdots 05}a^{6}-\frac{84\cdots 57}{64\cdots 05}a^{5}+\frac{40\cdots 64}{28\cdots 35}a^{4}-\frac{16\cdots 07}{64\cdots 05}a^{3}-\frac{90\cdots 91}{64\cdots 05}a^{2}+\frac{32\cdots 49}{64\cdots 05}a-\frac{18\cdots 52}{78\cdots 45}$, $\frac{10\cdots 10}{56\cdots 47}a^{15}-\frac{41\cdots 16}{28\cdots 35}a^{14}-\frac{94\cdots 89}{28\cdots 35}a^{13}+\frac{26\cdots 97}{28\cdots 35}a^{12}+\frac{65\cdots 16}{28\cdots 35}a^{11}+\frac{28\cdots 42}{28\cdots 35}a^{10}-\frac{69\cdots 72}{28\cdots 35}a^{9}-\frac{45\cdots 59}{28\cdots 35}a^{8}+\frac{38\cdots 02}{28\cdots 35}a^{7}-\frac{10\cdots 01}{28\cdots 35}a^{6}+\frac{51\cdots 68}{28\cdots 35}a^{5}+\frac{87\cdots 61}{28\cdots 35}a^{4}-\frac{14\cdots 19}{28\cdots 35}a^{3}-\frac{99\cdots 13}{56\cdots 47}a^{2}+\frac{45\cdots 96}{28\cdots 35}a-\frac{32\cdots 94}{33\cdots 15}$, $\frac{10\cdots 96}{64\cdots 05}a^{15}-\frac{14\cdots 88}{64\cdots 05}a^{14}+\frac{11\cdots 84}{12\cdots 81}a^{13}+\frac{68\cdots 58}{64\cdots 05}a^{12}-\frac{59\cdots 63}{12\cdots 81}a^{11}-\frac{27\cdots 74}{64\cdots 05}a^{10}+\frac{40\cdots 42}{12\cdots 81}a^{9}+\frac{17\cdots 10}{12\cdots 81}a^{8}-\frac{12\cdots 62}{12\cdots 81}a^{7}-\frac{67\cdots 10}{12\cdots 81}a^{6}+\frac{58\cdots 71}{16\cdots 39}a^{5}+\frac{20\cdots 62}{28\cdots 35}a^{4}-\frac{36\cdots 88}{64\cdots 05}a^{3}-\frac{33\cdots 96}{64\cdots 05}a^{2}+\frac{39\cdots 53}{64\cdots 05}a-\frac{11\cdots 58}{78\cdots 45}$, $\frac{21\cdots 46}{64\cdots 05}a^{15}-\frac{16\cdots 31}{64\cdots 05}a^{14}-\frac{49\cdots 77}{64\cdots 05}a^{13}+\frac{11\cdots 59}{64\cdots 05}a^{12}+\frac{31\cdots 68}{64\cdots 05}a^{11}+\frac{44\cdots 02}{64\cdots 05}a^{10}-\frac{31\cdots 46}{64\cdots 05}a^{9}-\frac{27\cdots 82}{64\cdots 05}a^{8}+\frac{36\cdots 56}{64\cdots 05}a^{7}+\frac{27\cdots 97}{64\cdots 05}a^{6}-\frac{20\cdots 16}{64\cdots 05}a^{5}+\frac{63\cdots 43}{28\cdots 35}a^{4}-\frac{24\cdots 98}{12\cdots 81}a^{3}-\frac{45\cdots 91}{64\cdots 05}a^{2}+\frac{40\cdots 76}{64\cdots 05}a+\frac{37\cdots 88}{15\cdots 89}$, $\frac{17\cdots 73}{56\cdots 47}a^{15}-\frac{59\cdots 36}{28\cdots 35}a^{14}-\frac{53\cdots 14}{28\cdots 35}a^{13}+\frac{43\cdots 17}{28\cdots 35}a^{12}+\frac{14\cdots 51}{28\cdots 35}a^{11}+\frac{10\cdots 02}{28\cdots 35}a^{10}-\frac{11\cdots 42}{28\cdots 35}a^{9}-\frac{16\cdots 14}{28\cdots 35}a^{8}+\frac{43\cdots 82}{28\cdots 35}a^{7}-\frac{37\cdots 81}{28\cdots 35}a^{6}-\frac{63\cdots 47}{28\cdots 35}a^{5}+\frac{17\cdots 66}{28\cdots 35}a^{4}-\frac{26\cdots 49}{28\cdots 35}a^{3}-\frac{75\cdots 05}{56\cdots 47}a^{2}+\frac{10\cdots 81}{28\cdots 35}a+\frac{76\cdots 51}{33\cdots 15}$ 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:  \( 1857575387.55 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 1857575387.55 \cdot 2}{2\cdot\sqrt{2200830136976503360666408203125}}\cr\approx \mathstrut & 0.499588720693 \end{aligned}\] (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 - 8*x^15 + x^14 + 57*x^13 + 113*x^12 - 40*x^11 - 1432*x^10 - 543*x^9 + 2294*x^8 - 682*x^7 - 234*x^6 + 3073*x^5 - 3074*x^4 - 4287*x^3 + 6343*x^2 - 628*x - 829) 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 - 8*x^15 + x^14 + 57*x^13 + 113*x^12 - 40*x^11 - 1432*x^10 - 543*x^9 + 2294*x^8 - 682*x^7 - 234*x^6 + 3073*x^5 - 3074*x^4 - 4287*x^3 + 6343*x^2 - 628*x - 829, 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 - 8*x^15 + x^14 + 57*x^13 + 113*x^12 - 40*x^11 - 1432*x^10 - 543*x^9 + 2294*x^8 - 682*x^7 - 234*x^6 + 3073*x^5 - 3074*x^4 - 4287*x^3 + 6343*x^2 - 628*x - 829); 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 - 8*x^15 + x^14 + 57*x^13 + 113*x^12 - 40*x^11 - 1432*x^10 - 543*x^9 + 2294*x^8 - 682*x^7 - 234*x^6 + 3073*x^5 - 3074*x^4 - 4287*x^3 + 6343*x^2 - 628*x - 829); 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

$C_2^6.\SD_{16}$ (as 16T1250):

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 1024
The 34 conjugacy class representatives for $C_2^6.\SD_{16}$
Character table for $C_2^6.\SD_{16}$

Intermediate fields

\(\Q(\sqrt{101}) \), 4.4.51005.1, 8.8.132690018825125.1

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 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.8.88033205479060134426656328125.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $16$ $16$ R $16$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{6}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ $16$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.1.4.3a1.4$x^{4} + 20$$4$$1$$3$$C_4$$$[\ ]_{4}$$
\(101\) Copy content Toggle raw display 101.1.4.3a1.3$x^{4} + 404$$4$$1$$3$$C_4$$$[\ ]_{4}$$
101.1.4.3a1.3$x^{4} + 404$$4$$1$$3$$C_4$$$[\ ]_{4}$$
101.2.4.6a1.2$x^{8} + 388 x^{7} + 56462 x^{6} + 3653020 x^{5} + 88755121 x^{4} + 7306040 x^{3} + 225848 x^{2} + 3104 x + 117$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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