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Properties

Label 16.8.907...000.2
Degree $16$
Signature $[8, 4]$
Discriminant $9.079\times 10^{25}$
Root discriminant \(41.92\)
Ramified primes $2,5,53$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $(C_2^2\times D_4):S_4$ (as 16T1054)

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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 - 2*x^15 - 32*x^14 + 36*x^13 + 412*x^12 - 184*x^11 - 2872*x^10 - 74*x^9 + 14487*x^8 + 13864*x^7 - 35744*x^6 - 99102*x^5 - 104826*x^4 - 55822*x^3 - 13854*x^2 - 1018*x + 101)
 
Copy content gp:K = bnfinit(y^16 - 2*y^15 - 32*y^14 + 36*y^13 + 412*y^12 - 184*y^11 - 2872*y^10 - 74*y^9 + 14487*y^8 + 13864*y^7 - 35744*y^6 - 99102*y^5 - 104826*y^4 - 55822*y^3 - 13854*y^2 - 1018*y + 101, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 32*x^14 + 36*x^13 + 412*x^12 - 184*x^11 - 2872*x^10 - 74*x^9 + 14487*x^8 + 13864*x^7 - 35744*x^6 - 99102*x^5 - 104826*x^4 - 55822*x^3 - 13854*x^2 - 1018*x + 101);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 - 32*x^14 + 36*x^13 + 412*x^12 - 184*x^11 - 2872*x^10 - 74*x^9 + 14487*x^8 + 13864*x^7 - 35744*x^6 - 99102*x^5 - 104826*x^4 - 55822*x^3 - 13854*x^2 - 1018*x + 101)
 

\( x^{16} - 2 x^{15} - 32 x^{14} + 36 x^{13} + 412 x^{12} - 184 x^{11} - 2872 x^{10} - 74 x^{9} + \cdots + 101 \) 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:   \(90785223184384000000000000\) \(\medspace = 2^{24}\cdot 5^{12}\cdot 53^{6}\) 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:  \(41.92\)
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:  $2^{43/24}5^{5/6}53^{1/2}\approx 96.3736588314662$
Ramified primes:   \(2\), \(5\), \(53\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{8745}a^{14}+\frac{791}{1749}a^{13}-\frac{68}{265}a^{12}+\frac{1238}{8745}a^{11}+\frac{1846}{8745}a^{10}-\frac{326}{2915}a^{9}+\frac{527}{1749}a^{8}+\frac{3412}{8745}a^{7}+\frac{2516}{8745}a^{6}+\frac{1299}{2915}a^{5}-\frac{802}{8745}a^{4}-\frac{8}{583}a^{3}+\frac{403}{8745}a^{2}-\frac{2476}{8745}a-\frac{2512}{8745}$, $\frac{1}{21\cdots 75}a^{15}-\frac{58\cdots 44}{21\cdots 75}a^{14}-\frac{87\cdots 88}{70\cdots 25}a^{13}+\frac{33\cdots 74}{21\cdots 75}a^{12}-\frac{75\cdots 01}{21\cdots 75}a^{11}-\frac{10\cdots 44}{70\cdots 25}a^{10}-\frac{30\cdots 08}{21\cdots 75}a^{9}-\frac{36\cdots 38}{17\cdots 75}a^{8}+\frac{95\cdots 73}{19\cdots 25}a^{7}-\frac{19\cdots 49}{70\cdots 25}a^{6}-\frac{22\cdots 97}{76\cdots 17}a^{5}+\frac{15\cdots 46}{70\cdots 25}a^{4}+\frac{82\cdots 88}{21\cdots 75}a^{3}-\frac{11\cdots 81}{39\cdots 75}a^{2}-\frac{41\cdots 83}{19\cdots 25}a+\frac{25\cdots 61}{70\cdots 25}$ 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:  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:  $C_{2}\times C_{2}$, which has order $4$ (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:   $\frac{1518316022818}{514968806770475}a^{15}-\frac{4717350667472}{514968806770475}a^{14}-\frac{43588052090552}{514968806770475}a^{13}+\frac{103281044235752}{514968806770475}a^{12}+\frac{518835023489492}{514968806770475}a^{11}-\frac{860383816769556}{514968806770475}a^{10}-\frac{35\cdots 54}{514968806770475}a^{9}+\frac{38\cdots 11}{514968806770475}a^{8}+\frac{18\cdots 44}{514968806770475}a^{7}+\frac{593658976130424}{514968806770475}a^{6}-\frac{11\cdots 62}{102993761354095}a^{5}-\frac{88\cdots 06}{514968806770475}a^{4}-\frac{50\cdots 66}{514968806770475}a^{3}-\frac{45\cdots 14}{514968806770475}a^{2}+\frac{40\cdots 46}{514968806770475}a+\frac{642711093715154}{514968806770475}$, $\frac{13\cdots 42}{70\cdots 25}a^{15}-\frac{11\cdots 29}{21\cdots 75}a^{14}-\frac{11\cdots 89}{21\cdots 75}a^{13}+\frac{82\cdots 88}{70\cdots 25}a^{12}+\frac{12\cdots 69}{21\cdots 75}a^{11}-\frac{18\cdots 67}{21\cdots 75}a^{10}-\frac{28\cdots 01}{70\cdots 25}a^{9}+\frac{58\cdots 62}{17\cdots 75}a^{8}+\frac{40\cdots 53}{19\cdots 25}a^{7}+\frac{14\cdots 18}{21\cdots 75}a^{6}-\frac{80\cdots 33}{12\cdots 95}a^{5}-\frac{25\cdots 17}{21\cdots 75}a^{4}-\frac{72\cdots 79}{70\cdots 25}a^{3}-\frac{93\cdots 48}{21\cdots 75}a^{2}-\frac{18\cdots 48}{19\cdots 25}a-\frac{47\cdots 47}{21\cdots 75}$, $\frac{62\cdots 37}{21\cdots 75}a^{15}-\frac{20\cdots 98}{21\cdots 75}a^{14}-\frac{17\cdots 93}{21\cdots 75}a^{13}+\frac{47\cdots 68}{21\cdots 75}a^{12}+\frac{69\cdots 51}{70\cdots 25}a^{11}-\frac{42\cdots 04}{21\cdots 75}a^{10}-\frac{14\cdots 86}{21\cdots 75}a^{9}+\frac{18\cdots 59}{19\cdots 25}a^{8}+\frac{42\cdots 29}{12\cdots 75}a^{7}-\frac{24\cdots 09}{21\cdots 75}a^{6}-\frac{46\cdots 18}{38\cdots 85}a^{5}-\frac{29\cdots 79}{21\cdots 75}a^{4}-\frac{72\cdots 94}{21\cdots 75}a^{3}+\frac{26\cdots 83}{70\cdots 25}a^{2}+\frac{76\cdots 03}{32\cdots 75}a+\frac{36\cdots 11}{21\cdots 75}$, $\frac{55\cdots 59}{42\cdots 35}a^{15}-\frac{19\cdots 81}{42\cdots 35}a^{14}-\frac{15\cdots 61}{42\cdots 35}a^{13}+\frac{42\cdots 56}{42\cdots 35}a^{12}+\frac{55\cdots 57}{14\cdots 45}a^{11}-\frac{35\cdots 63}{42\cdots 35}a^{10}-\frac{10\cdots 42}{42\cdots 35}a^{9}+\frac{14\cdots 58}{38\cdots 85}a^{8}+\frac{16\cdots 29}{12\cdots 95}a^{7}-\frac{85\cdots 58}{42\cdots 35}a^{6}-\frac{27\cdots 90}{69\cdots 47}a^{5}-\frac{26\cdots 33}{42\cdots 35}a^{4}-\frac{24\cdots 28}{42\cdots 35}a^{3}-\frac{47\cdots 39}{14\cdots 45}a^{2}-\frac{40\cdots 47}{38\cdots 85}a-\frac{18\cdots 73}{42\cdots 35}$, $\frac{12\cdots 08}{21\cdots 75}a^{15}-\frac{37\cdots 82}{21\cdots 75}a^{14}-\frac{38\cdots 87}{21\cdots 75}a^{13}+\frac{81\cdots 37}{21\cdots 75}a^{12}+\frac{15\cdots 59}{70\cdots 25}a^{11}-\frac{65\cdots 86}{21\cdots 75}a^{10}-\frac{30\cdots 99}{21\cdots 75}a^{9}+\frac{24\cdots 56}{19\cdots 25}a^{8}+\frac{44\cdots 03}{57\cdots 25}a^{7}+\frac{35\cdots 69}{21\cdots 75}a^{6}-\frac{88\cdots 47}{38\cdots 85}a^{5}-\frac{84\cdots 11}{21\cdots 75}a^{4}-\frac{62\cdots 21}{21\cdots 75}a^{3}-\frac{67\cdots 28}{70\cdots 25}a^{2}-\frac{22\cdots 34}{19\cdots 25}a+\frac{12\cdots 49}{21\cdots 75}$, $\frac{43\cdots 43}{42\cdots 35}a^{15}-\frac{20\cdots 47}{42\cdots 35}a^{14}-\frac{10\cdots 62}{42\cdots 35}a^{13}+\frac{47\cdots 47}{42\cdots 35}a^{12}+\frac{33\cdots 94}{14\cdots 45}a^{11}-\frac{45\cdots 41}{42\cdots 35}a^{10}-\frac{59\cdots 79}{42\cdots 35}a^{9}+\frac{20\cdots 96}{34\cdots 35}a^{8}+\frac{10\cdots 68}{12\cdots 95}a^{7}-\frac{69\cdots 11}{42\cdots 35}a^{6}-\frac{28\cdots 87}{76\cdots 17}a^{5}-\frac{24\cdots 26}{42\cdots 35}a^{4}+\frac{18\cdots 29}{42\cdots 35}a^{3}+\frac{63\cdots 17}{14\cdots 45}a^{2}+\frac{63\cdots 11}{38\cdots 85}a+\frac{12\cdots 49}{42\cdots 35}$, $\frac{26\cdots 16}{70\cdots 25}a^{15}-\frac{22\cdots 92}{21\cdots 75}a^{14}-\frac{23\cdots 72}{21\cdots 75}a^{13}+\frac{16\cdots 99}{70\cdots 25}a^{12}+\frac{28\cdots 12}{21\cdots 75}a^{11}-\frac{38\cdots 66}{21\cdots 75}a^{10}-\frac{65\cdots 98}{70\cdots 25}a^{9}+\frac{14\cdots 61}{19\cdots 25}a^{8}+\frac{93\cdots 44}{19\cdots 25}a^{7}+\frac{24\cdots 64}{21\cdots 75}a^{6}-\frac{18\cdots 94}{12\cdots 95}a^{5}-\frac{53\cdots 66}{21\cdots 75}a^{4}-\frac{12\cdots 92}{70\cdots 25}a^{3}-\frac{10\cdots 54}{21\cdots 75}a^{2}-\frac{57\cdots 54}{19\cdots 25}a-\frac{10\cdots 06}{21\cdots 75}$, $\frac{13\cdots 76}{19\cdots 25}a^{15}-\frac{10\cdots 88}{63\cdots 75}a^{14}-\frac{40\cdots 39}{19\cdots 25}a^{13}+\frac{61\cdots 29}{19\cdots 25}a^{12}+\frac{49\cdots 14}{19\cdots 25}a^{11}-\frac{36\cdots 77}{19\cdots 25}a^{10}-\frac{33\cdots 98}{19\cdots 25}a^{9}+\frac{19\cdots 34}{63\cdots 75}a^{8}+\frac{17\cdots 88}{19\cdots 25}a^{7}+\frac{14\cdots 33}{19\cdots 25}a^{6}-\frac{88\cdots 23}{38\cdots 85}a^{5}-\frac{11\cdots 72}{19\cdots 25}a^{4}-\frac{12\cdots 62}{19\cdots 25}a^{3}-\frac{58\cdots 98}{17\cdots 75}a^{2}-\frac{42\cdots 06}{63\cdots 75}a+\frac{75\cdots 73}{19\cdots 25}$, $\frac{30\cdots 76}{70\cdots 25}a^{15}-\frac{27\cdots 22}{21\cdots 75}a^{14}-\frac{26\cdots 92}{21\cdots 75}a^{13}+\frac{19\cdots 44}{70\cdots 25}a^{12}+\frac{31\cdots 27}{21\cdots 75}a^{11}-\frac{49\cdots 36}{21\cdots 75}a^{10}-\frac{70\cdots 93}{70\cdots 25}a^{9}+\frac{17\cdots 86}{17\cdots 75}a^{8}+\frac{10\cdots 14}{19\cdots 25}a^{7}+\frac{83\cdots 19}{21\cdots 75}a^{6}-\frac{20\cdots 77}{12\cdots 95}a^{5}-\frac{53\cdots 81}{21\cdots 75}a^{4}-\frac{11\cdots 37}{70\cdots 25}a^{3}-\frac{87\cdots 74}{21\cdots 75}a^{2}-\frac{38\cdots 84}{19\cdots 25}a-\frac{15\cdots 46}{21\cdots 75}$, $\frac{21\cdots 38}{21\cdots 75}a^{15}-\frac{24\cdots 44}{70\cdots 25}a^{14}-\frac{60\cdots 82}{21\cdots 75}a^{13}+\frac{17\cdots 77}{21\cdots 75}a^{12}+\frac{68\cdots 07}{21\cdots 75}a^{11}-\frac{15\cdots 01}{21\cdots 75}a^{10}-\frac{80\cdots 58}{39\cdots 75}a^{9}+\frac{21\cdots 97}{63\cdots 75}a^{8}+\frac{19\cdots 54}{19\cdots 25}a^{7}-\frac{62\cdots 46}{21\cdots 75}a^{6}-\frac{13\cdots 19}{38\cdots 85}a^{5}-\frac{10\cdots 86}{21\cdots 75}a^{4}-\frac{64\cdots 56}{21\cdots 75}a^{3}-\frac{20\cdots 39}{21\cdots 75}a^{2}-\frac{12\cdots 48}{63\cdots 75}a+\frac{26\cdots 74}{21\cdots 75}$, $\frac{18\cdots 59}{21\cdots 75}a^{15}-\frac{99\cdots 37}{39\cdots 75}a^{14}-\frac{17\cdots 42}{70\cdots 25}a^{13}+\frac{11\cdots 26}{21\cdots 75}a^{12}+\frac{12\cdots 82}{39\cdots 75}a^{11}-\frac{30\cdots 26}{70\cdots 25}a^{10}-\frac{43\cdots 52}{21\cdots 75}a^{9}+\frac{34\cdots 38}{19\cdots 25}a^{8}+\frac{20\cdots 77}{19\cdots 25}a^{7}+\frac{14\cdots 04}{70\cdots 25}a^{6}-\frac{12\cdots 91}{38\cdots 85}a^{5}-\frac{38\cdots 01}{70\cdots 25}a^{4}-\frac{83\cdots 58}{21\cdots 75}a^{3}-\frac{25\cdots 82}{21\cdots 75}a^{2}-\frac{22\cdots 32}{19\cdots 25}a+\frac{66\cdots 84}{70\cdots 25}$ 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:  \( 19525572.1304 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 19525572.1304 \cdot 1}{2\cdot\sqrt{90785223184384000000000000}}\cr\approx \mathstrut & 0.408813876291 \end{aligned}\] (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 - 2*x^15 - 32*x^14 + 36*x^13 + 412*x^12 - 184*x^11 - 2872*x^10 - 74*x^9 + 14487*x^8 + 13864*x^7 - 35744*x^6 - 99102*x^5 - 104826*x^4 - 55822*x^3 - 13854*x^2 - 1018*x + 101) 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 - 2*x^15 - 32*x^14 + 36*x^13 + 412*x^12 - 184*x^11 - 2872*x^10 - 74*x^9 + 14487*x^8 + 13864*x^7 - 35744*x^6 - 99102*x^5 - 104826*x^4 - 55822*x^3 - 13854*x^2 - 1018*x + 101, 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 - 2*x^15 - 32*x^14 + 36*x^13 + 412*x^12 - 184*x^11 - 2872*x^10 - 74*x^9 + 14487*x^8 + 13864*x^7 - 35744*x^6 - 99102*x^5 - 104826*x^4 - 55822*x^3 - 13854*x^2 - 1018*x + 101); 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 - 2*x^15 - 32*x^14 + 36*x^13 + 412*x^12 - 184*x^11 - 2872*x^10 - 74*x^9 + 14487*x^8 + 13864*x^7 - 35744*x^6 - 99102*x^5 - 104826*x^4 - 55822*x^3 - 13854*x^2 - 1018*x + 101); 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

$(C_2^2\times D_4):S_4$ (as 16T1054):

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 768
The 31 conjugacy class representatives for $(C_2^2\times D_4):S_4$
Character table for $(C_2^2\times D_4):S_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.21200.1, 8.8.11236000000.1

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: 16.0.58102542838005760000000000.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ R ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.2.8.24b2.5$x^{16} + 8 x^{15} + 38 x^{14} + 126 x^{13} + 322 x^{12} + 660 x^{11} + 1116 x^{10} + 1578 x^{9} + 1881 x^{8} + 1892 x^{7} + 1600 x^{6} + 1126 x^{5} + 650 x^{4} + 300 x^{3} + 108 x^{2} + 28 x + 7$$8$$2$$24$16T193$$[\frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{2}$$
\(5\) 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$$$[\ ]_{2}$$
5.2.6.10a1.2$x^{12} + 24 x^{11} + 252 x^{10} + 1520 x^{9} + 5820 x^{8} + 14784 x^{7} + 25376 x^{6} + 29568 x^{5} + 23280 x^{4} + 12160 x^{3} + 4032 x^{2} + 768 x + 69$$6$$2$$10$$D_6$$$[\ ]_{6}^{2}$$
\(53\) Copy content Toggle raw display 53.2.1.0a1.1$x^{2} + 49 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
53.2.1.0a1.1$x^{2} + 49 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
53.2.2.2a1.2$x^{4} + 98 x^{3} + 2405 x^{2} + 196 x + 57$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
53.2.2.2a1.2$x^{4} + 98 x^{3} + 2405 x^{2} + 196 x + 57$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
53.2.2.2a1.2$x^{4} + 98 x^{3} + 2405 x^{2} + 196 x + 57$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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