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Properties

Label 15.5.455...296.1
Degree $15$
Signature $[5, 5]$
Discriminant $-4.557\times 10^{28}$
Root discriminant \(81.39\)
Ramified primes $2,17,31,43$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_5^3:S_4$ (as 15T51)

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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^15 - 5*x^14 - 30*x^13 + 159*x^12 + 496*x^11 - 2579*x^10 - 4208*x^9 + 20521*x^8 + 13460*x^7 - 60845*x^6 + 48450*x^5 + 6957*x^4 - 403447*x^3 - 35034*x^2 - 86972*x - 710752)
 
Copy content gp:K = bnfinit(y^15 - 5*y^14 - 30*y^13 + 159*y^12 + 496*y^11 - 2579*y^10 - 4208*y^9 + 20521*y^8 + 13460*y^7 - 60845*y^6 + 48450*y^5 + 6957*y^4 - 403447*y^3 - 35034*y^2 - 86972*y - 710752, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 5*x^14 - 30*x^13 + 159*x^12 + 496*x^11 - 2579*x^10 - 4208*x^9 + 20521*x^8 + 13460*x^7 - 60845*x^6 + 48450*x^5 + 6957*x^4 - 403447*x^3 - 35034*x^2 - 86972*x - 710752);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 5*x^14 - 30*x^13 + 159*x^12 + 496*x^11 - 2579*x^10 - 4208*x^9 + 20521*x^8 + 13460*x^7 - 60845*x^6 + 48450*x^5 + 6957*x^4 - 403447*x^3 - 35034*x^2 - 86972*x - 710752)
 

\( x^{15} - 5 x^{14} - 30 x^{13} + 159 x^{12} + 496 x^{11} - 2579 x^{10} - 4208 x^{9} + 20521 x^{8} + \cdots - 710752 \) 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:  $15$
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:  $[5, 5]$
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:   \(-45574341341445246456603159296\) \(\medspace = -\,2^{8}\cdot 17^{5}\cdot 31^{8}\cdot 43^{5}\) 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:  \(81.39\)
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:  $2^{4/5}17^{1/2}31^{4/5}43^{1/2}\approx 734.2974838911407$
Ramified primes:   \(2\), \(17\), \(31\), \(43\) 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:  \(\Q(\sqrt{-731}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{14}a^{10}+\frac{1}{14}a^{9}+\frac{1}{14}a^{8}+\frac{1}{7}a^{7}+\frac{1}{7}a^{6}-\frac{3}{7}a^{5}+\frac{5}{14}a^{4}-\frac{5}{14}a^{3}+\frac{5}{14}a^{2}-\frac{3}{7}a$, $\frac{1}{28}a^{11}-\frac{1}{28}a^{10}+\frac{3}{14}a^{9}+\frac{5}{28}a^{7}-\frac{3}{28}a^{6}-\frac{1}{7}a^{5}+\frac{3}{14}a^{4}-\frac{13}{28}a^{3}+\frac{5}{28}a^{2}+\frac{3}{7}a$, $\frac{1}{196}a^{12}-\frac{3}{196}a^{11}+\frac{2}{49}a^{9}+\frac{39}{196}a^{8}-\frac{29}{196}a^{7}-\frac{3}{14}a^{6}+\frac{45}{98}a^{5}-\frac{93}{196}a^{4}+\frac{43}{196}a^{3}+\frac{8}{49}a^{2}-\frac{22}{49}a-\frac{2}{7}$, $\frac{1}{1372}a^{13}+\frac{1}{1372}a^{12}+\frac{4}{343}a^{11}-\frac{12}{343}a^{10}-\frac{181}{1372}a^{9}-\frac{97}{1372}a^{8}+\frac{159}{686}a^{7}-\frac{79}{343}a^{6}+\frac{617}{1372}a^{5}+\frac{27}{196}a^{4}-\frac{255}{686}a^{3}+\frac{69}{686}a^{2}+\frac{293}{686}a-\frac{22}{49}$, $\frac{1}{10\cdots 92}a^{14}+\frac{30\cdots 14}{25\cdots 23}a^{13}-\frac{98\cdots 83}{50\cdots 46}a^{12}+\frac{34\cdots 17}{10\cdots 92}a^{11}+\frac{22\cdots 41}{10\cdots 92}a^{10}+\frac{47\cdots 35}{25\cdots 23}a^{9}-\frac{16\cdots 38}{36\cdots 89}a^{8}-\frac{23\cdots 63}{10\cdots 92}a^{7}-\frac{94\cdots 79}{10\cdots 92}a^{6}+\frac{12\cdots 15}{50\cdots 46}a^{5}+\frac{23\cdots 57}{50\cdots 46}a^{4}-\frac{58\cdots 89}{10\cdots 92}a^{3}+\frac{18\cdots 29}{50\cdots 46}a^{2}+\frac{13\cdots 03}{50\cdots 46}a-\frac{72\cdots 77}{36\cdots 89}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

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:  $C_{2}$, which has order $2$ (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:  $9$
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:   $\frac{16\cdots 62}{36\cdots 89}a^{14}-\frac{11\cdots 97}{36\cdots 89}a^{13}-\frac{19\cdots 87}{36\cdots 89}a^{12}+\frac{24\cdots 53}{36\cdots 89}a^{11}+\frac{12\cdots 98}{36\cdots 89}a^{10}-\frac{27\cdots 09}{36\cdots 89}a^{9}+\frac{32\cdots 70}{52\cdots 27}a^{8}+\frac{12\cdots 99}{36\cdots 89}a^{7}-\frac{16\cdots 90}{36\cdots 89}a^{6}+\frac{14\cdots 61}{36\cdots 89}a^{5}-\frac{39\cdots 89}{36\cdots 89}a^{4}-\frac{45\cdots 41}{36\cdots 89}a^{3}+\frac{20\cdots 98}{36\cdots 89}a^{2}-\frac{60\cdots 68}{36\cdots 89}a-\frac{23\cdots 95}{52\cdots 27}$, $\frac{59\cdots 07}{50\cdots 46}a^{14}+\frac{32\cdots 49}{10\cdots 92}a^{13}-\frac{39\cdots 03}{10\cdots 92}a^{12}-\frac{32\cdots 51}{10\cdots 92}a^{11}+\frac{81\cdots 53}{10\cdots 92}a^{10}+\frac{80\cdots 67}{10\cdots 92}a^{9}-\frac{11\cdots 57}{14\cdots 56}a^{8}-\frac{10\cdots 53}{10\cdots 92}a^{7}+\frac{67\cdots 75}{10\cdots 92}a^{6}+\frac{67\cdots 03}{10\cdots 92}a^{5}-\frac{39\cdots 33}{10\cdots 92}a^{4}-\frac{17\cdots 75}{10\cdots 92}a^{3}+\frac{14\cdots 13}{10\cdots 92}a^{2}-\frac{82\cdots 86}{25\cdots 23}a-\frac{17\cdots 27}{36\cdots 89}$, $\frac{10\cdots 31}{10\cdots 92}a^{14}-\frac{35\cdots 13}{50\cdots 46}a^{13}-\frac{16\cdots 29}{10\cdots 92}a^{12}+\frac{19\cdots 25}{10\cdots 92}a^{11}+\frac{32\cdots 51}{25\cdots 23}a^{10}-\frac{14\cdots 29}{50\cdots 46}a^{9}+\frac{16\cdots 27}{14\cdots 56}a^{8}+\frac{17\cdots 97}{10\cdots 92}a^{7}-\frac{48\cdots 91}{25\cdots 23}a^{6}-\frac{91\cdots 69}{50\cdots 46}a^{5}+\frac{72\cdots 23}{10\cdots 92}a^{4}-\frac{14\cdots 43}{10\cdots 92}a^{3}-\frac{10\cdots 85}{10\cdots 92}a^{2}+\frac{31\cdots 75}{25\cdots 23}a-\frac{12\cdots 25}{36\cdots 89}$, $\frac{26\cdots 41}{50\cdots 46}a^{14}-\frac{11\cdots 31}{10\cdots 92}a^{13}-\frac{45\cdots 50}{25\cdots 23}a^{12}+\frac{33\cdots 37}{10\cdots 92}a^{11}+\frac{77\cdots 25}{25\cdots 23}a^{10}-\frac{49\cdots 35}{10\cdots 92}a^{9}-\frac{97\cdots 68}{36\cdots 89}a^{8}+\frac{34\cdots 95}{10\cdots 92}a^{7}+\frac{37\cdots 27}{50\cdots 46}a^{6}-\frac{16\cdots 67}{10\cdots 92}a^{5}+\frac{83\cdots 03}{50\cdots 46}a^{4}+\frac{72\cdots 65}{10\cdots 92}a^{3}-\frac{43\cdots 14}{25\cdots 23}a^{2}+\frac{27\cdots 01}{50\cdots 46}a+\frac{61\cdots 87}{36\cdots 89}$, $\frac{22\cdots 73}{10\cdots 92}a^{14}-\frac{30\cdots 16}{25\cdots 23}a^{13}-\frac{32\cdots 27}{50\cdots 46}a^{12}+\frac{39\cdots 85}{10\cdots 92}a^{11}+\frac{10\cdots 13}{10\cdots 92}a^{10}-\frac{34\cdots 13}{50\cdots 46}a^{9}-\frac{59\cdots 05}{72\cdots 78}a^{8}+\frac{61\cdots 23}{10\cdots 92}a^{7}+\frac{23\cdots 77}{10\cdots 92}a^{6}-\frac{13\cdots 21}{50\cdots 46}a^{5}+\frac{40\cdots 75}{50\cdots 46}a^{4}+\frac{50\cdots 53}{10\cdots 92}a^{3}-\frac{14\cdots 18}{25\cdots 23}a^{2}-\frac{34\cdots 22}{25\cdots 23}a+\frac{40\cdots 99}{36\cdots 89}$, $\frac{18\cdots 91}{50\cdots 46}a^{14}-\frac{83\cdots 33}{10\cdots 92}a^{13}-\frac{26\cdots 44}{25\cdots 23}a^{12}+\frac{27\cdots 19}{10\cdots 92}a^{11}+\frac{15\cdots 43}{50\cdots 46}a^{10}-\frac{47\cdots 71}{10\cdots 92}a^{9}-\frac{39\cdots 67}{72\cdots 78}a^{8}+\frac{41\cdots 73}{10\cdots 92}a^{7}+\frac{16\cdots 18}{25\cdots 23}a^{6}-\frac{11\cdots 79}{10\cdots 92}a^{5}-\frac{12\cdots 59}{50\cdots 46}a^{4}-\frac{73\cdots 89}{10\cdots 92}a^{3}-\frac{78\cdots 39}{25\cdots 23}a^{2}-\frac{22\cdots 78}{25\cdots 23}a-\frac{20\cdots 79}{36\cdots 89}$, $\frac{35\cdots 77}{25\cdots 23}a^{14}-\frac{50\cdots 67}{50\cdots 46}a^{13}-\frac{23\cdots 79}{10\cdots 92}a^{12}+\frac{30\cdots 01}{10\cdots 92}a^{11}-\frac{19\cdots 74}{25\cdots 23}a^{10}-\frac{17\cdots 29}{50\cdots 46}a^{9}+\frac{52\cdots 83}{14\cdots 56}a^{8}+\frac{16\cdots 73}{10\cdots 92}a^{7}-\frac{12\cdots 63}{50\cdots 46}a^{6}-\frac{25\cdots 09}{50\cdots 46}a^{5}+\frac{70\cdots 79}{10\cdots 92}a^{4}-\frac{17\cdots 43}{10\cdots 92}a^{3}-\frac{39\cdots 81}{50\cdots 46}a^{2}+\frac{32\cdots 05}{25\cdots 23}a-\frac{14\cdots 93}{36\cdots 89}$, $\frac{15\cdots 57}{50\cdots 46}a^{14}-\frac{53\cdots 89}{25\cdots 23}a^{13}-\frac{40\cdots 83}{10\cdots 92}a^{12}+\frac{55\cdots 85}{10\cdots 92}a^{11}+\frac{98\cdots 93}{50\cdots 46}a^{10}-\frac{19\cdots 68}{25\cdots 23}a^{9}+\frac{84\cdots 69}{14\cdots 56}a^{8}+\frac{45\cdots 01}{10\cdots 92}a^{7}-\frac{34\cdots 69}{50\cdots 46}a^{6}-\frac{77\cdots 73}{50\cdots 46}a^{5}+\frac{19\cdots 09}{10\cdots 92}a^{4}-\frac{42\cdots 93}{10\cdots 92}a^{3}-\frac{61\cdots 29}{50\cdots 46}a^{2}+\frac{12\cdots 27}{50\cdots 46}a-\frac{29\cdots 77}{36\cdots 89}$, $\frac{92\cdots 51}{50\cdots 46}a^{14}-\frac{29\cdots 14}{25\cdots 23}a^{13}-\frac{17\cdots 47}{50\cdots 46}a^{12}+\frac{32\cdots 41}{10\cdots 92}a^{11}+\frac{35\cdots 01}{10\cdots 92}a^{10}-\frac{11\cdots 50}{25\cdots 23}a^{9}+\frac{32\cdots 05}{72\cdots 78}a^{8}+\frac{30\cdots 51}{10\cdots 92}a^{7}-\frac{26\cdots 91}{10\cdots 92}a^{6}-\frac{17\cdots 61}{50\cdots 46}a^{5}+\frac{59\cdots 67}{50\cdots 46}a^{4}-\frac{22\cdots 33}{10\cdots 92}a^{3}-\frac{16\cdots 05}{10\cdots 92}a^{2}+\frac{44\cdots 31}{25\cdots 23}a-\frac{16\cdots 73}{36\cdots 89}$ 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:  \( 1950706707.66 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 1950706707.66 \cdot 2}{2\cdot\sqrt{45574341341445246456603159296}}\cr\approx \mathstrut & 2.86339391069 \end{aligned}\] (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^15 - 5*x^14 - 30*x^13 + 159*x^12 + 496*x^11 - 2579*x^10 - 4208*x^9 + 20521*x^8 + 13460*x^7 - 60845*x^6 + 48450*x^5 + 6957*x^4 - 403447*x^3 - 35034*x^2 - 86972*x - 710752) 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^15 - 5*x^14 - 30*x^13 + 159*x^12 + 496*x^11 - 2579*x^10 - 4208*x^9 + 20521*x^8 + 13460*x^7 - 60845*x^6 + 48450*x^5 + 6957*x^4 - 403447*x^3 - 35034*x^2 - 86972*x - 710752, 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^15 - 5*x^14 - 30*x^13 + 159*x^12 + 496*x^11 - 2579*x^10 - 4208*x^9 + 20521*x^8 + 13460*x^7 - 60845*x^6 + 48450*x^5 + 6957*x^4 - 403447*x^3 - 35034*x^2 - 86972*x - 710752); 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^15 - 5*x^14 - 30*x^13 + 159*x^12 + 496*x^11 - 2579*x^10 - 4208*x^9 + 20521*x^8 + 13460*x^7 - 60845*x^6 + 48450*x^5 + 6957*x^4 - 403447*x^3 - 35034*x^2 - 86972*x - 710752); 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

$C_5^3:S_4$ (as 15T51):

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 3000
The 38 conjugacy class representatives for $C_5^3:S_4$
Character table for $C_5^3:S_4$

Intermediate fields

3.1.731.1

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 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 R $15$ $15$ ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{7}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }$ $15$ R ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ R $15$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ R $15$ $15$ $15$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.5.8a1.1$x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 3$$5$$2$$8$$F_5$$$[\ ]_{5}^{4}$$
\(17\) Copy content Toggle raw display 17.1.2.1a1.1$x^{2} + 17$$2$$1$$1$$C_2$$$[\ ]_{2}$$
17.2.2.2a1.2$x^{4} + 32 x^{3} + 262 x^{2} + 96 x + 26$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
17.2.2.2a1.2$x^{4} + 32 x^{3} + 262 x^{2} + 96 x + 26$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
17.5.1.0a1.1$x^{5} + x + 14$$1$$5$$0$$C_5$$$[\ ]^{5}$$
\(31\) Copy content Toggle raw display 31.5.1.0a1.1$x^{5} + 7 x + 28$$1$$5$$0$$C_5$$$[\ ]^{5}$$
31.2.5.8a1.2$x^{10} + 145 x^{9} + 8425 x^{8} + 245630 x^{7} + 3612185 x^{6} + 21982319 x^{5} + 10836555 x^{4} + 2210670 x^{3} + 227475 x^{2} + 11745 x + 274$$5$$2$$8$$C_{10}$$$[\ ]_{5}^{2}$$
\(43\) Copy content Toggle raw display 43.1.2.1a1.1$x^{2} + 43$$2$$1$$1$$C_2$$$[\ ]_{2}$$
43.2.2.2a1.2$x^{4} + 84 x^{3} + 1770 x^{2} + 252 x + 52$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
43.2.2.2a1.2$x^{4} + 84 x^{3} + 1770 x^{2} + 252 x + 52$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
43.5.1.0a1.1$x^{5} + 8 x + 40$$1$$5$$0$$C_5$$$[\ ]^{5}$$

Spectrum of ring of integers

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