Properties

Label 15.1.394...000.1
Degree $15$
Signature $(1, 7)$
Discriminant $-3.943\times 10^{47}$
Root discriminant \(1489.56\)
Ramified primes $2,5,43,149$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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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 + 220*x^13 - 1490*x^12 + 21465*x^11 - 164089*x^10 + 1279530*x^9 - 9297720*x^8 + 55190240*x^7 - 271383040*x^6 + 1210610432*x^5 - 3732515920*x^4 + 15334623680*x^3 - 28645458560*x^2 + 73655450560*x + 43700529728)
 
Copy content gp:K = bnfinit(y^15 - 5*y^14 + 220*y^13 - 1490*y^12 + 21465*y^11 - 164089*y^10 + 1279530*y^9 - 9297720*y^8 + 55190240*y^7 - 271383040*y^6 + 1210610432*y^5 - 3732515920*y^4 + 15334623680*y^3 - 28645458560*y^2 + 73655450560*y + 43700529728, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 5*x^14 + 220*x^13 - 1490*x^12 + 21465*x^11 - 164089*x^10 + 1279530*x^9 - 9297720*x^8 + 55190240*x^7 - 271383040*x^6 + 1210610432*x^5 - 3732515920*x^4 + 15334623680*x^3 - 28645458560*x^2 + 73655450560*x + 43700529728);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 5*x^14 + 220*x^13 - 1490*x^12 + 21465*x^11 - 164089*x^10 + 1279530*x^9 - 9297720*x^8 + 55190240*x^7 - 271383040*x^6 + 1210610432*x^5 - 3732515920*x^4 + 15334623680*x^3 - 28645458560*x^2 + 73655450560*x + 43700529728)
 

\( x^{15} - 5 x^{14} + 220 x^{13} - 1490 x^{12} + 21465 x^{11} - 164089 x^{10} + 1279530 x^{9} + \cdots + 43700529728 \) 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:  $(1, 7)$
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:   \(-394340412390135315125796995225021875000000000000\) \(\medspace = -\,2^{12}\cdot 5^{17}\cdot 43^{13}\cdot 149^{5}\) 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:  \(1489.56\)
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:  $2^{4/5}5^{71/60}43^{9/10}149^{1/2}\approx 4213.599423113526$
Ramified primes:   \(2\), \(5\), \(43\), \(149\) 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:  \(\Q(\sqrt{-32035}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{10}a^{5}-\frac{1}{2}a^{3}+\frac{1}{5}$, $\frac{1}{10}a^{6}-\frac{1}{2}a^{2}+\frac{1}{5}a$, $\frac{1}{10}a^{7}-\frac{1}{2}a^{3}+\frac{1}{5}a^{2}$, $\frac{1}{20}a^{8}-\frac{1}{4}a^{4}-\frac{2}{5}a^{3}$, $\frac{1}{100}a^{9}-\frac{1}{50}a^{8}+\frac{1}{25}a^{7}+\frac{1}{50}a^{6}+\frac{1}{100}a^{5}+\frac{1}{50}a^{4}+\frac{23}{50}a^{3}-\frac{21}{50}a^{2}+\frac{1}{25}a-\frac{12}{25}$, $\frac{1}{8600}a^{10}-\frac{1}{200}a^{9}-\frac{67}{4300}a^{8}+\frac{3}{100}a^{7}-\frac{291}{8600}a^{6}+\frac{7}{200}a^{5}+\frac{191}{2150}a^{4}-\frac{19}{50}a^{3}+\frac{829}{2150}a^{2}-\frac{3}{25}a+\frac{231}{1075}$, $\frac{1}{17200}a^{11}-\frac{1}{17200}a^{10}-\frac{3}{1075}a^{9}+\frac{153}{8600}a^{8}-\frac{807}{17200}a^{7}-\frac{397}{17200}a^{6}+\frac{339}{8600}a^{5}-\frac{449}{4300}a^{4}-\frac{811}{2150}a^{3}+\frac{467}{2150}a^{2}+\frac{919}{2150}a+\frac{7}{215}$, $\frac{1}{86000}a^{12}-\frac{1}{86000}a^{11}+\frac{1}{21500}a^{10}-\frac{21}{8600}a^{9}-\frac{351}{17200}a^{8}-\frac{2461}{86000}a^{7}+\frac{513}{43000}a^{6}-\frac{209}{5375}a^{5}+\frac{329}{4300}a^{4}+\frac{231}{2150}a^{3}+\frac{1239}{5375}a^{2}-\frac{309}{5375}a+\frac{1706}{5375}$, $\frac{1}{172000}a^{13}-\frac{1}{172000}a^{12}+\frac{1}{43000}a^{11}-\frac{1}{17200}a^{10}-\frac{7}{34400}a^{9}+\frac{1699}{172000}a^{8}+\frac{3953}{86000}a^{7}-\frac{813}{21500}a^{6}-\frac{3}{1720}a^{5}+\frac{267}{4300}a^{4}-\frac{1607}{21500}a^{3}-\frac{1037}{5375}a^{2}-\frac{2157}{5375}a-\frac{141}{1075}$, $\frac{1}{19\cdots 00}a^{14}+\frac{41\cdots 13}{19\cdots 00}a^{13}-\frac{43\cdots 81}{19\cdots 00}a^{12}-\frac{60\cdots 33}{49\cdots 00}a^{11}-\frac{50\cdots 41}{19\cdots 00}a^{10}+\frac{36\cdots 07}{98\cdots 00}a^{9}+\frac{66\cdots 33}{78\cdots 40}a^{8}+\frac{65\cdots 66}{14\cdots 75}a^{7}+\frac{18\cdots 23}{49\cdots 00}a^{6}+\frac{22\cdots 99}{49\cdots 00}a^{5}-\frac{17\cdots 47}{24\cdots 00}a^{4}+\frac{17\cdots 47}{49\cdots 00}a^{3}+\frac{11\cdots 33}{61\cdots 25}a^{2}-\frac{30\cdots 34}{61\cdots 25}a-\frac{14\cdots 92}{61\cdots 25}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

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:  $7$
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:   $\frac{34\cdots 27}{45\cdots 00}a^{14}+\frac{31\cdots 99}{22\cdots 00}a^{13}+\frac{43\cdots 41}{45\cdots 00}a^{12}+\frac{96\cdots 91}{45\cdots 20}a^{11}-\frac{27\cdots 57}{45\cdots 00}a^{10}+\frac{29\cdots 59}{22\cdots 00}a^{9}-\frac{58\cdots 53}{45\cdots 00}a^{8}+\frac{13\cdots 57}{22\cdots 00}a^{7}-\frac{27\cdots 37}{45\cdots 20}a^{6}+\frac{43\cdots 63}{11\cdots 00}a^{5}-\frac{21\cdots 67}{28\cdots 75}a^{4}+\frac{17\cdots 47}{28\cdots 75}a^{3}-\frac{56\cdots 67}{57\cdots 50}a^{2}+\frac{34\cdots 69}{57\cdots 15}a+\frac{89\cdots 27}{28\cdots 75}$, $\frac{17\cdots 03}{11\cdots 00}a^{14}+\frac{74\cdots 68}{28\cdots 75}a^{13}-\frac{12\cdots 79}{28\cdots 75}a^{12}+\frac{54\cdots 85}{11\cdots 63}a^{11}-\frac{72\cdots 37}{28\cdots 75}a^{10}+\frac{19\cdots 51}{57\cdots 50}a^{9}-\frac{16\cdots 21}{57\cdots 50}a^{8}+\frac{46\cdots 84}{28\cdots 75}a^{7}-\frac{29\cdots 03}{22\cdots 60}a^{6}+\frac{42\cdots 39}{57\cdots 50}a^{5}-\frac{97\cdots 29}{57\cdots 50}a^{4}+\frac{33\cdots 32}{28\cdots 75}a^{3}-\frac{59\cdots 66}{28\cdots 75}a^{2}+\frac{57\cdots 16}{57\cdots 15}a+\frac{15\cdots 37}{28\cdots 75}$, $\frac{57\cdots 09}{10\cdots 40}a^{14}-\frac{10\cdots 53}{53\cdots 00}a^{13}+\frac{56\cdots 61}{53\cdots 00}a^{12}-\frac{16\cdots 13}{26\cdots 00}a^{11}+\frac{57\cdots 44}{66\cdots 25}a^{10}-\frac{67\cdots 11}{10\cdots 40}a^{9}+\frac{22\cdots 33}{53\cdots 00}a^{8}-\frac{40\cdots 59}{13\cdots 50}a^{7}+\frac{83\cdots 21}{53\cdots 00}a^{6}-\frac{13\cdots 61}{26\cdots 00}a^{5}+\frac{23\cdots 99}{13\cdots 05}a^{4}-\frac{32\cdots 09}{66\cdots 25}a^{3}+\frac{99\cdots 48}{66\cdots 25}a^{2}+\frac{20\cdots 87}{66\cdots 25}a+\frac{93\cdots 31}{66\cdots 25}$, $\frac{13\cdots 83}{19\cdots 00}a^{14}-\frac{35\cdots 73}{98\cdots 00}a^{13}+\frac{14\cdots 33}{98\cdots 00}a^{12}-\frac{10\cdots 37}{98\cdots 00}a^{11}+\frac{29\cdots 61}{19\cdots 00}a^{10}-\frac{91\cdots 77}{78\cdots 44}a^{9}+\frac{89\cdots 03}{98\cdots 00}a^{8}-\frac{15\cdots 81}{22\cdots 00}a^{7}+\frac{49\cdots 29}{12\cdots 50}a^{6}-\frac{19\cdots 89}{98\cdots 00}a^{5}+\frac{11\cdots 48}{12\cdots 25}a^{4}-\frac{17\cdots 47}{61\cdots 25}a^{3}+\frac{14\cdots 19}{12\cdots 50}a^{2}-\frac{15\cdots 43}{61\cdots 25}a+\frac{30\cdots 36}{49\cdots 09}$, $\frac{32\cdots 83}{39\cdots 00}a^{14}-\frac{13\cdots 63}{19\cdots 00}a^{13}+\frac{13\cdots 63}{78\cdots 40}a^{12}-\frac{26\cdots 07}{19\cdots 00}a^{11}+\frac{48\cdots 41}{39\cdots 00}a^{10}-\frac{13\cdots 89}{19\cdots 00}a^{9}+\frac{95\cdots 41}{39\cdots 00}a^{8}-\frac{29\cdots 51}{91\cdots 40}a^{7}-\frac{11\cdots 17}{49\cdots 00}a^{6}+\frac{12\cdots 23}{49\cdots 00}a^{5}-\frac{42\cdots 11}{49\cdots 00}a^{4}+\frac{32\cdots 67}{49\cdots 00}a^{3}-\frac{69\cdots 79}{49\cdots 90}a^{2}+\frac{57\cdots 62}{12\cdots 25}a+\frac{32\cdots 47}{12\cdots 25}$, $\frac{62\cdots 81}{22\cdots 00}a^{14}-\frac{57\cdots 77}{11\cdots 00}a^{13}+\frac{14\cdots 63}{22\cdots 00}a^{12}-\frac{83\cdots 29}{11\cdots 00}a^{11}+\frac{13\cdots 79}{22\cdots 00}a^{10}-\frac{29\cdots 69}{57\cdots 00}a^{9}+\frac{84\cdots 09}{22\cdots 00}a^{8}-\frac{24\cdots 49}{11\cdots 00}a^{7}+\frac{60\cdots 67}{57\cdots 00}a^{6}-\frac{48\cdots 67}{11\cdots 00}a^{5}+\frac{21\cdots 64}{14\cdots 75}a^{4}-\frac{71\cdots 01}{14\cdots 75}a^{3}+\frac{14\cdots 97}{14\cdots 75}a^{2}-\frac{25\cdots 77}{14\cdots 75}a-\frac{16\cdots 74}{14\cdots 75}$, $\frac{41\cdots 97}{19\cdots 00}a^{14}-\frac{84\cdots 07}{98\cdots 00}a^{13}+\frac{78\cdots 31}{19\cdots 00}a^{12}-\frac{65\cdots 13}{24\cdots 00}a^{11}+\frac{64\cdots 59}{19\cdots 00}a^{10}-\frac{26\cdots 11}{98\cdots 00}a^{9}+\frac{33\cdots 89}{19\cdots 00}a^{8}-\frac{14\cdots 53}{11\cdots 00}a^{7}+\frac{36\cdots 41}{49\cdots 00}a^{6}-\frac{31\cdots 39}{12\cdots 50}a^{5}+\frac{13\cdots 23}{12\cdots 50}a^{4}-\frac{86\cdots 27}{24\cdots 00}a^{3}+\frac{62\cdots 17}{61\cdots 25}a^{2}-\frac{76\cdots 08}{61\cdots 25}a-\frac{53\cdots 87}{61\cdots 25}$ 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:  \( 3847491857052816400 \) (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)
 
Unit signature rank:  \( 1 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 3847491857052816400 \cdot 1}{2\cdot\sqrt{394340412390135315125796995225021875000000000000}}\cr\approx \mathstrut & 2.36865137509985 \end{aligned}\] (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 + 220*x^13 - 1490*x^12 + 21465*x^11 - 164089*x^10 + 1279530*x^9 - 9297720*x^8 + 55190240*x^7 - 271383040*x^6 + 1210610432*x^5 - 3732515920*x^4 + 15334623680*x^3 - 28645458560*x^2 + 73655450560*x + 43700529728) 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 + 220*x^13 - 1490*x^12 + 21465*x^11 - 164089*x^10 + 1279530*x^9 - 9297720*x^8 + 55190240*x^7 - 271383040*x^6 + 1210610432*x^5 - 3732515920*x^4 + 15334623680*x^3 - 28645458560*x^2 + 73655450560*x + 43700529728, 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 + 220*x^13 - 1490*x^12 + 21465*x^11 - 164089*x^10 + 1279530*x^9 - 9297720*x^8 + 55190240*x^7 - 271383040*x^6 + 1210610432*x^5 - 3732515920*x^4 + 15334623680*x^3 - 28645458560*x^2 + 73655450560*x + 43700529728); 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 + 220*x^13 - 1490*x^12 + 21465*x^11 - 164089*x^10 + 1279530*x^9 - 9297720*x^8 + 55190240*x^7 - 271383040*x^6 + 1210610432*x^5 - 3732515920*x^4 + 15334623680*x^3 - 28645458560*x^2 + 73655450560*x + 43700529728); 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

$S_3\times F_5$ (as 15T11):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
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 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.160175.1, 5.1.170940050000.2

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 30 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 R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ R ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ $15$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ R ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 2.1.5.4a1.1$x^{5} + 2$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
2.1.5.4a1.1$x^{5} + 2$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
2.1.5.4a1.1$x^{5} + 2$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
\(5\) Copy content Toggle raw display 5.1.15.17a1.1$x^{15} + 5 x^{3} + 5$$15$$1$$17$$F_5 \times S_3$$$[\frac{5}{4}]_{12}^{2}$$
\(43\) Copy content Toggle raw display 43.1.5.4a1.1$x^{5} + 43$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
43.1.10.9a1.2$x^{10} + 129$$10$$1$$9$$F_{5}\times C_2$$$[\ ]_{10}^{4}$$
\(149\) Copy content Toggle raw display $\Q_{149}$$x + 147$$1$$1$$0$Trivial$$[\ ]$$
149.2.1.0a1.1$x^{2} + 145 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
149.2.1.0a1.1$x^{2} + 145 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
149.1.2.1a1.2$x^{2} + 298$$2$$1$$1$$C_2$$$[\ ]_{2}$$
149.2.2.2a1.2$x^{4} + 290 x^{3} + 21029 x^{2} + 580 x + 153$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
149.2.2.2a1.2$x^{4} + 290 x^{3} + 21029 x^{2} + 580 x + 153$$2$$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)