Properties

Label 16.0.150...856.1
Degree $16$
Signature $(0, 8)$
Discriminant $1.509\times 10^{39}$
Root discriminant \(280.98\)
Ramified primes $2,3,11,43$
Class number $488256$ (GRH)
Class group [2, 2, 2, 2, 30516] (GRH)
Galois group $C_2^6.(C_2\times D_4)$ (as 16T1143)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 288*x^13 + 1492*x^12 - 4512*x^11 + 31344*x^10 - 146976*x^9 + 883002*x^8 - 2430912*x^7 + 4234848*x^6 - 20351904*x^5 + 40473724*x^4 - 50965728*x^3 - 11526000*x^2 + 290739552*x + 281522905)
 
Copy content gp:K = bnfinit(y^16 - 288*y^13 + 1492*y^12 - 4512*y^11 + 31344*y^10 - 146976*y^9 + 883002*y^8 - 2430912*y^7 + 4234848*y^6 - 20351904*y^5 + 40473724*y^4 - 50965728*y^3 - 11526000*y^2 + 290739552*y + 281522905, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 288*x^13 + 1492*x^12 - 4512*x^11 + 31344*x^10 - 146976*x^9 + 883002*x^8 - 2430912*x^7 + 4234848*x^6 - 20351904*x^5 + 40473724*x^4 - 50965728*x^3 - 11526000*x^2 + 290739552*x + 281522905);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 288*x^13 + 1492*x^12 - 4512*x^11 + 31344*x^10 - 146976*x^9 + 883002*x^8 - 2430912*x^7 + 4234848*x^6 - 20351904*x^5 + 40473724*x^4 - 50965728*x^3 - 11526000*x^2 + 290739552*x + 281522905)
 

\( x^{16} - 288 x^{13} + 1492 x^{12} - 4512 x^{11} + 31344 x^{10} - 146976 x^{9} + 883002 x^{8} + \cdots + 281522905 \) 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:  $(0, 8)$
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:   \(1509221451981748002510926376155643641856\) \(\medspace = 2^{58}\cdot 3^{10}\cdot 11^{10}\cdot 43^{4}\) 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:  \(280.98\)
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:  $2^{31/8}3^{3/4}11^{3/4}43^{1/2}\approx 1324.6798063293725$
Ramified primes:   \(2\), \(3\), \(11\), \(43\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 a CM field.
Reflex fields:  unavailable$^{128}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8}a^{8}-\frac{1}{2}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{8}$, $\frac{1}{8}a^{9}-\frac{1}{2}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{8}a$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{8}a^{2}-\frac{1}{2}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{16}a^{8}+\frac{1}{8}a^{7}+\frac{3}{8}a^{6}-\frac{1}{8}a^{5}+\frac{3}{8}a^{4}-\frac{5}{16}a^{3}-\frac{3}{16}a^{2}-\frac{3}{16}a+\frac{5}{16}$, $\frac{1}{48}a^{12}+\frac{1}{48}a^{11}-\frac{1}{48}a^{10}-\frac{1}{48}a^{9}+\frac{1}{24}a^{7}+\frac{11}{24}a^{6}-\frac{1}{24}a^{5}+\frac{5}{16}a^{4}-\frac{5}{48}a^{3}+\frac{5}{48}a^{2}-\frac{19}{48}a-\frac{7}{24}$, $\frac{1}{48}a^{13}+\frac{1}{48}a^{11}-\frac{1}{16}a^{10}-\frac{1}{24}a^{9}-\frac{1}{48}a^{8}-\frac{11}{24}a^{7}-\frac{1}{8}a^{6}+\frac{11}{48}a^{5}-\frac{1}{24}a^{4}-\frac{5}{48}a^{3}+\frac{5}{16}a^{2}-\frac{1}{12}a-\frac{19}{48}$, $\frac{1}{4560}a^{14}+\frac{37}{4560}a^{13}-\frac{7}{2280}a^{12}-\frac{107}{4560}a^{11}-\frac{11}{456}a^{10}-\frac{13}{380}a^{9}-\frac{233}{4560}a^{8}-\frac{467}{2280}a^{7}+\frac{1043}{4560}a^{6}-\frac{103}{1520}a^{5}+\frac{359}{1140}a^{4}-\frac{397}{912}a^{3}+\frac{71}{285}a^{2}+\frac{167}{2280}a-\frac{17}{48}$, $\frac{1}{86\cdots 80}a^{15}+\frac{48\cdots 71}{86\cdots 80}a^{14}-\frac{47\cdots 71}{86\cdots 80}a^{13}+\frac{15\cdots 57}{86\cdots 80}a^{12}-\frac{63\cdots 49}{43\cdots 40}a^{11}+\frac{34\cdots 57}{43\cdots 40}a^{10}-\frac{59\cdots 47}{14\cdots 80}a^{9}+\frac{95\cdots 77}{43\cdots 40}a^{8}-\frac{72\cdots 53}{86\cdots 80}a^{7}-\frac{14\cdots 07}{86\cdots 80}a^{6}-\frac{16\cdots 91}{57\cdots 52}a^{5}+\frac{14\cdots 39}{86\cdots 80}a^{4}+\frac{15\cdots 37}{38\cdots 60}a^{3}-\frac{86\cdots 81}{72\cdots 40}a^{2}+\frac{64\cdots 79}{21\cdots 20}a+\frac{12\cdots 02}{57\cdots 39}$ 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:  $C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{30516}$, which has order $488256$ (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:  $C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{30516}$, which has order $488256$ (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);
 
Relative class number:   $122064$ (assuming GRH)

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:   $\frac{38\cdots 49}{16\cdots 20}a^{15}-\frac{21\cdots 31}{19\cdots 40}a^{14}+\frac{11\cdots 17}{99\cdots 20}a^{13}-\frac{11\cdots 43}{19\cdots 40}a^{12}+\frac{51\cdots 31}{83\cdots 10}a^{11}-\frac{20\cdots 91}{66\cdots 80}a^{10}+\frac{37\cdots 73}{33\cdots 40}a^{9}-\frac{15\cdots 93}{34\cdots 20}a^{8}+\frac{51\cdots 91}{16\cdots 20}a^{7}-\frac{47\cdots 19}{34\cdots 20}a^{6}+\frac{74\cdots 31}{33\cdots 40}a^{5}-\frac{30\cdots 69}{66\cdots 80}a^{4}-\frac{18\cdots 19}{41\cdots 05}a^{3}+\frac{40\cdots 71}{39\cdots 08}a^{2}+\frac{20\cdots 11}{99\cdots 20}a+\frac{16\cdots 17}{20\cdots 32}$, $\frac{14\cdots 77}{16\cdots 20}a^{15}+\frac{43\cdots 27}{19\cdots 40}a^{14}+\frac{40\cdots 51}{99\cdots 20}a^{13}-\frac{48\cdots 49}{19\cdots 40}a^{12}+\frac{27\cdots 59}{41\cdots 05}a^{11}-\frac{12\cdots 53}{66\cdots 80}a^{10}+\frac{71\cdots 39}{33\cdots 40}a^{9}-\frac{25\cdots 39}{34\cdots 20}a^{8}+\frac{94\cdots 23}{16\cdots 20}a^{7}-\frac{22\cdots 57}{34\cdots 20}a^{6}+\frac{48\cdots 53}{33\cdots 40}a^{5}-\frac{10\cdots 07}{66\cdots 80}a^{4}-\frac{88\cdots 89}{83\cdots 10}a^{3}-\frac{21\cdots 91}{39\cdots 08}a^{2}-\frac{15\cdots 27}{99\cdots 20}a-\frac{23\cdots 09}{20\cdots 32}$, $\frac{56\cdots 91}{49\cdots 60}a^{15}-\frac{19\cdots 61}{24\cdots 30}a^{14}+\frac{23\cdots 49}{24\cdots 30}a^{13}-\frac{47\cdots 47}{16\cdots 20}a^{12}+\frac{30\cdots 31}{16\cdots 20}a^{11}-\frac{31\cdots 69}{33\cdots 40}a^{10}+\frac{63\cdots 07}{16\cdots 20}a^{9}-\frac{29\cdots 27}{17\cdots 60}a^{8}+\frac{17\cdots 33}{16\cdots 20}a^{7}-\frac{32\cdots 53}{87\cdots 80}a^{6}+\frac{28\cdots 36}{41\cdots 05}a^{5}-\frac{65\cdots 59}{83\cdots 10}a^{4}-\frac{46\cdots 39}{49\cdots 60}a^{3}+\frac{94\cdots 39}{19\cdots 04}a^{2}-\frac{86\cdots 41}{49\cdots 60}a-\frac{13\cdots 61}{34\cdots 72}$, $\frac{88\cdots 59}{14\cdots 80}a^{15}-\frac{91\cdots 83}{45\cdots 20}a^{14}+\frac{13\cdots 11}{21\cdots 20}a^{13}-\frac{13\cdots 43}{86\cdots 80}a^{12}+\frac{41\cdots 97}{28\cdots 76}a^{11}-\frac{89\cdots 57}{15\cdots 40}a^{10}+\frac{18\cdots 49}{72\cdots 40}a^{9}-\frac{33\cdots 97}{28\cdots 60}a^{8}+\frac{99\cdots 67}{14\cdots 80}a^{7}-\frac{76\cdots 87}{28\cdots 60}a^{6}+\frac{34\cdots 97}{72\cdots 40}a^{5}-\frac{20\cdots 21}{57\cdots 52}a^{4}-\frac{85\cdots 11}{14\cdots 80}a^{3}+\frac{12\cdots 71}{86\cdots 80}a^{2}+\frac{57\cdots 39}{43\cdots 64}a-\frac{31\cdots 53}{91\cdots 24}$, $\frac{49\cdots 03}{54\cdots 05}a^{15}+\frac{14\cdots 87}{21\cdots 20}a^{14}-\frac{74\cdots 19}{72\cdots 40}a^{13}-\frac{11\cdots 77}{43\cdots 40}a^{12}+\frac{96\cdots 23}{72\cdots 40}a^{11}-\frac{67\cdots 81}{18\cdots 35}a^{10}+\frac{18\cdots 27}{72\cdots 40}a^{9}-\frac{17\cdots 53}{14\cdots 80}a^{8}+\frac{26\cdots 39}{36\cdots 70}a^{7}-\frac{13\cdots 33}{72\cdots 40}a^{6}+\frac{34\cdots 81}{14\cdots 88}a^{5}-\frac{19\cdots 23}{14\cdots 80}a^{4}+\frac{50\cdots 87}{21\cdots 20}a^{3}-\frac{26\cdots 63}{57\cdots 90}a^{2}-\frac{85\cdots 41}{72\cdots 40}a+\frac{21\cdots 51}{45\cdots 12}$, $\frac{17\cdots 53}{10\cdots 10}a^{15}-\frac{39\cdots 27}{72\cdots 40}a^{14}+\frac{91\cdots 73}{43\cdots 64}a^{13}-\frac{38\cdots 37}{91\cdots 24}a^{12}+\frac{55\cdots 69}{14\cdots 80}a^{11}-\frac{23\cdots 49}{14\cdots 80}a^{10}+\frac{48\cdots 21}{72\cdots 94}a^{9}-\frac{17\cdots 69}{57\cdots 52}a^{8}+\frac{26\cdots 13}{14\cdots 88}a^{7}-\frac{13\cdots 19}{18\cdots 35}a^{6}+\frac{94\cdots 51}{72\cdots 40}a^{5}-\frac{24\cdots 69}{28\cdots 60}a^{4}-\frac{88\cdots 63}{43\cdots 40}a^{3}+\frac{55\cdots 89}{14\cdots 80}a^{2}+\frac{52\cdots 49}{10\cdots 10}a-\frac{44\cdots 49}{91\cdots 24}$, $\frac{14\cdots 31}{43\cdots 40}a^{15}-\frac{59\cdots 69}{57\cdots 52}a^{14}-\frac{83\cdots 19}{21\cdots 20}a^{13}-\frac{67\cdots 43}{86\cdots 80}a^{12}+\frac{10\cdots 83}{14\cdots 80}a^{11}-\frac{46\cdots 19}{28\cdots 60}a^{10}+\frac{41\cdots 49}{72\cdots 40}a^{9}-\frac{85\cdots 27}{28\cdots 60}a^{8}+\frac{77\cdots 77}{14\cdots 80}a^{7}-\frac{57\cdots 17}{57\cdots 52}a^{6}+\frac{21\cdots 19}{72\cdots 40}a^{5}-\frac{46\cdots 93}{28\cdots 60}a^{4}-\frac{13\cdots 39}{43\cdots 40}a^{3}+\frac{12\cdots 13}{28\cdots 60}a^{2}+\frac{22\cdots 11}{21\cdots 20}a-\frac{53\cdots 63}{91\cdots 24}$ 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:  \( 64598388.8936 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 64598388.8936 \cdot 488256}{2\cdot\sqrt{1509221451981748002510926376155643641856}}\cr\approx \mathstrut & 0.98605636758 \end{aligned}\] (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 - 288*x^13 + 1492*x^12 - 4512*x^11 + 31344*x^10 - 146976*x^9 + 883002*x^8 - 2430912*x^7 + 4234848*x^6 - 20351904*x^5 + 40473724*x^4 - 50965728*x^3 - 11526000*x^2 + 290739552*x + 281522905) 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 - 288*x^13 + 1492*x^12 - 4512*x^11 + 31344*x^10 - 146976*x^9 + 883002*x^8 - 2430912*x^7 + 4234848*x^6 - 20351904*x^5 + 40473724*x^4 - 50965728*x^3 - 11526000*x^2 + 290739552*x + 281522905, 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 - 288*x^13 + 1492*x^12 - 4512*x^11 + 31344*x^10 - 146976*x^9 + 883002*x^8 - 2430912*x^7 + 4234848*x^6 - 20351904*x^5 + 40473724*x^4 - 50965728*x^3 - 11526000*x^2 + 290739552*x + 281522905); 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^16 - 288*x^13 + 1492*x^12 - 4512*x^11 + 31344*x^10 - 146976*x^9 + 883002*x^8 - 2430912*x^7 + 4234848*x^6 - 20351904*x^5 + 40473724*x^4 - 50965728*x^3 - 11526000*x^2 + 290739552*x + 281522905); 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

$C_2^6.(C_2\times D_4)$ (as 16T1143):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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 1024
The 58 conjugacy class representatives for $C_2^6.(C_2\times D_4)$
Character table for $C_2^6.(C_2\times D_4)$

Intermediate fields

\(\Q(\sqrt{66}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{6}, \sqrt{11})\), 8.8.36788541182705664.1

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 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.0.222220411196339044718009575937465647104.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{5}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ R ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ R ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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 2.1.16.58n1.779$x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 14$$16$$1$$58$16T333$$[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$$
\(3\) Copy content Toggle raw display 3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
3.2.4.6a1.3$x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$$4$$2$$6$$Q_8$$$[\ ]_{4}^{2}$$
\(11\) Copy content Toggle raw display 11.2.2.2a1.2$x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
11.2.2.2a1.2$x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
11.2.4.6a1.2$x^{8} + 28 x^{7} + 302 x^{6} + 1540 x^{5} + 3601 x^{4} + 3080 x^{3} + 1208 x^{2} + 224 x + 27$$4$$2$$6$$D_4$$$[\ ]_{4}^{2}$$
\(43\) Copy content Toggle raw display $\Q_{43}$$x + 40$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{43}$$x + 40$$1$$1$$0$Trivial$$[\ ]$$
43.1.2.1a1.2$x^{2} + 129$$2$$1$$1$$C_2$$$[\ ]_{2}$$
43.2.1.0a1.1$x^{2} + 42 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
43.1.2.1a1.2$x^{2} + 129$$2$$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$$$[\ ]_{2}^{2}$$
43.4.1.0a1.1$x^{4} + 5 x^{2} + 42 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$

Spectrum of ring of integers

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