Properties

Label 16.4.153...104.2
Degree $16$
Signature $(4, 6)$
Discriminant $1.532\times 10^{23}$
Root discriminant \(28.12\)
Ramified primes $2,3$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^4:Q_8$ (as 16T333)

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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 - 8*x^14 - 8*x^13 + 16*x^12 + 24*x^11 - 80*x^10 - 336*x^9 - 314*x^8 + 944*x^7 + 3520*x^6 + 5776*x^5 + 6164*x^4 + 4864*x^3 + 2728*x^2 + 1264*x + 478)
 
Copy content gp:K = bnfinit(y^16 - 8*y^14 - 8*y^13 + 16*y^12 + 24*y^11 - 80*y^10 - 336*y^9 - 314*y^8 + 944*y^7 + 3520*y^6 + 5776*y^5 + 6164*y^4 + 4864*y^3 + 2728*y^2 + 1264*y + 478, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 - 8*x^13 + 16*x^12 + 24*x^11 - 80*x^10 - 336*x^9 - 314*x^8 + 944*x^7 + 3520*x^6 + 5776*x^5 + 6164*x^4 + 4864*x^3 + 2728*x^2 + 1264*x + 478);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^14 - 8*x^13 + 16*x^12 + 24*x^11 - 80*x^10 - 336*x^9 - 314*x^8 + 944*x^7 + 3520*x^6 + 5776*x^5 + 6164*x^4 + 4864*x^3 + 2728*x^2 + 1264*x + 478)
 

\( x^{16} - 8 x^{14} - 8 x^{13} + 16 x^{12} + 24 x^{11} - 80 x^{10} - 336 x^{9} - 314 x^{8} + 944 x^{7} + \cdots + 478 \) 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:  $(4, 6)$
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:   \(153177439332441840943104\) \(\medspace = 2^{58}\cdot 3^{12}\) 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:  \(28.12\)
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:  $2^{31/8}3^{3/4}\approx 33.44507500385779$
Ramified primes:   \(2\), \(3\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
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 not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{17}a^{13}-\frac{8}{17}a^{12}-\frac{5}{17}a^{11}+\frac{3}{17}a^{10}-\frac{4}{17}a^{9}-\frac{7}{17}a^{8}-\frac{5}{17}a^{7}+\frac{6}{17}a^{6}-\frac{8}{17}a^{5}-\frac{3}{17}a^{4}-\frac{5}{17}a^{3}+\frac{3}{17}a^{2}+\frac{6}{17}a-\frac{7}{17}$, $\frac{1}{20519}a^{14}-\frac{130}{20519}a^{13}+\frac{6598}{20519}a^{12}+\frac{8535}{20519}a^{11}-\frac{3430}{20519}a^{10}-\frac{2426}{20519}a^{9}+\frac{2736}{20519}a^{8}-\frac{5572}{20519}a^{7}-\frac{6435}{20519}a^{6}+\frac{3778}{20519}a^{5}-\frac{9125}{20519}a^{4}-\frac{5303}{20519}a^{3}-\frac{5324}{20519}a^{2}+\frac{7880}{20519}a+\frac{3285}{20519}$, $\frac{1}{77\cdots 63}a^{15}-\frac{64219694587202}{77\cdots 63}a^{14}-\frac{21\cdots 64}{77\cdots 63}a^{13}-\frac{72\cdots 51}{77\cdots 63}a^{12}-\frac{16\cdots 59}{77\cdots 63}a^{11}+\frac{32\cdots 55}{77\cdots 63}a^{10}-\frac{43\cdots 91}{77\cdots 63}a^{9}-\frac{653669446699727}{89\cdots 01}a^{8}-\frac{21\cdots 80}{77\cdots 63}a^{7}+\frac{95\cdots 28}{77\cdots 63}a^{6}-\frac{14\cdots 66}{77\cdots 63}a^{5}-\frac{23\cdots 75}{77\cdots 63}a^{4}+\frac{31\cdots 08}{77\cdots 63}a^{3}-\frac{22\cdots 39}{10\cdots 53}a^{2}+\frac{16\cdots 05}{77\cdots 63}a-\frac{22\cdots 22}{77\cdots 63}$ 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:  $C_{2}$, 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:   $\frac{14823728735910}{26\cdots 67}a^{15}-\frac{955374050414374}{45\cdots 39}a^{14}-\frac{21\cdots 86}{45\cdots 39}a^{13}+\frac{71\cdots 94}{45\cdots 39}a^{12}+\frac{10\cdots 02}{45\cdots 39}a^{11}-\frac{18\cdots 06}{45\cdots 39}a^{10}-\frac{43\cdots 92}{45\cdots 39}a^{9}+\frac{15325873504787}{526080778454953}a^{8}+\frac{26\cdots 44}{45\cdots 39}a^{7}+\frac{42\cdots 14}{45\cdots 39}a^{6}-\frac{34\cdots 44}{45\cdots 39}a^{5}-\frac{19\cdots 57}{45\cdots 39}a^{4}-\frac{27\cdots 76}{45\cdots 39}a^{3}-\frac{19\cdots 02}{45\cdots 39}a^{2}-\frac{84\cdots 56}{45\cdots 39}a-\frac{50\cdots 43}{45\cdots 39}$, $\frac{9066839178560}{15\cdots 51}a^{15}-\frac{7815617982058}{15\cdots 51}a^{14}-\frac{66684261559068}{15\cdots 51}a^{13}-\frac{13481080027469}{15\cdots 51}a^{12}+\frac{161490267423636}{15\cdots 51}a^{11}+\frac{77413543420074}{15\cdots 51}a^{10}-\frac{812579894835296}{15\cdots 51}a^{9}-\frac{2710809714725}{1820348714377}a^{8}-\frac{721917614006508}{15\cdots 51}a^{7}+\frac{93\cdots 20}{15\cdots 51}a^{6}+\frac{23\cdots 08}{15\cdots 51}a^{5}+\frac{31\cdots 86}{15\cdots 51}a^{4}+\frac{27\cdots 32}{15\cdots 51}a^{3}+\frac{19\cdots 60}{15\cdots 51}a^{2}+\frac{74\cdots 32}{15\cdots 51}a+\frac{40\cdots 29}{15\cdots 51}$, $\frac{26\cdots 54}{77\cdots 63}a^{15}-\frac{509842498143137}{77\cdots 63}a^{14}-\frac{24\cdots 45}{77\cdots 63}a^{13}-\frac{17\cdots 92}{77\cdots 63}a^{12}+\frac{64\cdots 82}{77\cdots 63}a^{11}+\frac{47\cdots 20}{77\cdots 63}a^{10}-\frac{25\cdots 86}{77\cdots 63}a^{9}-\frac{10\cdots 22}{89\cdots 01}a^{8}-\frac{60\cdots 77}{77\cdots 63}a^{7}+\frac{31\cdots 18}{77\cdots 63}a^{6}+\frac{91\cdots 90}{77\cdots 63}a^{5}+\frac{12\cdots 91}{77\cdots 63}a^{4}+\frac{11\cdots 82}{77\cdots 63}a^{3}+\frac{71\cdots 16}{77\cdots 63}a^{2}+\frac{35\cdots 58}{77\cdots 63}a+\frac{16\cdots 73}{77\cdots 63}$, $\frac{15\cdots 38}{77\cdots 63}a^{15}+\frac{592394853707336}{10\cdots 53}a^{14}-\frac{21\cdots 09}{77\cdots 63}a^{13}-\frac{39\cdots 68}{77\cdots 63}a^{12}+\frac{61\cdots 54}{77\cdots 63}a^{11}+\frac{13\cdots 82}{77\cdots 63}a^{10}-\frac{20\cdots 02}{77\cdots 63}a^{9}-\frac{11\cdots 38}{89\cdots 01}a^{8}-\frac{10\cdots 49}{77\cdots 63}a^{7}+\frac{27\cdots 88}{77\cdots 63}a^{6}+\frac{10\cdots 16}{77\cdots 63}a^{5}+\frac{13\cdots 91}{77\cdots 63}a^{4}+\frac{76\cdots 74}{77\cdots 63}a^{3}+\frac{93\cdots 38}{77\cdots 63}a^{2}-\frac{11\cdots 70}{77\cdots 63}a-\frac{99\cdots 21}{77\cdots 63}$, $\frac{31\cdots 55}{77\cdots 63}a^{15}+\frac{10\cdots 76}{77\cdots 63}a^{14}-\frac{27\cdots 09}{77\cdots 63}a^{13}-\frac{37\cdots 57}{77\cdots 63}a^{12}+\frac{65\cdots 58}{77\cdots 63}a^{11}+\frac{12\cdots 65}{77\cdots 63}a^{10}-\frac{42\cdots 03}{10\cdots 53}a^{9}-\frac{14\cdots 51}{89\cdots 01}a^{8}-\frac{11\cdots 97}{77\cdots 63}a^{7}+\frac{35\cdots 63}{77\cdots 63}a^{6}+\frac{13\cdots 44}{77\cdots 63}a^{5}+\frac{19\cdots 12}{77\cdots 63}a^{4}+\frac{15\cdots 92}{77\cdots 63}a^{3}+\frac{95\cdots 09}{77\cdots 63}a^{2}+\frac{53\cdots 28}{77\cdots 63}a+\frac{18\cdots 39}{77\cdots 63}$, $\frac{63597246274404}{45\cdots 39}a^{15}-\frac{344757946047226}{45\cdots 39}a^{14}-\frac{329733203108437}{45\cdots 39}a^{13}-\frac{58474681236984}{45\cdots 39}a^{12}+\frac{62\cdots 41}{45\cdots 39}a^{11}+\frac{10\cdots 31}{45\cdots 39}a^{10}-\frac{14\cdots 67}{26\cdots 67}a^{9}-\frac{34778102006195}{526080778454953}a^{8}+\frac{11\cdots 81}{45\cdots 39}a^{7}+\frac{30\cdots 97}{45\cdots 39}a^{6}+\frac{25\cdots 92}{45\cdots 39}a^{5}-\frac{99\cdots 97}{45\cdots 39}a^{4}-\frac{34\cdots 64}{45\cdots 39}a^{3}-\frac{44\cdots 97}{45\cdots 39}a^{2}-\frac{35\cdots 34}{45\cdots 39}a-\frac{16\cdots 57}{45\cdots 39}$, $\frac{26\cdots 47}{77\cdots 63}a^{15}+\frac{58\cdots 85}{77\cdots 63}a^{14}-\frac{24\cdots 72}{77\cdots 63}a^{13}-\frac{25\cdots 65}{77\cdots 63}a^{12}+\frac{64\cdots 82}{77\cdots 63}a^{11}+\frac{94\cdots 00}{77\cdots 63}a^{10}-\frac{27\cdots 00}{77\cdots 63}a^{9}-\frac{11\cdots 72}{89\cdots 01}a^{8}-\frac{69\cdots 29}{77\cdots 63}a^{7}+\frac{33\cdots 23}{77\cdots 63}a^{6}+\frac{10\cdots 34}{77\cdots 63}a^{5}+\frac{13\cdots 55}{77\cdots 63}a^{4}+\frac{88\cdots 92}{77\cdots 63}a^{3}+\frac{31\cdots 94}{77\cdots 63}a^{2}+\frac{70\cdots 60}{77\cdots 63}a-\frac{68\cdots 83}{77\cdots 63}$, $\frac{19\cdots 46}{77\cdots 63}a^{15}+\frac{30\cdots 20}{77\cdots 63}a^{14}-\frac{25\cdots 16}{77\cdots 63}a^{13}-\frac{30\cdots 99}{77\cdots 63}a^{12}-\frac{27\cdots 63}{77\cdots 63}a^{11}+\frac{79\cdots 50}{77\cdots 63}a^{10}+\frac{24\cdots 23}{77\cdots 63}a^{9}-\frac{45\cdots 27}{89\cdots 01}a^{8}-\frac{99\cdots 84}{77\cdots 63}a^{7}-\frac{59\cdots 13}{77\cdots 63}a^{6}+\frac{46\cdots 02}{77\cdots 63}a^{5}+\frac{10\cdots 26}{77\cdots 63}a^{4}+\frac{11\cdots 68}{77\cdots 63}a^{3}+\frac{74\cdots 75}{77\cdots 63}a^{2}+\frac{42\cdots 50}{77\cdots 63}a+\frac{20\cdots 43}{77\cdots 63}$, $\frac{54\cdots 43}{77\cdots 63}a^{15}-\frac{73\cdots 92}{77\cdots 63}a^{14}-\frac{32\cdots 69}{77\cdots 63}a^{13}-\frac{62\cdots 10}{77\cdots 63}a^{12}+\frac{89\cdots 19}{77\cdots 63}a^{11}+\frac{26\cdots 61}{77\cdots 63}a^{10}-\frac{44\cdots 64}{77\cdots 63}a^{9}-\frac{14\cdots 12}{89\cdots 01}a^{8}-\frac{16\cdots 12}{77\cdots 63}a^{7}+\frac{51\cdots 00}{77\cdots 63}a^{6}+\frac{12\cdots 28}{77\cdots 63}a^{5}+\frac{16\cdots 53}{77\cdots 63}a^{4}+\frac{13\cdots 08}{77\cdots 63}a^{3}+\frac{76\cdots 67}{77\cdots 63}a^{2}+\frac{26\cdots 24}{77\cdots 63}a+\frac{94\cdots 37}{77\cdots 63}$ 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:  \( 283940.790469 \) (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:  \( 3 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 283940.790469 \cdot 1}{2\cdot\sqrt{153177439332441840943104}}\cr\approx \mathstrut & 0.35710800178 \end{aligned}\] (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^14 - 8*x^13 + 16*x^12 + 24*x^11 - 80*x^10 - 336*x^9 - 314*x^8 + 944*x^7 + 3520*x^6 + 5776*x^5 + 6164*x^4 + 4864*x^3 + 2728*x^2 + 1264*x + 478) 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^14 - 8*x^13 + 16*x^12 + 24*x^11 - 80*x^10 - 336*x^9 - 314*x^8 + 944*x^7 + 3520*x^6 + 5776*x^5 + 6164*x^4 + 4864*x^3 + 2728*x^2 + 1264*x + 478, 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^14 - 8*x^13 + 16*x^12 + 24*x^11 - 80*x^10 - 336*x^9 - 314*x^8 + 944*x^7 + 3520*x^6 + 5776*x^5 + 6164*x^4 + 4864*x^3 + 2728*x^2 + 1264*x + 478); 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 - 8*x^14 - 8*x^13 + 16*x^12 + 24*x^11 - 80*x^10 - 336*x^9 - 314*x^8 + 944*x^7 + 3520*x^6 + 5776*x^5 + 6164*x^4 + 4864*x^3 + 2728*x^2 + 1264*x + 478); 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

$C_2^4:Q_8$ (as 16T333):

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 128
The 26 conjugacy class representatives for $C_2^4:Q_8$
Character table for $C_2^4:Q_8$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\zeta_{24})^+\), 8.2.24461180928.9, 8.8.12230590464.1, 8.2.2717908992.2

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 16 siblings: 16.8.153177439332441840943104.4, 16.4.153177439332441840943104.15, 16.4.153177439332441840943104.6, 16.4.38294359833110460235776.8, 16.8.38294359833110460235776.8, 16.4.38294359833110460235776.35, 16.0.38294359833110460235776.18, 16.0.153177439332441840943104.107, 16.0.153177439332441840943104.142, some data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.12.153177439332441840943104.1

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 3.4.4.12a1.3$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 19$$4$$4$$12$$C_4:C_4$$$[\ ]_{4}^{4}$$

Spectrum of ring of integers

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