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Properties

Label 16.8.821...649.2
Degree $16$
Signature $[8, 4]$
Discriminant $8.216\times 10^{23}$
Root discriminant \(31.24\)
Ramified primes $17,47$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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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 - 3*x^15 + x^14 - 40*x^13 - 136*x^12 - 203*x^11 - 694*x^10 + 178*x^9 + 427*x^8 + 2206*x^7 + 2572*x^6 + 2059*x^5 + 1224*x^4 + 116*x^3 - 115*x^2 - 11*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 3*y^15 + y^14 - 40*y^13 - 136*y^12 - 203*y^11 - 694*y^10 + 178*y^9 + 427*y^8 + 2206*y^7 + 2572*y^6 + 2059*y^5 + 1224*y^4 + 116*y^3 - 115*y^2 - 11*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + x^14 - 40*x^13 - 136*x^12 - 203*x^11 - 694*x^10 + 178*x^9 + 427*x^8 + 2206*x^7 + 2572*x^6 + 2059*x^5 + 1224*x^4 + 116*x^3 - 115*x^2 - 11*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + x^14 - 40*x^13 - 136*x^12 - 203*x^11 - 694*x^10 + 178*x^9 + 427*x^8 + 2206*x^7 + 2572*x^6 + 2059*x^5 + 1224*x^4 + 116*x^3 - 115*x^2 - 11*x + 1)
 

\( x^{16} - 3 x^{15} + x^{14} - 40 x^{13} - 136 x^{12} - 203 x^{11} - 694 x^{10} + 178 x^{9} + 427 x^{8} + \cdots + 1 \) 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:   \(821630081083204084623649\) \(\medspace = 17^{14}\cdot 47^{4}\) 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:  \(31.24\)
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:  $17^{7/8}47^{1/2}\approx 81.7884043055344$
Ramified primes:   \(17\), \(47\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2\times C_4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{83\cdots 98}a^{15}-\frac{10\cdots 49}{83\cdots 98}a^{14}+\frac{76\cdots 53}{41\cdots 49}a^{13}+\frac{44\cdots 58}{41\cdots 49}a^{12}+\frac{11\cdots 43}{83\cdots 98}a^{11}+\frac{11\cdots 21}{41\cdots 49}a^{10}-\frac{19\cdots 57}{41\cdots 49}a^{9}-\frac{44\cdots 09}{83\cdots 98}a^{8}+\frac{15\cdots 42}{41\cdots 49}a^{7}-\frac{94\cdots 10}{41\cdots 49}a^{6}+\frac{23\cdots 35}{83\cdots 98}a^{5}-\frac{15\cdots 43}{41\cdots 49}a^{4}+\frac{13\cdots 65}{41\cdots 49}a^{3}-\frac{10\cdots 45}{83\cdots 98}a^{2}+\frac{18\cdots 67}{41\cdots 49}a+\frac{21\cdots 51}{83\cdots 98}$ 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:  $C_{2}$, 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:  $11$
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:   $\frac{55\cdots 81}{83\cdots 98}a^{15}+\frac{32\cdots 15}{83\cdots 98}a^{14}-\frac{16\cdots 97}{83\cdots 98}a^{13}-\frac{85\cdots 91}{83\cdots 98}a^{12}-\frac{27\cdots 33}{83\cdots 98}a^{11}-\frac{67\cdots 27}{83\cdots 98}a^{10}-\frac{11\cdots 63}{83\cdots 98}a^{9}-\frac{29\cdots 27}{83\cdots 98}a^{8}+\frac{25\cdots 83}{83\cdots 98}a^{7}+\frac{19\cdots 19}{83\cdots 98}a^{6}+\frac{11\cdots 95}{83\cdots 98}a^{5}+\frac{85\cdots 51}{83\cdots 98}a^{4}+\frac{77\cdots 25}{83\cdots 98}a^{3}+\frac{32\cdots 39}{83\cdots 98}a^{2}-\frac{47\cdots 81}{83\cdots 98}a-\frac{39\cdots 26}{41\cdots 49}$, $\frac{46\cdots 95}{83\cdots 98}a^{15}-\frac{13\cdots 63}{83\cdots 98}a^{14}+\frac{14\cdots 12}{41\cdots 49}a^{13}-\frac{91\cdots 00}{41\cdots 49}a^{12}-\frac{65\cdots 85}{83\cdots 98}a^{11}-\frac{49\cdots 29}{41\cdots 49}a^{10}-\frac{16\cdots 64}{41\cdots 49}a^{9}+\frac{61\cdots 13}{83\cdots 98}a^{8}+\frac{11\cdots 39}{41\cdots 49}a^{7}+\frac{51\cdots 33}{41\cdots 49}a^{6}+\frac{12\cdots 93}{83\cdots 98}a^{5}+\frac{46\cdots 75}{41\cdots 49}a^{4}+\frac{28\cdots 50}{41\cdots 49}a^{3}-\frac{64\cdots 25}{83\cdots 98}a^{2}-\frac{45\cdots 23}{41\cdots 49}a-\frac{48\cdots 87}{83\cdots 98}$, $\frac{71\cdots 43}{83\cdots 98}a^{15}-\frac{13\cdots 59}{41\cdots 49}a^{14}+\frac{14\cdots 81}{41\cdots 49}a^{13}-\frac{15\cdots 83}{41\cdots 49}a^{12}-\frac{35\cdots 13}{41\cdots 49}a^{11}-\frac{38\cdots 58}{41\cdots 49}a^{10}-\frac{20\cdots 15}{41\cdots 49}a^{9}+\frac{25\cdots 96}{41\cdots 49}a^{8}-\frac{23\cdots 23}{41\cdots 49}a^{7}+\frac{76\cdots 59}{41\cdots 49}a^{6}+\frac{27\cdots 52}{41\cdots 49}a^{5}+\frac{33\cdots 68}{41\cdots 49}a^{4}+\frac{59\cdots 42}{41\cdots 49}a^{3}-\frac{11\cdots 84}{41\cdots 49}a^{2}+\frac{10\cdots 80}{41\cdots 49}a+\frac{13\cdots 17}{83\cdots 98}$, $\frac{45\cdots 31}{83\cdots 98}a^{15}-\frac{91\cdots 95}{41\cdots 49}a^{14}+\frac{10\cdots 92}{41\cdots 49}a^{13}-\frac{96\cdots 45}{41\cdots 49}a^{12}-\frac{21\cdots 27}{41\cdots 49}a^{11}-\frac{17\cdots 35}{41\cdots 49}a^{10}-\frac{11\cdots 61}{41\cdots 49}a^{9}+\frac{19\cdots 44}{41\cdots 49}a^{8}+\frac{43\cdots 24}{41\cdots 49}a^{7}+\frac{44\cdots 47}{41\cdots 49}a^{6}+\frac{78\cdots 50}{41\cdots 49}a^{5}+\frac{38\cdots 41}{41\cdots 49}a^{4}-\frac{10\cdots 63}{41\cdots 49}a^{3}-\frac{16\cdots 42}{41\cdots 49}a^{2}-\frac{21\cdots 57}{41\cdots 49}a+\frac{24\cdots 17}{83\cdots 98}$, $\frac{36\cdots 02}{41\cdots 49}a^{15}-\frac{13\cdots 96}{41\cdots 49}a^{14}+\frac{24\cdots 25}{83\cdots 98}a^{13}-\frac{30\cdots 53}{83\cdots 98}a^{12}-\frac{39\cdots 99}{41\cdots 49}a^{11}-\frac{90\cdots 79}{83\cdots 98}a^{10}-\frac{43\cdots 51}{83\cdots 98}a^{9}+\frac{22\cdots 69}{41\cdots 49}a^{8}+\frac{77\cdots 03}{83\cdots 98}a^{7}+\frac{15\cdots 33}{83\cdots 98}a^{6}+\frac{39\cdots 22}{41\cdots 49}a^{5}+\frac{74\cdots 87}{83\cdots 98}a^{4}+\frac{20\cdots 21}{83\cdots 98}a^{3}-\frac{90\cdots 05}{41\cdots 49}a^{2}-\frac{32\cdots 89}{83\cdots 98}a+\frac{78\cdots 39}{83\cdots 98}$, $\frac{19\cdots 45}{83\cdots 98}a^{15}-\frac{25\cdots 29}{83\cdots 98}a^{14}-\frac{84\cdots 79}{83\cdots 98}a^{13}-\frac{34\cdots 13}{41\cdots 49}a^{12}-\frac{38\cdots 63}{83\cdots 98}a^{11}-\frac{76\cdots 95}{83\cdots 98}a^{10}-\frac{91\cdots 16}{41\cdots 49}a^{9}-\frac{16\cdots 19}{83\cdots 98}a^{8}+\frac{21\cdots 59}{83\cdots 98}a^{7}+\frac{25\cdots 03}{41\cdots 49}a^{6}+\frac{11\cdots 47}{83\cdots 98}a^{5}+\frac{88\cdots 89}{83\cdots 98}a^{4}+\frac{34\cdots 79}{41\cdots 49}a^{3}+\frac{17\cdots 03}{83\cdots 98}a^{2}-\frac{66\cdots 19}{83\cdots 98}a-\frac{55\cdots 61}{83\cdots 98}$, $\frac{16\cdots 25}{83\cdots 98}a^{15}-\frac{27\cdots 09}{41\cdots 49}a^{14}+\frac{33\cdots 81}{83\cdots 98}a^{13}-\frac{32\cdots 11}{41\cdots 49}a^{12}-\frac{96\cdots 51}{41\cdots 49}a^{11}-\frac{22\cdots 31}{83\cdots 98}a^{10}-\frac{47\cdots 95}{41\cdots 49}a^{9}+\frac{40\cdots 27}{41\cdots 49}a^{8}+\frac{77\cdots 25}{83\cdots 98}a^{7}+\frac{15\cdots 51}{41\cdots 49}a^{6}+\frac{12\cdots 96}{41\cdots 49}a^{5}+\frac{91\cdots 89}{83\cdots 98}a^{4}+\frac{42\cdots 35}{41\cdots 49}a^{3}-\frac{46\cdots 39}{41\cdots 49}a^{2}-\frac{43\cdots 87}{83\cdots 98}a+\frac{47\cdots 11}{83\cdots 98}$, $\frac{54\cdots 59}{83\cdots 98}a^{15}-\frac{88\cdots 73}{83\cdots 98}a^{14}-\frac{21\cdots 33}{83\cdots 98}a^{13}-\frac{19\cdots 35}{83\cdots 98}a^{12}-\frac{10\cdots 77}{83\cdots 98}a^{11}-\frac{18\cdots 75}{83\cdots 98}a^{10}-\frac{46\cdots 25}{83\cdots 98}a^{9}-\frac{33\cdots 49}{83\cdots 98}a^{8}+\frac{67\cdots 31}{83\cdots 98}a^{7}+\frac{12\cdots 57}{83\cdots 98}a^{6}+\frac{29\cdots 81}{83\cdots 98}a^{5}+\frac{19\cdots 25}{83\cdots 98}a^{4}+\frac{14\cdots 03}{83\cdots 98}a^{3}+\frac{27\cdots 65}{83\cdots 98}a^{2}-\frac{24\cdots 95}{83\cdots 98}a-\frac{46\cdots 19}{41\cdots 49}$, $\frac{11\cdots 72}{41\cdots 49}a^{15}-\frac{39\cdots 73}{41\cdots 49}a^{14}+\frac{50\cdots 69}{83\cdots 98}a^{13}-\frac{48\cdots 97}{41\cdots 49}a^{12}-\frac{14\cdots 57}{41\cdots 49}a^{11}-\frac{39\cdots 49}{83\cdots 98}a^{10}-\frac{77\cdots 00}{41\cdots 49}a^{9}+\frac{44\cdots 84}{41\cdots 49}a^{8}+\frac{63\cdots 91}{83\cdots 98}a^{7}+\frac{25\cdots 92}{41\cdots 49}a^{6}+\frac{22\cdots 46}{41\cdots 49}a^{5}+\frac{38\cdots 39}{83\cdots 98}a^{4}+\frac{10\cdots 69}{41\cdots 49}a^{3}-\frac{54\cdots 98}{41\cdots 49}a^{2}-\frac{56\cdots 37}{83\cdots 98}a+\frac{43\cdots 38}{41\cdots 49}$, $\frac{44\cdots 25}{41\cdots 49}a^{15}-\frac{28\cdots 81}{83\cdots 98}a^{14}+\frac{20\cdots 85}{83\cdots 98}a^{13}-\frac{37\cdots 39}{83\cdots 98}a^{12}-\frac{10\cdots 71}{83\cdots 98}a^{11}-\frac{16\cdots 17}{83\cdots 98}a^{10}-\frac{61\cdots 19}{83\cdots 98}a^{9}+\frac{26\cdots 83}{83\cdots 98}a^{8}+\frac{63\cdots 73}{83\cdots 98}a^{7}+\frac{20\cdots 19}{83\cdots 98}a^{6}+\frac{17\cdots 49}{83\cdots 98}a^{5}+\frac{21\cdots 07}{83\cdots 98}a^{4}+\frac{12\cdots 69}{83\cdots 98}a^{3}+\frac{47\cdots 65}{83\cdots 98}a^{2}+\frac{16\cdots 55}{83\cdots 98}a-\frac{12\cdots 05}{83\cdots 98}$, $\frac{26\cdots 34}{41\cdots 49}a^{15}-\frac{17\cdots 49}{83\cdots 98}a^{14}+\frac{86\cdots 87}{83\cdots 98}a^{13}-\frac{10\cdots 80}{41\cdots 49}a^{12}-\frac{66\cdots 47}{83\cdots 98}a^{11}-\frac{86\cdots 81}{83\cdots 98}a^{10}-\frac{16\cdots 69}{41\cdots 49}a^{9}+\frac{19\cdots 97}{83\cdots 98}a^{8}+\frac{23\cdots 95}{83\cdots 98}a^{7}+\frac{54\cdots 65}{41\cdots 49}a^{6}+\frac{10\cdots 51}{83\cdots 98}a^{5}+\frac{63\cdots 47}{83\cdots 98}a^{4}+\frac{11\cdots 50}{41\cdots 49}a^{3}-\frac{22\cdots 77}{83\cdots 98}a^{2}-\frac{16\cdots 51}{83\cdots 98}a-\frac{10\cdots 53}{41\cdots 49}$ 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:  \( 864723.662197 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 864723.662197 \cdot 1}{2\cdot\sqrt{821630081083204084623649}}\cr\approx \mathstrut & 0.190313072466 \end{aligned}\] (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 - 3*x^15 + x^14 - 40*x^13 - 136*x^12 - 203*x^11 - 694*x^10 + 178*x^9 + 427*x^8 + 2206*x^7 + 2572*x^6 + 2059*x^5 + 1224*x^4 + 116*x^3 - 115*x^2 - 11*x + 1) 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 - 3*x^15 + x^14 - 40*x^13 - 136*x^12 - 203*x^11 - 694*x^10 + 178*x^9 + 427*x^8 + 2206*x^7 + 2572*x^6 + 2059*x^5 + 1224*x^4 + 116*x^3 - 115*x^2 - 11*x + 1, 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 - 3*x^15 + x^14 - 40*x^13 - 136*x^12 - 203*x^11 - 694*x^10 + 178*x^9 + 427*x^8 + 2206*x^7 + 2572*x^6 + 2059*x^5 + 1224*x^4 + 116*x^3 - 115*x^2 - 11*x + 1); 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 - 3*x^15 + x^14 - 40*x^13 - 136*x^12 - 203*x^11 - 694*x^10 + 178*x^9 + 427*x^8 + 2206*x^7 + 2572*x^6 + 2059*x^5 + 1224*x^4 + 116*x^3 - 115*x^2 - 11*x + 1); 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

$C_2^2:C_8$ (as 16T24):

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 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.2.230911.1, 4.2.13583.1, 8.4.906438128657.1, \(\Q(\zeta_{17})^+\), 8.4.53319889921.1

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

Galois closure: deg 32
Degree 16 sibling: 16.0.4009292695690170390860412175969.16
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 ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ R ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ R ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(17\) Copy content Toggle raw display 17.1.8.7a1.1$x^{8} + 17$$8$$1$$7$$C_8$$$[\ ]_{8}$$
17.1.8.7a1.1$x^{8} + 17$$8$$1$$7$$C_8$$$[\ ]_{8}$$
\(47\) Copy content Toggle raw display 47.2.1.0a1.1$x^{2} + 45 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
47.2.1.0a1.1$x^{2} + 45 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
47.2.1.0a1.1$x^{2} + 45 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
47.2.1.0a1.1$x^{2} + 45 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
47.2.2.2a1.2$x^{4} + 90 x^{3} + 2035 x^{2} + 450 x + 72$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
47.2.2.2a1.2$x^{4} + 90 x^{3} + 2035 x^{2} + 450 x + 72$$2$$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)