↑

Properties

Label 16.12.860...064.1
Degree $16$
Signature $[12, 2]$
Discriminant $8.605\times 10^{26}$
Root discriminant \(48.24\)
Ramified primes $2,43,2777$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^7.C_2\wr S_4$ (as 16T1850)

Related objects

Downloads

Learn more

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 - 2*x^15 - 35*x^14 + 72*x^13 + 448*x^12 - 984*x^11 - 2681*x^10 + 6698*x^9 + 8782*x^8 - 26594*x^7 - 20622*x^6 + 62972*x^5 + 44263*x^4 - 74042*x^3 - 55712*x^2 + 32314*x + 25763)
 
Copy content gp:K = bnfinit(y^16 - 2*y^15 - 35*y^14 + 72*y^13 + 448*y^12 - 984*y^11 - 2681*y^10 + 6698*y^9 + 8782*y^8 - 26594*y^7 - 20622*y^6 + 62972*y^5 + 44263*y^4 - 74042*y^3 - 55712*y^2 + 32314*y + 25763, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 35*x^14 + 72*x^13 + 448*x^12 - 984*x^11 - 2681*x^10 + 6698*x^9 + 8782*x^8 - 26594*x^7 - 20622*x^6 + 62972*x^5 + 44263*x^4 - 74042*x^3 - 55712*x^2 + 32314*x + 25763);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 - 35*x^14 + 72*x^13 + 448*x^12 - 984*x^11 - 2681*x^10 + 6698*x^9 + 8782*x^8 - 26594*x^7 - 20622*x^6 + 62972*x^5 + 44263*x^4 - 74042*x^3 - 55712*x^2 + 32314*x + 25763)
 

\( x^{16} - 2 x^{15} - 35 x^{14} + 72 x^{13} + 448 x^{12} - 984 x^{11} - 2681 x^{10} + 6698 x^{9} + \cdots + 25763 \) 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:  $[12, 2]$
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:   \(860527385904312057534808064\) \(\medspace = 2^{16}\cdot 43^{3}\cdot 2777^{5}\) 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:  \(48.24\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{63/32}43^{3/4}2777^{1/2}\approx 3463.714482798763$
Ramified primes:   \(2\), \(43\), \(2777\) 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:  \(\Q(\sqrt{119411}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{46}a^{14}+\frac{5}{46}a^{13}-\frac{5}{46}a^{12}+\frac{6}{23}a^{11}+\frac{5}{46}a^{10}+\frac{3}{46}a^{9}-\frac{17}{46}a^{8}-\frac{7}{23}a^{7}-\frac{17}{46}a^{6}-\frac{9}{46}a^{5}+\frac{4}{23}a^{4}+\frac{7}{46}a^{3}+\frac{10}{23}a^{2}-\frac{15}{46}a+\frac{19}{46}$, $\frac{1}{36\cdots 42}a^{15}-\frac{92\cdots 99}{15\cdots 54}a^{14}+\frac{37\cdots 79}{36\cdots 42}a^{13}-\frac{53\cdots 23}{18\cdots 21}a^{12}-\frac{11\cdots 63}{36\cdots 42}a^{11}-\frac{12\cdots 33}{36\cdots 42}a^{10}+\frac{16\cdots 41}{36\cdots 42}a^{9}+\frac{48\cdots 71}{18\cdots 21}a^{8}-\frac{11\cdots 53}{36\cdots 42}a^{7}-\frac{34\cdots 77}{36\cdots 42}a^{6}+\frac{84\cdots 41}{18\cdots 21}a^{5}+\frac{17\cdots 03}{36\cdots 42}a^{4}+\frac{11\cdots 40}{18\cdots 21}a^{3}+\frac{63\cdots 79}{15\cdots 54}a^{2}-\frac{12\cdots 53}{36\cdots 42}a+\frac{29\cdots 52}{18\cdots 21}$ 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:  $C_{2}\times C_{2}$, which has order $4$ (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:  $13$
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:   $\frac{80\cdots 21}{36\cdots 42}a^{15}-\frac{77\cdots 10}{78\cdots 27}a^{14}-\frac{11\cdots 68}{18\cdots 21}a^{13}+\frac{12\cdots 97}{36\cdots 42}a^{12}+\frac{21\cdots 57}{36\cdots 42}a^{11}-\frac{75\cdots 62}{18\cdots 21}a^{10}-\frac{17\cdots 57}{18\cdots 21}a^{9}+\frac{90\cdots 31}{36\cdots 42}a^{8}-\frac{34\cdots 93}{36\cdots 42}a^{7}-\frac{16\cdots 75}{18\cdots 21}a^{6}+\frac{19\cdots 35}{36\cdots 42}a^{5}+\frac{77\cdots 41}{36\cdots 42}a^{4}-\frac{44\cdots 73}{36\cdots 42}a^{3}-\frac{51\cdots 17}{15\cdots 54}a^{2}+\frac{15\cdots 71}{18\cdots 21}a+\frac{69\cdots 33}{36\cdots 42}$, $\frac{11\cdots 61}{36\cdots 42}a^{15}-\frac{50\cdots 65}{78\cdots 27}a^{14}-\frac{19\cdots 34}{18\cdots 21}a^{13}+\frac{87\cdots 47}{36\cdots 42}a^{12}+\frac{51\cdots 17}{36\cdots 42}a^{11}-\frac{62\cdots 66}{18\cdots 21}a^{10}-\frac{14\cdots 25}{18\cdots 21}a^{9}+\frac{89\cdots 19}{36\cdots 42}a^{8}+\frac{77\cdots 15}{36\cdots 42}a^{7}-\frac{17\cdots 33}{18\cdots 21}a^{6}-\frac{10\cdots 73}{36\cdots 42}a^{5}+\frac{80\cdots 65}{36\cdots 42}a^{4}+\frac{13\cdots 03}{36\cdots 42}a^{3}-\frac{39\cdots 03}{15\cdots 54}a^{2}-\frac{37\cdots 09}{18\cdots 21}a+\frac{43\cdots 67}{36\cdots 42}$, $\frac{23\cdots 15}{18\cdots 21}a^{15}-\frac{25\cdots 68}{78\cdots 27}a^{14}-\frac{76\cdots 68}{18\cdots 21}a^{13}+\frac{20\cdots 23}{18\cdots 21}a^{12}+\frac{86\cdots 25}{18\cdots 21}a^{11}-\frac{26\cdots 55}{18\cdots 21}a^{10}-\frac{39\cdots 86}{18\cdots 21}a^{9}+\frac{16\cdots 72}{18\cdots 21}a^{8}+\frac{65\cdots 17}{18\cdots 21}a^{7}-\frac{54\cdots 81}{18\cdots 21}a^{6}-\frac{28\cdots 33}{18\cdots 21}a^{5}+\frac{10\cdots 12}{18\cdots 21}a^{4}+\frac{11\cdots 45}{18\cdots 21}a^{3}-\frac{41\cdots 84}{78\cdots 27}a^{2}-\frac{13\cdots 20}{18\cdots 21}a+\frac{26\cdots 93}{18\cdots 21}$, $\frac{15\cdots 01}{18\cdots 21}a^{15}-\frac{18\cdots 69}{78\cdots 27}a^{14}-\frac{49\cdots 04}{18\cdots 21}a^{13}+\frac{14\cdots 06}{18\cdots 21}a^{12}+\frac{54\cdots 57}{18\cdots 21}a^{11}-\frac{18\cdots 60}{18\cdots 21}a^{10}-\frac{23\cdots 54}{18\cdots 21}a^{9}+\frac{10\cdots 50}{18\cdots 21}a^{8}+\frac{31\cdots 83}{18\cdots 21}a^{7}-\frac{36\cdots 21}{18\cdots 21}a^{6}+\frac{88\cdots 01}{18\cdots 21}a^{5}+\frac{72\cdots 65}{18\cdots 21}a^{4}+\frac{43\cdots 97}{18\cdots 21}a^{3}-\frac{29\cdots 91}{78\cdots 27}a^{2}-\frac{97\cdots 14}{18\cdots 21}a+\frac{19\cdots 62}{18\cdots 21}$, $\frac{37\cdots 41}{36\cdots 42}a^{15}-\frac{26\cdots 80}{78\cdots 27}a^{14}-\frac{57\cdots 86}{18\cdots 21}a^{13}+\frac{41\cdots 63}{36\cdots 42}a^{12}+\frac{11\cdots 49}{36\cdots 42}a^{11}-\frac{25\cdots 87}{18\cdots 21}a^{10}-\frac{16\cdots 55}{18\cdots 21}a^{9}+\frac{28\cdots 15}{36\cdots 42}a^{8}-\frac{45\cdots 29}{36\cdots 42}a^{7}-\frac{44\cdots 79}{18\cdots 21}a^{6}+\frac{37\cdots 99}{36\cdots 42}a^{5}+\frac{17\cdots 57}{36\cdots 42}a^{4}-\frac{58\cdots 89}{36\cdots 42}a^{3}-\frac{79\cdots 99}{15\cdots 54}a^{2}+\frac{17\cdots 81}{18\cdots 21}a+\frac{72\cdots 45}{36\cdots 42}$, $\frac{42\cdots 63}{36\cdots 42}a^{15}-\frac{23\cdots 34}{78\cdots 27}a^{14}-\frac{69\cdots 38}{18\cdots 21}a^{13}+\frac{38\cdots 59}{36\cdots 42}a^{12}+\frac{16\cdots 31}{36\cdots 42}a^{11}-\frac{24\cdots 26}{18\cdots 21}a^{10}-\frac{37\cdots 79}{18\cdots 21}a^{9}+\frac{30\cdots 19}{36\cdots 42}a^{8}+\frac{14\cdots 81}{36\cdots 42}a^{7}-\frac{54\cdots 54}{18\cdots 21}a^{6}-\frac{84\cdots 71}{36\cdots 42}a^{5}+\frac{22\cdots 95}{36\cdots 42}a^{4}+\frac{22\cdots 97}{36\cdots 42}a^{3}-\frac{98\cdots 85}{15\cdots 54}a^{2}-\frac{13\cdots 23}{18\cdots 21}a+\frac{79\cdots 49}{36\cdots 42}$, $\frac{35\cdots 18}{78\cdots 27}a^{15}-\frac{72\cdots 99}{78\cdots 27}a^{14}-\frac{11\cdots 68}{78\cdots 27}a^{13}+\frac{25\cdots 79}{78\cdots 27}a^{12}+\frac{13\cdots 16}{78\cdots 27}a^{11}-\frac{34\cdots 65}{78\cdots 27}a^{10}-\frac{69\cdots 84}{78\cdots 27}a^{9}+\frac{22\cdots 14}{78\cdots 27}a^{8}+\frac{14\cdots 58}{78\cdots 27}a^{7}-\frac{77\cdots 20}{78\cdots 27}a^{6}-\frac{16\cdots 58}{78\cdots 27}a^{5}+\frac{14\cdots 89}{78\cdots 27}a^{4}+\frac{29\cdots 76}{78\cdots 27}a^{3}-\frac{12\cdots 93}{78\cdots 27}a^{2}-\frac{16\cdots 22}{78\cdots 27}a+\frac{39\cdots 24}{78\cdots 27}$, $\frac{24\cdots 53}{36\cdots 42}a^{15}-\frac{16\cdots 36}{78\cdots 27}a^{14}-\frac{38\cdots 88}{18\cdots 21}a^{13}+\frac{26\cdots 79}{36\cdots 42}a^{12}+\frac{77\cdots 73}{36\cdots 42}a^{11}-\frac{15\cdots 96}{18\cdots 21}a^{10}-\frac{13\cdots 91}{18\cdots 21}a^{9}+\frac{17\cdots 59}{36\cdots 42}a^{8}-\frac{14\cdots 89}{36\cdots 42}a^{7}-\frac{27\cdots 77}{18\cdots 21}a^{6}+\frac{14\cdots 59}{36\cdots 42}a^{5}+\frac{10\cdots 87}{36\cdots 42}a^{4}-\frac{15\cdots 09}{36\cdots 42}a^{3}-\frac{43\cdots 99}{15\cdots 54}a^{2}-\frac{12\cdots 14}{18\cdots 21}a+\frac{28\cdots 75}{36\cdots 42}$, $\frac{32\cdots 04}{18\cdots 21}a^{15}-\frac{27\cdots 25}{36\cdots 42}a^{14}-\frac{18\cdots 35}{36\cdots 42}a^{13}+\frac{91\cdots 79}{36\cdots 42}a^{12}+\frac{68\cdots 72}{18\cdots 21}a^{11}-\frac{10\cdots 63}{36\cdots 42}a^{10}+\frac{84\cdots 91}{36\cdots 42}a^{9}+\frac{54\cdots 95}{36\cdots 42}a^{8}-\frac{20\cdots 93}{18\cdots 21}a^{7}-\frac{15\cdots 33}{36\cdots 42}a^{6}+\frac{15\cdots 85}{36\cdots 42}a^{5}+\frac{13\cdots 36}{18\cdots 21}a^{4}-\frac{23\cdots 71}{36\cdots 42}a^{3}-\frac{15\cdots 59}{18\cdots 21}a^{2}+\frac{12\cdots 01}{36\cdots 42}a+\frac{13\cdots 93}{36\cdots 42}$, $\frac{39\cdots 06}{18\cdots 21}a^{15}-\frac{25\cdots 15}{36\cdots 42}a^{14}-\frac{24\cdots 39}{36\cdots 42}a^{13}+\frac{86\cdots 25}{36\cdots 42}a^{12}+\frac{13\cdots 37}{18\cdots 21}a^{11}-\frac{10\cdots 97}{36\cdots 42}a^{10}-\frac{94\cdots 49}{36\cdots 42}a^{9}+\frac{65\cdots 25}{36\cdots 42}a^{8}-\frac{95\cdots 65}{18\cdots 21}a^{7}-\frac{21\cdots 73}{36\cdots 42}a^{6}+\frac{77\cdots 17}{36\cdots 42}a^{5}+\frac{22\cdots 82}{18\cdots 21}a^{4}-\frac{13\cdots 81}{36\cdots 42}a^{3}-\frac{24\cdots 47}{18\cdots 21}a^{2}+\frac{77\cdots 11}{36\cdots 42}a+\frac{20\cdots 37}{36\cdots 42}$, $\frac{24\cdots 05}{36\cdots 42}a^{15}-\frac{25\cdots 35}{36\cdots 42}a^{14}-\frac{86\cdots 87}{36\cdots 42}a^{13}+\frac{45\cdots 92}{18\cdots 21}a^{12}+\frac{11\cdots 71}{36\cdots 42}a^{11}-\frac{12\cdots 47}{36\cdots 42}a^{10}-\frac{68\cdots 93}{36\cdots 42}a^{9}+\frac{40\cdots 47}{18\cdots 21}a^{8}+\frac{24\cdots 67}{36\cdots 42}a^{7}-\frac{30\cdots 27}{36\cdots 42}a^{6}-\frac{34\cdots 58}{18\cdots 21}a^{5}+\frac{51\cdots 85}{36\cdots 42}a^{4}+\frac{67\cdots 87}{18\cdots 21}a^{3}+\frac{12\cdots 93}{36\cdots 42}a^{2}-\frac{77\cdots 21}{36\cdots 42}a-\frac{16\cdots 32}{18\cdots 21}$, $\frac{41\cdots 56}{78\cdots 27}a^{15}-\frac{10\cdots 51}{36\cdots 42}a^{14}-\frac{51\cdots 83}{36\cdots 42}a^{13}+\frac{35\cdots 65}{36\cdots 42}a^{12}+\frac{15\cdots 67}{18\cdots 21}a^{11}-\frac{44\cdots 33}{36\cdots 42}a^{10}+\frac{18\cdots 83}{36\cdots 42}a^{9}+\frac{25\cdots 45}{36\cdots 42}a^{8}-\frac{13\cdots 58}{18\cdots 21}a^{7}-\frac{79\cdots 49}{36\cdots 42}a^{6}+\frac{11\cdots 07}{36\cdots 42}a^{5}+\frac{81\cdots 45}{18\cdots 21}a^{4}-\frac{22\cdots 61}{36\cdots 42}a^{3}-\frac{11\cdots 06}{18\cdots 21}a^{2}+\frac{15\cdots 69}{36\cdots 42}a+\frac{15\cdots 91}{36\cdots 42}$, $\frac{37\cdots 41}{36\cdots 42}a^{15}-\frac{23\cdots 96}{18\cdots 21}a^{14}-\frac{63\cdots 09}{18\cdots 21}a^{13}+\frac{17\cdots 13}{36\cdots 42}a^{12}+\frac{15\cdots 93}{36\cdots 42}a^{11}-\frac{56\cdots 59}{78\cdots 27}a^{10}-\frac{40\cdots 21}{18\cdots 21}a^{9}+\frac{17\cdots 95}{36\cdots 42}a^{8}+\frac{22\cdots 89}{36\cdots 42}a^{7}-\frac{33\cdots 81}{18\cdots 21}a^{6}-\frac{42\cdots 41}{36\cdots 42}a^{5}+\frac{13\cdots 71}{36\cdots 42}a^{4}+\frac{49\cdots 65}{36\cdots 42}a^{3}-\frac{13\cdots 15}{36\cdots 42}a^{2}-\frac{12\cdots 34}{18\cdots 21}a+\frac{57\cdots 03}{36\cdots 42}$ 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:  \( 83571859.9339 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 83571859.9339 \cdot 1}{2\cdot\sqrt{860527385904312057534808064}}\cr\approx \mathstrut & 0.230338963740 \end{aligned}\] (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 - 2*x^15 - 35*x^14 + 72*x^13 + 448*x^12 - 984*x^11 - 2681*x^10 + 6698*x^9 + 8782*x^8 - 26594*x^7 - 20622*x^6 + 62972*x^5 + 44263*x^4 - 74042*x^3 - 55712*x^2 + 32314*x + 25763) 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 - 2*x^15 - 35*x^14 + 72*x^13 + 448*x^12 - 984*x^11 - 2681*x^10 + 6698*x^9 + 8782*x^8 - 26594*x^7 - 20622*x^6 + 62972*x^5 + 44263*x^4 - 74042*x^3 - 55712*x^2 + 32314*x + 25763, 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 - 2*x^15 - 35*x^14 + 72*x^13 + 448*x^12 - 984*x^11 - 2681*x^10 + 6698*x^9 + 8782*x^8 - 26594*x^7 - 20622*x^6 + 62972*x^5 + 44263*x^4 - 74042*x^3 - 55712*x^2 + 32314*x + 25763); 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 = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 35*x^14 + 72*x^13 + 448*x^12 - 984*x^11 - 2681*x^10 + 6698*x^9 + 8782*x^8 - 26594*x^7 - 20622*x^6 + 62972*x^5 + 44263*x^4 - 74042*x^3 - 55712*x^2 + 32314*x + 25763); 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

$C_2^7.C_2\wr S_4$ (as 16T1850):

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); G, transitive_group_identification(G)
 
A solvable group of order 49152
The 104 conjugacy class representatives for $C_2^7.C_2\wr S_4$
Character table for $C_2^7.C_2\wr S_4$

Intermediate fields

4.4.2777.1, 8.8.1326417388.1

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(b)]
 
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: 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 ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ R ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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 2.2.2.4a1.1$x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 5$$2$$2$$4$$C_2^2$$$[2]^{2}$$
2.6.2.12a8.2$x^{12} + 2 x^{11} + 2 x^{10} + 4 x^{9} + 5 x^{8} + 4 x^{7} + 7 x^{6} + 6 x^{5} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 7$$2$$6$$12$12T134$$[2, 2, 2, 2, 2, 2]^{6}$$
\(43\) Copy content Toggle raw display 43.3.1.0a1.1$x^{3} + x + 40$$1$$3$$0$$C_3$$$[\ ]^{3}$$
43.3.1.0a1.1$x^{3} + x + 40$$1$$3$$0$$C_3$$$[\ ]^{3}$$
43.1.4.3a1.2$x^{4} + 129$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
43.6.1.0a1.1$x^{6} + 19 x^{3} + 28 x^{2} + 21 x + 3$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(2777\) Copy content Toggle raw display $\Q_{2777}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{2777}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{2777}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{2777}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{2777}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{2777}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$2$$2$$2$
Deg $4$$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)