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Properties

Label 16.12.281...733.3
Degree $16$
Signature $[12, 2]$
Discriminant $2.815\times 10^{31}$
Root discriminant \(92.38\)
Ramified primes $13,61$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^6.\SD_{16}$ (as 16T1250)

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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 - 6*x^15 - 31*x^14 + 151*x^13 + 516*x^12 - 590*x^11 - 6688*x^10 - 7131*x^9 + 37373*x^8 + 37037*x^7 - 29952*x^6 + 88844*x^5 - 42918*x^4 - 202319*x^3 - 104234*x^2 - 16731*x - 845)
 
Copy content gp:K = bnfinit(y^16 - 6*y^15 - 31*y^14 + 151*y^13 + 516*y^12 - 590*y^11 - 6688*y^10 - 7131*y^9 + 37373*y^8 + 37037*y^7 - 29952*y^6 + 88844*y^5 - 42918*y^4 - 202319*y^3 - 104234*y^2 - 16731*y - 845, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 - 31*x^14 + 151*x^13 + 516*x^12 - 590*x^11 - 6688*x^10 - 7131*x^9 + 37373*x^8 + 37037*x^7 - 29952*x^6 + 88844*x^5 - 42918*x^4 - 202319*x^3 - 104234*x^2 - 16731*x - 845);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 6*x^15 - 31*x^14 + 151*x^13 + 516*x^12 - 590*x^11 - 6688*x^10 - 7131*x^9 + 37373*x^8 + 37037*x^7 - 29952*x^6 + 88844*x^5 - 42918*x^4 - 202319*x^3 - 104234*x^2 - 16731*x - 845)
 

\( x^{16} - 6 x^{15} - 31 x^{14} + 151 x^{13} + 516 x^{12} - 590 x^{11} - 6688 x^{10} - 7131 x^{9} + \cdots - 845 \) 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:   \(28148042033661945531328523232733\) \(\medspace = 13^{9}\cdot 61^{12}\) 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:  \(92.38\)
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:  $13^{3/4}61^{3/4}\approx 149.4358817512091$
Ramified primes:   \(13\), \(61\) 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:  \(\Q(\sqrt{13}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{247}a^{14}+\frac{85}{247}a^{13}+\frac{86}{247}a^{12}-\frac{70}{247}a^{11}-\frac{30}{247}a^{10}-\frac{122}{247}a^{9}+\frac{59}{247}a^{8}-\frac{98}{247}a^{7}-\frac{119}{247}a^{6}-\frac{7}{19}a^{5}+\frac{8}{19}a^{4}-\frac{89}{247}a^{3}-\frac{31}{247}a^{2}+\frac{8}{19}a+\frac{8}{19}$, $\frac{1}{57\cdots 29}a^{15}+\frac{57\cdots 18}{57\cdots 29}a^{14}+\frac{27\cdots 23}{57\cdots 29}a^{13}-\frac{10\cdots 05}{57\cdots 29}a^{12}+\frac{25\cdots 15}{57\cdots 29}a^{11}-\frac{92\cdots 83}{44\cdots 33}a^{10}-\frac{18\cdots 11}{57\cdots 29}a^{9}+\frac{26\cdots 88}{57\cdots 29}a^{8}-\frac{17\cdots 52}{57\cdots 29}a^{7}+\frac{96\cdots 82}{57\cdots 29}a^{6}+\frac{86\cdots 07}{44\cdots 33}a^{5}-\frac{13\cdots 26}{57\cdots 29}a^{4}+\frac{17\cdots 55}{57\cdots 29}a^{3}+\frac{28\cdots 06}{57\cdots 29}a^{2}+\frac{13\cdots 95}{44\cdots 33}a-\frac{98\cdots 79}{44\cdots 33}$ 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:  $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:   $\frac{53\cdots 71}{31\cdots 03}a^{15}-\frac{32\cdots 89}{31\cdots 03}a^{14}-\frac{16\cdots 64}{31\cdots 03}a^{13}+\frac{82\cdots 38}{31\cdots 03}a^{12}+\frac{26\cdots 15}{31\cdots 03}a^{11}-\frac{34\cdots 43}{31\cdots 03}a^{10}-\frac{35\cdots 85}{31\cdots 03}a^{9}-\frac{33\cdots 40}{31\cdots 03}a^{8}+\frac{20\cdots 78}{31\cdots 03}a^{7}+\frac{13\cdots 58}{23\cdots 31}a^{6}-\frac{13\cdots 13}{23\cdots 31}a^{5}+\frac{49\cdots 16}{31\cdots 03}a^{4}-\frac{28\cdots 97}{31\cdots 03}a^{3}-\frac{80\cdots 86}{23\cdots 31}a^{2}-\frac{33\cdots 25}{23\cdots 31}a-\frac{29\cdots 53}{23\cdots 31}$, $\frac{69\cdots 11}{57\cdots 29}a^{15}-\frac{42\cdots 45}{57\cdots 29}a^{14}-\frac{21\cdots 31}{57\cdots 29}a^{13}+\frac{10\cdots 36}{57\cdots 29}a^{12}+\frac{34\cdots 50}{57\cdots 29}a^{11}-\frac{45\cdots 38}{57\cdots 29}a^{10}-\frac{46\cdots 83}{57\cdots 29}a^{9}-\frac{44\cdots 50}{57\cdots 29}a^{8}+\frac{26\cdots 89}{57\cdots 29}a^{7}+\frac{22\cdots 48}{57\cdots 29}a^{6}-\frac{18\cdots 55}{44\cdots 33}a^{5}+\frac{65\cdots 98}{57\cdots 29}a^{4}-\frac{37\cdots 95}{57\cdots 29}a^{3}-\frac{13\cdots 35}{57\cdots 29}a^{2}-\frac{43\cdots 67}{44\cdots 33}a-\frac{38\cdots 92}{44\cdots 33}$, $\frac{58\cdots 21}{31\cdots 03}a^{15}-\frac{35\cdots 42}{31\cdots 03}a^{14}-\frac{17\cdots 92}{31\cdots 03}a^{13}+\frac{89\cdots 03}{31\cdots 03}a^{12}+\frac{29\cdots 18}{31\cdots 03}a^{11}-\frac{37\cdots 30}{31\cdots 03}a^{10}-\frac{38\cdots 75}{31\cdots 03}a^{9}-\frac{37\cdots 57}{31\cdots 03}a^{8}+\frac{22\cdots 32}{31\cdots 03}a^{7}+\frac{14\cdots 86}{23\cdots 31}a^{6}-\frac{15\cdots 35}{23\cdots 31}a^{5}+\frac{53\cdots 81}{31\cdots 03}a^{4}-\frac{30\cdots 28}{31\cdots 03}a^{3}-\frac{88\cdots 98}{23\cdots 31}a^{2}-\frac{37\cdots 90}{23\cdots 31}a-\frac{32\cdots 57}{23\cdots 31}$, $\frac{61\cdots 12}{59\cdots 57}a^{15}-\frac{36\cdots 20}{59\cdots 57}a^{14}-\frac{18\cdots 71}{59\cdots 57}a^{13}+\frac{93\cdots 36}{59\cdots 57}a^{12}+\frac{31\cdots 79}{59\cdots 57}a^{11}-\frac{37\cdots 62}{59\cdots 57}a^{10}-\frac{40\cdots 98}{59\cdots 57}a^{9}-\frac{41\cdots 38}{59\cdots 57}a^{8}+\frac{23\cdots 16}{59\cdots 57}a^{7}+\frac{16\cdots 81}{45\cdots 89}a^{6}-\frac{15\cdots 51}{45\cdots 89}a^{5}+\frac{55\cdots 06}{59\cdots 57}a^{4}-\frac{28\cdots 90}{59\cdots 57}a^{3}-\frac{95\cdots 16}{45\cdots 89}a^{2}-\frac{22\cdots 99}{23\cdots 31}a-\frac{38\cdots 73}{45\cdots 89}$, $\frac{42\cdots 64}{57\cdots 29}a^{15}-\frac{20\cdots 29}{44\cdots 33}a^{14}-\frac{12\cdots 05}{57\cdots 29}a^{13}+\frac{65\cdots 38}{57\cdots 29}a^{12}+\frac{20\cdots 33}{57\cdots 29}a^{11}-\frac{28\cdots 30}{57\cdots 29}a^{10}-\frac{27\cdots 02}{57\cdots 29}a^{9}-\frac{25\cdots 82}{57\cdots 29}a^{8}+\frac{16\cdots 30}{57\cdots 29}a^{7}+\frac{12\cdots 57}{57\cdots 29}a^{6}-\frac{11\cdots 86}{44\cdots 33}a^{5}+\frac{40\cdots 51}{57\cdots 29}a^{4}-\frac{24\cdots 20}{57\cdots 29}a^{3}-\frac{80\cdots 39}{57\cdots 29}a^{2}-\frac{24\cdots 41}{44\cdots 33}a-\frac{20\cdots 36}{44\cdots 33}$, $\frac{40\cdots 47}{57\cdots 29}a^{15}-\frac{24\cdots 40}{57\cdots 29}a^{14}-\frac{12\cdots 48}{57\cdots 29}a^{13}+\frac{62\cdots 36}{57\cdots 29}a^{12}+\frac{20\cdots 21}{57\cdots 29}a^{11}-\frac{26\cdots 80}{57\cdots 29}a^{10}-\frac{26\cdots 05}{57\cdots 29}a^{9}-\frac{25\cdots 49}{57\cdots 29}a^{8}+\frac{11\cdots 45}{44\cdots 33}a^{7}+\frac{13\cdots 86}{57\cdots 29}a^{6}-\frac{10\cdots 55}{44\cdots 33}a^{5}+\frac{37\cdots 44}{57\cdots 29}a^{4}-\frac{16\cdots 83}{44\cdots 33}a^{3}-\frac{80\cdots 08}{57\cdots 29}a^{2}-\frac{25\cdots 22}{44\cdots 33}a-\frac{22\cdots 14}{44\cdots 33}$, $\frac{90\cdots 37}{31\cdots 03}a^{15}-\frac{36\cdots 24}{31\cdots 03}a^{14}-\frac{39\cdots 39}{31\cdots 03}a^{13}+\frac{88\cdots 86}{31\cdots 03}a^{12}+\frac{75\cdots 95}{31\cdots 03}a^{11}+\frac{20\cdots 05}{31\cdots 03}a^{10}-\frac{74\cdots 69}{31\cdots 03}a^{9}-\frac{17\cdots 29}{31\cdots 03}a^{8}+\frac{27\cdots 14}{31\cdots 03}a^{7}+\frac{78\cdots 61}{23\cdots 31}a^{6}+\frac{55\cdots 58}{23\cdots 31}a^{5}+\frac{98\cdots 08}{31\cdots 03}a^{4}+\frac{15\cdots 98}{31\cdots 03}a^{3}-\frac{27\cdots 49}{23\cdots 31}a^{2}-\frac{27\cdots 81}{23\cdots 31}a-\frac{28\cdots 21}{23\cdots 31}$, $\frac{12\cdots 37}{57\cdots 29}a^{15}-\frac{78\cdots 12}{57\cdots 29}a^{14}-\frac{37\cdots 76}{57\cdots 29}a^{13}+\frac{19\cdots 82}{57\cdots 29}a^{12}+\frac{61\cdots 24}{57\cdots 29}a^{11}-\frac{86\cdots 15}{57\cdots 29}a^{10}-\frac{64\cdots 86}{44\cdots 33}a^{9}-\frac{75\cdots 59}{57\cdots 29}a^{8}+\frac{48\cdots 56}{57\cdots 29}a^{7}+\frac{37\cdots 59}{57\cdots 29}a^{6}-\frac{33\cdots 36}{44\cdots 33}a^{5}+\frac{12\cdots 97}{57\cdots 29}a^{4}-\frac{80\cdots 55}{57\cdots 29}a^{3}-\frac{23\cdots 21}{57\cdots 29}a^{2}-\frac{72\cdots 88}{44\cdots 33}a-\frac{60\cdots 08}{44\cdots 33}$, $\frac{23\cdots 22}{57\cdots 29}a^{15}-\frac{14\cdots 58}{57\cdots 29}a^{14}-\frac{71\cdots 68}{57\cdots 29}a^{13}+\frac{36\cdots 15}{57\cdots 29}a^{12}+\frac{90\cdots 05}{44\cdots 33}a^{11}-\frac{15\cdots 55}{57\cdots 29}a^{10}-\frac{15\cdots 42}{57\cdots 29}a^{9}-\frac{14\cdots 09}{57\cdots 29}a^{8}+\frac{89\cdots 25}{57\cdots 29}a^{7}+\frac{76\cdots 76}{57\cdots 29}a^{6}-\frac{61\cdots 45}{44\cdots 33}a^{5}+\frac{21\cdots 53}{57\cdots 29}a^{4}-\frac{12\cdots 15}{57\cdots 29}a^{3}-\frac{46\cdots 00}{57\cdots 29}a^{2}-\frac{76\cdots 50}{23\cdots 07}a-\frac{12\cdots 73}{44\cdots 33}$, $\frac{10\cdots 58}{57\cdots 29}a^{15}-\frac{73\cdots 94}{57\cdots 29}a^{14}-\frac{27\cdots 25}{57\cdots 29}a^{13}+\frac{18\cdots 93}{57\cdots 29}a^{12}+\frac{39\cdots 23}{57\cdots 29}a^{11}-\frac{10\cdots 83}{57\cdots 29}a^{10}-\frac{63\cdots 19}{57\cdots 29}a^{9}-\frac{15\cdots 27}{44\cdots 33}a^{8}+\frac{42\cdots 73}{57\cdots 29}a^{7}+\frac{30\cdots 45}{57\cdots 29}a^{6}-\frac{31\cdots 74}{44\cdots 33}a^{5}+\frac{12\cdots 51}{57\cdots 29}a^{4}-\frac{80\cdots 80}{30\cdots 91}a^{3}-\frac{10\cdots 39}{57\cdots 29}a^{2}-\frac{35\cdots 90}{44\cdots 33}a+\frac{50\cdots 96}{44\cdots 33}$, $\frac{90\cdots 73}{30\cdots 91}a^{15}-\frac{50\cdots 61}{30\cdots 91}a^{14}-\frac{30\cdots 75}{30\cdots 91}a^{13}+\frac{12\cdots 85}{30\cdots 91}a^{12}+\frac{53\cdots 50}{30\cdots 91}a^{11}-\frac{29\cdots 57}{23\cdots 07}a^{10}-\frac{63\cdots 96}{30\cdots 91}a^{9}-\frac{87\cdots 56}{30\cdots 91}a^{8}+\frac{32\cdots 12}{30\cdots 91}a^{7}+\frac{48\cdots 69}{30\cdots 91}a^{6}-\frac{17\cdots 95}{23\cdots 07}a^{5}+\frac{61\cdots 34}{30\cdots 91}a^{4}+\frac{34\cdots 49}{30\cdots 91}a^{3}-\frac{21\cdots 61}{30\cdots 91}a^{2}-\frac{11\cdots 53}{23\cdots 07}a-\frac{11\cdots 89}{23\cdots 07}$, $\frac{79\cdots 53}{30\cdots 91}a^{15}-\frac{48\cdots 12}{30\cdots 91}a^{14}-\frac{23\cdots 69}{30\cdots 91}a^{13}+\frac{12\cdots 86}{30\cdots 91}a^{12}+\frac{39\cdots 86}{30\cdots 91}a^{11}-\frac{51\cdots 87}{30\cdots 91}a^{10}-\frac{52\cdots 70}{30\cdots 91}a^{9}-\frac{50\cdots 87}{30\cdots 91}a^{8}+\frac{30\cdots 02}{30\cdots 91}a^{7}+\frac{25\cdots 02}{30\cdots 91}a^{6}-\frac{20\cdots 33}{23\cdots 07}a^{5}+\frac{73\cdots 97}{30\cdots 91}a^{4}-\frac{42\cdots 49}{30\cdots 91}a^{3}-\frac{15\cdots 16}{30\cdots 91}a^{2}-\frac{49\cdots 01}{23\cdots 07}a-\frac{43\cdots 56}{23\cdots 07}$, $\frac{19\cdots 36}{57\cdots 29}a^{15}-\frac{11\cdots 81}{57\cdots 29}a^{14}-\frac{30\cdots 38}{30\cdots 91}a^{13}+\frac{29\cdots 09}{57\cdots 29}a^{12}+\frac{96\cdots 03}{57\cdots 29}a^{11}-\frac{53\cdots 03}{23\cdots 07}a^{10}-\frac{12\cdots 16}{57\cdots 29}a^{9}-\frac{11\cdots 77}{57\cdots 29}a^{8}+\frac{75\cdots 79}{57\cdots 29}a^{7}+\frac{63\cdots 11}{57\cdots 29}a^{6}-\frac{58\cdots 09}{44\cdots 33}a^{5}+\frac{17\cdots 02}{57\cdots 29}a^{4}-\frac{12\cdots 54}{57\cdots 29}a^{3}-\frac{41\cdots 51}{57\cdots 29}a^{2}-\frac{13\cdots 31}{44\cdots 33}a-\frac{11\cdots 17}{44\cdots 33}$ 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:  \( 20079785198.0 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 20079785198.0 \cdot 1}{2\cdot\sqrt{28148042033661945531328523232733}}\cr\approx \mathstrut & 0.306002275667 \end{aligned}\] (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 - 6*x^15 - 31*x^14 + 151*x^13 + 516*x^12 - 590*x^11 - 6688*x^10 - 7131*x^9 + 37373*x^8 + 37037*x^7 - 29952*x^6 + 88844*x^5 - 42918*x^4 - 202319*x^3 - 104234*x^2 - 16731*x - 845) 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 - 6*x^15 - 31*x^14 + 151*x^13 + 516*x^12 - 590*x^11 - 6688*x^10 - 7131*x^9 + 37373*x^8 + 37037*x^7 - 29952*x^6 + 88844*x^5 - 42918*x^4 - 202319*x^3 - 104234*x^2 - 16731*x - 845, 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 - 6*x^15 - 31*x^14 + 151*x^13 + 516*x^12 - 590*x^11 - 6688*x^10 - 7131*x^9 + 37373*x^8 + 37037*x^7 - 29952*x^6 + 88844*x^5 - 42918*x^4 - 202319*x^3 - 104234*x^2 - 16731*x - 845); 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 - 6*x^15 - 31*x^14 + 151*x^13 + 516*x^12 - 590*x^11 - 6688*x^10 - 7131*x^9 + 37373*x^8 + 37037*x^7 - 29952*x^6 + 88844*x^5 - 42918*x^4 - 202319*x^3 - 104234*x^2 - 16731*x - 845); 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

$C_2^6.\SD_{16}$ (as 16T1250):

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 1024
The 34 conjugacy class representatives for $C_2^6.\SD_{16}$
Character table for $C_2^6.\SD_{16}$

Intermediate fields

\(\Q(\sqrt{61}) \), 4.4.48373.1, 8.8.113190262471117.1

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: 16.8.166556461737644648114369959957.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $16$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ $16$ $16$ R ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ $16$

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
\(13\) Copy content Toggle raw display 13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.1.4.3a1.3$x^{4} + 52$$4$$1$$3$$C_4$$$[\ ]_{4}$$
13.2.2.2a1.2$x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
13.1.4.3a1.1$x^{4} + 13$$4$$1$$3$$C_4$$$[\ ]_{4}$$
\(61\) Copy content Toggle raw display 61.1.4.3a1.3$x^{4} + 244$$4$$1$$3$$C_4$$$[\ ]_{4}$$
61.1.4.3a1.3$x^{4} + 244$$4$$1$$3$$C_4$$$[\ ]_{4}$$
61.2.4.6a1.2$x^{8} + 240 x^{7} + 21608 x^{6} + 865440 x^{5} + 13046424 x^{4} + 1730880 x^{3} + 86432 x^{2} + 1920 x + 77$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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