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Properties

Label 20.4.125...816.1
Degree $20$
Signature $(4, 8)$
Discriminant $1.256\times 10^{32}$
Root discriminant \(40.27\)
Ramified primes $2,7,13,347$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^8.(C_3\times S_5)$ (as 20T753)

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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^20 - 8*x^18 - 22*x^17 - 44*x^16 + 48*x^15 + 390*x^14 + 878*x^13 + 1192*x^12 - 238*x^11 - 5046*x^10 - 12784*x^9 - 21285*x^8 - 24460*x^7 - 18100*x^6 - 7404*x^5 + 738*x^4 + 2406*x^3 - 38*x^2 + 136*x - 67)
 
Copy content gp:K = bnfinit(y^20 - 8*y^18 - 22*y^17 - 44*y^16 + 48*y^15 + 390*y^14 + 878*y^13 + 1192*y^12 - 238*y^11 - 5046*y^10 - 12784*y^9 - 21285*y^8 - 24460*y^7 - 18100*y^6 - 7404*y^5 + 738*y^4 + 2406*y^3 - 38*y^2 + 136*y - 67, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 8*x^18 - 22*x^17 - 44*x^16 + 48*x^15 + 390*x^14 + 878*x^13 + 1192*x^12 - 238*x^11 - 5046*x^10 - 12784*x^9 - 21285*x^8 - 24460*x^7 - 18100*x^6 - 7404*x^5 + 738*x^4 + 2406*x^3 - 38*x^2 + 136*x - 67);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 8*x^18 - 22*x^17 - 44*x^16 + 48*x^15 + 390*x^14 + 878*x^13 + 1192*x^12 - 238*x^11 - 5046*x^10 - 12784*x^9 - 21285*x^8 - 24460*x^7 - 18100*x^6 - 7404*x^5 + 738*x^4 + 2406*x^3 - 38*x^2 + 136*x - 67)
 

\( x^{20} - 8 x^{18} - 22 x^{17} - 44 x^{16} + 48 x^{15} + 390 x^{14} + 878 x^{13} + 1192 x^{12} + \cdots - 67 \) 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:  $20$
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:  $(4, 8)$
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:   \(125594776797687739037790100258816\) \(\medspace = 2^{30}\cdot 7^{10}\cdot 13^{4}\cdot 347^{4}\) 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:  \(40.27\)
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:  not computed
Ramified primes:   \(2\), \(7\), \(13\), \(347\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{61\cdots 57}a^{19}+\frac{69\cdots 37}{61\cdots 57}a^{18}-\frac{20\cdots 59}{61\cdots 57}a^{17}+\frac{52\cdots 71}{61\cdots 57}a^{16}-\frac{28\cdots 18}{61\cdots 57}a^{15}-\frac{19\cdots 16}{61\cdots 57}a^{14}+\frac{45\cdots 85}{61\cdots 57}a^{13}-\frac{15\cdots 32}{61\cdots 57}a^{12}+\frac{10\cdots 18}{61\cdots 57}a^{11}+\frac{75\cdots 80}{61\cdots 57}a^{10}-\frac{12\cdots 46}{61\cdots 57}a^{9}+\frac{26\cdots 73}{61\cdots 57}a^{8}-\frac{10\cdots 15}{61\cdots 57}a^{7}+\frac{18\cdots 25}{61\cdots 57}a^{6}-\frac{70\cdots 20}{61\cdots 57}a^{5}-\frac{23\cdots 96}{61\cdots 57}a^{4}-\frac{21\cdots 99}{61\cdots 57}a^{3}+\frac{19\cdots 22}{61\cdots 57}a^{2}+\frac{30\cdots 23}{61\cdots 57}a-\frac{13\cdots 50}{61\cdots 57}$ 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:  Trivial group, which has order $1$ (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:   $\frac{17\cdots 88}{61\cdots 57}a^{19}-\frac{12\cdots 90}{61\cdots 57}a^{18}-\frac{14\cdots 98}{61\cdots 57}a^{17}-\frac{36\cdots 04}{61\cdots 57}a^{16}-\frac{72\cdots 16}{61\cdots 57}a^{15}+\frac{92\cdots 32}{61\cdots 57}a^{14}+\frac{67\cdots 12}{61\cdots 57}a^{13}+\frac{14\cdots 12}{61\cdots 57}a^{12}+\frac{19\cdots 60}{61\cdots 57}a^{11}-\frac{67\cdots 04}{61\cdots 57}a^{10}-\frac{88\cdots 28}{61\cdots 57}a^{9}-\frac{21\cdots 34}{61\cdots 57}a^{8}-\frac{34\cdots 50}{61\cdots 57}a^{7}-\frac{38\cdots 16}{61\cdots 57}a^{6}-\frac{26\cdots 66}{61\cdots 57}a^{5}-\frac{90\cdots 41}{61\cdots 57}a^{4}+\frac{27\cdots 96}{61\cdots 57}a^{3}+\frac{44\cdots 56}{61\cdots 57}a^{2}-\frac{27\cdots 42}{61\cdots 57}a+\frac{58\cdots 12}{61\cdots 57}$, $\frac{25\cdots 52}{61\cdots 57}a^{19}+\frac{20\cdots 28}{61\cdots 57}a^{18}-\frac{20\cdots 08}{61\cdots 57}a^{17}-\frac{57\cdots 24}{61\cdots 57}a^{16}-\frac{11\cdots 00}{61\cdots 57}a^{15}+\frac{11\cdots 20}{61\cdots 57}a^{14}+\frac{10\cdots 76}{61\cdots 57}a^{13}+\frac{22\cdots 12}{61\cdots 57}a^{12}+\frac{31\cdots 24}{61\cdots 57}a^{11}-\frac{48\cdots 16}{61\cdots 57}a^{10}-\frac{13\cdots 20}{61\cdots 57}a^{9}-\frac{33\cdots 79}{61\cdots 57}a^{8}-\frac{55\cdots 40}{61\cdots 57}a^{7}-\frac{64\cdots 20}{61\cdots 57}a^{6}-\frac{47\cdots 20}{61\cdots 57}a^{5}-\frac{18\cdots 40}{61\cdots 57}a^{4}+\frac{31\cdots 00}{61\cdots 57}a^{3}+\frac{75\cdots 60}{61\cdots 57}a^{2}+\frac{58\cdots 24}{61\cdots 57}a-\frac{16\cdots 81}{61\cdots 57}$, $\frac{12\cdots 64}{61\cdots 57}a^{19}-\frac{28\cdots 26}{61\cdots 57}a^{18}-\frac{94\cdots 94}{61\cdots 57}a^{17}-\frac{24\cdots 61}{61\cdots 57}a^{16}-\frac{47\cdots 64}{61\cdots 57}a^{15}+\frac{69\cdots 80}{61\cdots 57}a^{14}+\frac{45\cdots 04}{61\cdots 57}a^{13}+\frac{96\cdots 36}{61\cdots 57}a^{12}+\frac{12\cdots 32}{61\cdots 57}a^{11}-\frac{61\cdots 24}{61\cdots 57}a^{10}-\frac{60\cdots 72}{61\cdots 57}a^{9}-\frac{14\cdots 72}{61\cdots 57}a^{8}-\frac{22\cdots 22}{61\cdots 57}a^{7}-\frac{23\cdots 72}{61\cdots 57}a^{6}-\frac{14\cdots 38}{61\cdots 57}a^{5}-\frac{26\cdots 79}{61\cdots 57}a^{4}+\frac{41\cdots 08}{61\cdots 57}a^{3}+\frac{25\cdots 84}{61\cdots 57}a^{2}-\frac{11\cdots 10}{61\cdots 57}a-\frac{99\cdots 25}{61\cdots 57}$, $\frac{21\cdots 73}{61\cdots 57}a^{19}-\frac{18\cdots 97}{61\cdots 57}a^{18}-\frac{16\cdots 05}{61\cdots 57}a^{17}-\frac{45\cdots 76}{61\cdots 57}a^{16}-\frac{90\cdots 23}{61\cdots 57}a^{15}+\frac{10\cdots 19}{61\cdots 57}a^{14}+\frac{80\cdots 45}{61\cdots 57}a^{13}+\frac{17\cdots 62}{61\cdots 57}a^{12}+\frac{24\cdots 35}{61\cdots 57}a^{11}-\frac{53\cdots 14}{61\cdots 57}a^{10}-\frac{10\cdots 85}{61\cdots 57}a^{9}-\frac{26\cdots 90}{61\cdots 57}a^{8}-\frac{43\cdots 78}{61\cdots 57}a^{7}-\frac{50\cdots 77}{61\cdots 57}a^{6}-\frac{38\cdots 53}{61\cdots 57}a^{5}-\frac{17\cdots 16}{61\cdots 57}a^{4}-\frac{96\cdots 98}{61\cdots 57}a^{3}+\frac{35\cdots 29}{61\cdots 57}a^{2}-\frac{60\cdots 27}{61\cdots 57}a+\frac{11\cdots 48}{61\cdots 57}$, $\frac{50\cdots 30}{61\cdots 57}a^{19}+\frac{14\cdots 12}{61\cdots 57}a^{18}-\frac{39\cdots 79}{61\cdots 57}a^{17}-\frac{12\cdots 14}{61\cdots 57}a^{16}-\frac{25\cdots 92}{61\cdots 57}a^{15}+\frac{16\cdots 69}{61\cdots 57}a^{14}+\frac{20\cdots 32}{61\cdots 57}a^{13}+\frac{49\cdots 69}{61\cdots 57}a^{12}+\frac{74\cdots 25}{61\cdots 57}a^{11}+\frac{97\cdots 73}{61\cdots 57}a^{10}-\frac{25\cdots 68}{61\cdots 57}a^{9}-\frac{71\cdots 57}{61\cdots 57}a^{8}-\frac{12\cdots 61}{61\cdots 57}a^{7}-\frac{15\cdots 23}{61\cdots 57}a^{6}-\frac{13\cdots 72}{61\cdots 57}a^{5}-\frac{77\cdots 70}{61\cdots 57}a^{4}-\frac{18\cdots 36}{61\cdots 57}a^{3}+\frac{67\cdots 07}{61\cdots 57}a^{2}+\frac{18\cdots 94}{61\cdots 57}a+\frac{12\cdots 24}{61\cdots 57}$, $\frac{36\cdots 18}{61\cdots 57}a^{19}+\frac{12\cdots 86}{61\cdots 57}a^{18}-\frac{28\cdots 56}{61\cdots 57}a^{17}-\frac{90\cdots 56}{61\cdots 57}a^{16}-\frac{19\cdots 02}{61\cdots 57}a^{15}+\frac{11\cdots 17}{61\cdots 57}a^{14}+\frac{14\cdots 76}{61\cdots 57}a^{13}+\frac{37\cdots 57}{61\cdots 57}a^{12}+\frac{56\cdots 79}{61\cdots 57}a^{11}+\frac{10\cdots 67}{61\cdots 57}a^{10}-\frac{18\cdots 43}{61\cdots 57}a^{9}-\frac{52\cdots 23}{61\cdots 57}a^{8}-\frac{95\cdots 20}{61\cdots 57}a^{7}-\frac{12\cdots 52}{61\cdots 57}a^{6}-\frac{10\cdots 06}{61\cdots 57}a^{5}-\frac{62\cdots 69}{61\cdots 57}a^{4}-\frac{18\cdots 49}{61\cdots 57}a^{3}+\frac{25\cdots 75}{61\cdots 57}a^{2}+\frac{56\cdots 14}{61\cdots 57}a+\frac{67\cdots 36}{61\cdots 57}$, $\frac{10\cdots 36}{61\cdots 57}a^{19}+\frac{18\cdots 73}{61\cdots 57}a^{18}-\frac{81\cdots 13}{61\cdots 57}a^{17}-\frac{24\cdots 36}{61\cdots 57}a^{16}-\frac{51\cdots 60}{61\cdots 57}a^{15}+\frac{39\cdots 44}{61\cdots 57}a^{14}+\frac{41\cdots 91}{61\cdots 57}a^{13}+\frac{10\cdots 96}{61\cdots 57}a^{12}+\frac{14\cdots 28}{61\cdots 57}a^{11}+\frac{82\cdots 18}{61\cdots 57}a^{10}-\frac{51\cdots 17}{61\cdots 57}a^{9}-\frac{14\cdots 43}{61\cdots 57}a^{8}-\frac{25\cdots 85}{61\cdots 57}a^{7}-\frac{31\cdots 15}{61\cdots 57}a^{6}-\frac{26\cdots 07}{61\cdots 57}a^{5}-\frac{13\cdots 17}{61\cdots 57}a^{4}-\frac{24\cdots 57}{61\cdots 57}a^{3}+\frac{19\cdots 43}{61\cdots 57}a^{2}+\frac{44\cdots 82}{61\cdots 57}a+\frac{19\cdots 08}{61\cdots 57}$, $\frac{26\cdots 69}{61\cdots 57}a^{19}-\frac{16\cdots 62}{61\cdots 57}a^{18}-\frac{18\cdots 56}{61\cdots 57}a^{17}-\frac{50\cdots 96}{61\cdots 57}a^{16}-\frac{99\cdots 24}{61\cdots 57}a^{15}+\frac{16\cdots 40}{61\cdots 57}a^{14}+\frac{89\cdots 73}{61\cdots 57}a^{13}+\frac{19\cdots 19}{61\cdots 57}a^{12}+\frac{25\cdots 89}{61\cdots 57}a^{11}-\frac{11\cdots 17}{61\cdots 57}a^{10}-\frac{11\cdots 38}{61\cdots 57}a^{9}-\frac{28\cdots 99}{61\cdots 57}a^{8}-\frac{46\cdots 65}{61\cdots 57}a^{7}-\frac{52\cdots 32}{61\cdots 57}a^{6}-\frac{40\cdots 16}{61\cdots 57}a^{5}-\frac{20\cdots 47}{61\cdots 57}a^{4}-\frac{32\cdots 28}{61\cdots 57}a^{3}+\frac{92\cdots 19}{61\cdots 57}a^{2}-\frac{22\cdots 05}{61\cdots 57}a+\frac{57\cdots 47}{61\cdots 57}$, $\frac{33\cdots 84}{61\cdots 57}a^{19}+\frac{66\cdots 34}{61\cdots 57}a^{18}-\frac{24\cdots 53}{61\cdots 57}a^{17}-\frac{13\cdots 74}{61\cdots 57}a^{16}-\frac{31\cdots 08}{61\cdots 57}a^{15}-\frac{13\cdots 02}{61\cdots 57}a^{14}+\frac{16\cdots 75}{61\cdots 57}a^{13}+\frac{58\cdots 63}{61\cdots 57}a^{12}+\frac{10\cdots 20}{61\cdots 57}a^{11}+\frac{72\cdots 64}{61\cdots 57}a^{10}-\frac{19\cdots 14}{61\cdots 57}a^{9}-\frac{81\cdots 67}{61\cdots 57}a^{8}-\frac{16\cdots 44}{61\cdots 57}a^{7}-\frac{22\cdots 99}{61\cdots 57}a^{6}-\frac{22\cdots 81}{61\cdots 57}a^{5}-\frac{13\cdots 59}{61\cdots 57}a^{4}-\frac{32\cdots 17}{61\cdots 57}a^{3}+\frac{80\cdots 44}{61\cdots 57}a^{2}+\frac{26\cdots 61}{61\cdots 57}a+\frac{17\cdots 89}{61\cdots 57}$, $\frac{19\cdots 18}{61\cdots 57}a^{19}+\frac{65\cdots 62}{61\cdots 57}a^{18}-\frac{15\cdots 91}{61\cdots 57}a^{17}-\frac{47\cdots 73}{61\cdots 57}a^{16}-\frac{10\cdots 61}{61\cdots 57}a^{15}+\frac{57\cdots 39}{61\cdots 57}a^{14}+\frac{76\cdots 89}{61\cdots 57}a^{13}+\frac{19\cdots 37}{61\cdots 57}a^{12}+\frac{29\cdots 31}{61\cdots 57}a^{11}+\frac{57\cdots 44}{61\cdots 57}a^{10}-\frac{94\cdots 25}{61\cdots 57}a^{9}-\frac{27\cdots 19}{61\cdots 57}a^{8}-\frac{50\cdots 65}{61\cdots 57}a^{7}-\frac{64\cdots 65}{61\cdots 57}a^{6}-\frac{57\cdots 79}{61\cdots 57}a^{5}-\frac{33\cdots 00}{61\cdots 57}a^{4}-\frac{10\cdots 83}{61\cdots 57}a^{3}+\frac{12\cdots 70}{61\cdots 57}a^{2}+\frac{37\cdots 61}{61\cdots 57}a+\frac{38\cdots 55}{61\cdots 57}$, $\frac{28\cdots 89}{61\cdots 57}a^{19}+\frac{10\cdots 68}{61\cdots 57}a^{18}-\frac{22\cdots 40}{61\cdots 57}a^{17}-\frac{71\cdots 47}{61\cdots 57}a^{16}-\frac{15\cdots 96}{61\cdots 57}a^{15}+\frac{83\cdots 99}{61\cdots 57}a^{14}+\frac{11\cdots 59}{61\cdots 57}a^{13}+\frac{29\cdots 17}{61\cdots 57}a^{12}+\frac{44\cdots 21}{61\cdots 57}a^{11}+\frac{89\cdots 67}{61\cdots 57}a^{10}-\frac{14\cdots 59}{61\cdots 57}a^{9}-\frac{41\cdots 12}{61\cdots 57}a^{8}-\frac{75\cdots 09}{61\cdots 57}a^{7}-\frac{96\cdots 94}{61\cdots 57}a^{6}-\frac{85\cdots 65}{61\cdots 57}a^{5}-\frac{51\cdots 22}{61\cdots 57}a^{4}-\frac{15\cdots 12}{61\cdots 57}a^{3}+\frac{15\cdots 66}{61\cdots 57}a^{2}+\frac{58\cdots 92}{61\cdots 57}a+\frac{59\cdots 18}{61\cdots 57}$ 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:  \( 158343620.216 \) (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:  \( 4 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 158343620.216 \cdot 1}{2\cdot\sqrt{125594776797687739037790100258816}}\cr\approx \mathstrut & 0.274564076738 \end{aligned}\] (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^20 - 8*x^18 - 22*x^17 - 44*x^16 + 48*x^15 + 390*x^14 + 878*x^13 + 1192*x^12 - 238*x^11 - 5046*x^10 - 12784*x^9 - 21285*x^8 - 24460*x^7 - 18100*x^6 - 7404*x^5 + 738*x^4 + 2406*x^3 - 38*x^2 + 136*x - 67) 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^20 - 8*x^18 - 22*x^17 - 44*x^16 + 48*x^15 + 390*x^14 + 878*x^13 + 1192*x^12 - 238*x^11 - 5046*x^10 - 12784*x^9 - 21285*x^8 - 24460*x^7 - 18100*x^6 - 7404*x^5 + 738*x^4 + 2406*x^3 - 38*x^2 + 136*x - 67, 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^20 - 8*x^18 - 22*x^17 - 44*x^16 + 48*x^15 + 390*x^14 + 878*x^13 + 1192*x^12 - 238*x^11 - 5046*x^10 - 12784*x^9 - 21285*x^8 - 24460*x^7 - 18100*x^6 - 7404*x^5 + 738*x^4 + 2406*x^3 - 38*x^2 + 136*x - 67); 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^20 - 8*x^18 - 22*x^17 - 44*x^16 + 48*x^15 + 390*x^14 + 878*x^13 + 1192*x^12 - 238*x^11 - 5046*x^10 - 12784*x^9 - 21285*x^8 - 24460*x^7 - 18100*x^6 - 7404*x^5 + 738*x^4 + 2406*x^3 - 38*x^2 + 136*x - 67); 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

$C_2^8.(C_3\times S_5)$ (as 20T753):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
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 non-solvable group of order 92160
The 45 conjugacy class representatives for $C_2^8.(C_3\times S_5)$
Character table for $C_2^8.(C_3\times S_5)$

Intermediate fields

5.3.4511.1

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 30 siblings: data not computed
Degree 40 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 $15{,}\,{\href{/padicField/3.5.0.1}{5} }$ $15{,}\,{\href{/padicField/5.5.0.1}{5} }$ R ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ R ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.5.0.1}{5} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.5.4.30a64.1$x^{20} + 2 x^{19} + 4 x^{17} + 8 x^{16} + 6 x^{15} + 12 x^{14} + 12 x^{13} + 20 x^{12} + 22 x^{11} + 22 x^{10} + 28 x^{9} + 19 x^{8} + 34 x^{7} + 20 x^{6} + 20 x^{5} + 20 x^{4} + 6 x^{3} + 16 x^{2} + 2 x + 7$$4$$5$$30$20T67not computed
\(7\) Copy content Toggle raw display 7.5.1.0a1.1$x^{5} + x + 4$$1$$5$$0$$C_5$$$[\ ]^{5}$$
7.5.3.10a1.3$x^{15} + 3 x^{11} + 12 x^{10} + 3 x^{7} + 24 x^{6} + 48 x^{5} + x^{3} + 12 x^{2} + 48 x + 71$$3$$5$$10$$C_{15}$$$[\ ]_{3}^{5}$$
\(13\) 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$$$[\ ]^{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
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$$$[\ ]^{2}$$
13.4.1.0a1.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
13.4.1.0a1.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
\(347\) Copy content Toggle raw display $\Q_{347}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $6$$2$$3$$3$

Spectrum of ring of integers

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