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Properties

Label 20.4.459...336.2
Degree $20$
Signature $[4, 8]$
Discriminant $4.600\times 10^{30}$
Root discriminant \(34.13\)
Ramified primes $2,3,401$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^9.C_2\wr D_5$ (as 20T853)

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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 - 6*x^19 - 4*x^18 + 74*x^17 - 318*x^15 - 193*x^14 + 612*x^13 + 1819*x^12 + 254*x^11 - 2818*x^10 - 1246*x^9 - 1626*x^8 - 190*x^7 + 2594*x^6 - 300*x^5 + 982*x^4 - 472*x^3 + 29*x^2 - 18*x - 3)
 
Copy content gp:K = bnfinit(y^20 - 6*y^19 - 4*y^18 + 74*y^17 - 318*y^15 - 193*y^14 + 612*y^13 + 1819*y^12 + 254*y^11 - 2818*y^10 - 1246*y^9 - 1626*y^8 - 190*y^7 + 2594*y^6 - 300*y^5 + 982*y^4 - 472*y^3 + 29*y^2 - 18*y - 3, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^19 - 4*x^18 + 74*x^17 - 318*x^15 - 193*x^14 + 612*x^13 + 1819*x^12 + 254*x^11 - 2818*x^10 - 1246*x^9 - 1626*x^8 - 190*x^7 + 2594*x^6 - 300*x^5 + 982*x^4 - 472*x^3 + 29*x^2 - 18*x - 3);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 6*x^19 - 4*x^18 + 74*x^17 - 318*x^15 - 193*x^14 + 612*x^13 + 1819*x^12 + 254*x^11 - 2818*x^10 - 1246*x^9 - 1626*x^8 - 190*x^7 + 2594*x^6 - 300*x^5 + 982*x^4 - 472*x^3 + 29*x^2 - 18*x - 3)
 

\( x^{20} - 6 x^{19} - 4 x^{18} + 74 x^{17} - 318 x^{15} - 193 x^{14} + 612 x^{13} + 1819 x^{12} + 254 x^{11} + \cdots - 3 \) 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:   \(4599651543326335737037335822336\) \(\medspace = 2^{20}\cdot 3^{8}\cdot 401^{8}\) 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:  \(34.13\)
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:  $2^{31/16}3^{3/4}401^{1/2}\approx 174.84722497415262$
Ramified primes:   \(2\), \(3\), \(401\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{139}a^{18}+\frac{21}{139}a^{17}+\frac{68}{139}a^{16}-\frac{6}{139}a^{15}-\frac{45}{139}a^{14}+\frac{47}{139}a^{13}-\frac{1}{139}a^{12}-\frac{23}{139}a^{11}+\frac{25}{139}a^{10}-\frac{57}{139}a^{9}-\frac{52}{139}a^{8}-\frac{11}{139}a^{7}+\frac{48}{139}a^{6}+\frac{18}{139}a^{5}+\frac{31}{139}a^{4}-\frac{33}{139}a^{3}+\frac{36}{139}a^{2}+\frac{16}{139}a+\frac{16}{139}$, $\frac{1}{47\cdots 77}a^{19}-\frac{57\cdots 76}{47\cdots 77}a^{18}-\frac{28\cdots 16}{47\cdots 77}a^{17}-\frac{17\cdots 77}{47\cdots 77}a^{16}+\frac{10\cdots 66}{47\cdots 77}a^{15}-\frac{67\cdots 35}{47\cdots 77}a^{14}+\frac{53\cdots 25}{47\cdots 77}a^{13}+\frac{64\cdots 85}{47\cdots 77}a^{12}-\frac{14\cdots 48}{47\cdots 77}a^{11}+\frac{11\cdots 07}{47\cdots 77}a^{10}+\frac{14\cdots 57}{47\cdots 77}a^{9}+\frac{21\cdots 23}{47\cdots 77}a^{8}+\frac{12\cdots 36}{47\cdots 77}a^{7}+\frac{12\cdots 28}{47\cdots 77}a^{6}-\frac{30\cdots 57}{47\cdots 77}a^{5}-\frac{50\cdots 41}{47\cdots 77}a^{4}-\frac{10\cdots 94}{47\cdots 77}a^{3}-\frac{64\cdots 25}{47\cdots 77}a^{2}-\frac{11\cdots 08}{47\cdots 77}a-\frac{19\cdots 91}{47\cdots 77}$ 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:  $C_{2}$, which has order $2$ (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:  $11$
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:   $\frac{25\cdots 58}{48\cdots 41}a^{19}-\frac{15\cdots 25}{48\cdots 41}a^{18}-\frac{72\cdots 98}{48\cdots 41}a^{17}+\frac{19\cdots 30}{48\cdots 41}a^{16}-\frac{38\cdots 14}{48\cdots 41}a^{15}-\frac{82\cdots 75}{48\cdots 41}a^{14}-\frac{31\cdots 88}{48\cdots 41}a^{13}+\frac{17\cdots 45}{48\cdots 41}a^{12}+\frac{43\cdots 26}{48\cdots 41}a^{11}-\frac{42\cdots 30}{48\cdots 41}a^{10}-\frac{75\cdots 90}{48\cdots 41}a^{9}-\frac{15\cdots 15}{48\cdots 41}a^{8}-\frac{29\cdots 34}{48\cdots 41}a^{7}+\frac{27\cdots 24}{48\cdots 41}a^{6}+\frac{67\cdots 28}{48\cdots 41}a^{5}-\frac{21\cdots 75}{48\cdots 41}a^{4}+\frac{21\cdots 94}{48\cdots 41}a^{3}-\frac{14\cdots 85}{48\cdots 41}a^{2}+\frac{20\cdots 50}{48\cdots 41}a+\frac{28\cdots 76}{48\cdots 41}$, $\frac{25\cdots 58}{48\cdots 41}a^{19}-\frac{15\cdots 25}{48\cdots 41}a^{18}-\frac{72\cdots 98}{48\cdots 41}a^{17}+\frac{19\cdots 30}{48\cdots 41}a^{16}-\frac{38\cdots 14}{48\cdots 41}a^{15}-\frac{82\cdots 75}{48\cdots 41}a^{14}-\frac{31\cdots 88}{48\cdots 41}a^{13}+\frac{17\cdots 45}{48\cdots 41}a^{12}+\frac{43\cdots 26}{48\cdots 41}a^{11}-\frac{42\cdots 30}{48\cdots 41}a^{10}-\frac{75\cdots 90}{48\cdots 41}a^{9}-\frac{15\cdots 15}{48\cdots 41}a^{8}-\frac{29\cdots 34}{48\cdots 41}a^{7}+\frac{27\cdots 24}{48\cdots 41}a^{6}+\frac{67\cdots 28}{48\cdots 41}a^{5}-\frac{21\cdots 75}{48\cdots 41}a^{4}+\frac{21\cdots 94}{48\cdots 41}a^{3}-\frac{14\cdots 85}{48\cdots 41}a^{2}+\frac{20\cdots 50}{48\cdots 41}a+\frac{77\cdots 17}{48\cdots 41}$, $\frac{60\cdots 86}{47\cdots 77}a^{19}-\frac{31\cdots 56}{47\cdots 77}a^{18}-\frac{58\cdots 73}{47\cdots 77}a^{17}+\frac{44\cdots 52}{47\cdots 77}a^{16}+\frac{41\cdots 52}{47\cdots 77}a^{15}-\frac{21\cdots 18}{47\cdots 77}a^{14}-\frac{30\cdots 52}{47\cdots 77}a^{13}+\frac{34\cdots 48}{47\cdots 77}a^{12}+\frac{15\cdots 15}{47\cdots 77}a^{11}+\frac{10\cdots 01}{47\cdots 77}a^{10}-\frac{20\cdots 61}{47\cdots 77}a^{9}-\frac{25\cdots 66}{47\cdots 77}a^{8}-\frac{96\cdots 06}{47\cdots 77}a^{7}-\frac{35\cdots 23}{47\cdots 77}a^{6}+\frac{16\cdots 98}{47\cdots 77}a^{5}+\frac{15\cdots 71}{47\cdots 77}a^{4}+\frac{86\cdots 86}{47\cdots 77}a^{3}+\frac{71\cdots 30}{47\cdots 77}a^{2}-\frac{63\cdots 45}{47\cdots 77}a-\frac{18\cdots 93}{47\cdots 77}$, $\frac{89\cdots 29}{47\cdots 77}a^{19}-\frac{53\cdots 79}{47\cdots 77}a^{18}-\frac{36\cdots 17}{47\cdots 77}a^{17}+\frac{65\cdots 16}{47\cdots 77}a^{16}+\frac{60\cdots 47}{47\cdots 77}a^{15}-\frac{28\cdots 30}{47\cdots 77}a^{14}-\frac{17\cdots 52}{47\cdots 77}a^{13}+\frac{53\cdots 30}{47\cdots 77}a^{12}+\frac{16\cdots 12}{47\cdots 77}a^{11}+\frac{25\cdots 84}{47\cdots 77}a^{10}-\frac{24\cdots 46}{47\cdots 77}a^{9}-\frac{11\cdots 95}{47\cdots 77}a^{8}-\frac{14\cdots 78}{47\cdots 77}a^{7}-\frac{18\cdots 52}{47\cdots 77}a^{6}+\frac{22\cdots 38}{47\cdots 77}a^{5}-\frac{26\cdots 17}{47\cdots 77}a^{4}+\frac{91\cdots 48}{47\cdots 77}a^{3}-\frac{43\cdots 09}{47\cdots 77}a^{2}+\frac{68\cdots 48}{47\cdots 77}a-\frac{24\cdots 12}{47\cdots 77}$, $\frac{73\cdots 51}{47\cdots 77}a^{19}-\frac{34\cdots 95}{47\cdots 77}a^{18}-\frac{90\cdots 95}{47\cdots 77}a^{17}+\frac{51\cdots 03}{47\cdots 77}a^{16}+\frac{74\cdots 32}{47\cdots 77}a^{15}-\frac{24\cdots 75}{47\cdots 77}a^{14}-\frac{45\cdots 29}{47\cdots 77}a^{13}+\frac{29\cdots 32}{47\cdots 77}a^{12}+\frac{19\cdots 72}{47\cdots 77}a^{11}+\frac{19\cdots 67}{47\cdots 77}a^{10}-\frac{20\cdots 24}{47\cdots 77}a^{9}-\frac{36\cdots 73}{47\cdots 77}a^{8}-\frac{20\cdots 41}{47\cdots 77}a^{7}-\frac{16\cdots 82}{47\cdots 77}a^{6}+\frac{17\cdots 06}{47\cdots 77}a^{5}+\frac{21\cdots 66}{47\cdots 77}a^{4}+\frac{14\cdots 51}{47\cdots 77}a^{3}+\frac{68\cdots 84}{47\cdots 77}a^{2}-\frac{34\cdots 96}{47\cdots 77}a+\frac{46\cdots 07}{47\cdots 77}$, $\frac{24\cdots 28}{48\cdots 41}a^{19}-\frac{14\cdots 95}{48\cdots 41}a^{18}-\frac{10\cdots 10}{48\cdots 41}a^{17}+\frac{17\cdots 39}{48\cdots 41}a^{16}+\frac{85\cdots 70}{48\cdots 41}a^{15}-\frac{75\cdots 84}{48\cdots 41}a^{14}-\frac{51\cdots 48}{48\cdots 41}a^{13}+\frac{13\cdots 87}{48\cdots 41}a^{12}+\frac{44\cdots 14}{48\cdots 41}a^{11}+\frac{10\cdots 81}{48\cdots 41}a^{10}-\frac{63\cdots 38}{48\cdots 41}a^{9}-\frac{32\cdots 51}{48\cdots 41}a^{8}-\frac{48\cdots 12}{48\cdots 41}a^{7}-\frac{10\cdots 75}{48\cdots 41}a^{6}+\frac{58\cdots 84}{48\cdots 41}a^{5}-\frac{54\cdots 17}{48\cdots 41}a^{4}+\frac{28\cdots 84}{48\cdots 41}a^{3}-\frac{11\cdots 49}{48\cdots 41}a^{2}+\frac{22\cdots 64}{48\cdots 41}a-\frac{17\cdots 05}{48\cdots 41}$, $\frac{12\cdots 12}{48\cdots 41}a^{19}-\frac{85\cdots 74}{48\cdots 41}a^{18}+\frac{64\cdots 98}{48\cdots 41}a^{17}+\frac{10\cdots 70}{48\cdots 41}a^{16}-\frac{73\cdots 10}{48\cdots 41}a^{15}-\frac{43\cdots 05}{48\cdots 41}a^{14}+\frac{79\cdots 50}{48\cdots 41}a^{13}+\frac{11\cdots 50}{48\cdots 41}a^{12}+\frac{17\cdots 48}{48\cdots 41}a^{11}-\frac{18\cdots 38}{48\cdots 41}a^{10}-\frac{47\cdots 80}{48\cdots 41}a^{9}+\frac{11\cdots 04}{48\cdots 41}a^{8}+\frac{53\cdots 60}{48\cdots 41}a^{7}+\frac{21\cdots 39}{48\cdots 41}a^{6}+\frac{42\cdots 64}{48\cdots 41}a^{5}-\frac{28\cdots 69}{48\cdots 41}a^{4}+\frac{38\cdots 78}{48\cdots 41}a^{3}-\frac{14\cdots 08}{48\cdots 41}a^{2}+\frac{10\cdots 24}{48\cdots 41}a+\frac{54\cdots 75}{48\cdots 41}$, $\frac{34\cdots 06}{47\cdots 77}a^{19}-\frac{21\cdots 44}{47\cdots 77}a^{18}-\frac{10\cdots 11}{47\cdots 77}a^{17}+\frac{26\cdots 28}{47\cdots 77}a^{16}-\frac{44\cdots 43}{47\cdots 77}a^{15}-\frac{11\cdots 05}{47\cdots 77}a^{14}-\frac{46\cdots 45}{47\cdots 77}a^{13}+\frac{23\cdots 37}{47\cdots 77}a^{12}+\frac{59\cdots 86}{47\cdots 77}a^{11}-\frac{50\cdots 02}{47\cdots 77}a^{10}-\frac{10\cdots 53}{47\cdots 77}a^{9}-\frac{25\cdots 11}{47\cdots 77}a^{8}-\frac{36\cdots 48}{47\cdots 77}a^{7}+\frac{61\cdots 67}{47\cdots 77}a^{6}+\frac{93\cdots 69}{47\cdots 77}a^{5}-\frac{27\cdots 61}{47\cdots 77}a^{4}+\frac{25\cdots 46}{47\cdots 77}a^{3}-\frac{19\cdots 77}{47\cdots 77}a^{2}+\frac{25\cdots 26}{47\cdots 77}a+\frac{98\cdots 00}{47\cdots 77}$, $\frac{38\cdots 93}{47\cdots 77}a^{19}-\frac{23\cdots 51}{47\cdots 77}a^{18}-\frac{14\cdots 54}{47\cdots 77}a^{17}+\frac{28\cdots 40}{47\cdots 77}a^{16}-\frac{11\cdots 50}{47\cdots 77}a^{15}-\frac{12\cdots 13}{47\cdots 77}a^{14}-\frac{68\cdots 49}{47\cdots 77}a^{13}+\frac{24\cdots 31}{47\cdots 77}a^{12}+\frac{69\cdots 89}{47\cdots 77}a^{11}+\frac{54\cdots 81}{47\cdots 77}a^{10}-\frac{11\cdots 52}{47\cdots 77}a^{9}-\frac{41\cdots 87}{47\cdots 77}a^{8}-\frac{54\cdots 37}{47\cdots 77}a^{7}-\frac{49\cdots 93}{47\cdots 77}a^{6}+\frac{10\cdots 32}{47\cdots 77}a^{5}-\frac{16\cdots 30}{47\cdots 77}a^{4}+\frac{34\cdots 10}{47\cdots 77}a^{3}-\frac{18\cdots 06}{47\cdots 77}a^{2}+\frac{21\cdots 63}{47\cdots 77}a-\frac{72\cdots 99}{47\cdots 77}$, $\frac{37\cdots 40}{47\cdots 77}a^{19}-\frac{21\cdots 20}{47\cdots 77}a^{18}-\frac{19\cdots 90}{47\cdots 77}a^{17}+\frac{27\cdots 30}{47\cdots 77}a^{16}+\frac{62\cdots 33}{47\cdots 77}a^{15}-\frac{11\cdots 53}{47\cdots 77}a^{14}-\frac{98\cdots 50}{47\cdots 77}a^{13}+\frac{19\cdots 85}{47\cdots 77}a^{12}+\frac{71\cdots 32}{47\cdots 77}a^{11}+\frac{28\cdots 38}{47\cdots 77}a^{10}-\frac{91\cdots 61}{47\cdots 77}a^{9}-\frac{64\cdots 93}{47\cdots 77}a^{8}-\frac{87\cdots 33}{47\cdots 77}a^{7}-\frac{36\cdots 34}{47\cdots 77}a^{6}+\frac{82\cdots 21}{47\cdots 77}a^{5}+\frac{42\cdots 22}{47\cdots 77}a^{4}+\frac{45\cdots 45}{47\cdots 77}a^{3}-\frac{44\cdots 91}{47\cdots 77}a^{2}+\frac{20\cdots 78}{47\cdots 77}a-\frac{61\cdots 89}{47\cdots 77}$, $\frac{10\cdots 80}{47\cdots 77}a^{19}-\frac{61\cdots 19}{47\cdots 77}a^{18}-\frac{35\cdots 05}{47\cdots 77}a^{17}+\frac{76\cdots 18}{47\cdots 77}a^{16}-\frac{75\cdots 85}{47\cdots 77}a^{15}-\frac{33\cdots 30}{47\cdots 77}a^{14}-\frac{16\cdots 09}{47\cdots 77}a^{13}+\frac{71\cdots 72}{47\cdots 77}a^{12}+\frac{18\cdots 64}{47\cdots 77}a^{11}-\frac{92\cdots 92}{47\cdots 77}a^{10}-\frac{32\cdots 23}{47\cdots 77}a^{9}-\frac{99\cdots 56}{47\cdots 77}a^{8}-\frac{81\cdots 50}{47\cdots 77}a^{7}+\frac{34\cdots 39}{47\cdots 77}a^{6}+\frac{27\cdots 53}{47\cdots 77}a^{5}-\frac{63\cdots 86}{47\cdots 77}a^{4}+\frac{47\cdots 28}{47\cdots 77}a^{3}-\frac{55\cdots 32}{47\cdots 77}a^{2}+\frac{60\cdots 76}{47\cdots 77}a+\frac{11\cdots 05}{47\cdots 77}$ 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:  \( 28351202.0412 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 28351202.0412 \cdot 2}{2\cdot\sqrt{4599651543326335737037335822336}}\cr\approx \mathstrut & 0.513769052500 \end{aligned}\] (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 - 6*x^19 - 4*x^18 + 74*x^17 - 318*x^15 - 193*x^14 + 612*x^13 + 1819*x^12 + 254*x^11 - 2818*x^10 - 1246*x^9 - 1626*x^8 - 190*x^7 + 2594*x^6 - 300*x^5 + 982*x^4 - 472*x^3 + 29*x^2 - 18*x - 3) 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 - 6*x^19 - 4*x^18 + 74*x^17 - 318*x^15 - 193*x^14 + 612*x^13 + 1819*x^12 + 254*x^11 - 2818*x^10 - 1246*x^9 - 1626*x^8 - 190*x^7 + 2594*x^6 - 300*x^5 + 982*x^4 - 472*x^3 + 29*x^2 - 18*x - 3, 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 - 6*x^19 - 4*x^18 + 74*x^17 - 318*x^15 - 193*x^14 + 612*x^13 + 1819*x^12 + 254*x^11 - 2818*x^10 - 1246*x^9 - 1626*x^8 - 190*x^7 + 2594*x^6 - 300*x^5 + 982*x^4 - 472*x^3 + 29*x^2 - 18*x - 3); 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 - 6*x^19 - 4*x^18 + 74*x^17 - 318*x^15 - 193*x^14 + 612*x^13 + 1819*x^12 + 254*x^11 - 2818*x^10 - 1246*x^9 - 1626*x^8 - 190*x^7 + 2594*x^6 - 300*x^5 + 982*x^4 - 472*x^3 + 29*x^2 - 18*x - 3); 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

$C_2^9.C_2\wr D_5$ (as 20T853):

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 163840
The 280 conjugacy class representatives for $C_2^9.C_2\wr D_5$
Character table for $C_2^9.C_2\wr D_5$

Intermediate fields

5.5.160801.1, 10.8.238297758114816.4

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 20 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 20.2.4491847210279624743200523264.2

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 ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{3}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.10.0.1}{10} }^{2}$ ${\href{/padicField/47.10.0.1}{10} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.5.2.10a3.1$x^{10} + 4 x^{7} + 4 x^{5} + 3 x^{4} + 6 x^{2} + 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$$[2, 2, 2, 2, 2]^{5}$$
2.5.2.10a3.1$x^{10} + 4 x^{7} + 4 x^{5} + 3 x^{4} + 6 x^{2} + 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$$[2, 2, 2, 2, 2]^{5}$$
\(3\) Copy content Toggle raw display 3.1.2.1a1.1$x^{2} + 3$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.8.1.0a1.1$x^{8} + 2 x^{5} + x^{4} + 2 x^{2} + 2 x + 2$$1$$8$$0$$C_8$$$[\ ]^{8}$$
3.2.4.6a1.3$x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$$4$$2$$6$$Q_8$$$[\ ]_{4}^{2}$$
\(401\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$2$$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)