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Properties

Label 20.20.828...125.1
Degree $20$
Signature $(20, 0)$
Discriminant $8.280\times 10^{31}$
Root discriminant \(39.44\)
Ramified primes $3,5,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{20}$ (as 20T1)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 39*x^18 + 43*x^17 + 579*x^16 - 664*x^15 - 4199*x^14 + 4764*x^13 + 16062*x^12 - 17268*x^11 - 32674*x^10 + 31502*x^9 + 34487*x^8 - 26929*x^7 - 18180*x^6 + 9222*x^5 + 4671*x^4 - 978*x^3 - 468*x^2 - 21*x + 1)
 
Copy content gp:K = bnfinit(y^20 - y^19 - 39*y^18 + 43*y^17 + 579*y^16 - 664*y^15 - 4199*y^14 + 4764*y^13 + 16062*y^12 - 17268*y^11 - 32674*y^10 + 31502*y^9 + 34487*y^8 - 26929*y^7 - 18180*y^6 + 9222*y^5 + 4671*y^4 - 978*y^3 - 468*y^2 - 21*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 - 39*x^18 + 43*x^17 + 579*x^16 - 664*x^15 - 4199*x^14 + 4764*x^13 + 16062*x^12 - 17268*x^11 - 32674*x^10 + 31502*x^9 + 34487*x^8 - 26929*x^7 - 18180*x^6 + 9222*x^5 + 4671*x^4 - 978*x^3 - 468*x^2 - 21*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - x^19 - 39*x^18 + 43*x^17 + 579*x^16 - 664*x^15 - 4199*x^14 + 4764*x^13 + 16062*x^12 - 17268*x^11 - 32674*x^10 + 31502*x^9 + 34487*x^8 - 26929*x^7 - 18180*x^6 + 9222*x^5 + 4671*x^4 - 978*x^3 - 468*x^2 - 21*x + 1)
 

\( x^{20} - x^{19} - 39 x^{18} + 43 x^{17} + 579 x^{16} - 664 x^{15} - 4199 x^{14} + 4764 x^{13} + 16062 x^{12} + \cdots + 1 \) 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:  $20$
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:  $(20, 0)$
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:   \(82802905234194108120391845703125\) \(\medspace = 3^{10}\cdot 5^{15}\cdot 11^{16}\) 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:  \(39.44\)
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:  $3^{1/2}5^{3/4}11^{4/5}\approx 39.436855462307015$
Ramified primes:   \(3\), \(5\), \(11\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{20}$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(165=3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{165}(64,·)$, $\chi_{165}(1,·)$, $\chi_{165}(4,·)$, $\chi_{165}(136,·)$, $\chi_{165}(137,·)$, $\chi_{165}(16,·)$, $\chi_{165}(23,·)$, $\chi_{165}(152,·)$, $\chi_{165}(91,·)$, $\chi_{165}(92,·)$, $\chi_{165}(158,·)$, $\chi_{165}(31,·)$, $\chi_{165}(34,·)$, $\chi_{165}(38,·)$, $\chi_{165}(49,·)$, $\chi_{165}(47,·)$, $\chi_{165}(113,·)$, $\chi_{165}(53,·)$, $\chi_{165}(122,·)$, $\chi_{165}(124,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{20\cdots 31}a^{19}-\frac{63\cdots 01}{20\cdots 31}a^{18}+\frac{15\cdots 49}{20\cdots 31}a^{17}-\frac{99\cdots 43}{20\cdots 31}a^{16}-\frac{39\cdots 79}{20\cdots 31}a^{15}+\frac{45\cdots 78}{20\cdots 31}a^{14}-\frac{69\cdots 59}{20\cdots 31}a^{13}+\frac{32\cdots 52}{20\cdots 31}a^{12}+\frac{23\cdots 52}{20\cdots 31}a^{11}-\frac{30\cdots 35}{20\cdots 31}a^{10}-\frac{23\cdots 87}{20\cdots 31}a^{9}+\frac{13\cdots 04}{20\cdots 31}a^{8}-\frac{58\cdots 05}{20\cdots 31}a^{7}+\frac{76\cdots 82}{20\cdots 31}a^{6}+\frac{69\cdots 08}{20\cdots 31}a^{5}+\frac{10\cdots 04}{20\cdots 31}a^{4}-\frac{37\cdots 17}{20\cdots 31}a^{3}-\frac{39\cdots 14}{20\cdots 31}a^{2}-\frac{60\cdots 08}{20\cdots 31}a+\frac{65\cdots 69}{20\cdots 31}$ 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:  $C_{2}$, 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:  $19$
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:   $\frac{17\cdots 94}{14\cdots 39}a^{19}-\frac{21\cdots 77}{14\cdots 39}a^{18}-\frac{66\cdots 51}{14\cdots 39}a^{17}+\frac{88\cdots 77}{14\cdots 39}a^{16}+\frac{97\cdots 12}{14\cdots 39}a^{15}-\frac{13\cdots 09}{14\cdots 39}a^{14}-\frac{69\cdots 28}{14\cdots 39}a^{13}+\frac{96\cdots 87}{14\cdots 39}a^{12}+\frac{25\cdots 00}{14\cdots 39}a^{11}-\frac{34\cdots 21}{14\cdots 39}a^{10}-\frac{48\cdots 71}{14\cdots 39}a^{9}+\frac{63\cdots 61}{14\cdots 39}a^{8}+\frac{44\cdots 60}{14\cdots 39}a^{7}-\frac{53\cdots 12}{14\cdots 39}a^{6}-\frac{19\cdots 88}{14\cdots 39}a^{5}+\frac{17\cdots 75}{14\cdots 39}a^{4}+\frac{42\cdots 48}{14\cdots 39}a^{3}-\frac{18\cdots 88}{14\cdots 39}a^{2}-\frac{40\cdots 51}{14\cdots 39}a+\frac{40\cdots 05}{14\cdots 39}$, $\frac{65\cdots 50}{20\cdots 31}a^{19}+\frac{44\cdots 60}{20\cdots 31}a^{18}-\frac{25\cdots 90}{20\cdots 31}a^{17}-\frac{15\cdots 90}{20\cdots 31}a^{16}+\frac{40\cdots 30}{20\cdots 31}a^{15}+\frac{22\cdots 90}{20\cdots 31}a^{14}-\frac{31\cdots 80}{20\cdots 31}a^{13}-\frac{18\cdots 70}{20\cdots 31}a^{12}+\frac{13\cdots 80}{20\cdots 31}a^{11}+\frac{90\cdots 70}{20\cdots 31}a^{10}-\frac{31\cdots 10}{20\cdots 31}a^{9}-\frac{24\cdots 95}{20\cdots 31}a^{8}+\frac{40\cdots 30}{20\cdots 31}a^{7}+\frac{34\cdots 40}{20\cdots 31}a^{6}-\frac{28\cdots 76}{20\cdots 31}a^{5}-\frac{22\cdots 05}{20\cdots 31}a^{4}+\frac{10\cdots 70}{20\cdots 31}a^{3}+\frac{45\cdots 65}{20\cdots 31}a^{2}-\frac{17\cdots 65}{20\cdots 31}a-\frac{12\cdots 16}{20\cdots 31}$, $\frac{50\cdots 16}{20\cdots 31}a^{19}-\frac{51\cdots 71}{20\cdots 31}a^{18}-\frac{20\cdots 68}{20\cdots 31}a^{17}+\frac{22\cdots 76}{20\cdots 31}a^{16}+\frac{30\cdots 09}{20\cdots 31}a^{15}-\frac{35\cdots 56}{20\cdots 31}a^{14}-\frac{23\cdots 29}{20\cdots 31}a^{13}+\frac{26\cdots 58}{20\cdots 31}a^{12}+\frac{93\cdots 93}{20\cdots 31}a^{11}-\frac{10\cdots 60}{20\cdots 31}a^{10}-\frac{21\cdots 27}{20\cdots 31}a^{9}+\frac{20\cdots 88}{20\cdots 31}a^{8}+\frac{26\cdots 69}{20\cdots 31}a^{7}-\frac{21\cdots 15}{20\cdots 31}a^{6}-\frac{17\cdots 98}{20\cdots 31}a^{5}+\frac{10\cdots 19}{20\cdots 31}a^{4}+\frac{53\cdots 73}{20\cdots 31}a^{3}-\frac{15\cdots 67}{20\cdots 31}a^{2}-\frac{64\cdots 54}{20\cdots 31}a-\frac{15\cdots 68}{20\cdots 31}$, $\frac{50\cdots 16}{20\cdots 31}a^{19}-\frac{51\cdots 71}{20\cdots 31}a^{18}-\frac{20\cdots 68}{20\cdots 31}a^{17}+\frac{22\cdots 76}{20\cdots 31}a^{16}+\frac{30\cdots 09}{20\cdots 31}a^{15}-\frac{35\cdots 56}{20\cdots 31}a^{14}-\frac{23\cdots 29}{20\cdots 31}a^{13}+\frac{26\cdots 58}{20\cdots 31}a^{12}+\frac{93\cdots 93}{20\cdots 31}a^{11}-\frac{10\cdots 60}{20\cdots 31}a^{10}-\frac{21\cdots 27}{20\cdots 31}a^{9}+\frac{20\cdots 88}{20\cdots 31}a^{8}+\frac{26\cdots 69}{20\cdots 31}a^{7}-\frac{21\cdots 15}{20\cdots 31}a^{6}-\frac{17\cdots 98}{20\cdots 31}a^{5}+\frac{10\cdots 19}{20\cdots 31}a^{4}+\frac{53\cdots 73}{20\cdots 31}a^{3}-\frac{15\cdots 67}{20\cdots 31}a^{2}-\frac{64\cdots 54}{20\cdots 31}a-\frac{36\cdots 99}{20\cdots 31}$, $\frac{18\cdots 84}{20\cdots 31}a^{19}-\frac{22\cdots 24}{20\cdots 31}a^{18}-\frac{69\cdots 17}{20\cdots 31}a^{17}+\frac{96\cdots 34}{20\cdots 31}a^{16}+\frac{10\cdots 37}{20\cdots 31}a^{15}-\frac{14\cdots 79}{20\cdots 31}a^{14}-\frac{70\cdots 81}{20\cdots 31}a^{13}+\frac{10\cdots 17}{20\cdots 31}a^{12}+\frac{24\cdots 67}{20\cdots 31}a^{11}-\frac{37\cdots 67}{20\cdots 31}a^{10}-\frac{43\cdots 68}{20\cdots 31}a^{9}+\frac{65\cdots 42}{20\cdots 31}a^{8}+\frac{34\cdots 71}{20\cdots 31}a^{7}-\frac{52\cdots 35}{20\cdots 31}a^{6}-\frac{95\cdots 22}{20\cdots 31}a^{5}+\frac{15\cdots 96}{20\cdots 31}a^{4}+\frac{75\cdots 47}{20\cdots 31}a^{3}-\frac{14\cdots 58}{20\cdots 31}a^{2}-\frac{43\cdots 56}{20\cdots 31}a+\frac{18\cdots 62}{20\cdots 31}$, $\frac{76\cdots 86}{20\cdots 31}a^{19}-\frac{12\cdots 93}{20\cdots 31}a^{18}-\frac{29\cdots 39}{20\cdots 31}a^{17}+\frac{51\cdots 88}{20\cdots 31}a^{16}+\frac{41\cdots 88}{20\cdots 31}a^{15}-\frac{77\cdots 41}{20\cdots 31}a^{14}-\frac{28\cdots 32}{20\cdots 31}a^{13}+\frac{55\cdots 13}{20\cdots 31}a^{12}+\frac{97\cdots 20}{20\cdots 31}a^{11}-\frac{19\cdots 23}{20\cdots 31}a^{10}-\frac{16\cdots 09}{20\cdots 31}a^{9}+\frac{34\cdots 24}{20\cdots 31}a^{8}+\frac{11\cdots 00}{20\cdots 31}a^{7}-\frac{28\cdots 38}{20\cdots 31}a^{6}-\frac{32\cdots 70}{20\cdots 31}a^{5}+\frac{80\cdots 50}{20\cdots 31}a^{4}+\frac{14\cdots 62}{20\cdots 31}a^{3}-\frac{58\cdots 57}{20\cdots 31}a^{2}-\frac{34\cdots 90}{20\cdots 31}a-\frac{51\cdots 72}{20\cdots 31}$, $\frac{57\cdots 33}{20\cdots 31}a^{19}-\frac{62\cdots 33}{20\cdots 31}a^{18}-\frac{21\cdots 59}{20\cdots 31}a^{17}+\frac{26\cdots 29}{20\cdots 31}a^{16}+\frac{31\cdots 79}{20\cdots 31}a^{15}-\frac{39\cdots 28}{20\cdots 31}a^{14}-\frac{21\cdots 61}{20\cdots 31}a^{13}+\frac{26\cdots 60}{20\cdots 31}a^{12}+\frac{73\cdots 49}{20\cdots 31}a^{11}-\frac{86\cdots 27}{20\cdots 31}a^{10}-\frac{11\cdots 51}{20\cdots 31}a^{9}+\frac{12\cdots 81}{20\cdots 31}a^{8}+\frac{75\cdots 94}{20\cdots 31}a^{7}-\frac{54\cdots 90}{20\cdots 31}a^{6}-\frac{29\cdots 45}{20\cdots 31}a^{5}-\frac{21\cdots 06}{20\cdots 31}a^{4}-\frac{66\cdots 54}{20\cdots 31}a^{3}+\frac{84\cdots 15}{20\cdots 31}a^{2}+\frac{19\cdots 92}{20\cdots 31}a+\frac{19\cdots 97}{20\cdots 31}$, $\frac{22\cdots 24}{20\cdots 31}a^{19}-\frac{16\cdots 47}{20\cdots 31}a^{18}-\frac{84\cdots 29}{20\cdots 31}a^{17}+\frac{71\cdots 52}{20\cdots 31}a^{16}+\frac{12\cdots 84}{20\cdots 31}a^{15}-\frac{11\cdots 99}{20\cdots 31}a^{14}-\frac{85\cdots 88}{20\cdots 31}a^{13}+\frac{76\cdots 90}{20\cdots 31}a^{12}+\frac{30\cdots 82}{20\cdots 31}a^{11}-\frac{25\cdots 79}{20\cdots 31}a^{10}-\frac{52\cdots 76}{20\cdots 31}a^{9}+\frac{39\cdots 57}{20\cdots 31}a^{8}+\frac{38\cdots 86}{20\cdots 31}a^{7}-\frac{21\cdots 93}{20\cdots 31}a^{6}-\frac{66\cdots 94}{20\cdots 31}a^{5}-\frac{32\cdots 55}{20\cdots 31}a^{4}-\frac{14\cdots 85}{20\cdots 31}a^{3}+\frac{29\cdots 58}{20\cdots 31}a^{2}+\frac{52\cdots 72}{20\cdots 31}a-\frac{17\cdots 76}{20\cdots 31}$, $\frac{21\cdots 58}{20\cdots 31}a^{19}-\frac{16\cdots 31}{20\cdots 31}a^{18}-\frac{84\cdots 89}{20\cdots 31}a^{17}+\frac{73\cdots 75}{20\cdots 31}a^{16}+\frac{12\cdots 98}{20\cdots 31}a^{15}-\frac{11\cdots 52}{20\cdots 31}a^{14}-\frac{92\cdots 39}{20\cdots 31}a^{13}+\frac{82\cdots 06}{20\cdots 31}a^{12}+\frac{36\cdots 99}{20\cdots 31}a^{11}-\frac{29\cdots 66}{20\cdots 31}a^{10}-\frac{75\cdots 01}{20\cdots 31}a^{9}+\frac{53\cdots 92}{20\cdots 31}a^{8}+\frac{82\cdots 80}{20\cdots 31}a^{7}-\frac{43\cdots 50}{20\cdots 31}a^{6}-\frac{43\cdots 30}{20\cdots 31}a^{5}+\frac{13\cdots 03}{20\cdots 31}a^{4}+\frac{97\cdots 74}{20\cdots 31}a^{3}-\frac{11\cdots 80}{20\cdots 31}a^{2}-\frac{75\cdots 57}{20\cdots 31}a-\frac{45\cdots 55}{20\cdots 31}$, $\frac{19\cdots 10}{20\cdots 31}a^{19}-\frac{24\cdots 62}{20\cdots 31}a^{18}-\frac{74\cdots 11}{20\cdots 31}a^{17}+\frac{10\cdots 57}{20\cdots 31}a^{16}+\frac{10\cdots 39}{20\cdots 31}a^{15}-\frac{15\cdots 05}{20\cdots 31}a^{14}-\frac{75\cdots 83}{20\cdots 31}a^{13}+\frac{11\cdots 65}{20\cdots 31}a^{12}+\frac{26\cdots 07}{20\cdots 31}a^{11}-\frac{39\cdots 49}{20\cdots 31}a^{10}-\frac{47\cdots 32}{20\cdots 31}a^{9}+\frac{69\cdots 81}{20\cdots 31}a^{8}+\frac{37\cdots 71}{20\cdots 31}a^{7}-\frac{54\cdots 33}{20\cdots 31}a^{6}-\frac{10\cdots 54}{20\cdots 31}a^{5}+\frac{14\cdots 56}{20\cdots 31}a^{4}+\frac{68\cdots 19}{20\cdots 31}a^{3}-\frac{10\cdots 85}{20\cdots 31}a^{2}+\frac{65\cdots 75}{20\cdots 31}a+\frac{21\cdots 13}{20\cdots 31}$, $\frac{15\cdots 94}{20\cdots 31}a^{19}-\frac{90\cdots 84}{20\cdots 31}a^{18}-\frac{53\cdots 37}{20\cdots 31}a^{17}+\frac{35\cdots 79}{20\cdots 31}a^{16}+\frac{57\cdots 67}{20\cdots 31}a^{15}-\frac{52\cdots 09}{20\cdots 31}a^{14}-\frac{14\cdots 21}{20\cdots 31}a^{13}+\frac{37\cdots 77}{20\cdots 31}a^{12}-\frac{10\cdots 61}{20\cdots 31}a^{11}-\frac{13\cdots 71}{20\cdots 31}a^{10}+\frac{76\cdots 62}{20\cdots 31}a^{9}+\frac{24\cdots 72}{20\cdots 31}a^{8}-\frac{17\cdots 99}{20\cdots 31}a^{7}-\frac{21\cdots 01}{20\cdots 31}a^{6}+\frac{14\cdots 46}{20\cdots 31}a^{5}+\frac{76\cdots 96}{20\cdots 31}a^{4}-\frac{39\cdots 03}{20\cdots 31}a^{3}-\frac{88\cdots 59}{20\cdots 31}a^{2}+\frac{24\cdots 77}{20\cdots 31}a+\frac{19\cdots 03}{20\cdots 31}$, $\frac{18\cdots 34}{20\cdots 31}a^{19}-\frac{26\cdots 24}{20\cdots 31}a^{18}-\frac{70\cdots 77}{20\cdots 31}a^{17}+\frac{10\cdots 24}{20\cdots 31}a^{16}+\frac{10\cdots 27}{20\cdots 31}a^{15}-\frac{16\cdots 29}{20\cdots 31}a^{14}-\frac{71\cdots 01}{20\cdots 31}a^{13}+\frac{11\cdots 87}{20\cdots 31}a^{12}+\frac{25\cdots 19}{20\cdots 31}a^{11}-\frac{43\cdots 57}{20\cdots 31}a^{10}-\frac{44\cdots 88}{20\cdots 31}a^{9}+\frac{79\cdots 17}{20\cdots 31}a^{8}+\frac{35\cdots 41}{20\cdots 31}a^{7}-\frac{69\cdots 11}{20\cdots 31}a^{6}-\frac{94\cdots 36}{20\cdots 31}a^{5}+\frac{24\cdots 21}{20\cdots 31}a^{4}+\frac{59\cdots 27}{20\cdots 31}a^{3}-\frac{29\cdots 54}{20\cdots 31}a^{2}-\frac{14\cdots 43}{20\cdots 31}a+\frac{34\cdots 58}{20\cdots 31}$, $\frac{86\cdots 58}{20\cdots 31}a^{19}-\frac{11\cdots 25}{20\cdots 31}a^{18}-\frac{33\cdots 63}{20\cdots 31}a^{17}+\frac{47\cdots 33}{20\cdots 31}a^{16}+\frac{47\cdots 34}{20\cdots 31}a^{15}-\frac{72\cdots 38}{20\cdots 31}a^{14}-\frac{32\cdots 44}{20\cdots 31}a^{13}+\frac{50\cdots 11}{20\cdots 31}a^{12}+\frac{10\cdots 11}{20\cdots 31}a^{11}-\frac{17\cdots 85}{20\cdots 31}a^{10}-\frac{16\cdots 53}{20\cdots 31}a^{9}+\frac{30\cdots 23}{20\cdots 31}a^{8}+\frac{93\cdots 86}{20\cdots 31}a^{7}-\frac{23\cdots 16}{20\cdots 31}a^{6}+\frac{14\cdots 90}{20\cdots 31}a^{5}+\frac{54\cdots 09}{20\cdots 31}a^{4}-\frac{15\cdots 34}{20\cdots 31}a^{3}-\frac{80\cdots 21}{20\cdots 31}a^{2}+\frac{20\cdots 89}{20\cdots 31}a-\frac{24\cdots 41}{20\cdots 31}$, $\frac{25\cdots 70}{20\cdots 31}a^{19}-\frac{74\cdots 22}{20\cdots 31}a^{18}-\frac{92\cdots 71}{20\cdots 31}a^{17}+\frac{29\cdots 12}{20\cdots 31}a^{16}+\frac{11\cdots 79}{20\cdots 31}a^{15}-\frac{42\cdots 85}{20\cdots 31}a^{14}-\frac{54\cdots 03}{20\cdots 31}a^{13}+\frac{28\cdots 55}{20\cdots 31}a^{12}+\frac{32\cdots 27}{20\cdots 31}a^{11}-\frac{95\cdots 63}{20\cdots 31}a^{10}+\frac{51\cdots 18}{20\cdots 31}a^{9}+\frac{14\cdots 36}{20\cdots 31}a^{8}-\frac{14\cdots 69}{20\cdots 31}a^{7}-\frac{65\cdots 23}{20\cdots 31}a^{6}+\frac{13\cdots 28}{20\cdots 31}a^{5}-\frac{20\cdots 69}{20\cdots 31}a^{4}-\frac{38\cdots 11}{20\cdots 31}a^{3}+\frac{97\cdots 10}{20\cdots 31}a^{2}+\frac{30\cdots 64}{20\cdots 31}a-\frac{14\cdots 73}{20\cdots 31}$, $\frac{97\cdots 74}{20\cdots 31}a^{19}-\frac{13\cdots 51}{20\cdots 31}a^{18}-\frac{37\cdots 17}{20\cdots 31}a^{17}+\frac{56\cdots 17}{20\cdots 31}a^{16}+\frac{54\cdots 06}{20\cdots 31}a^{15}-\frac{85\cdots 60}{20\cdots 31}a^{14}-\frac{37\cdots 67}{20\cdots 31}a^{13}+\frac{61\cdots 15}{20\cdots 31}a^{12}+\frac{13\cdots 49}{20\cdots 31}a^{11}-\frac{21\cdots 03}{20\cdots 31}a^{10}-\frac{24\cdots 08}{20\cdots 31}a^{9}+\frac{39\cdots 65}{20\cdots 31}a^{8}+\frac{20\cdots 93}{20\cdots 31}a^{7}-\frac{32\cdots 24}{20\cdots 31}a^{6}-\frac{74\cdots 53}{20\cdots 31}a^{5}+\frac{10\cdots 60}{20\cdots 31}a^{4}+\frac{19\cdots 66}{20\cdots 31}a^{3}-\frac{11\cdots 39}{20\cdots 31}a^{2}-\frac{28\cdots 27}{20\cdots 31}a+\frac{31\cdots 65}{20\cdots 31}$, $a$, $\frac{15\cdots 74}{20\cdots 31}a^{19}-\frac{21\cdots 94}{20\cdots 31}a^{18}-\frac{61\cdots 93}{20\cdots 31}a^{17}+\frac{88\cdots 74}{20\cdots 31}a^{16}+\frac{88\cdots 37}{20\cdots 31}a^{15}-\frac{13\cdots 79}{20\cdots 31}a^{14}-\frac{61\cdots 01}{20\cdots 31}a^{13}+\frac{94\cdots 28}{20\cdots 31}a^{12}+\frac{21\cdots 38}{20\cdots 31}a^{11}-\frac{33\cdots 67}{20\cdots 31}a^{10}-\frac{38\cdots 68}{20\cdots 31}a^{9}+\frac{59\cdots 02}{20\cdots 31}a^{8}+\frac{29\cdots 88}{20\cdots 31}a^{7}-\frac{46\cdots 60}{20\cdots 31}a^{6}-\frac{81\cdots 48}{20\cdots 31}a^{5}+\frac{12\cdots 01}{20\cdots 31}a^{4}+\frac{11\cdots 32}{20\cdots 31}a^{3}-\frac{80\cdots 92}{20\cdots 31}a^{2}-\frac{17\cdots 41}{20\cdots 31}a-\frac{14\cdots 00}{20\cdots 31}$, $\frac{82\cdots 56}{20\cdots 31}a^{19}-\frac{16\cdots 35}{20\cdots 31}a^{18}-\frac{33\cdots 04}{20\cdots 31}a^{17}+\frac{66\cdots 85}{20\cdots 31}a^{16}+\frac{50\cdots 28}{20\cdots 31}a^{15}-\frac{10\cdots 56}{20\cdots 31}a^{14}-\frac{38\cdots 34}{20\cdots 31}a^{13}+\frac{71\cdots 05}{20\cdots 31}a^{12}+\frac{15\cdots 13}{20\cdots 31}a^{11}-\frac{26\cdots 51}{20\cdots 31}a^{10}-\frac{35\cdots 14}{20\cdots 31}a^{9}+\frac{48\cdots 83}{20\cdots 31}a^{8}+\frac{40\cdots 55}{20\cdots 31}a^{7}-\frac{42\cdots 98}{20\cdots 31}a^{6}-\frac{17\cdots 63}{20\cdots 31}a^{5}+\frac{15\cdots 72}{20\cdots 31}a^{4}-\frac{53\cdots 00}{20\cdots 31}a^{3}-\frac{25\cdots 02}{20\cdots 31}a^{2}+\frac{10\cdots 33}{20\cdots 31}a+\frac{17\cdots 20}{20\cdots 31}$, $\frac{22\cdots 79}{20\cdots 31}a^{19}-\frac{32\cdots 81}{20\cdots 31}a^{18}-\frac{86\cdots 05}{20\cdots 31}a^{17}+\frac{13\cdots 41}{20\cdots 31}a^{16}+\frac{12\cdots 24}{20\cdots 31}a^{15}-\frac{20\cdots 03}{20\cdots 31}a^{14}-\frac{84\cdots 65}{20\cdots 31}a^{13}+\frac{14\cdots 91}{20\cdots 31}a^{12}+\frac{29\cdots 30}{20\cdots 31}a^{11}-\frac{51\cdots 15}{20\cdots 31}a^{10}-\frac{49\cdots 19}{20\cdots 31}a^{9}+\frac{91\cdots 57}{20\cdots 31}a^{8}+\frac{34\cdots 61}{20\cdots 31}a^{7}-\frac{73\cdots 50}{20\cdots 31}a^{6}-\frac{57\cdots 84}{20\cdots 31}a^{5}+\frac{21\cdots 17}{20\cdots 31}a^{4}-\frac{45\cdots 45}{20\cdots 31}a^{3}-\frac{19\cdots 78}{20\cdots 31}a^{2}+\frac{55\cdots 49}{20\cdots 31}a+\frac{97\cdots 30}{20\cdots 31}$ 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:  \( 2427056473.64 \) (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:  \( 19 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 2427056473.64 \cdot 1}{2\cdot\sqrt{82802905234194108120391845703125}}\cr\approx \mathstrut & 0.139838582178 \end{aligned}\] (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 - x^19 - 39*x^18 + 43*x^17 + 579*x^16 - 664*x^15 - 4199*x^14 + 4764*x^13 + 16062*x^12 - 17268*x^11 - 32674*x^10 + 31502*x^9 + 34487*x^8 - 26929*x^7 - 18180*x^6 + 9222*x^5 + 4671*x^4 - 978*x^3 - 468*x^2 - 21*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^20 - x^19 - 39*x^18 + 43*x^17 + 579*x^16 - 664*x^15 - 4199*x^14 + 4764*x^13 + 16062*x^12 - 17268*x^11 - 32674*x^10 + 31502*x^9 + 34487*x^8 - 26929*x^7 - 18180*x^6 + 9222*x^5 + 4671*x^4 - 978*x^3 - 468*x^2 - 21*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^20 - x^19 - 39*x^18 + 43*x^17 + 579*x^16 - 664*x^15 - 4199*x^14 + 4764*x^13 + 16062*x^12 - 17268*x^11 - 32674*x^10 + 31502*x^9 + 34487*x^8 - 26929*x^7 - 18180*x^6 + 9222*x^5 + 4671*x^4 - 978*x^3 - 468*x^2 - 21*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^20 - x^19 - 39*x^18 + 43*x^17 + 579*x^16 - 664*x^15 - 4199*x^14 + 4764*x^13 + 16062*x^12 - 17268*x^11 - 32674*x^10 + 31502*x^9 + 34487*x^8 - 26929*x^7 - 18180*x^6 + 9222*x^5 + 4671*x^4 - 978*x^3 - 468*x^2 - 21*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

$C_{20}$ (as 20T1):

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 cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $20$ R R $20$ R $20$ $20$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{5}$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

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
\(3\) Copy content Toggle raw display 3.10.2.10a1.1$x^{20} + 4 x^{16} + 4 x^{15} + 4 x^{14} + 4 x^{12} + 10 x^{11} + 16 x^{10} + 8 x^{9} + 4 x^{8} + 4 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + x^{2} + 7 x + 4$$2$$10$$10$20T1$$[\ ]_{2}^{10}$$
\(5\) Copy content Toggle raw display 5.5.4.15a1.2$x^{20} + 16 x^{16} + 12 x^{15} + 96 x^{12} + 144 x^{11} + 54 x^{10} + 256 x^{8} + 576 x^{7} + 432 x^{6} + 108 x^{5} + 256 x^{4} + 768 x^{3} + 869 x^{2} + 432 x + 81$$4$$5$$15$20T1not computed
\(11\) Copy content Toggle raw display 11.2.5.8a1.2$x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31367 x^{5} + 29970 x^{4} + 14840 x^{3} + 4000 x^{2} + 560 x + 43$$5$$2$$8$$C_{10}$$$[\ ]_{5}^{2}$$
11.2.5.8a1.2$x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31367 x^{5} + 29970 x^{4} + 14840 x^{3} + 4000 x^{2} + 560 x + 43$$5$$2$$8$$C_{10}$$$[\ ]_{5}^{2}$$

Spectrum of ring of integers

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