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Properties

Label 20.20.191...125.1
Degree $20$
Signature $(20, 0)$
Discriminant $1.918\times 10^{34}$
Root discriminant \(51.78\)
Ramified primes $3,5,239$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4\times D_5$ (as 20T6)

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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 - 8*x^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429)
 
Copy content gp:K = bnfinit(y^20 - 8*y^19 - 20*y^18 + 284*y^17 - 21*y^16 - 4198*y^15 + 3737*y^14 + 33772*y^13 - 42360*y^12 - 162403*y^11 + 230822*y^10 + 483926*y^9 - 710001*y^8 - 898678*y^7 + 1266895*y^6 + 1017051*y^5 - 1261123*y^4 - 651709*y^3 + 616317*y^2 + 185386*y - 101429, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 8*x^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 8*x^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429)
 

\( x^{20} - 8 x^{19} - 20 x^{18} + 284 x^{17} - 21 x^{16} - 4198 x^{15} + 3737 x^{14} + 33772 x^{13} + \cdots - 101429 \) 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:   \(19184293930982457734868438720703125\) \(\medspace = 3^{10}\cdot 5^{15}\cdot 239^{8}\) 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:  \(51.78\)
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:  $3^{1/2}5^{3/4}239^{1/2}\approx 89.53381316204927$
Ramified primes:   \(3\), \(5\), \(239\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{239}a^{17}-\frac{22}{239}a^{16}-\frac{74}{239}a^{15}-\frac{68}{239}a^{14}-\frac{40}{239}a^{13}+\frac{89}{239}a^{12}-\frac{41}{239}a^{11}+\frac{22}{239}a^{10}+\frac{7}{239}a^{9}+\frac{18}{239}a^{8}+\frac{65}{239}a^{7}-\frac{44}{239}a^{6}+\frac{18}{239}a^{5}+\frac{62}{239}a^{3}+\frac{114}{239}a^{2}-\frac{60}{239}a-\frac{3}{239}$, $\frac{1}{207691}a^{18}-\frac{94}{207691}a^{17}-\frac{7333}{207691}a^{16}+\frac{26531}{207691}a^{15}-\frac{558}{18881}a^{14}-\frac{64668}{207691}a^{13}-\frac{87231}{207691}a^{12}-\frac{95972}{207691}a^{11}+\frac{27581}{207691}a^{10}-\frac{77444}{207691}a^{9}+\frac{31990}{207691}a^{8}+\frac{37340}{207691}a^{7}-\frac{51545}{207691}a^{6}-\frac{55310}{207691}a^{5}-\frac{18819}{207691}a^{4}+\frac{4428}{18881}a^{3}-\frac{2077}{18881}a^{2}-\frac{16954}{207691}a-\frac{90604}{207691}$, $\frac{1}{11\cdots 59}a^{19}-\frac{17\cdots 60}{11\cdots 59}a^{18}-\frac{56\cdots 88}{10\cdots 69}a^{17}+\frac{23\cdots 83}{11\cdots 59}a^{16}+\frac{43\cdots 09}{11\cdots 59}a^{15}+\frac{50\cdots 20}{11\cdots 59}a^{14}-\frac{78\cdots 89}{62\cdots 61}a^{13}-\frac{51\cdots 84}{11\cdots 59}a^{12}-\frac{33\cdots 64}{11\cdots 59}a^{11}+\frac{26\cdots 91}{11\cdots 59}a^{10}+\frac{20\cdots 86}{11\cdots 59}a^{9}-\frac{12\cdots 21}{10\cdots 69}a^{8}+\frac{54\cdots 27}{11\cdots 59}a^{7}-\frac{46\cdots 96}{11\cdots 59}a^{6}-\frac{51\cdots 89}{11\cdots 59}a^{5}+\frac{96\cdots 23}{11\cdots 59}a^{4}+\frac{41\cdots 47}{10\cdots 69}a^{3}-\frac{26\cdots 05}{11\cdots 59}a^{2}-\frac{35\cdots 86}{11\cdots 59}a+\frac{37\cdots 91}{11\cdots 59}$ 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:  $19$
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:   $\frac{18\cdots 31}{20\cdots 01}a^{19}-\frac{11\cdots 13}{20\cdots 01}a^{18}-\frac{56\cdots 50}{20\cdots 01}a^{17}+\frac{42\cdots 15}{20\cdots 01}a^{16}+\frac{67\cdots 73}{20\cdots 01}a^{15}-\frac{65\cdots 79}{20\cdots 01}a^{14}-\frac{41\cdots 68}{20\cdots 01}a^{13}+\frac{55\cdots 45}{20\cdots 01}a^{12}+\frac{14\cdots 57}{20\cdots 01}a^{11}-\frac{27\cdots 02}{20\cdots 01}a^{10}-\frac{34\cdots 90}{20\cdots 01}a^{9}+\frac{83\cdots 09}{20\cdots 01}a^{8}+\frac{85\cdots 62}{20\cdots 01}a^{7}-\frac{15\cdots 48}{20\cdots 01}a^{6}-\frac{19\cdots 83}{20\cdots 01}a^{5}+\frac{15\cdots 47}{20\cdots 01}a^{4}+\frac{26\cdots 81}{20\cdots 01}a^{3}-\frac{76\cdots 01}{20\cdots 01}a^{2}-\frac{13\cdots 44}{20\cdots 01}a+\frac{11\cdots 53}{20\cdots 01}$, $\frac{64\cdots 11}{10\cdots 69}a^{19}-\frac{45\cdots 75}{11\cdots 59}a^{18}-\frac{21\cdots 63}{11\cdots 59}a^{17}+\frac{16\cdots 14}{11\cdots 59}a^{16}+\frac{26\cdots 44}{11\cdots 59}a^{15}-\frac{23\cdots 67}{10\cdots 69}a^{14}-\frac{86\cdots 29}{62\cdots 61}a^{13}+\frac{21\cdots 37}{11\cdots 59}a^{12}+\frac{57\cdots 51}{11\cdots 59}a^{11}-\frac{10\cdots 19}{11\cdots 59}a^{10}-\frac{14\cdots 46}{11\cdots 59}a^{9}+\frac{32\cdots 82}{11\cdots 59}a^{8}+\frac{35\cdots 75}{11\cdots 59}a^{7}-\frac{58\cdots 71}{11\cdots 59}a^{6}-\frac{78\cdots 86}{11\cdots 59}a^{5}+\frac{60\cdots 31}{11\cdots 59}a^{4}+\frac{94\cdots 01}{10\cdots 69}a^{3}-\frac{26\cdots 89}{10\cdots 69}a^{2}-\frac{53\cdots 41}{11\cdots 59}a+\frac{43\cdots 25}{11\cdots 59}$, $\frac{22\cdots 49}{10\cdots 69}a^{19}-\frac{16\cdots 29}{11\cdots 59}a^{18}-\frac{77\cdots 58}{11\cdots 59}a^{17}+\frac{59\cdots 62}{11\cdots 59}a^{16}+\frac{92\cdots 88}{11\cdots 59}a^{15}-\frac{82\cdots 11}{10\cdots 69}a^{14}-\frac{29\cdots 90}{62\cdots 61}a^{13}+\frac{76\cdots 15}{11\cdots 59}a^{12}+\frac{18\cdots 07}{11\cdots 59}a^{11}-\frac{38\cdots 39}{11\cdots 59}a^{10}-\frac{43\cdots 03}{11\cdots 59}a^{9}+\frac{11\cdots 68}{11\cdots 59}a^{8}+\frac{10\cdots 14}{11\cdots 59}a^{7}-\frac{21\cdots 22}{11\cdots 59}a^{6}-\frac{24\cdots 41}{11\cdots 59}a^{5}+\frac{21\cdots 77}{11\cdots 59}a^{4}+\frac{31\cdots 23}{10\cdots 69}a^{3}-\frac{96\cdots 57}{10\cdots 69}a^{2}-\frac{18\cdots 19}{11\cdots 59}a+\frac{15\cdots 14}{11\cdots 59}$, $\frac{23\cdots 03}{11\cdots 59}a^{19}-\frac{14\cdots 61}{11\cdots 59}a^{18}-\frac{71\cdots 54}{11\cdots 59}a^{17}+\frac{54\cdots 66}{11\cdots 59}a^{16}+\frac{86\cdots 19}{11\cdots 59}a^{15}-\frac{83\cdots 70}{11\cdots 59}a^{14}-\frac{28\cdots 22}{62\cdots 61}a^{13}+\frac{69\cdots 12}{11\cdots 59}a^{12}+\frac{19\cdots 34}{11\cdots 59}a^{11}-\frac{34\cdots 40}{11\cdots 59}a^{10}-\frac{49\cdots 62}{11\cdots 59}a^{9}+\frac{10\cdots 51}{11\cdots 59}a^{8}+\frac{12\cdots 93}{11\cdots 59}a^{7}-\frac{18\cdots 80}{11\cdots 59}a^{6}-\frac{27\cdots 39}{11\cdots 59}a^{5}+\frac{19\cdots 68}{11\cdots 59}a^{4}+\frac{31\cdots 96}{10\cdots 69}a^{3}-\frac{95\cdots 26}{11\cdots 59}a^{2}-\frac{17\cdots 42}{11\cdots 59}a+\frac{14\cdots 73}{11\cdots 59}$, $\frac{84\cdots 95}{11\cdots 59}a^{19}-\frac{53\cdots 18}{11\cdots 59}a^{18}-\frac{25\cdots 04}{11\cdots 59}a^{17}+\frac{19\cdots 52}{11\cdots 59}a^{16}+\frac{30\cdots 30}{11\cdots 59}a^{15}-\frac{30\cdots 97}{11\cdots 59}a^{14}-\frac{98\cdots 16}{62\cdots 61}a^{13}+\frac{25\cdots 74}{11\cdots 59}a^{12}+\frac{63\cdots 00}{11\cdots 59}a^{11}-\frac{12\cdots 22}{11\cdots 59}a^{10}-\frac{14\cdots 96}{11\cdots 59}a^{9}+\frac{38\cdots 34}{11\cdots 59}a^{8}+\frac{36\cdots 57}{11\cdots 59}a^{7}-\frac{69\cdots 41}{11\cdots 59}a^{6}-\frac{87\cdots 38}{11\cdots 59}a^{5}+\frac{71\cdots 15}{11\cdots 59}a^{4}+\frac{10\cdots 73}{10\cdots 69}a^{3}-\frac{35\cdots 30}{11\cdots 59}a^{2}-\frac{63\cdots 65}{11\cdots 59}a+\frac{51\cdots 98}{11\cdots 59}$, $\frac{15\cdots 10}{11\cdots 59}a^{19}-\frac{10\cdots 83}{11\cdots 59}a^{18}-\frac{49\cdots 65}{11\cdots 59}a^{17}+\frac{37\cdots 86}{11\cdots 59}a^{16}+\frac{59\cdots 03}{11\cdots 59}a^{15}-\frac{57\cdots 45}{11\cdots 59}a^{14}-\frac{17\cdots 16}{57\cdots 51}a^{13}+\frac{47\cdots 74}{11\cdots 59}a^{12}+\frac{13\cdots 43}{11\cdots 59}a^{11}-\frac{21\cdots 56}{10\cdots 69}a^{10}-\frac{33\cdots 37}{11\cdots 59}a^{9}+\frac{72\cdots 04}{11\cdots 59}a^{8}+\frac{82\cdots 35}{11\cdots 59}a^{7}-\frac{13\cdots 21}{11\cdots 59}a^{6}-\frac{16\cdots 65}{10\cdots 69}a^{5}+\frac{13\cdots 21}{11\cdots 59}a^{4}+\frac{21\cdots 61}{10\cdots 69}a^{3}-\frac{66\cdots 79}{11\cdots 59}a^{2}-\frac{11\cdots 10}{10\cdots 69}a+\frac{98\cdots 42}{11\cdots 59}$, $\frac{38\cdots 32}{10\cdots 69}a^{19}-\frac{27\cdots 49}{11\cdots 59}a^{18}-\frac{13\cdots 91}{11\cdots 59}a^{17}+\frac{99\cdots 75}{11\cdots 59}a^{16}+\frac{15\cdots 88}{11\cdots 59}a^{15}-\frac{13\cdots 02}{10\cdots 69}a^{14}-\frac{51\cdots 58}{62\cdots 61}a^{13}+\frac{12\cdots 86}{11\cdots 59}a^{12}+\frac{34\cdots 19}{11\cdots 59}a^{11}-\frac{63\cdots 98}{11\cdots 59}a^{10}-\frac{87\cdots 38}{11\cdots 59}a^{9}+\frac{19\cdots 83}{11\cdots 59}a^{8}+\frac{21\cdots 82}{11\cdots 59}a^{7}-\frac{35\cdots 44}{11\cdots 59}a^{6}-\frac{47\cdots 32}{11\cdots 59}a^{5}+\frac{36\cdots 14}{11\cdots 59}a^{4}+\frac{56\cdots 71}{10\cdots 69}a^{3}-\frac{16\cdots 51}{10\cdots 69}a^{2}-\frac{32\cdots 64}{11\cdots 59}a+\frac{26\cdots 39}{11\cdots 59}$, $\frac{19\cdots 63}{11\cdots 59}a^{19}-\frac{12\cdots 27}{11\cdots 59}a^{18}-\frac{59\cdots 90}{11\cdots 59}a^{17}+\frac{44\cdots 86}{11\cdots 59}a^{16}+\frac{73\cdots 13}{11\cdots 59}a^{15}-\frac{68\cdots 94}{11\cdots 59}a^{14}-\frac{24\cdots 82}{62\cdots 61}a^{13}+\frac{57\cdots 61}{11\cdots 59}a^{12}+\frac{17\cdots 85}{11\cdots 59}a^{11}-\frac{28\cdots 63}{11\cdots 59}a^{10}-\frac{48\cdots 04}{11\cdots 59}a^{9}+\frac{87\cdots 00}{11\cdots 59}a^{8}+\frac{12\cdots 57}{11\cdots 59}a^{7}-\frac{14\cdots 65}{10\cdots 69}a^{6}-\frac{25\cdots 62}{11\cdots 59}a^{5}+\frac{16\cdots 07}{11\cdots 59}a^{4}+\frac{28\cdots 07}{10\cdots 69}a^{3}-\frac{81\cdots 38}{11\cdots 59}a^{2}-\frac{15\cdots 60}{11\cdots 59}a+\frac{12\cdots 52}{11\cdots 59}$, $\frac{12\cdots 34}{11\cdots 59}a^{19}-\frac{76\cdots 14}{11\cdots 59}a^{18}-\frac{37\cdots 34}{11\cdots 59}a^{17}+\frac{28\cdots 73}{11\cdots 59}a^{16}+\frac{44\cdots 81}{11\cdots 59}a^{15}-\frac{43\cdots 54}{11\cdots 59}a^{14}-\frac{14\cdots 76}{62\cdots 61}a^{13}+\frac{36\cdots 61}{11\cdots 59}a^{12}+\frac{95\cdots 01}{11\cdots 59}a^{11}-\frac{18\cdots 27}{11\cdots 59}a^{10}-\frac{23\cdots 49}{11\cdots 59}a^{9}+\frac{54\cdots 99}{11\cdots 59}a^{8}+\frac{56\cdots 43}{11\cdots 59}a^{7}-\frac{90\cdots 07}{10\cdots 69}a^{6}-\frac{12\cdots 36}{11\cdots 59}a^{5}+\frac{10\cdots 09}{11\cdots 59}a^{4}+\frac{15\cdots 58}{10\cdots 69}a^{3}-\frac{49\cdots 14}{11\cdots 59}a^{2}-\frac{90\cdots 96}{11\cdots 59}a+\frac{73\cdots 20}{11\cdots 59}$, $\frac{40\cdots 16}{11\cdots 59}a^{19}-\frac{21\cdots 98}{10\cdots 69}a^{18}-\frac{13\cdots 10}{11\cdots 59}a^{17}+\frac{85\cdots 74}{11\cdots 59}a^{16}+\frac{19\cdots 67}{11\cdots 59}a^{15}-\frac{12\cdots 38}{11\cdots 59}a^{14}-\frac{82\cdots 39}{62\cdots 61}a^{13}+\frac{10\cdots 01}{11\cdots 59}a^{12}+\frac{75\cdots 50}{10\cdots 69}a^{11}-\frac{50\cdots 62}{11\cdots 59}a^{10}-\frac{27\cdots 11}{10\cdots 69}a^{9}+\frac{14\cdots 48}{11\cdots 59}a^{8}+\frac{75\cdots 78}{11\cdots 59}a^{7}-\frac{24\cdots 81}{11\cdots 59}a^{6}-\frac{11\cdots 00}{11\cdots 59}a^{5}+\frac{22\cdots 19}{11\cdots 59}a^{4}+\frac{82\cdots 75}{10\cdots 69}a^{3}-\frac{10\cdots 43}{11\cdots 59}a^{2}-\frac{29\cdots 36}{11\cdots 59}a+\frac{17\cdots 90}{11\cdots 59}$, $\frac{30\cdots 68}{11\cdots 59}a^{19}-\frac{19\cdots 14}{11\cdots 59}a^{18}-\frac{94\cdots 78}{11\cdots 59}a^{17}+\frac{70\cdots 63}{11\cdots 59}a^{16}+\frac{11\cdots 10}{11\cdots 59}a^{15}-\frac{10\cdots 70}{11\cdots 59}a^{14}-\frac{38\cdots 19}{62\cdots 61}a^{13}+\frac{91\cdots 62}{11\cdots 59}a^{12}+\frac{26\cdots 71}{11\cdots 59}a^{11}-\frac{45\cdots 74}{11\cdots 59}a^{10}-\frac{71\cdots 30}{11\cdots 59}a^{9}+\frac{13\cdots 65}{11\cdots 59}a^{8}+\frac{17\cdots 09}{11\cdots 59}a^{7}-\frac{24\cdots 54}{11\cdots 59}a^{6}-\frac{37\cdots 49}{11\cdots 59}a^{5}+\frac{25\cdots 12}{11\cdots 59}a^{4}+\frac{42\cdots 72}{10\cdots 69}a^{3}-\frac{12\cdots 47}{11\cdots 59}a^{2}-\frac{23\cdots 20}{11\cdots 59}a+\frac{18\cdots 46}{11\cdots 59}$, $\frac{49\cdots 37}{11\cdots 59}a^{19}-\frac{31\cdots 11}{11\cdots 59}a^{18}-\frac{15\cdots 72}{11\cdots 59}a^{17}+\frac{10\cdots 18}{10\cdots 69}a^{16}+\frac{18\cdots 45}{11\cdots 59}a^{15}-\frac{17\cdots 95}{11\cdots 59}a^{14}-\frac{58\cdots 85}{62\cdots 61}a^{13}+\frac{13\cdots 52}{10\cdots 69}a^{12}+\frac{38\cdots 99}{11\cdots 59}a^{11}-\frac{72\cdots 14}{11\cdots 59}a^{10}-\frac{93\cdots 68}{11\cdots 59}a^{9}+\frac{22\cdots 33}{11\cdots 59}a^{8}+\frac{22\cdots 41}{11\cdots 59}a^{7}-\frac{39\cdots 45}{11\cdots 59}a^{6}-\frac{51\cdots 77}{11\cdots 59}a^{5}+\frac{40\cdots 37}{11\cdots 59}a^{4}+\frac{62\cdots 45}{10\cdots 69}a^{3}-\frac{19\cdots 72}{11\cdots 59}a^{2}-\frac{35\cdots 40}{11\cdots 59}a+\frac{29\cdots 38}{11\cdots 59}$, $\frac{10\cdots 24}{11\cdots 59}a^{19}-\frac{75\cdots 03}{11\cdots 59}a^{18}-\frac{26\cdots 66}{11\cdots 59}a^{17}+\frac{26\cdots 02}{11\cdots 59}a^{16}+\frac{16\cdots 18}{10\cdots 69}a^{15}-\frac{37\cdots 35}{11\cdots 59}a^{14}+\frac{25\cdots 30}{62\cdots 61}a^{13}+\frac{29\cdots 93}{11\cdots 59}a^{12}-\frac{12\cdots 62}{11\cdots 59}a^{11}-\frac{13\cdots 74}{11\cdots 59}a^{10}+\frac{64\cdots 93}{11\cdots 59}a^{9}+\frac{36\cdots 92}{11\cdots 59}a^{8}-\frac{15\cdots 95}{11\cdots 59}a^{7}-\frac{60\cdots 13}{11\cdots 59}a^{6}+\frac{16\cdots 35}{11\cdots 59}a^{5}+\frac{51\cdots 16}{10\cdots 69}a^{4}-\frac{48\cdots 66}{10\cdots 69}a^{3}-\frac{25\cdots 83}{11\cdots 59}a^{2}-\frac{14\cdots 16}{11\cdots 59}a+\frac{27\cdots 37}{10\cdots 69}$, $\frac{63\cdots 37}{11\cdots 59}a^{19}-\frac{45\cdots 69}{11\cdots 59}a^{18}-\frac{17\cdots 67}{11\cdots 59}a^{17}+\frac{16\cdots 46}{11\cdots 59}a^{16}+\frac{15\cdots 39}{11\cdots 59}a^{15}-\frac{26\cdots 41}{11\cdots 59}a^{14}-\frac{98\cdots 95}{62\cdots 61}a^{13}+\frac{22\cdots 22}{11\cdots 59}a^{12}-\frac{47\cdots 82}{11\cdots 59}a^{11}-\frac{11\cdots 58}{11\cdots 59}a^{10}+\frac{29\cdots 21}{11\cdots 59}a^{9}+\frac{38\cdots 09}{11\cdots 59}a^{8}-\frac{65\cdots 86}{11\cdots 59}a^{7}-\frac{76\cdots 25}{11\cdots 59}a^{6}+\frac{29\cdots 82}{11\cdots 59}a^{5}+\frac{85\cdots 11}{11\cdots 59}a^{4}+\frac{73\cdots 66}{10\cdots 69}a^{3}-\frac{46\cdots 74}{11\cdots 59}a^{2}-\frac{77\cdots 35}{11\cdots 59}a+\frac{69\cdots 41}{11\cdots 59}$, $\frac{88\cdots 15}{11\cdots 59}a^{19}-\frac{55\cdots 99}{11\cdots 59}a^{18}-\frac{27\cdots 21}{11\cdots 59}a^{17}+\frac{20\cdots 82}{11\cdots 59}a^{16}+\frac{31\cdots 55}{10\cdots 69}a^{15}-\frac{31\cdots 10}{11\cdots 59}a^{14}-\frac{11\cdots 47}{62\cdots 61}a^{13}+\frac{26\cdots 77}{11\cdots 59}a^{12}+\frac{85\cdots 69}{11\cdots 59}a^{11}-\frac{13\cdots 79}{11\cdots 59}a^{10}-\frac{23\cdots 28}{11\cdots 59}a^{9}+\frac{40\cdots 01}{11\cdots 59}a^{8}+\frac{54\cdots 37}{11\cdots 59}a^{7}-\frac{74\cdots 75}{11\cdots 59}a^{6}-\frac{11\cdots 75}{11\cdots 59}a^{5}+\frac{69\cdots 60}{10\cdots 69}a^{4}+\frac{12\cdots 89}{10\cdots 69}a^{3}-\frac{37\cdots 27}{11\cdots 59}a^{2}-\frac{68\cdots 34}{11\cdots 59}a+\frac{50\cdots 72}{10\cdots 69}$, $\frac{18\cdots 32}{11\cdots 59}a^{19}-\frac{14\cdots 38}{11\cdots 59}a^{18}-\frac{33\cdots 03}{11\cdots 59}a^{17}+\frac{50\cdots 19}{11\cdots 59}a^{16}-\frac{10\cdots 49}{11\cdots 59}a^{15}-\frac{71\cdots 07}{11\cdots 59}a^{14}+\frac{37\cdots 71}{62\cdots 61}a^{13}+\frac{54\cdots 02}{11\cdots 59}a^{12}-\frac{71\cdots 55}{11\cdots 59}a^{11}-\frac{23\cdots 32}{11\cdots 59}a^{10}+\frac{34\cdots 38}{11\cdots 59}a^{9}+\frac{63\cdots 45}{11\cdots 59}a^{8}-\frac{80\cdots 91}{10\cdots 69}a^{7}-\frac{98\cdots 13}{11\cdots 59}a^{6}+\frac{12\cdots 73}{11\cdots 59}a^{5}+\frac{81\cdots 79}{11\cdots 59}a^{4}-\frac{69\cdots 40}{10\cdots 69}a^{3}-\frac{26\cdots 74}{11\cdots 59}a^{2}+\frac{14\cdots 27}{11\cdots 59}a-\frac{19\cdots 91}{11\cdots 59}$, $\frac{45\cdots 61}{11\cdots 59}a^{19}-\frac{30\cdots 49}{11\cdots 59}a^{18}-\frac{13\cdots 90}{11\cdots 59}a^{17}+\frac{11\cdots 32}{11\cdots 59}a^{16}+\frac{13\cdots 96}{11\cdots 59}a^{15}-\frac{17\cdots 07}{11\cdots 59}a^{14}-\frac{27\cdots 00}{57\cdots 51}a^{13}+\frac{14\cdots 20}{11\cdots 59}a^{12}-\frac{43\cdots 44}{11\cdots 59}a^{11}-\frac{64\cdots 77}{10\cdots 69}a^{10}+\frac{76\cdots 77}{11\cdots 59}a^{9}+\frac{21\cdots 41}{11\cdots 59}a^{8}-\frac{20\cdots 76}{11\cdots 59}a^{7}-\frac{39\cdots 32}{11\cdots 59}a^{6}+\frac{95\cdots 60}{10\cdots 69}a^{5}+\frac{40\cdots 35}{11\cdots 59}a^{4}+\frac{23\cdots 96}{10\cdots 69}a^{3}-\frac{19\cdots 62}{11\cdots 59}a^{2}-\frac{23\cdots 02}{10\cdots 69}a+\frac{27\cdots 79}{11\cdots 59}$, $\frac{27\cdots 50}{11\cdots 59}a^{19}-\frac{17\cdots 71}{11\cdots 59}a^{18}-\frac{84\cdots 22}{11\cdots 59}a^{17}+\frac{64\cdots 16}{11\cdots 59}a^{16}+\frac{10\cdots 06}{11\cdots 59}a^{15}-\frac{98\cdots 26}{11\cdots 59}a^{14}-\frac{33\cdots 35}{62\cdots 61}a^{13}+\frac{82\cdots 82}{11\cdots 59}a^{12}+\frac{22\cdots 82}{11\cdots 59}a^{11}-\frac{41\cdots 05}{11\cdots 59}a^{10}-\frac{56\cdots 80}{11\cdots 59}a^{9}+\frac{12\cdots 88}{11\cdots 59}a^{8}+\frac{13\cdots 98}{11\cdots 59}a^{7}-\frac{22\cdots 88}{11\cdots 59}a^{6}-\frac{30\cdots 90}{11\cdots 59}a^{5}+\frac{23\cdots 19}{11\cdots 59}a^{4}+\frac{36\cdots 80}{10\cdots 69}a^{3}-\frac{11\cdots 16}{11\cdots 59}a^{2}-\frac{20\cdots 80}{11\cdots 59}a+\frac{16\cdots 77}{11\cdots 59}$, $\frac{10\cdots 27}{11\cdots 59}a^{19}-\frac{65\cdots 59}{11\cdots 59}a^{18}-\frac{35\cdots 10}{11\cdots 59}a^{17}+\frac{24\cdots 96}{11\cdots 59}a^{16}+\frac{49\cdots 56}{11\cdots 59}a^{15}-\frac{39\cdots 46}{11\cdots 59}a^{14}-\frac{19\cdots 99}{62\cdots 61}a^{13}+\frac{33\cdots 77}{11\cdots 59}a^{12}+\frac{18\cdots 73}{11\cdots 59}a^{11}-\frac{17\cdots 59}{11\cdots 59}a^{10}-\frac{67\cdots 75}{11\cdots 59}a^{9}+\frac{55\cdots 17}{11\cdots 59}a^{8}+\frac{18\cdots 85}{11\cdots 59}a^{7}-\frac{10\cdots 15}{11\cdots 59}a^{6}-\frac{32\cdots 82}{11\cdots 59}a^{5}+\frac{11\cdots 26}{11\cdots 59}a^{4}+\frac{31\cdots 63}{10\cdots 69}a^{3}-\frac{61\cdots 24}{11\cdots 59}a^{2}-\frac{14\cdots 89}{11\cdots 59}a+\frac{99\cdots 35}{11\cdots 59}$ 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:  \( 62134578326.1 \) (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:  \( 17 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 62134578326.1 \cdot 1}{2\cdot\sqrt{19184293930982457734868438720703125}}\cr\approx \mathstrut & 0.235196233782 \end{aligned}\] (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^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429) 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^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429, 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^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429); 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^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429); 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

$C_4\times D_5$ (as 20T6):

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 solvable group of order 40
The 16 conjugacy class representatives for $C_4\times D_5$
Character table for $C_4\times D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 5.5.12852225.1, 10.10.825898437253125.1

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

Galois closure: deg 40
Degree 20 sibling: deg 20
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 $20$ R R $20$ ${\href{/padicField/11.2.0.1}{2} }^{10}$ $20$ $20$ ${\href{/padicField/19.2.0.1}{2} }^{10}$ ${\href{/padicField/23.4.0.1}{4} }^{5}$ ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/padicField/47.4.0.1}{4} }^{5}$ ${\href{/padicField/53.4.0.1}{4} }^{5}$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

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 3.2.2.2a1.1$x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(5\) Copy content Toggle raw display 5.1.4.3a1.4$x^{4} + 20$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(239\) Copy content Toggle raw display $\Q_{239}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{239}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{239}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{239}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_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)