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Properties

Label 20.8.154...000.1
Degree $20$
Signature $[8, 6]$
Discriminant $1.542\times 10^{29}$
Root discriminant \(28.80\)
Ramified primes $2,5,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.C_{10}$ (as 20T427)

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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 + 28*x^18 - 38*x^17 - 59*x^16 + 288*x^15 - 216*x^14 - 692*x^13 + 1105*x^12 + 1352*x^11 - 3784*x^10 - 430*x^9 + 6057*x^8 - 2062*x^7 - 2114*x^6 + 2022*x^5 - 906*x^4 + 72*x^3 + 64*x^2 - 20*x + 1)
 
Copy content gp:K = bnfinit(y^20 - 8*y^19 + 28*y^18 - 38*y^17 - 59*y^16 + 288*y^15 - 216*y^14 - 692*y^13 + 1105*y^12 + 1352*y^11 - 3784*y^10 - 430*y^9 + 6057*y^8 - 2062*y^7 - 2114*y^6 + 2022*y^5 - 906*y^4 + 72*y^3 + 64*y^2 - 20*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 8*x^19 + 28*x^18 - 38*x^17 - 59*x^16 + 288*x^15 - 216*x^14 - 692*x^13 + 1105*x^12 + 1352*x^11 - 3784*x^10 - 430*x^9 + 6057*x^8 - 2062*x^7 - 2114*x^6 + 2022*x^5 - 906*x^4 + 72*x^3 + 64*x^2 - 20*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 8*x^19 + 28*x^18 - 38*x^17 - 59*x^16 + 288*x^15 - 216*x^14 - 692*x^13 + 1105*x^12 + 1352*x^11 - 3784*x^10 - 430*x^9 + 6057*x^8 - 2062*x^7 - 2114*x^6 + 2022*x^5 - 906*x^4 + 72*x^3 + 64*x^2 - 20*x + 1)
 

\( x^{20} - 8 x^{19} + 28 x^{18} - 38 x^{17} - 59 x^{16} + 288 x^{15} - 216 x^{14} - 692 x^{13} + 1105 x^{12} + \cdots + 1 \) 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:  $[8, 6]$
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:   \(154181708612560135336755200000\) \(\medspace = 2^{30}\cdot 5^{5}\cdot 11^{16}\) 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:  \(28.80\)
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:   \(2\), \(5\), \(11\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11}a^{16}-\frac{4}{11}a^{15}-\frac{1}{11}a^{14}-\frac{5}{11}a^{13}+\frac{4}{11}a^{12}+\frac{3}{11}a^{11}-\frac{4}{11}a^{10}+\frac{1}{11}a^{9}+\frac{1}{11}a^{8}+\frac{3}{11}a^{7}-\frac{1}{11}a^{6}-\frac{5}{11}a^{5}-\frac{2}{11}a^{4}-\frac{2}{11}a^{3}+\frac{4}{11}a^{2}-\frac{2}{11}a+\frac{1}{11}$, $\frac{1}{11}a^{17}+\frac{5}{11}a^{15}+\frac{2}{11}a^{14}-\frac{5}{11}a^{13}-\frac{3}{11}a^{12}-\frac{3}{11}a^{11}-\frac{4}{11}a^{10}+\frac{5}{11}a^{9}-\frac{4}{11}a^{8}+\frac{2}{11}a^{6}+\frac{1}{11}a^{4}-\frac{4}{11}a^{3}+\frac{3}{11}a^{2}+\frac{4}{11}a+\frac{4}{11}$, $\frac{1}{11}a^{18}-\frac{1}{11}a^{12}+\frac{3}{11}a^{11}+\frac{3}{11}a^{10}+\frac{2}{11}a^{9}-\frac{5}{11}a^{8}-\frac{2}{11}a^{7}+\frac{5}{11}a^{6}+\frac{4}{11}a^{5}-\frac{5}{11}a^{4}+\frac{2}{11}a^{3}-\frac{5}{11}a^{2}+\frac{3}{11}a-\frac{5}{11}$, $\frac{1}{49\cdots 47}a^{19}+\frac{73\cdots 88}{49\cdots 47}a^{18}-\frac{17\cdots 50}{49\cdots 47}a^{17}-\frac{47\cdots 03}{45\cdots 83}a^{16}-\frac{78\cdots 95}{49\cdots 47}a^{15}-\frac{42\cdots 38}{49\cdots 47}a^{14}-\frac{17\cdots 95}{44\cdots 77}a^{13}-\frac{21\cdots 93}{49\cdots 47}a^{12}+\frac{10\cdots 23}{49\cdots 47}a^{11}-\frac{16\cdots 35}{49\cdots 47}a^{10}+\frac{19\cdots 85}{49\cdots 47}a^{9}-\frac{80\cdots 09}{49\cdots 47}a^{8}+\frac{18\cdots 84}{49\cdots 47}a^{7}-\frac{23\cdots 50}{49\cdots 47}a^{6}-\frac{18\cdots 58}{49\cdots 47}a^{5}+\frac{15\cdots 90}{49\cdots 47}a^{4}-\frac{19\cdots 65}{49\cdots 47}a^{3}+\frac{23\cdots 82}{49\cdots 47}a^{2}-\frac{23\cdots 62}{49\cdots 47}a-\frac{21\cdots 12}{49\cdots 47}$ 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:  $13$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{40\cdots 92}{49\cdots 47}a^{19}-\frac{33\cdots 23}{49\cdots 47}a^{18}+\frac{12\cdots 42}{49\cdots 47}a^{17}-\frac{17\cdots 18}{45\cdots 83}a^{16}-\frac{17\cdots 92}{44\cdots 77}a^{15}+\frac{12\cdots 57}{49\cdots 47}a^{14}-\frac{10\cdots 83}{44\cdots 77}a^{13}-\frac{25\cdots 52}{49\cdots 47}a^{12}+\frac{52\cdots 77}{49\cdots 47}a^{11}+\frac{42\cdots 38}{49\cdots 47}a^{10}-\frac{16\cdots 51}{49\cdots 47}a^{9}+\frac{26\cdots 87}{49\cdots 47}a^{8}+\frac{25\cdots 31}{49\cdots 47}a^{7}-\frac{15\cdots 55}{49\cdots 47}a^{6}-\frac{69\cdots 52}{49\cdots 47}a^{5}+\frac{11\cdots 05}{49\cdots 47}a^{4}-\frac{54\cdots 47}{49\cdots 47}a^{3}+\frac{82\cdots 63}{49\cdots 47}a^{2}+\frac{35\cdots 10}{49\cdots 47}a-\frac{10\cdots 33}{49\cdots 47}$, $\frac{12\cdots 86}{49\cdots 47}a^{19}-\frac{95\cdots 58}{49\cdots 47}a^{18}+\frac{32\cdots 41}{49\cdots 47}a^{17}-\frac{37\cdots 68}{45\cdots 83}a^{16}-\frac{70\cdots 40}{44\cdots 77}a^{15}+\frac{33\cdots 35}{49\cdots 47}a^{14}-\frac{20\cdots 85}{49\cdots 47}a^{13}-\frac{86\cdots 12}{49\cdots 47}a^{12}+\frac{11\cdots 54}{49\cdots 47}a^{11}+\frac{18\cdots 99}{49\cdots 47}a^{10}-\frac{42\cdots 65}{49\cdots 47}a^{9}-\frac{11\cdots 24}{49\cdots 47}a^{8}+\frac{70\cdots 17}{49\cdots 47}a^{7}-\frac{13\cdots 43}{49\cdots 47}a^{6}-\frac{26\cdots 66}{49\cdots 47}a^{5}+\frac{19\cdots 35}{49\cdots 47}a^{4}-\frac{80\cdots 61}{49\cdots 47}a^{3}-\frac{16\cdots 23}{49\cdots 47}a^{2}+\frac{55\cdots 45}{44\cdots 77}a-\frac{13\cdots 13}{49\cdots 47}$, $\frac{56\cdots 79}{49\cdots 47}a^{19}-\frac{37\cdots 74}{49\cdots 47}a^{18}+\frac{10\cdots 14}{49\cdots 47}a^{17}-\frac{25\cdots 55}{45\cdots 83}a^{16}-\frac{48\cdots 07}{44\cdots 77}a^{15}+\frac{10\cdots 13}{49\cdots 47}a^{14}+\frac{87\cdots 20}{49\cdots 47}a^{13}-\frac{48\cdots 73}{49\cdots 47}a^{12}+\frac{24\cdots 96}{49\cdots 47}a^{11}+\frac{14\cdots 97}{49\cdots 47}a^{10}-\frac{83\cdots 86}{49\cdots 47}a^{9}-\frac{29\cdots 96}{49\cdots 47}a^{8}+\frac{21\cdots 61}{49\cdots 47}a^{7}+\frac{39\cdots 94}{49\cdots 47}a^{6}-\frac{16\cdots 42}{49\cdots 47}a^{5}-\frac{17\cdots 37}{49\cdots 47}a^{4}+\frac{93\cdots 97}{49\cdots 47}a^{3}-\frac{64\cdots 47}{49\cdots 47}a^{2}-\frac{18\cdots 11}{49\cdots 47}a+\frac{38\cdots 33}{44\cdots 77}$, $\frac{39\cdots 46}{49\cdots 47}a^{19}-\frac{31\cdots 06}{49\cdots 47}a^{18}+\frac{10\cdots 30}{44\cdots 77}a^{17}-\frac{14\cdots 30}{45\cdots 83}a^{16}-\frac{21\cdots 34}{49\cdots 47}a^{15}+\frac{11\cdots 22}{49\cdots 47}a^{14}-\frac{98\cdots 30}{49\cdots 47}a^{13}-\frac{26\cdots 63}{49\cdots 47}a^{12}+\frac{46\cdots 56}{49\cdots 47}a^{11}+\frac{47\cdots 04}{49\cdots 47}a^{10}-\frac{15\cdots 06}{49\cdots 47}a^{9}+\frac{80\cdots 09}{49\cdots 47}a^{8}+\frac{23\cdots 52}{49\cdots 47}a^{7}-\frac{10\cdots 84}{49\cdots 47}a^{6}-\frac{70\cdots 68}{49\cdots 47}a^{5}+\frac{88\cdots 98}{49\cdots 47}a^{4}-\frac{46\cdots 60}{49\cdots 47}a^{3}+\frac{70\cdots 24}{49\cdots 47}a^{2}+\frac{13\cdots 96}{49\cdots 47}a-\frac{92\cdots 43}{49\cdots 47}$, $\frac{28\cdots 59}{49\cdots 47}a^{19}-\frac{20\cdots 86}{49\cdots 47}a^{18}+\frac{54\cdots 91}{44\cdots 77}a^{17}-\frac{36\cdots 45}{40\cdots 53}a^{16}-\frac{22\cdots 62}{44\cdots 77}a^{15}+\frac{59\cdots 64}{44\cdots 77}a^{14}+\frac{39\cdots 71}{49\cdots 47}a^{13}-\frac{23\cdots 78}{49\cdots 47}a^{12}+\frac{13\cdots 59}{49\cdots 47}a^{11}+\frac{59\cdots 31}{49\cdots 47}a^{10}-\frac{68\cdots 26}{49\cdots 47}a^{9}-\frac{92\cdots 07}{49\cdots 47}a^{8}+\frac{13\cdots 78}{49\cdots 47}a^{7}+\frac{74\cdots 17}{49\cdots 47}a^{6}-\frac{67\cdots 52}{49\cdots 47}a^{5}+\frac{76\cdots 26}{44\cdots 77}a^{4}+\frac{17\cdots 11}{49\cdots 47}a^{3}-\frac{10\cdots 08}{49\cdots 47}a^{2}+\frac{82\cdots 27}{49\cdots 47}a-\frac{26\cdots 24}{49\cdots 47}$, $\frac{56\cdots 79}{49\cdots 47}a^{19}-\frac{37\cdots 74}{49\cdots 47}a^{18}+\frac{10\cdots 14}{49\cdots 47}a^{17}-\frac{25\cdots 55}{45\cdots 83}a^{16}-\frac{48\cdots 07}{44\cdots 77}a^{15}+\frac{10\cdots 13}{49\cdots 47}a^{14}+\frac{87\cdots 20}{49\cdots 47}a^{13}-\frac{48\cdots 73}{49\cdots 47}a^{12}+\frac{24\cdots 96}{49\cdots 47}a^{11}+\frac{14\cdots 97}{49\cdots 47}a^{10}-\frac{83\cdots 86}{49\cdots 47}a^{9}-\frac{29\cdots 96}{49\cdots 47}a^{8}+\frac{21\cdots 61}{49\cdots 47}a^{7}+\frac{39\cdots 94}{49\cdots 47}a^{6}-\frac{16\cdots 42}{49\cdots 47}a^{5}-\frac{17\cdots 37}{49\cdots 47}a^{4}+\frac{93\cdots 97}{49\cdots 47}a^{3}-\frac{64\cdots 47}{49\cdots 47}a^{2}-\frac{18\cdots 11}{49\cdots 47}a+\frac{83\cdots 10}{44\cdots 77}$, $\frac{10\cdots 81}{49\cdots 47}a^{19}-\frac{82\cdots 79}{49\cdots 47}a^{18}+\frac{28\cdots 69}{49\cdots 47}a^{17}-\frac{34\cdots 55}{45\cdots 83}a^{16}-\frac{63\cdots 92}{49\cdots 47}a^{15}+\frac{29\cdots 35}{49\cdots 47}a^{14}-\frac{20\cdots 64}{49\cdots 47}a^{13}-\frac{72\cdots 35}{49\cdots 47}a^{12}+\frac{10\cdots 52}{49\cdots 47}a^{11}+\frac{14\cdots 17}{49\cdots 47}a^{10}-\frac{37\cdots 27}{49\cdots 47}a^{9}-\frac{63\cdots 66}{49\cdots 47}a^{8}+\frac{60\cdots 61}{49\cdots 47}a^{7}-\frac{18\cdots 90}{49\cdots 47}a^{6}-\frac{20\cdots 20}{49\cdots 47}a^{5}+\frac{20\cdots 69}{49\cdots 47}a^{4}-\frac{95\cdots 32}{49\cdots 47}a^{3}+\frac{14\cdots 03}{49\cdots 47}a^{2}+\frac{67\cdots 33}{49\cdots 47}a-\frac{11\cdots 29}{49\cdots 47}$, $\frac{40\cdots 92}{49\cdots 47}a^{19}-\frac{33\cdots 23}{49\cdots 47}a^{18}+\frac{12\cdots 42}{49\cdots 47}a^{17}-\frac{17\cdots 18}{45\cdots 83}a^{16}-\frac{17\cdots 92}{44\cdots 77}a^{15}+\frac{12\cdots 57}{49\cdots 47}a^{14}-\frac{10\cdots 83}{44\cdots 77}a^{13}-\frac{25\cdots 52}{49\cdots 47}a^{12}+\frac{52\cdots 77}{49\cdots 47}a^{11}+\frac{42\cdots 38}{49\cdots 47}a^{10}-\frac{16\cdots 51}{49\cdots 47}a^{9}+\frac{26\cdots 87}{49\cdots 47}a^{8}+\frac{25\cdots 31}{49\cdots 47}a^{7}-\frac{15\cdots 55}{49\cdots 47}a^{6}-\frac{69\cdots 52}{49\cdots 47}a^{5}+\frac{11\cdots 05}{49\cdots 47}a^{4}-\frac{54\cdots 47}{49\cdots 47}a^{3}+\frac{82\cdots 63}{49\cdots 47}a^{2}+\frac{35\cdots 10}{49\cdots 47}a-\frac{59\cdots 86}{49\cdots 47}$, $\frac{11\cdots 87}{49\cdots 47}a^{19}-\frac{94\cdots 66}{49\cdots 47}a^{18}+\frac{29\cdots 06}{44\cdots 77}a^{17}-\frac{40\cdots 24}{45\cdots 83}a^{16}-\frac{71\cdots 61}{49\cdots 47}a^{15}+\frac{33\cdots 41}{49\cdots 47}a^{14}-\frac{23\cdots 12}{49\cdots 47}a^{13}-\frac{83\cdots 17}{49\cdots 47}a^{12}+\frac{12\cdots 46}{49\cdots 47}a^{11}+\frac{17\cdots 23}{49\cdots 47}a^{10}-\frac{43\cdots 47}{49\cdots 47}a^{9}-\frac{91\cdots 87}{49\cdots 47}a^{8}+\frac{71\cdots 51}{49\cdots 47}a^{7}-\frac{15\cdots 36}{49\cdots 47}a^{6}-\frac{30\cdots 81}{49\cdots 47}a^{5}+\frac{17\cdots 95}{49\cdots 47}a^{4}-\frac{53\cdots 39}{49\cdots 47}a^{3}-\frac{96\cdots 22}{49\cdots 47}a^{2}+\frac{27\cdots 93}{49\cdots 47}a+\frac{40\cdots 92}{49\cdots 47}$, $\frac{22\cdots 64}{49\cdots 47}a^{19}-\frac{18\cdots 25}{49\cdots 47}a^{18}+\frac{65\cdots 68}{49\cdots 47}a^{17}-\frac{87\cdots 00}{45\cdots 83}a^{16}-\frac{11\cdots 19}{49\cdots 47}a^{15}+\frac{66\cdots 30}{49\cdots 47}a^{14}-\frac{59\cdots 08}{49\cdots 47}a^{13}-\frac{14\cdots 22}{49\cdots 47}a^{12}+\frac{26\cdots 95}{49\cdots 47}a^{11}+\frac{25\cdots 56}{49\cdots 47}a^{10}-\frac{87\cdots 05}{49\cdots 47}a^{9}+\frac{58\cdots 11}{49\cdots 47}a^{8}+\frac{12\cdots 47}{49\cdots 47}a^{7}-\frac{69\cdots 26}{49\cdots 47}a^{6}-\frac{25\cdots 95}{49\cdots 47}a^{5}+\frac{53\cdots 10}{49\cdots 47}a^{4}-\frac{28\cdots 33}{44\cdots 77}a^{3}+\frac{93\cdots 06}{49\cdots 47}a^{2}-\frac{15\cdots 35}{49\cdots 47}a-\frac{12\cdots 88}{49\cdots 47}$, $\frac{22\cdots 39}{49\cdots 47}a^{19}-\frac{18\cdots 58}{49\cdots 47}a^{18}+\frac{64\cdots 63}{49\cdots 47}a^{17}-\frac{81\cdots 38}{45\cdots 83}a^{16}-\frac{13\cdots 97}{49\cdots 47}a^{15}+\frac{66\cdots 42}{49\cdots 47}a^{14}-\frac{51\cdots 01}{49\cdots 47}a^{13}-\frac{15\cdots 30}{49\cdots 47}a^{12}+\frac{25\cdots 79}{49\cdots 47}a^{11}+\frac{30\cdots 08}{49\cdots 47}a^{10}-\frac{79\cdots 11}{44\cdots 77}a^{9}-\frac{76\cdots 39}{49\cdots 47}a^{8}+\frac{13\cdots 77}{49\cdots 47}a^{7}-\frac{45\cdots 64}{44\cdots 77}a^{6}-\frac{46\cdots 53}{49\cdots 47}a^{5}+\frac{46\cdots 10}{49\cdots 47}a^{4}-\frac{19\cdots 18}{44\cdots 77}a^{3}+\frac{25\cdots 90}{49\cdots 47}a^{2}+\frac{76\cdots 87}{49\cdots 47}a-\frac{26\cdots 95}{49\cdots 47}$, $\frac{16\cdots 91}{49\cdots 47}a^{19}-\frac{13\cdots 83}{49\cdots 47}a^{18}+\frac{45\cdots 02}{49\cdots 47}a^{17}-\frac{56\cdots 06}{45\cdots 83}a^{16}-\frac{88\cdots 16}{44\cdots 77}a^{15}+\frac{46\cdots 96}{49\cdots 47}a^{14}-\frac{34\cdots 03}{49\cdots 47}a^{13}-\frac{11\cdots 60}{49\cdots 47}a^{12}+\frac{17\cdots 91}{49\cdots 47}a^{11}+\frac{22\cdots 69}{49\cdots 47}a^{10}-\frac{61\cdots 99}{49\cdots 47}a^{9}-\frac{90\cdots 20}{49\cdots 47}a^{8}+\frac{99\cdots 31}{49\cdots 47}a^{7}-\frac{29\cdots 47}{49\cdots 47}a^{6}-\frac{38\cdots 78}{49\cdots 47}a^{5}+\frac{29\cdots 49}{49\cdots 47}a^{4}-\frac{12\cdots 14}{49\cdots 47}a^{3}+\frac{89\cdots 59}{49\cdots 47}a^{2}+\frac{90\cdots 04}{49\cdots 47}a-\frac{23\cdots 32}{49\cdots 47}$, $\frac{73\cdots 48}{49\cdots 47}a^{19}-\frac{59\cdots 68}{49\cdots 47}a^{18}+\frac{20\cdots 19}{49\cdots 47}a^{17}-\frac{27\cdots 45}{45\cdots 83}a^{16}-\frac{40\cdots 50}{49\cdots 47}a^{15}+\frac{21\cdots 89}{49\cdots 47}a^{14}-\frac{15\cdots 55}{44\cdots 77}a^{13}-\frac{48\cdots 03}{49\cdots 47}a^{12}+\frac{84\cdots 48}{49\cdots 47}a^{11}+\frac{91\cdots 62}{49\cdots 47}a^{10}-\frac{28\cdots 39}{49\cdots 47}a^{9}-\frac{76\cdots 33}{49\cdots 47}a^{8}+\frac{44\cdots 68}{49\cdots 47}a^{7}-\frac{18\cdots 99}{49\cdots 47}a^{6}-\frac{14\cdots 15}{49\cdots 47}a^{5}+\frac{14\cdots 16}{44\cdots 77}a^{4}-\frac{68\cdots 84}{49\cdots 47}a^{3}+\frac{54\cdots 20}{49\cdots 47}a^{2}+\frac{36\cdots 81}{49\cdots 47}a-\frac{93\cdots 67}{49\cdots 47}$ 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:  \( 9214414.27353 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{6}\cdot 9214414.27353 \cdot 1}{2\cdot\sqrt{154181708612560135336755200000}}\cr\approx \mathstrut & 0.184816391574 \end{aligned}\] (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 + 28*x^18 - 38*x^17 - 59*x^16 + 288*x^15 - 216*x^14 - 692*x^13 + 1105*x^12 + 1352*x^11 - 3784*x^10 - 430*x^9 + 6057*x^8 - 2062*x^7 - 2114*x^6 + 2022*x^5 - 906*x^4 + 72*x^3 + 64*x^2 - 20*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 - 8*x^19 + 28*x^18 - 38*x^17 - 59*x^16 + 288*x^15 - 216*x^14 - 692*x^13 + 1105*x^12 + 1352*x^11 - 3784*x^10 - 430*x^9 + 6057*x^8 - 2062*x^7 - 2114*x^6 + 2022*x^5 - 906*x^4 + 72*x^3 + 64*x^2 - 20*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 - 8*x^19 + 28*x^18 - 38*x^17 - 59*x^16 + 288*x^15 - 216*x^14 - 692*x^13 + 1105*x^12 + 1352*x^11 - 3784*x^10 - 430*x^9 + 6057*x^8 - 2062*x^7 - 2114*x^6 + 2022*x^5 - 906*x^4 + 72*x^3 + 64*x^2 - 20*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 - 8*x^19 + 28*x^18 - 38*x^17 - 59*x^16 + 288*x^15 - 216*x^14 - 692*x^13 + 1105*x^12 + 1352*x^11 - 3784*x^10 - 430*x^9 + 6057*x^8 - 2062*x^7 - 2114*x^6 + 2022*x^5 - 906*x^4 + 72*x^3 + 64*x^2 - 20*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

$C_2^{10}.C_{10}$ (as 20T427):

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 10240
The 136 conjugacy class representatives for $C_2^{10}.C_{10}$
Character table for $C_2^{10}.C_{10}$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.219503494144.1

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 20 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 20.6.470525233802978928640000000000.6

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $20$ R $20$ R ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ $20$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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 2.5.4.30a104.2$x^{20} + 2 x^{19} + 6 x^{17} + 8 x^{16} + 6 x^{15} + 18 x^{14} + 12 x^{13} + 26 x^{12} + 30 x^{11} + 22 x^{10} + 40 x^{9} + 25 x^{8} + 40 x^{7} + 30 x^{6} + 22 x^{5} + 26 x^{4} + 10 x^{3} + 18 x^{2} + 4 x + 11$$4$$5$$30$20T427not computed
\(5\) Copy content Toggle raw display 5.10.1.0a1.1$x^{10} + 3 x^{5} + 3 x^{4} + 2 x^{3} + 4 x^{2} + x + 2$$1$$10$$0$$C_{10}$$$[\ ]^{10}$$
5.5.2.5a1.1$x^{10} + 8 x^{6} + 6 x^{5} + 16 x^{2} + 29 x + 9$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
\(11\) 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)