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Properties

Label 20.4.108...125.3
Degree $20$
Signature $[4, 8]$
Discriminant $1.082\times 10^{55}$
Root discriminant \(564.56\)
Ramified primes $5,11,41$
Class number $125$ (GRH)
Class group [5, 5, 5] (GRH)
Galois group $C_2\times F_5$ (as 20T9)

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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 - 156*x^18 - 952*x^17 + 17527*x^16 + 241902*x^15 - 1136451*x^14 - 21143487*x^13 + 32888707*x^12 + 309949732*x^11 - 8399093196*x^10 + 50338298318*x^9 + 193793523407*x^8 - 3449484980348*x^7 - 29815059450616*x^6 + 19997778190588*x^5 - 750024162066003*x^4 + 2346903326902332*x^3 + 8078639162151899*x^2 + 16769387472122923*x + 53854635816438281)
 
Copy content gp:K = bnfinit(y^20 - 8*y^19 - 156*y^18 - 952*y^17 + 17527*y^16 + 241902*y^15 - 1136451*y^14 - 21143487*y^13 + 32888707*y^12 + 309949732*y^11 - 8399093196*y^10 + 50338298318*y^9 + 193793523407*y^8 - 3449484980348*y^7 - 29815059450616*y^6 + 19997778190588*y^5 - 750024162066003*y^4 + 2346903326902332*y^3 + 8078639162151899*y^2 + 16769387472122923*y + 53854635816438281, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 8*x^19 - 156*x^18 - 952*x^17 + 17527*x^16 + 241902*x^15 - 1136451*x^14 - 21143487*x^13 + 32888707*x^12 + 309949732*x^11 - 8399093196*x^10 + 50338298318*x^9 + 193793523407*x^8 - 3449484980348*x^7 - 29815059450616*x^6 + 19997778190588*x^5 - 750024162066003*x^4 + 2346903326902332*x^3 + 8078639162151899*x^2 + 16769387472122923*x + 53854635816438281);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 8*x^19 - 156*x^18 - 952*x^17 + 17527*x^16 + 241902*x^15 - 1136451*x^14 - 21143487*x^13 + 32888707*x^12 + 309949732*x^11 - 8399093196*x^10 + 50338298318*x^9 + 193793523407*x^8 - 3449484980348*x^7 - 29815059450616*x^6 + 19997778190588*x^5 - 750024162066003*x^4 + 2346903326902332*x^3 + 8078639162151899*x^2 + 16769387472122923*x + 53854635816438281)
 

\( x^{20} - 8 x^{19} - 156 x^{18} - 952 x^{17} + 17527 x^{16} + 241902 x^{15} - 1136451 x^{14} + \cdots + 53\!\cdots\!81 \) 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:   \(10818327022517881605935827576487172209050638458251953125\) \(\medspace = 5^{15}\cdot 11^{18}\cdot 41^{16}\) 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:  \(564.56\)
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:  $5^{3/4}11^{9/10}41^{4/5}\approx 564.557271214292$
Ramified primes:   \(5\), \(11\), \(41\) 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: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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{10}a^{12}+\frac{1}{5}a^{10}-\frac{1}{2}a^{9}-\frac{1}{10}a^{8}+\frac{1}{10}a^{6}-\frac{1}{10}a^{4}-\frac{1}{2}a^{3}-\frac{3}{10}a^{2}-\frac{1}{2}a-\frac{2}{5}$, $\frac{1}{20}a^{13}-\frac{1}{20}a^{12}-\frac{3}{20}a^{11}+\frac{3}{20}a^{10}+\frac{1}{5}a^{9}+\frac{3}{10}a^{8}-\frac{1}{5}a^{7}+\frac{9}{20}a^{6}-\frac{3}{10}a^{5}+\frac{3}{10}a^{4}-\frac{3}{20}a^{3}+\frac{3}{20}a^{2}+\frac{3}{10}a-\frac{1}{20}$, $\frac{1}{20}a^{14}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{10}a^{8}+\frac{1}{4}a^{7}+\frac{7}{20}a^{6}-\frac{1}{20}a^{4}-\frac{3}{20}a^{2}+\frac{1}{4}a+\frac{3}{20}$, $\frac{1}{20}a^{15}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}-\frac{1}{10}a^{9}+\frac{1}{4}a^{8}+\frac{7}{20}a^{7}-\frac{1}{20}a^{5}-\frac{3}{20}a^{3}+\frac{1}{4}a^{2}+\frac{3}{20}a$, $\frac{1}{80}a^{16}+\frac{1}{80}a^{15}-\frac{1}{80}a^{14}-\frac{1}{40}a^{13}+\frac{1}{80}a^{12}-\frac{9}{80}a^{11}-\frac{1}{16}a^{10}-\frac{7}{16}a^{9}-\frac{11}{40}a^{8}+\frac{3}{8}a^{7}+\frac{9}{40}a^{6}+\frac{11}{80}a^{5}+\frac{11}{40}a^{4}-\frac{2}{5}a^{3}+\frac{13}{80}a^{2}+\frac{13}{40}a-\frac{17}{80}$, $\frac{1}{80}a^{17}-\frac{1}{40}a^{15}-\frac{1}{80}a^{14}-\frac{1}{80}a^{13}+\frac{1}{40}a^{12}+\frac{1}{5}a^{11}-\frac{13}{40}a^{10}+\frac{37}{80}a^{9}+\frac{1}{4}a^{8}+\frac{1}{20}a^{7}-\frac{7}{16}a^{6}+\frac{7}{16}a^{5}-\frac{3}{40}a^{4}+\frac{17}{80}a^{3}-\frac{23}{80}a^{2}-\frac{27}{80}a-\frac{11}{80}$, $\frac{1}{22800}a^{18}+\frac{91}{22800}a^{17}+\frac{119}{22800}a^{16}+\frac{7}{2280}a^{15}-\frac{289}{22800}a^{14}+\frac{341}{22800}a^{13}-\frac{1001}{22800}a^{12}+\frac{1109}{4560}a^{11}-\frac{1739}{3800}a^{10}-\frac{41}{950}a^{9}+\frac{289}{3800}a^{8}-\frac{265}{912}a^{7}-\frac{203}{2850}a^{6}+\frac{4513}{11400}a^{5}-\frac{717}{7600}a^{4}+\frac{203}{570}a^{3}+\frac{567}{7600}a^{2}+\frac{257}{600}a-\frac{4103}{11400}$, $\frac{1}{30\cdots 00}a^{19}+\frac{59\cdots 17}{30\cdots 00}a^{18}+\frac{81\cdots 03}{20\cdots 20}a^{17}-\frac{60\cdots 77}{10\cdots 00}a^{16}+\frac{56\cdots 71}{32\cdots 50}a^{15}+\frac{30\cdots 41}{15\cdots 00}a^{14}+\frac{50\cdots 01}{20\cdots 20}a^{13}-\frac{22\cdots 29}{77\cdots 00}a^{12}-\frac{71\cdots 09}{30\cdots 00}a^{11}-\frac{17\cdots 91}{10\cdots 00}a^{10}-\frac{18\cdots 47}{41\cdots 64}a^{9}-\frac{46\cdots 31}{30\cdots 00}a^{8}-\frac{10\cdots 33}{38\cdots 00}a^{7}-\frac{47\cdots 41}{38\cdots 00}a^{6}+\frac{13\cdots 73}{15\cdots 40}a^{5}+\frac{92\cdots 23}{38\cdots 00}a^{4}+\frac{69\cdots 21}{30\cdots 00}a^{3}+\frac{14\cdots 97}{30\cdots 00}a^{2}+\frac{54\cdots 51}{12\cdots 20}a-\frac{24\cdots 69}{10\cdots 00}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  $C_{5}\times C_{5}\times C_{5}$, which has order $125$ (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:  $C_{10}\times C_{10}\times C_{5}$, which has order $500$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $11$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{17\cdots 21}{24\cdots 20}a^{19}-\frac{76\cdots 43}{24\cdots 20}a^{18}-\frac{26\cdots 07}{24\cdots 20}a^{17}-\frac{30\cdots 47}{24\cdots 20}a^{16}+\frac{46\cdots 73}{61\cdots 80}a^{15}+\frac{24\cdots 31}{12\cdots 88}a^{14}+\frac{22\cdots 19}{24\cdots 20}a^{13}-\frac{40\cdots 93}{30\cdots 40}a^{12}-\frac{11\cdots 97}{24\cdots 20}a^{11}+\frac{37\cdots 57}{12\cdots 80}a^{10}-\frac{24\cdots 21}{49\cdots 44}a^{9}+\frac{14\cdots 51}{12\cdots 80}a^{8}+\frac{47\cdots 43}{30\cdots 40}a^{7}-\frac{47\cdots 31}{61\cdots 80}a^{6}-\frac{18\cdots 71}{61\cdots 80}a^{5}-\frac{33\cdots 92}{38\cdots 55}a^{4}-\frac{47\cdots 19}{49\cdots 44}a^{3}-\frac{62\cdots 11}{24\cdots 20}a^{2}-\frac{30\cdots 97}{61\cdots 80}a+\frac{74\cdots 15}{49\cdots 44}$, $\frac{28\cdots 73}{38\cdots 00}a^{19}-\frac{12\cdots 89}{11\cdots 00}a^{18}-\frac{91\cdots 57}{11\cdots 00}a^{17}+\frac{42\cdots 13}{11\cdots 00}a^{16}+\frac{99\cdots 17}{57\cdots 00}a^{15}+\frac{14\cdots 91}{15\cdots 00}a^{14}-\frac{22\cdots 47}{11\cdots 00}a^{13}-\frac{59\cdots 61}{57\cdots 00}a^{12}+\frac{14\cdots 99}{11\cdots 00}a^{11}+\frac{16\cdots 43}{20\cdots 00}a^{10}-\frac{31\cdots 19}{38\cdots 00}a^{9}+\frac{15\cdots 59}{20\cdots 00}a^{8}-\frac{18\cdots 49}{28\cdots 00}a^{7}-\frac{10\cdots 31}{28\cdots 00}a^{6}-\frac{34\cdots 71}{57\cdots 00}a^{5}+\frac{15\cdots 59}{95\cdots 00}a^{4}-\frac{79\cdots 41}{11\cdots 00}a^{3}+\frac{16\cdots 47}{38\cdots 00}a^{2}-\frac{66\cdots 53}{28\cdots 00}a-\frac{51\cdots 57}{11\cdots 00}$, $\frac{74\cdots 63}{11\cdots 00}a^{19}-\frac{27\cdots 31}{38\cdots 00}a^{18}-\frac{96\cdots 39}{11\cdots 00}a^{17}-\frac{34\cdots 69}{11\cdots 00}a^{16}+\frac{36\cdots 77}{28\cdots 00}a^{15}+\frac{73\cdots 23}{62\cdots 25}a^{14}-\frac{13\cdots 29}{11\cdots 00}a^{13}-\frac{10\cdots 17}{95\cdots 00}a^{12}+\frac{78\cdots 53}{11\cdots 00}a^{11}+\frac{13\cdots 01}{20\cdots 00}a^{10}-\frac{25\cdots 03}{38\cdots 00}a^{9}+\frac{30\cdots 29}{60\cdots 00}a^{8}+\frac{21\cdots 69}{95\cdots 00}a^{7}-\frac{52\cdots 93}{19\cdots 00}a^{6}-\frac{62\cdots 07}{57\cdots 00}a^{5}+\frac{10\cdots 47}{14\cdots 00}a^{4}-\frac{79\cdots 47}{11\cdots 00}a^{3}+\frac{23\cdots 57}{11\cdots 00}a^{2}-\frac{15\cdots 07}{57\cdots 00}a-\frac{36\cdots 69}{11\cdots 00}$, $\frac{15\cdots 01}{16\cdots 00}a^{19}-\frac{12\cdots 51}{16\cdots 00}a^{18}-\frac{25\cdots 73}{16\cdots 00}a^{17}-\frac{21\cdots 33}{16\cdots 00}a^{16}+\frac{18\cdots 53}{84\cdots 00}a^{15}+\frac{26\cdots 57}{84\cdots 00}a^{14}-\frac{21\cdots 43}{16\cdots 00}a^{13}-\frac{27\cdots 09}{84\cdots 00}a^{12}-\frac{38\cdots 33}{56\cdots 00}a^{11}+\frac{10\cdots 23}{56\cdots 00}a^{10}+\frac{12\cdots 71}{29\cdots 00}a^{9}-\frac{66\cdots 03}{16\cdots 00}a^{8}-\frac{16\cdots 01}{42\cdots 00}a^{7}-\frac{67\cdots 73}{84\cdots 00}a^{6}-\frac{14\cdots 79}{14\cdots 00}a^{5}+\frac{77\cdots 59}{21\cdots 00}a^{4}+\frac{82\cdots 77}{56\cdots 00}a^{3}+\frac{58\cdots 69}{16\cdots 00}a^{2}+\frac{77\cdots 91}{84\cdots 00}a+\frac{52\cdots 21}{19\cdots 00}$, $\frac{31\cdots 51}{33\cdots 00}a^{19}-\frac{50\cdots 61}{33\cdots 00}a^{18}-\frac{15\cdots 71}{11\cdots 00}a^{17}+\frac{11\cdots 51}{59\cdots 00}a^{16}+\frac{12\cdots 53}{28\cdots 00}a^{15}+\frac{22\cdots 37}{16\cdots 00}a^{14}-\frac{52\cdots 01}{11\cdots 00}a^{13}-\frac{15\cdots 51}{42\cdots 00}a^{12}+\frac{10\cdots 61}{33\cdots 00}a^{11}+\frac{47\cdots 83}{11\cdots 00}a^{10}-\frac{20\cdots 61}{11\cdots 00}a^{9}-\frac{49\cdots 73}{33\cdots 00}a^{8}+\frac{44\cdots 17}{42\cdots 00}a^{7}+\frac{19\cdots 13}{42\cdots 00}a^{6}-\frac{35\cdots 81}{42\cdots 00}a^{5}+\frac{99\cdots 09}{42\cdots 00}a^{4}-\frac{12\cdots 49}{33\cdots 00}a^{3}-\frac{49\cdots 61}{33\cdots 00}a^{2}-\frac{22\cdots 81}{70\cdots 00}a-\frac{14\cdots 07}{11\cdots 00}$, $\frac{11\cdots 81}{11\cdots 00}a^{19}-\frac{50\cdots 43}{11\cdots 00}a^{18}-\frac{45\cdots 19}{22\cdots 20}a^{17}-\frac{16\cdots 71}{11\cdots 00}a^{16}+\frac{23\cdots 27}{14\cdots 00}a^{15}+\frac{18\cdots 11}{56\cdots 00}a^{14}-\frac{12\cdots 89}{22\cdots 20}a^{13}-\frac{82\cdots 29}{28\cdots 00}a^{12}-\frac{25\cdots 69}{11\cdots 00}a^{11}+\frac{87\cdots 47}{11\cdots 00}a^{10}-\frac{10\cdots 03}{11\cdots 80}a^{9}+\frac{11\cdots 89}{11\cdots 00}a^{8}+\frac{37\cdots 21}{70\cdots 00}a^{7}-\frac{94\cdots 87}{28\cdots 00}a^{6}-\frac{13\cdots 37}{28\cdots 40}a^{5}-\frac{44\cdots 87}{14\cdots 00}a^{4}-\frac{24\cdots 99}{11\cdots 00}a^{3}-\frac{21\cdots 83}{11\cdots 00}a^{2}+\frac{11\cdots 67}{56\cdots 80}a+\frac{20\cdots 31}{38\cdots 00}$, $\frac{63\cdots 41}{61\cdots 60}a^{19}-\frac{33\cdots 01}{41\cdots 64}a^{18}-\frac{20\cdots 43}{12\cdots 92}a^{17}-\frac{58\cdots 71}{61\cdots 60}a^{16}+\frac{28\cdots 63}{15\cdots 40}a^{15}+\frac{26\cdots 09}{10\cdots 60}a^{14}-\frac{76\cdots 89}{61\cdots 60}a^{13}-\frac{11\cdots 73}{51\cdots 80}a^{12}+\frac{12\cdots 97}{32\cdots 40}a^{11}+\frac{18\cdots 25}{41\cdots 64}a^{10}-\frac{18\cdots 91}{20\cdots 20}a^{9}+\frac{29\cdots 41}{61\cdots 60}a^{8}+\frac{12\cdots 87}{51\cdots 80}a^{7}-\frac{19\cdots 01}{51\cdots 80}a^{6}-\frac{49\cdots 73}{15\cdots 40}a^{5}+\frac{11\cdots 57}{30\cdots 48}a^{4}-\frac{74\cdots 91}{12\cdots 92}a^{3}+\frac{12\cdots 53}{61\cdots 60}a^{2}+\frac{18\cdots 81}{15\cdots 40}a+\frac{11\cdots 59}{21\cdots 40}$, $\frac{10\cdots 53}{25\cdots 00}a^{19}+\frac{13\cdots 59}{19\cdots 00}a^{18}-\frac{51\cdots 57}{77\cdots 00}a^{17}-\frac{52\cdots 17}{48\cdots 75}a^{16}-\frac{80\cdots 33}{38\cdots 00}a^{15}+\frac{43\cdots 13}{38\cdots 00}a^{14}+\frac{65\cdots 73}{77\cdots 00}a^{13}-\frac{17\cdots 77}{77\cdots 00}a^{12}-\frac{19\cdots 06}{48\cdots 75}a^{11}-\frac{90\cdots 63}{25\cdots 00}a^{10}-\frac{38\cdots 31}{64\cdots 00}a^{9}-\frac{66\cdots 49}{25\cdots 00}a^{8}+\frac{35\cdots 59}{77\cdots 00}a^{7}+\frac{55\cdots 01}{77\cdots 00}a^{6}+\frac{88\cdots 03}{77\cdots 00}a^{5}+\frac{68\cdots 61}{25\cdots 00}a^{4}-\frac{79\cdots 79}{19\cdots 00}a^{3}-\frac{43\cdots 43}{25\cdots 00}a^{2}-\frac{30\cdots 37}{77\cdots 00}a-\frac{18\cdots 29}{13\cdots 00}$, $\frac{57\cdots 87}{30\cdots 00}a^{19}+\frac{16\cdots 21}{10\cdots 00}a^{18}+\frac{48\cdots 59}{30\cdots 00}a^{17}-\frac{29\cdots 21}{30\cdots 00}a^{16}-\frac{90\cdots 23}{19\cdots 00}a^{15}-\frac{27\cdots 27}{51\cdots 00}a^{14}-\frac{73\cdots 91}{30\cdots 00}a^{13}-\frac{18\cdots 21}{32\cdots 50}a^{12}-\frac{38\cdots 83}{30\cdots 00}a^{11}-\frac{13\cdots 49}{10\cdots 00}a^{10}-\frac{10\cdots 37}{10\cdots 00}a^{9}-\frac{47\cdots 01}{30\cdots 00}a^{8}-\frac{30\cdots 99}{25\cdots 00}a^{7}-\frac{11\cdots 51}{25\cdots 00}a^{6}-\frac{88\cdots 01}{19\cdots 00}a^{5}-\frac{45\cdots 89}{77\cdots 00}a^{4}+\frac{16\cdots 47}{30\cdots 00}a^{3}+\frac{46\cdots 43}{30\cdots 00}a^{2}+\frac{25\cdots 91}{77\cdots 00}a+\frac{69\cdots 21}{10\cdots 00}$, $\frac{81\cdots 23}{61\cdots 60}a^{19}+\frac{60\cdots 13}{20\cdots 20}a^{18}-\frac{23\cdots 17}{12\cdots 92}a^{17}-\frac{21\cdots 33}{61\cdots 60}a^{16}-\frac{36\cdots 51}{30\cdots 48}a^{15}+\frac{26\cdots 73}{10\cdots 60}a^{14}+\frac{11\cdots 97}{61\cdots 60}a^{13}-\frac{67\cdots 95}{10\cdots 16}a^{12}-\frac{11\cdots 43}{61\cdots 60}a^{11}+\frac{15\cdots 27}{41\cdots 64}a^{10}-\frac{11\cdots 77}{20\cdots 20}a^{9}+\frac{12\cdots 11}{61\cdots 60}a^{8}+\frac{33\cdots 17}{51\cdots 80}a^{7}+\frac{18\cdots 03}{51\cdots 08}a^{6}+\frac{55\cdots 63}{19\cdots 30}a^{5}+\frac{19\cdots 53}{30\cdots 48}a^{4}-\frac{21\cdots 89}{61\cdots 60}a^{3}-\frac{66\cdots 97}{61\cdots 60}a^{2}-\frac{18\cdots 99}{77\cdots 20}a-\frac{12\cdots 47}{21\cdots 40}$, $\frac{65\cdots 13}{30\cdots 00}a^{19}-\frac{12\cdots 83}{30\cdots 00}a^{18}-\frac{62\cdots 19}{30\cdots 00}a^{17}+\frac{89\cdots 81}{30\cdots 00}a^{16}+\frac{53\cdots 17}{77\cdots 00}a^{15}+\frac{56\cdots 31}{15\cdots 00}a^{14}-\frac{31\cdots 29}{30\cdots 00}a^{13}-\frac{90\cdots 83}{38\cdots 00}a^{12}+\frac{87\cdots 21}{10\cdots 00}a^{11}+\frac{95\cdots 49}{10\cdots 00}a^{10}-\frac{46\cdots 63}{10\cdots 00}a^{9}+\frac{84\cdots 61}{30\cdots 00}a^{8}+\frac{12\cdots 91}{38\cdots 00}a^{7}-\frac{17\cdots 19}{96\cdots 50}a^{6}+\frac{21\cdots 21}{32\cdots 50}a^{5}+\frac{48\cdots 67}{38\cdots 00}a^{4}-\frac{14\cdots 89}{10\cdots 00}a^{3}-\frac{16\cdots 03}{30\cdots 00}a^{2}-\frac{61\cdots 23}{38\cdots 00}a-\frac{21\cdots 67}{35\cdots 00}$ 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:  \( 1189076316751181600 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 1189076316751181600 \cdot 125}{2\cdot\sqrt{10818327022517881605935827576487172209050638458251953125}}\cr\approx \mathstrut & 0.878149833589371 \end{aligned}\] (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 - 156*x^18 - 952*x^17 + 17527*x^16 + 241902*x^15 - 1136451*x^14 - 21143487*x^13 + 32888707*x^12 + 309949732*x^11 - 8399093196*x^10 + 50338298318*x^9 + 193793523407*x^8 - 3449484980348*x^7 - 29815059450616*x^6 + 19997778190588*x^5 - 750024162066003*x^4 + 2346903326902332*x^3 + 8078639162151899*x^2 + 16769387472122923*x + 53854635816438281) 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 - 156*x^18 - 952*x^17 + 17527*x^16 + 241902*x^15 - 1136451*x^14 - 21143487*x^13 + 32888707*x^12 + 309949732*x^11 - 8399093196*x^10 + 50338298318*x^9 + 193793523407*x^8 - 3449484980348*x^7 - 29815059450616*x^6 + 19997778190588*x^5 - 750024162066003*x^4 + 2346903326902332*x^3 + 8078639162151899*x^2 + 16769387472122923*x + 53854635816438281, 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 - 156*x^18 - 952*x^17 + 17527*x^16 + 241902*x^15 - 1136451*x^14 - 21143487*x^13 + 32888707*x^12 + 309949732*x^11 - 8399093196*x^10 + 50338298318*x^9 + 193793523407*x^8 - 3449484980348*x^7 - 29815059450616*x^6 + 19997778190588*x^5 - 750024162066003*x^4 + 2346903326902332*x^3 + 8078639162151899*x^2 + 16769387472122923*x + 53854635816438281); 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 - 156*x^18 - 952*x^17 + 17527*x^16 + 241902*x^15 - 1136451*x^14 - 21143487*x^13 + 32888707*x^12 + 309949732*x^11 - 8399093196*x^10 + 50338298318*x^9 + 193793523407*x^8 - 3449484980348*x^7 - 29815059450616*x^6 + 19997778190588*x^5 - 750024162066003*x^4 + 2346903326902332*x^3 + 8078639162151899*x^2 + 16769387472122923*x + 53854635816438281); 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

$C_2\times F_5$ (as 20T9):

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 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.15125.1, 5.1.5171495850125.3, 10.2.133721846639300482312578125.2

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 10 siblings: 10.0.294188062606461061087671875.1, deg 10
Degree 20 sibling: deg 20
Minimal sibling: 10.0.294188062606461061087671875.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{5}$ ${\href{/padicField/3.4.0.1}{4} }^{5}$ R ${\href{/padicField/7.4.0.1}{4} }^{5}$ R ${\href{/padicField/13.4.0.1}{4} }^{5}$ ${\href{/padicField/17.4.0.1}{4} }^{5}$ ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{5}$ ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{5}$ R ${\href{/padicField/43.4.0.1}{4} }^{5}$ ${\href{/padicField/47.4.0.1}{4} }^{5}$ ${\href{/padicField/53.4.0.1}{4} }^{5}$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

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
\(5\) Copy content Toggle raw display 5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
\(11\) Copy content Toggle raw display 11.1.10.9a1.4$x^{10} + 44$$10$$1$$9$$C_{10}$$$[\ ]_{10}$$
11.1.10.9a1.4$x^{10} + 44$$10$$1$$9$$C_{10}$$$[\ ]_{10}$$
\(41\) Copy content Toggle raw display 41.2.5.8a1.1$x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$$5$$2$$8$$C_{10}$$$[\ ]_{5}^{2}$$
41.2.5.8a1.1$x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$$5$$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)