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Properties

Label 20.0.496...000.3
Degree $20$
Signature $[0, 10]$
Discriminant $4.969\times 10^{45}$
Root discriminant \(192.67\)
Ramified primes $2,3,5,13$
Class number $18622568$ (GRH)
Class group [2, 2, 4655642] (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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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 - 45*x^18 - 20*x^17 + 1035*x^16 + 636*x^15 - 7840*x^14 + 2360*x^13 + 91710*x^12 + 74300*x^11 + 203160*x^10 - 37920*x^9 + 5611600*x^8 + 1269200*x^7 + 35937265*x^6 - 1871502*x^5 + 209760805*x^4 + 13888310*x^3 + 772864335*x^2 + 119407110*x + 1584355549)
 
Copy content gp:K = bnfinit(y^20 - 45*y^18 - 20*y^17 + 1035*y^16 + 636*y^15 - 7840*y^14 + 2360*y^13 + 91710*y^12 + 74300*y^11 + 203160*y^10 - 37920*y^9 + 5611600*y^8 + 1269200*y^7 + 35937265*y^6 - 1871502*y^5 + 209760805*y^4 + 13888310*y^3 + 772864335*y^2 + 119407110*y + 1584355549, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 45*x^18 - 20*x^17 + 1035*x^16 + 636*x^15 - 7840*x^14 + 2360*x^13 + 91710*x^12 + 74300*x^11 + 203160*x^10 - 37920*x^9 + 5611600*x^8 + 1269200*x^7 + 35937265*x^6 - 1871502*x^5 + 209760805*x^4 + 13888310*x^3 + 772864335*x^2 + 119407110*x + 1584355549);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 45*x^18 - 20*x^17 + 1035*x^16 + 636*x^15 - 7840*x^14 + 2360*x^13 + 91710*x^12 + 74300*x^11 + 203160*x^10 - 37920*x^9 + 5611600*x^8 + 1269200*x^7 + 35937265*x^6 - 1871502*x^5 + 209760805*x^4 + 13888310*x^3 + 772864335*x^2 + 119407110*x + 1584355549)
 

\( x^{20} - 45 x^{18} - 20 x^{17} + 1035 x^{16} + 636 x^{15} - 7840 x^{14} + 2360 x^{13} + \cdots + 1584355549 \) 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:  $[0, 10]$
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:   \(4968509573481201782226562500000000000000000000\) \(\medspace = 2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 13^{10}\) 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:  \(192.67\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2\cdot 3^{1/2}5^{17/10}13^{1/2}\approx 192.6687618815246$
Ramified primes:   \(2\), \(3\), \(5\), \(13\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_2\times C_{10}$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3900=2^{2}\cdot 3\cdot 5^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{3900}(1,·)$, $\chi_{3900}(389,·)$, $\chi_{3900}(571,·)$, $\chi_{3900}(1351,·)$, $\chi_{3900}(3719,·)$, $\chi_{3900}(781,·)$, $\chi_{3900}(1169,·)$, $\chi_{3900}(2131,·)$, $\chi_{3900}(599,·)$, $\chi_{3900}(1561,·)$, $\chi_{3900}(1949,·)$, $\chi_{3900}(2911,·)$, $\chi_{3900}(1379,·)$, $\chi_{3900}(2341,·)$, $\chi_{3900}(2729,·)$, $\chi_{3900}(3691,·)$, $\chi_{3900}(2159,·)$, $\chi_{3900}(3121,·)$, $\chi_{3900}(3509,·)$, $\chi_{3900}(2939,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{2}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{3}$, $\frac{1}{175}a^{10}-\frac{2}{35}a^{8}-\frac{2}{35}a^{7}-\frac{2}{5}a^{6}-\frac{1}{25}a^{5}+\frac{11}{35}a^{4}+\frac{1}{5}a^{3}+\frac{2}{35}a^{2}+\frac{2}{35}a-\frac{1}{25}$, $\frac{1}{175}a^{11}-\frac{2}{35}a^{9}-\frac{2}{35}a^{8}+\frac{1}{35}a^{7}-\frac{1}{25}a^{6}+\frac{11}{35}a^{5}+\frac{1}{5}a^{4}+\frac{2}{35}a^{3}+\frac{2}{35}a^{2}-\frac{82}{175}a$, $\frac{1}{1225}a^{12}+\frac{1}{1225}a^{11}+\frac{1}{1225}a^{10}-\frac{4}{245}a^{9}+\frac{17}{245}a^{8}-\frac{62}{1225}a^{7}-\frac{372}{1225}a^{6}+\frac{538}{1225}a^{5}-\frac{2}{49}a^{4}-\frac{94}{245}a^{3}+\frac{13}{1225}a^{2}+\frac{503}{1225}a+\frac{39}{175}$, $\frac{1}{1225}a^{13}-\frac{2}{35}a^{9}-\frac{1}{175}a^{8}+\frac{1}{245}a^{7}-\frac{16}{35}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{24}{175}a^{3}+\frac{2}{7}a^{2}-\frac{109}{245}a-\frac{12}{35}$, $\frac{1}{8575}a^{14}-\frac{1}{8575}a^{13}-\frac{2}{8575}a^{12}+\frac{12}{8575}a^{11}-\frac{16}{8575}a^{10}-\frac{387}{8575}a^{9}+\frac{542}{8575}a^{8}-\frac{83}{1225}a^{7}-\frac{3449}{8575}a^{6}+\frac{3222}{8575}a^{5}-\frac{572}{8575}a^{4}-\frac{3778}{8575}a^{3}+\frac{3279}{8575}a^{2}-\frac{594}{8575}a+\frac{276}{1225}$, $\frac{1}{8575}a^{15}-\frac{3}{8575}a^{13}+\frac{3}{8575}a^{12}-\frac{11}{8575}a^{11}-\frac{18}{8575}a^{10}+\frac{59}{1715}a^{9}+\frac{346}{8575}a^{8}-\frac{166}{8575}a^{7}+\frac{662}{8575}a^{6}-\frac{772}{1715}a^{5}+\frac{82}{1715}a^{4}-\frac{639}{8575}a^{3}+\frac{1614}{8575}a^{2}+\frac{2962}{8575}a-\frac{389}{1225}$, $\frac{1}{8575}a^{16}-\frac{3}{8575}a^{12}-\frac{17}{8575}a^{11}+\frac{16}{8575}a^{10}-\frac{121}{1715}a^{9}-\frac{107}{1715}a^{8}+\frac{256}{8575}a^{7}-\frac{1922}{8575}a^{6}-\frac{522}{8575}a^{5}-\frac{219}{1715}a^{4}+\frac{72}{1715}a^{3}-\frac{984}{8575}a^{2}+\frac{821}{1715}a+\frac{394}{1225}$, $\frac{1}{8575}a^{17}-\frac{3}{8575}a^{13}-\frac{3}{8575}a^{12}-\frac{19}{8575}a^{11}-\frac{3}{8575}a^{10}-\frac{13}{343}a^{9}-\frac{269}{8575}a^{8}-\frac{68}{1715}a^{7}-\frac{3672}{8575}a^{6}-\frac{374}{8575}a^{5}-\frac{803}{1715}a^{4}+\frac{3951}{8575}a^{3}-\frac{2573}{8575}a^{2}+\frac{52}{175}a-\frac{6}{175}$, $\frac{1}{20\cdots 75}a^{18}-\frac{347916039343}{20\cdots 75}a^{17}+\frac{823483440516}{20\cdots 75}a^{16}-\frac{102215307981}{29\cdots 25}a^{15}+\frac{148661584597}{20\cdots 75}a^{14}+\frac{6727995921543}{20\cdots 75}a^{13}+\frac{477964817927}{40\cdots 35}a^{12}-\frac{33800038161672}{20\cdots 75}a^{11}+\frac{39683221728739}{20\cdots 75}a^{10}-\frac{186469479958939}{20\cdots 75}a^{9}+\frac{457804812328923}{20\cdots 75}a^{8}+\frac{855188327173743}{20\cdots 75}a^{7}+\frac{70\cdots 98}{20\cdots 75}a^{6}+\frac{46064818040044}{202003057305175}a^{5}-\frac{44\cdots 39}{20\cdots 75}a^{4}+\frac{776661118538101}{40\cdots 35}a^{3}-\frac{27\cdots 68}{20\cdots 75}a^{2}+\frac{19\cdots 67}{40\cdots 35}a+\frac{46282465170911}{116584621644701}$, $\frac{1}{66\cdots 75}a^{19}+\frac{16\cdots 59}{66\cdots 75}a^{18}+\frac{47\cdots 07}{13\cdots 15}a^{17}+\frac{16\cdots 64}{66\cdots 75}a^{16}-\frac{62\cdots 81}{66\cdots 75}a^{15}+\frac{59\cdots 01}{66\cdots 75}a^{14}-\frac{18\cdots 66}{66\cdots 75}a^{13}-\frac{14\cdots 09}{66\cdots 75}a^{12}-\frac{11\cdots 46}{66\cdots 75}a^{11}+\frac{92\cdots 61}{13\cdots 15}a^{10}-\frac{11\cdots 73}{66\cdots 75}a^{9}-\frac{18\cdots 17}{66\cdots 75}a^{8}-\frac{29\cdots 97}{66\cdots 75}a^{7}-\frac{30\cdots 62}{66\cdots 75}a^{6}-\frac{31\cdots 79}{66\cdots 75}a^{5}+\frac{22\cdots 42}{94\cdots 25}a^{4}-\frac{22\cdots 94}{66\cdots 75}a^{3}-\frac{25\cdots 57}{66\cdots 75}a^{2}+\frac{12\cdots 81}{66\cdots 75}a-\frac{13\cdots 03}{94\cdots 25}$ 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:  $7$

Class group and class number

Ideal class group:  $C_{2}\times C_{2}\times C_{4655642}$, which has order $18622568$ (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:  $C_{2}\times C_{2}\times C_{4655642}$, which has order $18622568$ (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);
 
Relative class number:   $9311284$ (assuming GRH)

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:  $9$
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:   $\frac{64\cdots 88}{66\cdots 75}a^{19}+\frac{10\cdots 24}{13\cdots 15}a^{18}-\frac{31\cdots 02}{66\cdots 75}a^{17}-\frac{50\cdots 69}{13\cdots 15}a^{16}+\frac{68\cdots 32}{66\cdots 75}a^{15}+\frac{61\cdots 46}{66\cdots 75}a^{14}-\frac{42\cdots 24}{66\cdots 75}a^{13}-\frac{47\cdots 94}{66\cdots 75}a^{12}+\frac{84\cdots 52}{66\cdots 75}a^{11}+\frac{51\cdots 24}{66\cdots 75}a^{10}+\frac{42\cdots 34}{13\cdots 15}a^{9}+\frac{73\cdots 63}{66\cdots 75}a^{8}+\frac{50\cdots 24}{13\cdots 15}a^{7}+\frac{18\cdots 76}{66\cdots 75}a^{6}+\frac{20\cdots 98}{13\cdots 15}a^{5}+\frac{21\cdots 03}{94\cdots 25}a^{4}+\frac{67\cdots 78}{66\cdots 75}a^{3}+\frac{50\cdots 91}{66\cdots 75}a^{2}-\frac{32\cdots 24}{66\cdots 75}a+\frac{57\cdots 34}{21\cdots 75}$, $\frac{15\cdots 68}{66\cdots 75}a^{19}-\frac{17\cdots 02}{66\cdots 75}a^{18}-\frac{13\cdots 18}{13\cdots 15}a^{17}+\frac{62\cdots 14}{66\cdots 75}a^{16}+\frac{15\cdots 76}{66\cdots 75}a^{15}-\frac{14\cdots 34}{66\cdots 75}a^{14}-\frac{94\cdots 68}{66\cdots 75}a^{13}+\frac{32\cdots 17}{66\cdots 75}a^{12}+\frac{92\cdots 72}{66\cdots 75}a^{11}-\frac{19\cdots 03}{13\cdots 15}a^{10}+\frac{31\cdots 48}{26\cdots 43}a^{9}+\frac{47\cdots 43}{66\cdots 75}a^{8}+\frac{73\cdots 76}{66\cdots 75}a^{7}-\frac{15\cdots 13}{66\cdots 75}a^{6}+\frac{56\cdots 84}{66\cdots 75}a^{5}-\frac{55\cdots 87}{94\cdots 25}a^{4}+\frac{28\cdots 94}{66\cdots 75}a^{3}+\frac{17\cdots 11}{66\cdots 75}a^{2}+\frac{10\cdots 94}{13\cdots 15}a+\frac{14\cdots 01}{21\cdots 75}$, $\frac{18\cdots 12}{66\cdots 75}a^{19}-\frac{32\cdots 54}{66\cdots 75}a^{18}-\frac{85\cdots 92}{66\cdots 75}a^{17}+\frac{11\cdots 69}{66\cdots 75}a^{16}+\frac{20\cdots 04}{66\cdots 75}a^{15}-\frac{22\cdots 56}{66\cdots 75}a^{14}-\frac{18\cdots 44}{66\cdots 75}a^{13}+\frac{60\cdots 97}{13\cdots 15}a^{12}+\frac{16\cdots 12}{66\cdots 75}a^{11}-\frac{16\cdots 73}{66\cdots 75}a^{10}+\frac{51\cdots 94}{13\cdots 15}a^{9}-\frac{10\cdots 23}{66\cdots 75}a^{8}+\frac{97\cdots 12}{66\cdots 75}a^{7}-\frac{12\cdots 13}{66\cdots 75}a^{6}+\frac{42\cdots 62}{66\cdots 75}a^{5}-\frac{18\cdots 23}{94\cdots 25}a^{4}+\frac{31\cdots 04}{66\cdots 75}a^{3}-\frac{39\cdots 92}{66\cdots 75}a^{2}+\frac{20\cdots 78}{26\cdots 43}a-\frac{95\cdots 54}{43\cdots 15}$, $\frac{26\cdots 44}{66\cdots 75}a^{19}-\frac{15\cdots 52}{66\cdots 75}a^{18}-\frac{16\cdots 02}{66\cdots 75}a^{17}+\frac{10\cdots 51}{13\cdots 15}a^{16}+\frac{53\cdots 28}{66\cdots 75}a^{15}-\frac{77\cdots 22}{66\cdots 75}a^{14}-\frac{87\cdots 76}{66\cdots 75}a^{13}-\frac{20\cdots 32}{66\cdots 75}a^{12}+\frac{15\cdots 88}{13\cdots 15}a^{11}-\frac{67\cdots 58}{66\cdots 75}a^{10}-\frac{15\cdots 46}{13\cdots 15}a^{9}-\frac{10\cdots 66}{66\cdots 75}a^{8}+\frac{24\cdots 36}{66\cdots 75}a^{7}-\frac{44\cdots 08}{26\cdots 43}a^{6}-\frac{13\cdots 22}{66\cdots 75}a^{5}-\frac{18\cdots 36}{94\cdots 25}a^{4}+\frac{52\cdots 22}{13\cdots 15}a^{3}-\frac{41\cdots 03}{66\cdots 75}a^{2}-\frac{43\cdots 04}{13\cdots 15}a-\frac{40\cdots 96}{21\cdots 75}$, $\frac{90\cdots 32}{66\cdots 75}a^{19}+\frac{14\cdots 59}{66\cdots 75}a^{18}-\frac{35\cdots 17}{66\cdots 75}a^{17}-\frac{67\cdots 78}{66\cdots 75}a^{16}+\frac{67\cdots 71}{66\cdots 75}a^{15}+\frac{12\cdots 49}{66\cdots 75}a^{14}-\frac{12\cdots 18}{66\cdots 75}a^{13}+\frac{10\cdots 57}{66\cdots 75}a^{12}+\frac{57\cdots 58}{66\cdots 75}a^{11}+\frac{12\cdots 76}{66\cdots 75}a^{10}+\frac{29\cdots 10}{26\cdots 43}a^{9}+\frac{80\cdots 02}{66\cdots 75}a^{8}+\frac{36\cdots 93}{66\cdots 75}a^{7}+\frac{59\cdots 22}{66\cdots 75}a^{6}+\frac{21\cdots 22}{66\cdots 75}a^{5}+\frac{61\cdots 92}{94\cdots 25}a^{4}+\frac{11\cdots 56}{13\cdots 15}a^{3}+\frac{12\cdots 49}{66\cdots 75}a^{2}+\frac{69\cdots 06}{66\cdots 75}a+\frac{47\cdots 19}{21\cdots 75}$, $\frac{92\cdots 79}{66\cdots 75}a^{19}-\frac{31\cdots 61}{13\cdots 15}a^{18}-\frac{42\cdots 16}{66\cdots 75}a^{17}+\frac{62\cdots 42}{66\cdots 75}a^{16}+\frac{10\cdots 28}{66\cdots 75}a^{15}-\frac{14\cdots 57}{66\cdots 75}a^{14}-\frac{79\cdots 47}{66\cdots 75}a^{13}+\frac{23\cdots 23}{66\cdots 75}a^{12}+\frac{12\cdots 06}{13\cdots 15}a^{11}-\frac{13\cdots 88}{66\cdots 75}a^{10}+\frac{71\cdots 34}{13\cdots 15}a^{9}-\frac{26\cdots 81}{66\cdots 75}a^{8}+\frac{88\cdots 56}{13\cdots 15}a^{7}-\frac{25\cdots 92}{66\cdots 75}a^{6}+\frac{26\cdots 58}{66\cdots 75}a^{5}-\frac{13\cdots 96}{94\cdots 25}a^{4}+\frac{14\cdots 59}{66\cdots 75}a^{3}+\frac{14\cdots 53}{66\cdots 75}a^{2}+\frac{25\cdots 22}{66\cdots 75}a+\frac{11\cdots 99}{21\cdots 75}$, $\frac{64\cdots 98}{66\cdots 75}a^{19}+\frac{10\cdots 66}{66\cdots 75}a^{18}-\frac{40\cdots 26}{66\cdots 75}a^{17}-\frac{74\cdots 27}{66\cdots 75}a^{16}+\frac{11\cdots 93}{66\cdots 75}a^{15}+\frac{23\cdots 81}{66\cdots 75}a^{14}-\frac{17\cdots 97}{66\cdots 75}a^{13}-\frac{27\cdots 88}{66\cdots 75}a^{12}+\frac{17\cdots 64}{66\cdots 75}a^{11}+\frac{17\cdots 77}{66\cdots 75}a^{10}-\frac{24\cdots 43}{13\cdots 15}a^{9}-\frac{10\cdots 87}{66\cdots 75}a^{8}+\frac{44\cdots 17}{66\cdots 75}a^{7}+\frac{17\cdots 49}{66\cdots 75}a^{6}-\frac{38\cdots 77}{66\cdots 75}a^{5}-\frac{68\cdots 62}{94\cdots 25}a^{4}-\frac{23\cdots 63}{26\cdots 43}a^{3}-\frac{28\cdots 88}{26\cdots 43}a^{2}-\frac{67\cdots 54}{13\cdots 15}a-\frac{47\cdots 26}{21\cdots 75}$, $\frac{90\cdots 66}{18\cdots 45}a^{19}-\frac{72\cdots 17}{94\cdots 25}a^{18}-\frac{20\cdots 36}{94\cdots 25}a^{17}+\frac{25\cdots 17}{94\cdots 25}a^{16}+\frac{50\cdots 73}{94\cdots 25}a^{15}-\frac{10\cdots 92}{18\cdots 45}a^{14}-\frac{63\cdots 47}{13\cdots 75}a^{13}+\frac{31\cdots 60}{37\cdots 49}a^{12}+\frac{41\cdots 43}{94\cdots 25}a^{11}-\frac{38\cdots 62}{94\cdots 25}a^{10}+\frac{39\cdots 23}{18\cdots 45}a^{9}-\frac{37\cdots 82}{18\cdots 45}a^{8}+\frac{24\cdots 61}{94\cdots 25}a^{7}-\frac{19\cdots 33}{94\cdots 25}a^{6}+\frac{10\cdots 46}{94\cdots 25}a^{5}-\frac{83\cdots 45}{37\cdots 49}a^{4}+\frac{76\cdots 91}{94\cdots 25}a^{3}-\frac{51\cdots 11}{94\cdots 25}a^{2}+\frac{12\cdots 01}{94\cdots 25}a-\frac{23\cdots 88}{31\cdots 25}$, $\frac{25\cdots 76}{17\cdots 75}a^{19}+\frac{52\cdots 23}{34\cdots 15}a^{18}-\frac{27\cdots 04}{17\cdots 75}a^{17}-\frac{24\cdots 88}{34\cdots 15}a^{16}+\frac{46\cdots 14}{17\cdots 75}a^{15}+\frac{29\cdots 92}{17\cdots 75}a^{14}-\frac{11\cdots 77}{34\cdots 75}a^{13}-\frac{25\cdots 38}{17\cdots 75}a^{12}+\frac{16\cdots 29}{17\cdots 75}a^{11}+\frac{24\cdots 23}{17\cdots 75}a^{10}+\frac{45\cdots 93}{34\cdots 15}a^{9}+\frac{37\cdots 51}{17\cdots 75}a^{8}-\frac{11\cdots 77}{34\cdots 15}a^{7}+\frac{93\cdots 27}{17\cdots 75}a^{6}-\frac{16\cdots 99}{34\cdots 15}a^{5}+\frac{16\cdots 33}{34\cdots 75}a^{4}-\frac{51\cdots 44}{17\cdots 75}a^{3}+\frac{37\cdots 51}{24\cdots 25}a^{2}-\frac{16\cdots 73}{17\cdots 75}a+\frac{51\cdots 93}{56\cdots 75}$ 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:  \( 180801817.57689384 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 180801817.57689384 \cdot 18622568}{2\cdot\sqrt{4968509573481201782226562500000000000000000000}}\cr\approx \mathstrut & 2.29032958609776 \end{aligned}\] (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 - 45*x^18 - 20*x^17 + 1035*x^16 + 636*x^15 - 7840*x^14 + 2360*x^13 + 91710*x^12 + 74300*x^11 + 203160*x^10 - 37920*x^9 + 5611600*x^8 + 1269200*x^7 + 35937265*x^6 - 1871502*x^5 + 209760805*x^4 + 13888310*x^3 + 772864335*x^2 + 119407110*x + 1584355549) 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 - 45*x^18 - 20*x^17 + 1035*x^16 + 636*x^15 - 7840*x^14 + 2360*x^13 + 91710*x^12 + 74300*x^11 + 203160*x^10 - 37920*x^9 + 5611600*x^8 + 1269200*x^7 + 35937265*x^6 - 1871502*x^5 + 209760805*x^4 + 13888310*x^3 + 772864335*x^2 + 119407110*x + 1584355549, 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 - 45*x^18 - 20*x^17 + 1035*x^16 + 636*x^15 - 7840*x^14 + 2360*x^13 + 91710*x^12 + 74300*x^11 + 203160*x^10 - 37920*x^9 + 5611600*x^8 + 1269200*x^7 + 35937265*x^6 - 1871502*x^5 + 209760805*x^4 + 13888310*x^3 + 772864335*x^2 + 119407110*x + 1584355549); 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 - 45*x^18 - 20*x^17 + 1035*x^16 + 636*x^15 - 7840*x^14 + 2360*x^13 + 91710*x^12 + 74300*x^11 + 203160*x^10 - 37920*x^9 + 5611600*x^8 + 1269200*x^7 + 35937265*x^6 - 1871502*x^5 + 209760805*x^4 + 13888310*x^3 + 772864335*x^2 + 119407110*x + 1584355549); 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

$C_2\times C_{10}$ (as 20T3):

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)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-195}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-13}, \sqrt{15})\), 5.5.390625.1, 10.0.68835601043701171875.3, 10.0.58014531250000000000.3, 10.10.189843750000000000.1

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 R R R ${\href{/padicField/7.1.0.1}{1} }^{20}$ ${\href{/padicField/11.5.0.1}{5} }^{4}$ R ${\href{/padicField/17.5.0.1}{5} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }^{2}$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ ${\href{/padicField/37.10.0.1}{10} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{10}$ ${\href{/padicField/47.10.0.1}{10} }^{2}$ ${\href{/padicField/53.5.0.1}{5} }^{4}$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

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
\(2\) Copy content Toggle raw display 2.10.2.20a1.1$x^{20} + 2 x^{16} + 2 x^{15} + 2 x^{13} + 3 x^{12} + 4 x^{11} + 5 x^{10} + 2 x^{9} + 4 x^{8} + 4 x^{7} + 7 x^{6} + 6 x^{5} + 3 x^{4} + 6 x^{3} + 5 x^{2} + 4 x + 5$$2$$10$$20$20T3not computed
\(3\) Copy content Toggle raw display 3.10.2.10a1.2$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} + 4 x + 7$$2$$10$$10$20T3$$[\ ]_{2}^{10}$$
\(5\) Copy content Toggle raw display 5.2.10.34a3.1$x^{20} + 40 x^{19} + 740 x^{18} + 8400 x^{17} + 65480 x^{16} + 371968 x^{15} + 1596160 x^{14} + 5299200 x^{13} + 13850080 x^{12} + 28888320 x^{11} + 48612480 x^{10} + 66490880 x^{9} + 74071040 x^{8} + 66846720 x^{7} + 48220160 x^{6} + 27222016 x^{5} + 11680000 x^{4} + 3655680 x^{3} + 783360 x^{2} + 102400 x + 6149$$10$$2$$34$not computednot computed
\(13\) Copy content Toggle raw display 13.10.2.10a1.2$x^{20} + 14 x^{15} + 10 x^{14} + 16 x^{13} + 2 x^{12} + 2 x^{11} + 53 x^{10} + 70 x^{9} + 137 x^{8} + 94 x^{7} + 88 x^{6} + 54 x^{5} + 37 x^{4} + 34 x^{3} + 5 x^{2} + 4 x + 17$$2$$10$$10$20T3$$[\ ]_{2}^{10}$$

Spectrum of ring of integers

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