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Properties

Label 20.8.500...552.1
Degree $20$
Signature $[8, 6]$
Discriminant $5.002\times 10^{33}$
Root discriminant \(48.41\)
Ramified primes $2,61,397$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^9.S_5$ (as 20T674)

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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 + 4*x^18 - 33*x^17 + 24*x^16 - 339*x^15 - 59*x^14 - 1397*x^13 + 1772*x^12 + 1872*x^11 - 3190*x^10 + 26559*x^9 + 17509*x^8 - 53515*x^7 - 62851*x^6 + 10860*x^5 + 76191*x^4 + 20860*x^3 - 41472*x^2 - 10700*x + 6131)
 
Copy content gp:K = bnfinit(y^20 + 4*y^18 - 33*y^17 + 24*y^16 - 339*y^15 - 59*y^14 - 1397*y^13 + 1772*y^12 + 1872*y^11 - 3190*y^10 + 26559*y^9 + 17509*y^8 - 53515*y^7 - 62851*y^6 + 10860*y^5 + 76191*y^4 + 20860*y^3 - 41472*y^2 - 10700*y + 6131, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 4*x^18 - 33*x^17 + 24*x^16 - 339*x^15 - 59*x^14 - 1397*x^13 + 1772*x^12 + 1872*x^11 - 3190*x^10 + 26559*x^9 + 17509*x^8 - 53515*x^7 - 62851*x^6 + 10860*x^5 + 76191*x^4 + 20860*x^3 - 41472*x^2 - 10700*x + 6131);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 + 4*x^18 - 33*x^17 + 24*x^16 - 339*x^15 - 59*x^14 - 1397*x^13 + 1772*x^12 + 1872*x^11 - 3190*x^10 + 26559*x^9 + 17509*x^8 - 53515*x^7 - 62851*x^6 + 10860*x^5 + 76191*x^4 + 20860*x^3 - 41472*x^2 - 10700*x + 6131)
 

\( x^{20} + 4 x^{18} - 33 x^{17} + 24 x^{16} - 339 x^{15} - 59 x^{14} - 1397 x^{13} + 1772 x^{12} + \cdots + 6131 \) 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:   \(5001984585680627954608248429847552\) \(\medspace = 2^{10}\cdot 61^{7}\cdot 397^{7}\) 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:  \(48.41\)
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:  $2^{15/8}61^{3/4}397^{1/2}\approx 1595.232068334258$
Ramified primes:   \(2\), \(61\), \(397\) 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:  \(\Q(\sqrt{24217}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{12\cdots 79}a^{19}+\frac{27\cdots 40}{12\cdots 79}a^{18}+\frac{55\cdots 80}{12\cdots 79}a^{17}-\frac{14\cdots 93}{12\cdots 79}a^{16}-\frac{32\cdots 93}{12\cdots 79}a^{15}+\frac{65\cdots 28}{12\cdots 79}a^{14}-\frac{16\cdots 98}{12\cdots 79}a^{13}-\frac{76\cdots 13}{12\cdots 79}a^{12}+\frac{22\cdots 38}{12\cdots 79}a^{11}-\frac{61\cdots 04}{12\cdots 79}a^{10}-\frac{35\cdots 85}{12\cdots 79}a^{9}+\frac{51\cdots 77}{12\cdots 79}a^{8}+\frac{26\cdots 85}{12\cdots 79}a^{7}-\frac{32\cdots 75}{12\cdots 79}a^{6}-\frac{26\cdots 06}{12\cdots 79}a^{5}+\frac{54\cdots 85}{12\cdots 79}a^{4}-\frac{68\cdots 59}{12\cdots 79}a^{3}-\frac{43\cdots 34}{12\cdots 79}a^{2}+\frac{19\cdots 20}{12\cdots 79}a+\frac{37\cdots 86}{12\cdots 79}$ 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:  $C_{2}$, which has order $2$ (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:  $13$
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:   $\frac{38\cdots 90}{12\cdots 79}a^{19}-\frac{43\cdots 10}{12\cdots 79}a^{18}+\frac{19\cdots 81}{12\cdots 79}a^{17}-\frac{14\cdots 37}{12\cdots 79}a^{16}+\frac{25\cdots 68}{12\cdots 79}a^{15}-\frac{15\cdots 64}{12\cdots 79}a^{14}+\frac{14\cdots 48}{12\cdots 79}a^{13}-\frac{67\cdots 27}{12\cdots 79}a^{12}+\frac{14\cdots 33}{12\cdots 79}a^{11}-\frac{77\cdots 47}{12\cdots 79}a^{10}-\frac{68\cdots 71}{12\cdots 79}a^{9}+\frac{10\cdots 94}{12\cdots 79}a^{8}-\frac{56\cdots 18}{12\cdots 79}a^{7}-\frac{16\cdots 97}{12\cdots 79}a^{6}-\frac{39\cdots 43}{12\cdots 79}a^{5}+\frac{14\cdots 72}{12\cdots 79}a^{4}+\frac{15\cdots 48}{12\cdots 79}a^{3}-\frac{13\cdots 90}{12\cdots 79}a^{2}-\frac{63\cdots 86}{12\cdots 79}a+\frac{36\cdots 69}{12\cdots 79}$, $\frac{38\cdots 90}{12\cdots 79}a^{19}-\frac{43\cdots 10}{12\cdots 79}a^{18}+\frac{19\cdots 81}{12\cdots 79}a^{17}-\frac{14\cdots 37}{12\cdots 79}a^{16}+\frac{25\cdots 68}{12\cdots 79}a^{15}-\frac{15\cdots 64}{12\cdots 79}a^{14}+\frac{14\cdots 48}{12\cdots 79}a^{13}-\frac{67\cdots 27}{12\cdots 79}a^{12}+\frac{14\cdots 33}{12\cdots 79}a^{11}-\frac{77\cdots 47}{12\cdots 79}a^{10}-\frac{68\cdots 71}{12\cdots 79}a^{9}+\frac{10\cdots 94}{12\cdots 79}a^{8}-\frac{56\cdots 18}{12\cdots 79}a^{7}-\frac{16\cdots 97}{12\cdots 79}a^{6}-\frac{39\cdots 43}{12\cdots 79}a^{5}+\frac{14\cdots 72}{12\cdots 79}a^{4}+\frac{15\cdots 48}{12\cdots 79}a^{3}-\frac{13\cdots 90}{12\cdots 79}a^{2}-\frac{63\cdots 86}{12\cdots 79}a+\frac{48\cdots 48}{12\cdots 79}$, $\frac{17\cdots 66}{12\cdots 79}a^{19}-\frac{49\cdots 47}{12\cdots 79}a^{18}+\frac{12\cdots 10}{12\cdots 79}a^{17}-\frac{82\cdots 93}{12\cdots 79}a^{16}+\frac{23\cdots 31}{12\cdots 79}a^{15}-\frac{90\cdots 11}{12\cdots 79}a^{14}+\frac{18\cdots 53}{12\cdots 79}a^{13}-\frac{41\cdots 11}{12\cdots 79}a^{12}+\frac{11\cdots 37}{12\cdots 79}a^{11}-\frac{13\cdots 33}{12\cdots 79}a^{10}+\frac{38\cdots 86}{12\cdots 79}a^{9}+\frac{61\cdots 79}{12\cdots 79}a^{8}-\frac{12\cdots 85}{12\cdots 79}a^{7}-\frac{16\cdots 15}{12\cdots 79}a^{6}+\frac{89\cdots 45}{12\cdots 79}a^{5}+\frac{65\cdots 11}{12\cdots 79}a^{4}+\frac{38\cdots 88}{12\cdots 79}a^{3}-\frac{18\cdots 95}{12\cdots 79}a^{2}+\frac{13\cdots 68}{12\cdots 79}a+\frac{79\cdots 94}{12\cdots 79}$, $\frac{63\cdots 33}{12\cdots 79}a^{19}-\frac{76\cdots 20}{12\cdots 79}a^{18}+\frac{33\cdots 00}{12\cdots 79}a^{17}-\frac{25\cdots 52}{12\cdots 79}a^{16}+\frac{44\cdots 76}{12\cdots 79}a^{15}-\frac{26\cdots 87}{12\cdots 79}a^{14}+\frac{27\cdots 10}{12\cdots 79}a^{13}-\frac{11\cdots 09}{12\cdots 79}a^{12}+\frac{25\cdots 97}{12\cdots 79}a^{11}-\frac{15\cdots 82}{12\cdots 79}a^{10}-\frac{58\cdots 86}{12\cdots 79}a^{9}+\frac{17\cdots 54}{12\cdots 79}a^{8}-\frac{96\cdots 90}{12\cdots 79}a^{7}-\frac{26\cdots 04}{12\cdots 79}a^{6}-\frac{90\cdots 65}{12\cdots 79}a^{5}+\frac{25\cdots 80}{12\cdots 79}a^{4}+\frac{28\cdots 54}{12\cdots 79}a^{3}-\frac{21\cdots 13}{12\cdots 79}a^{2}-\frac{14\cdots 15}{12\cdots 79}a+\frac{59\cdots 64}{12\cdots 79}$, $\frac{10\cdots 09}{12\cdots 79}a^{19}-\frac{10\cdots 88}{12\cdots 79}a^{18}+\frac{52\cdots 46}{12\cdots 79}a^{17}-\frac{39\cdots 94}{12\cdots 79}a^{16}+\frac{65\cdots 12}{12\cdots 79}a^{15}-\frac{42\cdots 11}{12\cdots 79}a^{14}+\frac{36\cdots 10}{12\cdots 79}a^{13}-\frac{18\cdots 35}{12\cdots 79}a^{12}+\frac{37\cdots 54}{12\cdots 79}a^{11}-\frac{18\cdots 37}{12\cdots 79}a^{10}-\frac{14\cdots 17}{12\cdots 79}a^{9}+\frac{29\cdots 72}{12\cdots 79}a^{8}-\frac{11\cdots 19}{12\cdots 79}a^{7}-\frac{44\cdots 20}{12\cdots 79}a^{6}-\frac{21\cdots 87}{12\cdots 79}a^{5}+\frac{31\cdots 32}{12\cdots 79}a^{4}+\frac{46\cdots 19}{12\cdots 79}a^{3}-\frac{24\cdots 18}{12\cdots 79}a^{2}-\frac{16\cdots 68}{12\cdots 79}a+\frac{71\cdots 63}{12\cdots 79}$, $\frac{13\cdots 34}{12\cdots 79}a^{19}-\frac{19\cdots 17}{12\cdots 79}a^{18}+\frac{61\cdots 70}{12\cdots 79}a^{17}-\frac{46\cdots 53}{12\cdots 79}a^{16}+\frac{42\cdots 17}{12\cdots 79}a^{15}-\frac{49\cdots 38}{12\cdots 79}a^{14}+\frac{60\cdots 58}{12\cdots 79}a^{13}-\frac{22\cdots 99}{12\cdots 79}a^{12}+\frac{31\cdots 70}{12\cdots 79}a^{11}+\frac{57\cdots 33}{12\cdots 79}a^{10}-\frac{24\cdots 61}{12\cdots 79}a^{9}+\frac{32\cdots 35}{12\cdots 79}a^{8}+\frac{19\cdots 43}{12\cdots 79}a^{7}-\frac{52\cdots 11}{12\cdots 79}a^{6}-\frac{95\cdots 64}{12\cdots 79}a^{5}-\frac{14\cdots 57}{12\cdots 79}a^{4}+\frac{92\cdots 59}{12\cdots 79}a^{3}+\frac{45\cdots 02}{12\cdots 79}a^{2}-\frac{24\cdots 24}{12\cdots 79}a-\frac{27\cdots 67}{12\cdots 79}$, $\frac{54\cdots 10}{12\cdots 79}a^{19}-\frac{30\cdots 85}{12\cdots 79}a^{18}+\frac{25\cdots 16}{12\cdots 79}a^{17}-\frac{19\cdots 95}{12\cdots 79}a^{16}+\frac{25\cdots 53}{12\cdots 79}a^{15}-\frac{20\cdots 62}{12\cdots 79}a^{14}+\frac{95\cdots 58}{12\cdots 79}a^{13}-\frac{90\cdots 25}{12\cdots 79}a^{12}+\frac{15\cdots 27}{12\cdots 79}a^{11}-\frac{19\cdots 22}{12\cdots 79}a^{10}-\frac{93\cdots 92}{12\cdots 79}a^{9}+\frac{15\cdots 53}{12\cdots 79}a^{8}+\frac{44\cdots 14}{12\cdots 79}a^{7}-\frac{22\cdots 27}{12\cdots 79}a^{6}-\frac{21\cdots 86}{12\cdots 79}a^{5}+\frac{53\cdots 95}{12\cdots 79}a^{4}+\frac{27\cdots 24}{12\cdots 79}a^{3}+\frac{13\cdots 97}{12\cdots 79}a^{2}-\frac{46\cdots 77}{12\cdots 79}a+\frac{87\cdots 90}{12\cdots 79}$, $\frac{42\cdots 85}{12\cdots 79}a^{19}-\frac{46\cdots 72}{12\cdots 79}a^{18}+\frac{21\cdots 29}{12\cdots 79}a^{17}-\frac{16\cdots 25}{12\cdots 79}a^{16}+\frac{27\cdots 96}{12\cdots 79}a^{15}-\frac{17\cdots 00}{12\cdots 79}a^{14}+\frac{16\cdots 92}{12\cdots 79}a^{13}-\frac{75\cdots 61}{12\cdots 79}a^{12}+\frac{16\cdots 40}{12\cdots 79}a^{11}-\frac{91\cdots 67}{12\cdots 79}a^{10}-\frac{13\cdots 93}{12\cdots 79}a^{9}+\frac{11\cdots 44}{12\cdots 79}a^{8}-\frac{44\cdots 38}{12\cdots 79}a^{7}-\frac{19\cdots 45}{12\cdots 79}a^{6}-\frac{93\cdots 62}{12\cdots 79}a^{5}+\frac{15\cdots 85}{12\cdots 79}a^{4}+\frac{18\cdots 22}{12\cdots 79}a^{3}-\frac{87\cdots 97}{12\cdots 79}a^{2}-\frac{94\cdots 21}{12\cdots 79}a+\frac{29\cdots 38}{12\cdots 79}$, $\frac{16\cdots 23}{12\cdots 79}a^{19}-\frac{13\cdots 35}{12\cdots 79}a^{18}+\frac{79\cdots 96}{12\cdots 79}a^{17}-\frac{60\cdots 41}{12\cdots 79}a^{16}+\frac{92\cdots 86}{12\cdots 79}a^{15}-\frac{63\cdots 09}{12\cdots 79}a^{14}+\frac{46\cdots 59}{12\cdots 79}a^{13}-\frac{27\cdots 41}{12\cdots 79}a^{12}+\frac{53\cdots 67}{12\cdots 79}a^{11}-\frac{20\cdots 49}{12\cdots 79}a^{10}-\frac{23\cdots 06}{12\cdots 79}a^{9}+\frac{44\cdots 26}{12\cdots 79}a^{8}-\frac{10\cdots 33}{12\cdots 79}a^{7}-\frac{69\cdots 13}{12\cdots 79}a^{6}-\frac{43\cdots 94}{12\cdots 79}a^{5}+\frac{41\cdots 47}{12\cdots 79}a^{4}+\frac{77\cdots 62}{12\cdots 79}a^{3}-\frac{25\cdots 96}{12\cdots 79}a^{2}-\frac{29\cdots 68}{12\cdots 79}a+\frac{74\cdots 88}{12\cdots 79}$, $\frac{21\cdots 56}{12\cdots 79}a^{19}-\frac{24\cdots 58}{12\cdots 79}a^{18}+\frac{11\cdots 02}{12\cdots 79}a^{17}-\frac{85\cdots 40}{12\cdots 79}a^{16}+\frac{14\cdots 44}{12\cdots 79}a^{15}-\frac{90\cdots 02}{12\cdots 79}a^{14}+\frac{88\cdots 18}{12\cdots 79}a^{13}-\frac{40\cdots 41}{12\cdots 79}a^{12}+\frac{84\cdots 66}{12\cdots 79}a^{11}-\frac{53\cdots 37}{12\cdots 79}a^{10}-\frac{77\cdots 48}{12\cdots 79}a^{9}+\frac{58\cdots 64}{12\cdots 79}a^{8}-\frac{27\cdots 43}{12\cdots 79}a^{7}-\frac{85\cdots 90}{12\cdots 79}a^{6}-\frac{42\cdots 53}{12\cdots 79}a^{5}+\frac{70\cdots 80}{12\cdots 79}a^{4}+\frac{86\cdots 04}{12\cdots 79}a^{3}-\frac{52\cdots 95}{12\cdots 79}a^{2}-\frac{30\cdots 28}{12\cdots 79}a+\frac{11\cdots 18}{12\cdots 79}$, $\frac{78\cdots 88}{12\cdots 79}a^{19}-\frac{49\cdots 10}{12\cdots 79}a^{18}+\frac{34\cdots 52}{12\cdots 79}a^{17}-\frac{28\cdots 53}{12\cdots 79}a^{16}+\frac{36\cdots 11}{12\cdots 79}a^{15}-\frac{29\cdots 91}{12\cdots 79}a^{14}+\frac{13\cdots 28}{12\cdots 79}a^{13}-\frac{11\cdots 99}{12\cdots 79}a^{12}+\frac{21\cdots 07}{12\cdots 79}a^{11}+\frac{14\cdots 32}{12\cdots 79}a^{10}-\frac{27\cdots 49}{12\cdots 79}a^{9}+\frac{22\cdots 76}{12\cdots 79}a^{8}-\frac{71\cdots 45}{12\cdots 79}a^{7}-\frac{41\cdots 21}{12\cdots 79}a^{6}-\frac{22\cdots 43}{12\cdots 79}a^{5}+\frac{23\cdots 95}{12\cdots 79}a^{4}+\frac{45\cdots 94}{12\cdots 79}a^{3}-\frac{13\cdots 70}{12\cdots 79}a^{2}-\frac{24\cdots 85}{12\cdots 79}a+\frac{78\cdots 29}{12\cdots 79}$, $\frac{58\cdots 88}{12\cdots 79}a^{19}-\frac{70\cdots 69}{12\cdots 79}a^{18}+\frac{31\cdots 89}{12\cdots 79}a^{17}-\frac{23\cdots 24}{12\cdots 79}a^{16}+\frac{41\cdots 19}{12\cdots 79}a^{15}-\frac{24\cdots 42}{12\cdots 79}a^{14}+\frac{26\cdots 86}{12\cdots 79}a^{13}-\frac{11\cdots 45}{12\cdots 79}a^{12}+\frac{24\cdots 56}{12\cdots 79}a^{11}-\frac{18\cdots 74}{12\cdots 79}a^{10}+\frac{34\cdots 17}{12\cdots 79}a^{9}+\frac{15\cdots 12}{12\cdots 79}a^{8}-\frac{81\cdots 83}{12\cdots 79}a^{7}-\frac{21\cdots 66}{12\cdots 79}a^{6}-\frac{10\cdots 94}{12\cdots 79}a^{5}+\frac{19\cdots 58}{12\cdots 79}a^{4}+\frac{20\cdots 70}{12\cdots 79}a^{3}-\frac{13\cdots 73}{12\cdots 79}a^{2}-\frac{73\cdots 62}{12\cdots 79}a+\frac{28\cdots 65}{12\cdots 79}$, $\frac{57\cdots 72}{12\cdots 79}a^{19}-\frac{63\cdots 68}{12\cdots 79}a^{18}+\frac{28\cdots 78}{12\cdots 79}a^{17}-\frac{22\cdots 41}{12\cdots 79}a^{16}+\frac{37\cdots 10}{12\cdots 79}a^{15}-\frac{23\cdots 15}{12\cdots 79}a^{14}+\frac{21\cdots 36}{12\cdots 79}a^{13}-\frac{98\cdots 04}{12\cdots 79}a^{12}+\frac{21\cdots 80}{12\cdots 79}a^{11}-\frac{94\cdots 19}{12\cdots 79}a^{10}-\frac{85\cdots 94}{12\cdots 79}a^{9}+\frac{16\cdots 97}{12\cdots 79}a^{8}-\frac{75\cdots 17}{12\cdots 79}a^{7}-\frac{26\cdots 34}{12\cdots 79}a^{6}-\frac{11\cdots 49}{12\cdots 79}a^{5}+\frac{19\cdots 36}{12\cdots 79}a^{4}+\frac{27\cdots 89}{12\cdots 79}a^{3}-\frac{14\cdots 41}{12\cdots 79}a^{2}-\frac{10\cdots 88}{12\cdots 79}a+\frac{39\cdots 03}{12\cdots 79}$ 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:  \( 897877398.442 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{6}\cdot 897877398.442 \cdot 2}{2\cdot\sqrt{5001984585680627954608248429847552}}\cr\approx \mathstrut & 0.199970089268 \end{aligned}\] (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 + 4*x^18 - 33*x^17 + 24*x^16 - 339*x^15 - 59*x^14 - 1397*x^13 + 1772*x^12 + 1872*x^11 - 3190*x^10 + 26559*x^9 + 17509*x^8 - 53515*x^7 - 62851*x^6 + 10860*x^5 + 76191*x^4 + 20860*x^3 - 41472*x^2 - 10700*x + 6131) 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 + 4*x^18 - 33*x^17 + 24*x^16 - 339*x^15 - 59*x^14 - 1397*x^13 + 1772*x^12 + 1872*x^11 - 3190*x^10 + 26559*x^9 + 17509*x^8 - 53515*x^7 - 62851*x^6 + 10860*x^5 + 76191*x^4 + 20860*x^3 - 41472*x^2 - 10700*x + 6131, 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 + 4*x^18 - 33*x^17 + 24*x^16 - 339*x^15 - 59*x^14 - 1397*x^13 + 1772*x^12 + 1872*x^11 - 3190*x^10 + 26559*x^9 + 17509*x^8 - 53515*x^7 - 62851*x^6 + 10860*x^5 + 76191*x^4 + 20860*x^3 - 41472*x^2 - 10700*x + 6131); 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 + 4*x^18 - 33*x^17 + 24*x^16 - 339*x^15 - 59*x^14 - 1397*x^13 + 1772*x^12 + 1872*x^11 - 3190*x^10 + 26559*x^9 + 17509*x^8 - 53515*x^7 - 62851*x^6 + 10860*x^5 + 76191*x^4 + 20860*x^3 - 41472*x^2 - 10700*x + 6131); 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

$C_2^9.S_5$ (as 20T674):

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 non-solvable group of order 61440
The 74 conjugacy class representatives for $C_2^9.S_5$
Character table for $C_2^9.S_5$

Intermediate fields

10.10.14202376626313.1

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: 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 R ${\href{/padicField/3.5.0.1}{5} }^{4}$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.4.0.1}{4} }^{5}$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.5.2.10a5.2$x^{10} + 2 x^{8} + 2 x^{7} + 4 x^{5} + x^{4} + 2 x^{3} + 2 x^{2} + 4 x + 3$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$$[2, 2, 2, 2]^{10}$$
2.10.1.0a1.1$x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$$1$$10$$0$$C_{10}$$$[\ ]^{10}$$
\(61\) Copy content Toggle raw display $\Q_{61}$$x + 59$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{61}$$x + 59$$1$$1$$0$Trivial$$[\ ]$$
61.2.1.0a1.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
61.2.1.0a1.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
61.2.1.0a1.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
61.1.4.3a1.1$x^{4} + 61$$4$$1$$3$$C_4$$$[\ ]_{4}$$
61.2.2.2a1.2$x^{4} + 120 x^{3} + 3604 x^{2} + 240 x + 65$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
61.2.2.2a1.2$x^{4} + 120 x^{3} + 3604 x^{2} + 240 x + 65$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(397\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$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)