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Properties

Label 20.20.850...000.1
Degree $20$
Signature $(20, 0)$
Discriminant $8.501\times 10^{33}$
Root discriminant \(49.71\)
Ramified primes $2,5,103$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_5$ (as 20T32)

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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^19 - 33*x^18 + 120*x^17 + 468*x^16 - 1392*x^15 - 3799*x^14 + 8012*x^13 + 19008*x^12 - 23720*x^11 - 57969*x^10 + 31228*x^9 + 101057*x^8 - 964*x^7 - 86922*x^6 - 33928*x^5 + 22258*x^4 + 19672*x^3 + 5396*x^2 + 600*x + 20)
 
Copy content gp:K = bnfinit(y^20 - 4*y^19 - 33*y^18 + 120*y^17 + 468*y^16 - 1392*y^15 - 3799*y^14 + 8012*y^13 + 19008*y^12 - 23720*y^11 - 57969*y^10 + 31228*y^9 + 101057*y^8 - 964*y^7 - 86922*y^6 - 33928*y^5 + 22258*y^4 + 19672*y^3 + 5396*y^2 + 600*y + 20, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 - 33*x^18 + 120*x^17 + 468*x^16 - 1392*x^15 - 3799*x^14 + 8012*x^13 + 19008*x^12 - 23720*x^11 - 57969*x^10 + 31228*x^9 + 101057*x^8 - 964*x^7 - 86922*x^6 - 33928*x^5 + 22258*x^4 + 19672*x^3 + 5396*x^2 + 600*x + 20);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 4*x^19 - 33*x^18 + 120*x^17 + 468*x^16 - 1392*x^15 - 3799*x^14 + 8012*x^13 + 19008*x^12 - 23720*x^11 - 57969*x^10 + 31228*x^9 + 101057*x^8 - 964*x^7 - 86922*x^6 - 33928*x^5 + 22258*x^4 + 19672*x^3 + 5396*x^2 + 600*x + 20)
 

\( x^{20} - 4 x^{19} - 33 x^{18} + 120 x^{17} + 468 x^{16} - 1392 x^{15} - 3799 x^{14} + 8012 x^{13} + \cdots + 20 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $20$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(20, 0)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(8501150111111046013911040000000000\) \(\medspace = 2^{36}\cdot 5^{10}\cdot 103^{8}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(49.71\)
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:  $2^{13/6}5^{1/2}103^{1/2}\approx 101.89087030315206$
Ramified primes:   \(2\), \(5\), \(103\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{82\cdots 34}a^{19}-\frac{27\cdots 01}{27\cdots 78}a^{18}+\frac{13\cdots 77}{91\cdots 26}a^{17}+\frac{13\cdots 37}{13\cdots 89}a^{16}-\frac{29\cdots 02}{13\cdots 89}a^{15}-\frac{72\cdots 49}{27\cdots 78}a^{14}-\frac{10\cdots 37}{82\cdots 34}a^{13}-\frac{76\cdots 69}{82\cdots 34}a^{12}-\frac{18\cdots 86}{41\cdots 67}a^{11}+\frac{28\cdots 89}{82\cdots 34}a^{10}-\frac{72\cdots 43}{82\cdots 34}a^{9}+\frac{27\cdots 03}{91\cdots 26}a^{8}-\frac{25\cdots 57}{82\cdots 34}a^{7}+\frac{97\cdots 91}{41\cdots 67}a^{6}-\frac{79\cdots 94}{41\cdots 67}a^{5}-\frac{86\cdots 59}{27\cdots 78}a^{4}-\frac{10\cdots 03}{41\cdots 67}a^{3}+\frac{25\cdots 16}{41\cdots 67}a^{2}-\frac{75\cdots 82}{41\cdots 67}a+\frac{19\cdots 90}{41\cdots 67}$ 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:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $19$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{15\cdots 83}{36\cdots 33}a^{19}-\frac{94\cdots 17}{24\cdots 22}a^{18}-\frac{71\cdots 11}{40\cdots 37}a^{17}+\frac{12\cdots 50}{12\cdots 11}a^{16}-\frac{18\cdots 18}{12\cdots 11}a^{15}-\frac{24\cdots 17}{24\cdots 22}a^{14}+\frac{83\cdots 83}{36\cdots 33}a^{13}+\frac{36\cdots 25}{73\cdots 66}a^{12}-\frac{50\cdots 86}{36\cdots 33}a^{11}-\frac{96\cdots 37}{73\cdots 66}a^{10}+\frac{15\cdots 21}{36\cdots 33}a^{9}+\frac{16\cdots 99}{81\cdots 74}a^{8}-\frac{25\cdots 39}{36\cdots 33}a^{7}-\frac{70\cdots 41}{36\cdots 33}a^{6}+\frac{19\cdots 82}{36\cdots 33}a^{5}+\frac{35\cdots 07}{24\cdots 22}a^{4}-\frac{51\cdots 20}{36\cdots 33}a^{3}-\frac{25\cdots 86}{36\cdots 33}a^{2}-\frac{35\cdots 26}{36\cdots 33}a+\frac{94\cdots 71}{36\cdots 33}$, $\frac{33\cdots 82}{41\cdots 67}a^{19}-\frac{48\cdots 15}{13\cdots 89}a^{18}-\frac{24\cdots 15}{91\cdots 26}a^{17}+\frac{28\cdots 27}{27\cdots 78}a^{16}+\frac{98\cdots 71}{27\cdots 78}a^{15}-\frac{33\cdots 87}{27\cdots 78}a^{14}-\frac{11\cdots 59}{41\cdots 67}a^{13}+\frac{29\cdots 32}{41\cdots 67}a^{12}+\frac{11\cdots 51}{82\cdots 34}a^{11}-\frac{18\cdots 37}{82\cdots 34}a^{10}-\frac{17\cdots 67}{41\cdots 67}a^{9}+\frac{15\cdots 33}{45\cdots 63}a^{8}+\frac{62\cdots 33}{82\cdots 34}a^{7}-\frac{11\cdots 13}{82\cdots 34}a^{6}-\frac{56\cdots 39}{82\cdots 34}a^{5}-\frac{43\cdots 67}{27\cdots 78}a^{4}+\frac{87\cdots 30}{41\cdots 67}a^{3}+\frac{51\cdots 78}{41\cdots 67}a^{2}+\frac{85\cdots 72}{41\cdots 67}a+\frac{28\cdots 79}{41\cdots 67}$, $\frac{34\cdots 23}{41\cdots 67}a^{19}-\frac{45\cdots 22}{13\cdots 89}a^{18}-\frac{13\cdots 01}{45\cdots 63}a^{17}+\frac{14\cdots 77}{13\cdots 89}a^{16}+\frac{56\cdots 14}{13\cdots 89}a^{15}-\frac{16\cdots 47}{13\cdots 89}a^{14}-\frac{13\cdots 75}{41\cdots 67}a^{13}+\frac{30\cdots 92}{41\cdots 67}a^{12}+\frac{69\cdots 37}{41\cdots 67}a^{11}-\frac{98\cdots 39}{41\cdots 67}a^{10}-\frac{21\cdots 85}{41\cdots 67}a^{9}+\frac{17\cdots 79}{45\cdots 63}a^{8}+\frac{36\cdots 62}{41\cdots 67}a^{7}-\frac{86\cdots 11}{41\cdots 67}a^{6}-\frac{32\cdots 16}{41\cdots 67}a^{5}-\frac{16\cdots 73}{13\cdots 89}a^{4}+\frac{10\cdots 11}{41\cdots 67}a^{3}+\frac{49\cdots 48}{41\cdots 67}a^{2}+\frac{73\cdots 44}{41\cdots 67}a+\frac{28\cdots 01}{41\cdots 67}$, $\frac{23\cdots 55}{41\cdots 67}a^{19}-\frac{40\cdots 82}{13\cdots 89}a^{18}-\frac{71\cdots 63}{45\cdots 63}a^{17}+\frac{23\cdots 99}{27\cdots 78}a^{16}+\frac{23\cdots 98}{13\cdots 89}a^{15}-\frac{26\cdots 97}{27\cdots 78}a^{14}-\frac{43\cdots 15}{41\cdots 67}a^{13}+\frac{23\cdots 30}{41\cdots 67}a^{12}+\frac{18\cdots 07}{41\cdots 67}a^{11}-\frac{14\cdots 83}{82\cdots 34}a^{10}-\frac{53\cdots 12}{41\cdots 67}a^{9}+\frac{13\cdots 49}{45\cdots 63}a^{8}+\frac{97\cdots 68}{41\cdots 67}a^{7}-\frac{17\cdots 67}{82\cdots 34}a^{6}-\frac{93\cdots 13}{41\cdots 67}a^{5}+\frac{36\cdots 65}{27\cdots 78}a^{4}+\frac{31\cdots 22}{41\cdots 67}a^{3}+\frac{14\cdots 86}{41\cdots 67}a^{2}+\frac{29\cdots 62}{41\cdots 67}a+\frac{25\cdots 01}{41\cdots 67}$, $\frac{26\cdots 09}{13\cdots 89}a^{19}-\frac{56\cdots 31}{45\cdots 63}a^{18}-\frac{16\cdots 33}{45\cdots 63}a^{17}+\frac{31\cdots 27}{91\cdots 26}a^{16}+\frac{46\cdots 58}{45\cdots 63}a^{15}-\frac{32\cdots 49}{91\cdots 26}a^{14}+\frac{21\cdots 57}{13\cdots 89}a^{13}+\frac{24\cdots 03}{13\cdots 89}a^{12}-\frac{18\cdots 55}{13\cdots 89}a^{11}-\frac{13\cdots 71}{27\cdots 78}a^{10}+\frac{62\cdots 66}{13\cdots 89}a^{9}+\frac{27\cdots 08}{45\cdots 63}a^{8}-\frac{10\cdots 93}{13\cdots 89}a^{7}-\frac{40\cdots 31}{27\cdots 78}a^{6}+\frac{86\cdots 71}{13\cdots 89}a^{5}-\frac{27\cdots 55}{91\cdots 26}a^{4}-\frac{40\cdots 44}{13\cdots 89}a^{3}+\frac{19\cdots 43}{13\cdots 89}a^{2}+\frac{12\cdots 34}{13\cdots 89}a+\frac{13\cdots 11}{13\cdots 89}$, $\frac{18\cdots 43}{41\cdots 67}a^{19}-\frac{26\cdots 35}{13\cdots 89}a^{18}-\frac{63\cdots 64}{45\cdots 63}a^{17}+\frac{16\cdots 85}{27\cdots 78}a^{16}+\frac{25\cdots 53}{13\cdots 89}a^{15}-\frac{18\cdots 77}{27\cdots 78}a^{14}-\frac{58\cdots 23}{41\cdots 67}a^{13}+\frac{16\cdots 42}{41\cdots 67}a^{12}+\frac{28\cdots 62}{41\cdots 67}a^{11}-\frac{10\cdots 35}{82\cdots 34}a^{10}-\frac{85\cdots 14}{41\cdots 67}a^{9}+\frac{96\cdots 31}{45\cdots 63}a^{8}+\frac{15\cdots 90}{41\cdots 67}a^{7}-\frac{11\cdots 75}{82\cdots 34}a^{6}-\frac{13\cdots 14}{41\cdots 67}a^{5}-\frac{90\cdots 91}{27\cdots 78}a^{4}+\frac{44\cdots 24}{41\cdots 67}a^{3}+\frac{20\cdots 57}{41\cdots 67}a^{2}+\frac{29\cdots 10}{41\cdots 67}a+\frac{11\cdots 29}{41\cdots 67}$, $\frac{25\cdots 77}{13\cdots 89}a^{19}-\frac{39\cdots 42}{45\cdots 63}a^{18}-\frac{51\cdots 67}{91\cdots 26}a^{17}+\frac{23\cdots 33}{91\cdots 26}a^{16}+\frac{66\cdots 11}{91\cdots 26}a^{15}-\frac{27\cdots 03}{91\cdots 26}a^{14}-\frac{73\cdots 34}{13\cdots 89}a^{13}+\frac{23\cdots 31}{13\cdots 89}a^{12}+\frac{69\cdots 51}{27\cdots 78}a^{11}-\frac{15\cdots 57}{27\cdots 78}a^{10}-\frac{10\cdots 34}{13\cdots 89}a^{9}+\frac{41\cdots 08}{45\cdots 63}a^{8}+\frac{37\cdots 35}{27\cdots 78}a^{7}-\frac{15\cdots 23}{27\cdots 78}a^{6}-\frac{34\cdots 85}{27\cdots 78}a^{5}-\frac{98\cdots 43}{91\cdots 26}a^{4}+\frac{55\cdots 22}{13\cdots 89}a^{3}+\frac{25\cdots 59}{13\cdots 89}a^{2}+\frac{39\cdots 04}{13\cdots 89}a+\frac{15\cdots 21}{13\cdots 89}$, $\frac{29\cdots 37}{27\cdots 78}a^{19}-\frac{40\cdots 43}{91\cdots 26}a^{18}-\frac{31\cdots 23}{91\cdots 26}a^{17}+\frac{60\cdots 62}{45\cdots 63}a^{16}+\frac{22\cdots 34}{45\cdots 63}a^{15}-\frac{14\cdots 89}{91\cdots 26}a^{14}-\frac{10\cdots 33}{27\cdots 78}a^{13}+\frac{25\cdots 51}{27\cdots 78}a^{12}+\frac{26\cdots 11}{13\cdots 89}a^{11}-\frac{78\cdots 53}{27\cdots 78}a^{10}-\frac{16\cdots 93}{27\cdots 78}a^{9}+\frac{39\cdots 11}{91\cdots 26}a^{8}+\frac{28\cdots 59}{27\cdots 78}a^{7}-\frac{24\cdots 35}{13\cdots 89}a^{6}-\frac{12\cdots 60}{13\cdots 89}a^{5}-\frac{20\cdots 73}{91\cdots 26}a^{4}+\frac{39\cdots 47}{13\cdots 89}a^{3}+\frac{23\cdots 81}{13\cdots 89}a^{2}+\frac{42\cdots 51}{13\cdots 89}a+\frac{24\cdots 96}{13\cdots 89}$, $\frac{15\cdots 59}{41\cdots 67}a^{19}+\frac{31\cdots 28}{13\cdots 89}a^{18}-\frac{20\cdots 01}{91\cdots 26}a^{17}+\frac{11\cdots 19}{27\cdots 78}a^{16}+\frac{12\cdots 47}{27\cdots 78}a^{15}-\frac{48\cdots 35}{27\cdots 78}a^{14}-\frac{18\cdots 91}{41\cdots 67}a^{13}+\frac{66\cdots 92}{41\cdots 67}a^{12}+\frac{20\cdots 81}{82\cdots 34}a^{11}-\frac{44\cdots 75}{82\cdots 34}a^{10}-\frac{31\cdots 65}{41\cdots 67}a^{9}+\frac{11\cdots 38}{45\cdots 63}a^{8}+\frac{10\cdots 51}{82\cdots 34}a^{7}+\frac{17\cdots 01}{82\cdots 34}a^{6}-\frac{91\cdots 27}{82\cdots 34}a^{5}-\frac{11\cdots 51}{27\cdots 78}a^{4}+\frac{12\cdots 80}{41\cdots 67}a^{3}+\frac{86\cdots 52}{41\cdots 67}a^{2}+\frac{15\cdots 68}{41\cdots 67}a+\frac{60\cdots 12}{41\cdots 67}$, $\frac{50\cdots 36}{41\cdots 67}a^{19}-\frac{14\cdots 57}{27\cdots 78}a^{18}-\frac{35\cdots 15}{91\cdots 26}a^{17}+\frac{21\cdots 54}{13\cdots 89}a^{16}+\frac{14\cdots 93}{27\cdots 78}a^{15}-\frac{50\cdots 21}{27\cdots 78}a^{14}-\frac{17\cdots 52}{41\cdots 67}a^{13}+\frac{87\cdots 51}{82\cdots 34}a^{12}+\frac{16\cdots 83}{82\cdots 34}a^{11}-\frac{26\cdots 41}{82\cdots 34}a^{10}-\frac{25\cdots 76}{41\cdots 67}a^{9}+\frac{44\cdots 53}{91\cdots 26}a^{8}+\frac{91\cdots 29}{82\cdots 34}a^{7}-\frac{83\cdots 98}{41\cdots 67}a^{6}-\frac{82\cdots 41}{82\cdots 34}a^{5}-\frac{65\cdots 59}{27\cdots 78}a^{4}+\frac{12\cdots 96}{41\cdots 67}a^{3}+\frac{76\cdots 69}{41\cdots 67}a^{2}+\frac{13\cdots 52}{41\cdots 67}a+\frac{54\cdots 57}{41\cdots 67}$, $\frac{23\cdots 51}{82\cdots 34}a^{19}-\frac{33\cdots 15}{27\cdots 78}a^{18}-\frac{40\cdots 54}{45\cdots 63}a^{17}+\frac{10\cdots 29}{27\cdots 78}a^{16}+\frac{32\cdots 03}{27\cdots 78}a^{15}-\frac{59\cdots 00}{13\cdots 89}a^{14}-\frac{74\cdots 53}{82\cdots 34}a^{13}+\frac{21\cdots 19}{82\cdots 34}a^{12}+\frac{35\cdots 83}{82\cdots 34}a^{11}-\frac{34\cdots 08}{41\cdots 67}a^{10}-\frac{10\cdots 89}{82\cdots 34}a^{9}+\frac{12\cdots 37}{91\cdots 26}a^{8}+\frac{95\cdots 14}{41\cdots 67}a^{7}-\frac{70\cdots 81}{82\cdots 34}a^{6}-\frac{17\cdots 59}{82\cdots 34}a^{5}-\frac{28\cdots 09}{13\cdots 89}a^{4}+\frac{28\cdots 42}{41\cdots 67}a^{3}+\frac{12\cdots 64}{41\cdots 67}a^{2}+\frac{19\cdots 66}{41\cdots 67}a+\frac{85\cdots 11}{41\cdots 67}$, $\frac{38\cdots 11}{91\cdots 26}a^{19}-\frac{16\cdots 97}{91\cdots 26}a^{18}-\frac{59\cdots 27}{45\cdots 63}a^{17}+\frac{24\cdots 85}{45\cdots 63}a^{16}+\frac{16\cdots 41}{91\cdots 26}a^{15}-\frac{58\cdots 11}{91\cdots 26}a^{14}-\frac{12\cdots 17}{91\cdots 26}a^{13}+\frac{34\cdots 45}{91\cdots 26}a^{12}+\frac{60\cdots 27}{91\cdots 26}a^{11}-\frac{11\cdots 67}{91\cdots 26}a^{10}-\frac{18\cdots 81}{91\cdots 26}a^{9}+\frac{18\cdots 51}{91\cdots 26}a^{8}+\frac{16\cdots 68}{45\cdots 63}a^{7}-\frac{56\cdots 50}{45\cdots 63}a^{6}-\frac{28\cdots 33}{91\cdots 26}a^{5}-\frac{31\cdots 03}{91\cdots 26}a^{4}+\frac{46\cdots 89}{45\cdots 63}a^{3}+\frac{21\cdots 60}{45\cdots 63}a^{2}+\frac{32\cdots 69}{45\cdots 63}a+\frac{13\cdots 46}{45\cdots 63}$, $\frac{64\cdots 65}{27\cdots 78}a^{19}-\frac{89\cdots 51}{91\cdots 26}a^{18}-\frac{34\cdots 73}{45\cdots 63}a^{17}+\frac{13\cdots 73}{45\cdots 63}a^{16}+\frac{95\cdots 33}{91\cdots 26}a^{15}-\frac{31\cdots 57}{91\cdots 26}a^{14}-\frac{22\cdots 53}{27\cdots 78}a^{13}+\frac{56\cdots 85}{27\cdots 78}a^{12}+\frac{11\cdots 07}{27\cdots 78}a^{11}-\frac{17\cdots 89}{27\cdots 78}a^{10}-\frac{34\cdots 33}{27\cdots 78}a^{9}+\frac{93\cdots 37}{91\cdots 26}a^{8}+\frac{30\cdots 60}{13\cdots 89}a^{7}-\frac{74\cdots 74}{13\cdots 89}a^{6}-\frac{54\cdots 87}{27\cdots 78}a^{5}-\frac{30\cdots 41}{91\cdots 26}a^{4}+\frac{85\cdots 04}{13\cdots 89}a^{3}+\frac{43\cdots 98}{13\cdots 89}a^{2}+\frac{67\cdots 66}{13\cdots 89}a+\frac{26\cdots 36}{13\cdots 89}$, $\frac{51\cdots 24}{13\cdots 89}a^{19}-\frac{14\cdots 67}{91\cdots 26}a^{18}-\frac{54\cdots 08}{45\cdots 63}a^{17}+\frac{21\cdots 36}{45\cdots 63}a^{16}+\frac{74\cdots 88}{45\cdots 63}a^{15}-\frac{51\cdots 39}{91\cdots 26}a^{14}-\frac{17\cdots 52}{13\cdots 89}a^{13}+\frac{91\cdots 05}{27\cdots 78}a^{12}+\frac{85\cdots 22}{13\cdots 89}a^{11}-\frac{28\cdots 95}{27\cdots 78}a^{10}-\frac{25\cdots 79}{13\cdots 89}a^{9}+\frac{15\cdots 03}{91\cdots 26}a^{8}+\frac{45\cdots 31}{13\cdots 89}a^{7}-\frac{12\cdots 43}{13\cdots 89}a^{6}-\frac{41\cdots 59}{13\cdots 89}a^{5}-\frac{43\cdots 09}{91\cdots 26}a^{4}+\frac{13\cdots 18}{13\cdots 89}a^{3}+\frac{66\cdots 20}{13\cdots 89}a^{2}+\frac{10\cdots 29}{13\cdots 89}a+\frac{41\cdots 41}{13\cdots 89}$, $\frac{33\cdots 66}{13\cdots 89}a^{19}-\frac{29\cdots 04}{45\cdots 63}a^{18}-\frac{89\cdots 17}{91\cdots 26}a^{17}+\frac{19\cdots 73}{91\cdots 26}a^{16}+\frac{15\cdots 69}{91\cdots 26}a^{15}-\frac{24\cdots 85}{91\cdots 26}a^{14}-\frac{21\cdots 47}{13\cdots 89}a^{13}+\frac{22\cdots 74}{13\cdots 89}a^{12}+\frac{22\cdots 85}{27\cdots 78}a^{11}-\frac{13\cdots 53}{27\cdots 78}a^{10}-\frac{35\cdots 91}{13\cdots 89}a^{9}+\frac{28\cdots 14}{45\cdots 63}a^{8}+\frac{12\cdots 15}{27\cdots 78}a^{7}+\frac{65\cdots 25}{27\cdots 78}a^{6}-\frac{10\cdots 23}{27\cdots 78}a^{5}-\frac{10\cdots 73}{91\cdots 26}a^{4}+\frac{15\cdots 69}{13\cdots 89}a^{3}+\frac{95\cdots 45}{13\cdots 89}a^{2}+\frac{16\cdots 28}{13\cdots 89}a+\frac{84\cdots 58}{13\cdots 89}$, $\frac{28\cdots 37}{82\cdots 34}a^{19}-\frac{42\cdots 67}{27\cdots 78}a^{18}-\frac{49\cdots 42}{45\cdots 63}a^{17}+\frac{63\cdots 72}{13\cdots 89}a^{16}+\frac{39\cdots 61}{27\cdots 78}a^{15}-\frac{14\cdots 77}{27\cdots 78}a^{14}-\frac{90\cdots 09}{82\cdots 34}a^{13}+\frac{26\cdots 05}{82\cdots 34}a^{12}+\frac{43\cdots 83}{82\cdots 34}a^{11}-\frac{83\cdots 91}{82\cdots 34}a^{10}-\frac{13\cdots 11}{82\cdots 34}a^{9}+\frac{14\cdots 57}{91\cdots 26}a^{8}+\frac{11\cdots 06}{41\cdots 67}a^{7}-\frac{41\cdots 97}{41\cdots 67}a^{6}-\frac{21\cdots 81}{82\cdots 34}a^{5}-\frac{81\cdots 03}{27\cdots 78}a^{4}+\frac{33\cdots 92}{41\cdots 67}a^{3}+\frac{16\cdots 64}{41\cdots 67}a^{2}+\frac{24\cdots 37}{41\cdots 67}a+\frac{98\cdots 78}{41\cdots 67}$, $\frac{35\cdots 38}{41\cdots 67}a^{19}-\frac{39\cdots 59}{27\cdots 78}a^{18}-\frac{18\cdots 54}{45\cdots 63}a^{17}+\frac{15\cdots 75}{27\cdots 78}a^{16}+\frac{10\cdots 57}{13\cdots 89}a^{15}-\frac{11\cdots 00}{13\cdots 89}a^{14}-\frac{28\cdots 38}{41\cdots 67}a^{13}+\frac{46\cdots 81}{82\cdots 34}a^{12}+\frac{15\cdots 99}{41\cdots 67}a^{11}-\frac{78\cdots 16}{41\cdots 67}a^{10}-\frac{47\cdots 39}{41\cdots 67}a^{9}+\frac{24\cdots 13}{91\cdots 26}a^{8}+\frac{81\cdots 70}{41\cdots 67}a^{7}+\frac{35\cdots 73}{82\cdots 34}a^{6}-\frac{70\cdots 52}{41\cdots 67}a^{5}-\frac{59\cdots 54}{13\cdots 89}a^{4}+\frac{21\cdots 65}{41\cdots 67}a^{3}+\frac{11\cdots 88}{41\cdots 67}a^{2}+\frac{17\cdots 84}{41\cdots 67}a+\frac{60\cdots 93}{41\cdots 67}$, $\frac{27\cdots 11}{82\cdots 34}a^{19}-\frac{49\cdots 91}{27\cdots 78}a^{18}-\frac{38\cdots 31}{45\cdots 63}a^{17}+\frac{72\cdots 91}{13\cdots 89}a^{16}+\frac{23\cdots 39}{27\cdots 78}a^{15}-\frac{16\cdots 33}{27\cdots 78}a^{14}-\frac{36\cdots 87}{82\cdots 34}a^{13}+\frac{28\cdots 71}{82\cdots 34}a^{12}+\frac{12\cdots 07}{82\cdots 34}a^{11}-\frac{93\cdots 11}{82\cdots 34}a^{10}-\frac{32\cdots 07}{82\cdots 34}a^{9}+\frac{18\cdots 43}{91\cdots 26}a^{8}+\frac{30\cdots 26}{41\cdots 67}a^{7}-\frac{69\cdots 23}{41\cdots 67}a^{6}-\frac{65\cdots 05}{82\cdots 34}a^{5}+\frac{12\cdots 25}{27\cdots 78}a^{4}+\frac{13\cdots 02}{41\cdots 67}a^{3}+\frac{26\cdots 77}{41\cdots 67}a^{2}+\frac{38\cdots 87}{41\cdots 67}a+\frac{32\cdots 63}{41\cdots 67}$, $\frac{51\cdots 03}{82\cdots 34}a^{19}-\frac{35\cdots 62}{13\cdots 89}a^{18}-\frac{92\cdots 14}{45\cdots 63}a^{17}+\frac{10\cdots 15}{13\cdots 89}a^{16}+\frac{77\cdots 43}{27\cdots 78}a^{15}-\frac{12\cdots 91}{13\cdots 89}a^{14}-\frac{18\cdots 39}{82\cdots 34}a^{13}+\frac{22\cdots 49}{41\cdots 67}a^{12}+\frac{90\cdots 73}{82\cdots 34}a^{11}-\frac{73\cdots 67}{41\cdots 67}a^{10}-\frac{27\cdots 43}{82\cdots 34}a^{9}+\frac{13\cdots 08}{45\cdots 63}a^{8}+\frac{23\cdots 34}{41\cdots 67}a^{7}-\frac{69\cdots 14}{41\cdots 67}a^{6}-\frac{42\cdots 49}{82\cdots 34}a^{5}-\frac{98\cdots 79}{13\cdots 89}a^{4}+\frac{66\cdots 27}{41\cdots 67}a^{3}+\frac{33\cdots 79}{41\cdots 67}a^{2}+\frac{49\cdots 68}{41\cdots 67}a+\frac{13\cdots 51}{41\cdots 67}$ 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:  \( 70785617543.0 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 18 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 70785617543.0 \cdot 1}{2\cdot\sqrt{8501150111111046013911040000000000}}\cr\approx \mathstrut & 0.402509467940 \end{aligned}\] (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^19 - 33*x^18 + 120*x^17 + 468*x^16 - 1392*x^15 - 3799*x^14 + 8012*x^13 + 19008*x^12 - 23720*x^11 - 57969*x^10 + 31228*x^9 + 101057*x^8 - 964*x^7 - 86922*x^6 - 33928*x^5 + 22258*x^4 + 19672*x^3 + 5396*x^2 + 600*x + 20) 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^19 - 33*x^18 + 120*x^17 + 468*x^16 - 1392*x^15 - 3799*x^14 + 8012*x^13 + 19008*x^12 - 23720*x^11 - 57969*x^10 + 31228*x^9 + 101057*x^8 - 964*x^7 - 86922*x^6 - 33928*x^5 + 22258*x^4 + 19672*x^3 + 5396*x^2 + 600*x + 20, 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^19 - 33*x^18 + 120*x^17 + 468*x^16 - 1392*x^15 - 3799*x^14 + 8012*x^13 + 19008*x^12 - 23720*x^11 - 57969*x^10 + 31228*x^9 + 101057*x^8 - 964*x^7 - 86922*x^6 - 33928*x^5 + 22258*x^4 + 19672*x^3 + 5396*x^2 + 600*x + 20); 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^19 - 33*x^18 + 120*x^17 + 468*x^16 - 1392*x^15 - 3799*x^14 + 8012*x^13 + 19008*x^12 - 23720*x^11 - 57969*x^10 + 31228*x^9 + 101057*x^8 - 964*x^7 - 86922*x^6 - 33928*x^5 + 22258*x^4 + 19672*x^3 + 5396*x^2 + 600*x + 20); 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

$S_5$ (as 20T32):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.3688067268608000.1

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 5 sibling: 5.5.13579520.1
Degree 6 sibling: 6.6.1357952000.1
Degree 10 siblings: 10.10.3688067268608000.1, 10.10.23050420428800000.1
Degree 12 sibling: 12.12.1844033634304000000.1
Degree 15 sibling: deg 15
Degree 20 siblings: deg 20, deg 20
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed
Minimal sibling: 5.5.13579520.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.2.0.1}{2} }$ R ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.3.0.1}{3} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.2.0.1}{2} }^{10}$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

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 2.2.4.16a1.1$x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 19 x^{4} + 16 x^{3} + 14 x^{2} + 8 x + 7$$4$$2$$16$$S_4$$$[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$$
2.2.6.20a1.45$x^{12} + 6 x^{11} + 23 x^{10} + 60 x^{9} + 120 x^{8} + 186 x^{7} + 231 x^{6} + 228 x^{5} + 180 x^{4} + 110 x^{3} + 55 x^{2} + 20 x + 9$$6$$2$$20$$S_4$$$[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$$
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(103\) Copy content Toggle raw display 103.2.1.0a1.1$x^{2} + 102 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
103.2.1.0a1.1$x^{2} + 102 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
103.2.2.2a1.2$x^{4} + 204 x^{3} + 10414 x^{2} + 1020 x + 128$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
103.2.2.2a1.2$x^{4} + 204 x^{3} + 10414 x^{2} + 1020 x + 128$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
103.2.2.2a1.2$x^{4} + 204 x^{3} + 10414 x^{2} + 1020 x + 128$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
103.2.2.2a1.2$x^{4} + 204 x^{3} + 10414 x^{2} + 1020 x + 128$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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