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Properties

Label 20.20.488...704.1
Degree $20$
Signature $(20, 0)$
Discriminant $4.884\times 10^{45}$
Root discriminant \(192.50\)
Ramified primes $2,17,53$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_{20}:C_4$ (as 20T18)

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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 - 10*x^19 - 116*x^18 + 1313*x^17 + 4736*x^16 - 67464*x^15 - 71774*x^14 + 1744748*x^13 - 126850*x^12 - 24570618*x^11 + 14939336*x^10 + 193331394*x^9 - 163802315*x^8 - 826609978*x^7 + 787082462*x^6 + 1707884929*x^5 - 1813914636*x^4 - 1179787402*x^3 + 1644305872*x^2 - 315124556*x - 58612856)
 
Copy content gp:K = bnfinit(y^20 - 10*y^19 - 116*y^18 + 1313*y^17 + 4736*y^16 - 67464*y^15 - 71774*y^14 + 1744748*y^13 - 126850*y^12 - 24570618*y^11 + 14939336*y^10 + 193331394*y^9 - 163802315*y^8 - 826609978*y^7 + 787082462*y^6 + 1707884929*y^5 - 1813914636*y^4 - 1179787402*y^3 + 1644305872*y^2 - 315124556*y - 58612856, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 10*x^19 - 116*x^18 + 1313*x^17 + 4736*x^16 - 67464*x^15 - 71774*x^14 + 1744748*x^13 - 126850*x^12 - 24570618*x^11 + 14939336*x^10 + 193331394*x^9 - 163802315*x^8 - 826609978*x^7 + 787082462*x^6 + 1707884929*x^5 - 1813914636*x^4 - 1179787402*x^3 + 1644305872*x^2 - 315124556*x - 58612856);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 10*x^19 - 116*x^18 + 1313*x^17 + 4736*x^16 - 67464*x^15 - 71774*x^14 + 1744748*x^13 - 126850*x^12 - 24570618*x^11 + 14939336*x^10 + 193331394*x^9 - 163802315*x^8 - 826609978*x^7 + 787082462*x^6 + 1707884929*x^5 - 1813914636*x^4 - 1179787402*x^3 + 1644305872*x^2 - 315124556*x - 58612856)
 

\( x^{20} - 10 x^{19} - 116 x^{18} + 1313 x^{17} + 4736 x^{16} - 67464 x^{15} - 71774 x^{14} + \cdots - 58612856 \) 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:   \(4884274415246183319344904689389832005326536704\) \(\medspace = 2^{16}\cdot 17^{15}\cdot 53^{13}\) 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:  \(192.50\)
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:  $2\cdot 17^{3/4}53^{3/4}\approx 328.90735776316194$
Ramified primes:   \(2\), \(17\), \(53\) 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:  \(\Q(\sqrt{901}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{10}a^{10}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{2}{5}a^{6}+\frac{3}{10}a^{4}-\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{10}a^{11}+\frac{1}{5}a^{9}-\frac{1}{10}a^{8}-\frac{2}{5}a^{7}+\frac{3}{10}a^{5}-\frac{2}{5}a^{3}-\frac{1}{10}a^{2}-\frac{1}{5}a$, $\frac{1}{10}a^{12}-\frac{1}{10}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{1}{10}a^{6}-\frac{1}{10}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{10}a^{13}+\frac{1}{5}a^{9}-\frac{1}{10}a^{8}-\frac{1}{2}a^{7}-\frac{2}{5}a^{6}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{3}{10}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{10}a^{14}-\frac{1}{10}a^{9}+\frac{1}{10}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{3}{10}a^{3}-\frac{2}{5}a^{2}+\frac{2}{5}$, $\frac{1}{530}a^{15}-\frac{11}{530}a^{14}-\frac{21}{530}a^{13}-\frac{17}{530}a^{12}-\frac{23}{530}a^{11}+\frac{13}{530}a^{10}+\frac{61}{530}a^{9}-\frac{11}{53}a^{8}+\frac{249}{530}a^{7}-\frac{13}{106}a^{6}+\frac{59}{530}a^{5}-\frac{177}{530}a^{4}-\frac{77}{265}a^{3}-\frac{109}{530}a^{2}-\frac{3}{265}a+\frac{56}{265}$, $\frac{1}{1060}a^{16}-\frac{9}{265}a^{14}-\frac{9}{265}a^{13}+\frac{1}{530}a^{12}-\frac{7}{265}a^{11}-\frac{2}{265}a^{10}-\frac{117}{530}a^{9}+\frac{23}{530}a^{8}-\frac{20}{53}a^{7}-\frac{63}{530}a^{6}+\frac{13}{53}a^{5}-\frac{193}{1060}a^{4}-\frac{239}{530}a^{3}-\frac{41}{106}a^{2}+\frac{91}{265}a+\frac{96}{265}$, $\frac{1}{1060}a^{17}-\frac{2}{265}a^{14}-\frac{3}{265}a^{13}-\frac{1}{265}a^{12}+\frac{3}{265}a^{11}+\frac{11}{530}a^{10}+\frac{61}{530}a^{9}+\frac{23}{265}a^{8}+\frac{63}{265}a^{7}-\frac{43}{265}a^{6}+\frac{23}{1060}a^{5}+\frac{179}{530}a^{4}-\frac{9}{530}a^{3}+\frac{64}{265}a^{2}-\frac{117}{265}a-\frac{52}{265}$, $\frac{1}{10\cdots 40}a^{18}-\frac{89\cdots 27}{10\cdots 40}a^{17}-\frac{11\cdots 53}{10\cdots 14}a^{16}+\frac{32\cdots 79}{26\cdots 85}a^{15}+\frac{71\cdots 12}{26\cdots 85}a^{14}+\frac{61\cdots 27}{53\cdots 70}a^{13}+\frac{75\cdots 87}{53\cdots 57}a^{12}+\frac{24\cdots 57}{53\cdots 70}a^{11}+\frac{18\cdots 41}{53\cdots 70}a^{10}+\frac{13\cdots 47}{26\cdots 85}a^{9}+\frac{17\cdots 69}{10\cdots 14}a^{8}+\frac{38\cdots 43}{53\cdots 70}a^{7}+\frac{46\cdots 59}{10\cdots 40}a^{6}-\frac{54\cdots 19}{20\cdots 80}a^{5}-\frac{77\cdots 09}{53\cdots 57}a^{4}-\frac{59\cdots 08}{26\cdots 85}a^{3}-\frac{41\cdots 01}{53\cdots 70}a^{2}-\frac{42\cdots 21}{26\cdots 85}a+\frac{84\cdots 87}{26\cdots 85}$, $\frac{1}{92\cdots 20}a^{19}+\frac{65\cdots 03}{46\cdots 10}a^{18}+\frac{37\cdots 87}{46\cdots 10}a^{17}-\frac{20\cdots 94}{23\cdots 55}a^{16}+\frac{40\cdots 09}{46\cdots 10}a^{15}+\frac{41\cdots 76}{23\cdots 55}a^{14}-\frac{16\cdots 44}{46\cdots 51}a^{13}+\frac{52\cdots 91}{23\cdots 55}a^{12}-\frac{31\cdots 56}{23\cdots 55}a^{11}-\frac{10\cdots 59}{46\cdots 10}a^{10}-\frac{75\cdots 99}{92\cdots 02}a^{9}+\frac{55\cdots 37}{23\cdots 55}a^{8}-\frac{74\cdots 49}{18\cdots 04}a^{7}+\frac{21\cdots 21}{46\cdots 10}a^{6}+\frac{91\cdots 49}{92\cdots 02}a^{5}-\frac{11\cdots 53}{46\cdots 10}a^{4}+\frac{13\cdots 98}{46\cdots 51}a^{3}+\frac{73\cdots 56}{23\cdots 55}a^{2}-\frac{10\cdots 06}{23\cdots 55}a+\frac{10\cdots 47}{23\cdots 55}$ 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:  $C_{2}$, 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:  $C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:   $\frac{45\cdots 64}{87\cdots 67}a^{19}-\frac{42\cdots 67}{87\cdots 67}a^{18}-\frac{10\cdots 31}{17\cdots 34}a^{17}+\frac{11\cdots 61}{17\cdots 34}a^{16}+\frac{24\cdots 90}{87\cdots 67}a^{15}-\frac{29\cdots 11}{87\cdots 67}a^{14}-\frac{48\cdots 47}{87\cdots 67}a^{13}+\frac{75\cdots 55}{87\cdots 67}a^{12}+\frac{36\cdots 24}{87\cdots 67}a^{11}-\frac{10\cdots 19}{87\cdots 67}a^{10}+\frac{67\cdots 75}{87\cdots 67}a^{9}+\frac{17\cdots 77}{17\cdots 34}a^{8}-\frac{24\cdots 77}{87\cdots 67}a^{7}-\frac{38\cdots 08}{87\cdots 67}a^{6}+\frac{26\cdots 41}{17\cdots 34}a^{5}+\frac{16\cdots 03}{17\cdots 34}a^{4}-\frac{31\cdots 65}{87\cdots 67}a^{3}-\frac{14\cdots 65}{17\cdots 34}a^{2}+\frac{30\cdots 03}{87\cdots 67}a+\frac{43\cdots 33}{87\cdots 67}$, $\frac{18\cdots 47}{87\cdots 67}a^{19}-\frac{38\cdots 19}{17\cdots 34}a^{18}-\frac{41\cdots 09}{17\cdots 34}a^{17}+\frac{99\cdots 45}{35\cdots 68}a^{16}+\frac{77\cdots 04}{87\cdots 67}a^{15}-\frac{12\cdots 21}{87\cdots 67}a^{14}-\frac{86\cdots 84}{87\cdots 67}a^{13}+\frac{64\cdots 77}{17\cdots 34}a^{12}-\frac{12\cdots 95}{87\cdots 67}a^{11}-\frac{89\cdots 79}{17\cdots 34}a^{10}+\frac{78\cdots 63}{17\cdots 34}a^{9}+\frac{34\cdots 32}{87\cdots 67}a^{8}-\frac{36\cdots 87}{87\cdots 67}a^{7}-\frac{14\cdots 53}{87\cdots 67}a^{6}+\frac{30\cdots 75}{17\cdots 34}a^{5}+\frac{12\cdots 81}{35\cdots 68}a^{4}-\frac{59\cdots 63}{17\cdots 34}a^{3}-\frac{27\cdots 09}{87\cdots 67}a^{2}+\frac{21\cdots 57}{87\cdots 67}a+\frac{51\cdots 45}{16\cdots 39}$, $\frac{40\cdots 91}{87\cdots 70}a^{19}-\frac{77\cdots 41}{17\cdots 40}a^{18}-\frac{96\cdots 23}{17\cdots 40}a^{17}+\frac{10\cdots 89}{17\cdots 40}a^{16}+\frac{10\cdots 57}{43\cdots 35}a^{15}-\frac{26\cdots 97}{87\cdots 67}a^{14}-\frac{20\cdots 09}{43\cdots 35}a^{13}+\frac{13\cdots 47}{17\cdots 34}a^{12}+\frac{52\cdots 19}{17\cdots 34}a^{11}-\frac{96\cdots 13}{87\cdots 70}a^{10}+\frac{12\cdots 78}{87\cdots 67}a^{9}+\frac{38\cdots 01}{43\cdots 35}a^{8}-\frac{25\cdots 53}{87\cdots 70}a^{7}-\frac{67\cdots 49}{17\cdots 40}a^{6}+\frac{24\cdots 93}{17\cdots 40}a^{5}+\frac{14\cdots 09}{17\cdots 40}a^{4}-\frac{28\cdots 02}{87\cdots 67}a^{3}-\frac{62\cdots 69}{87\cdots 67}a^{2}+\frac{26\cdots 75}{87\cdots 67}a+\frac{19\cdots 83}{43\cdots 35}$, $\frac{10\cdots 83}{43\cdots 35}a^{19}-\frac{10\cdots 31}{43\cdots 35}a^{18}-\frac{26\cdots 02}{87\cdots 67}a^{17}+\frac{26\cdots 24}{87\cdots 67}a^{16}+\frac{58\cdots 92}{43\cdots 35}a^{15}-\frac{13\cdots 83}{87\cdots 70}a^{14}-\frac{23\cdots 11}{87\cdots 70}a^{13}+\frac{17\cdots 16}{43\cdots 35}a^{12}+\frac{37\cdots 75}{17\cdots 34}a^{11}-\frac{25\cdots 06}{43\cdots 35}a^{10}+\frac{12\cdots 29}{87\cdots 70}a^{9}+\frac{41\cdots 63}{87\cdots 70}a^{8}-\frac{20\cdots 81}{17\cdots 34}a^{7}-\frac{91\cdots 21}{43\cdots 35}a^{6}+\frac{58\cdots 19}{87\cdots 70}a^{5}+\frac{20\cdots 22}{43\cdots 35}a^{4}-\frac{14\cdots 29}{87\cdots 70}a^{3}-\frac{34\cdots 59}{87\cdots 67}a^{2}+\frac{72\cdots 57}{43\cdots 35}a+\frac{10\cdots 11}{43\cdots 35}$, $\frac{29\cdots 17}{87\cdots 67}a^{19}-\frac{56\cdots 57}{17\cdots 34}a^{18}-\frac{36\cdots 61}{87\cdots 67}a^{17}+\frac{73\cdots 99}{17\cdots 34}a^{16}+\frac{16\cdots 60}{87\cdots 67}a^{15}-\frac{19\cdots 47}{87\cdots 67}a^{14}-\frac{33\cdots 25}{87\cdots 67}a^{13}+\frac{49\cdots 78}{87\cdots 67}a^{12}+\frac{26\cdots 49}{87\cdots 67}a^{11}-\frac{71\cdots 77}{87\cdots 67}a^{10}+\frac{51\cdots 00}{87\cdots 67}a^{9}+\frac{57\cdots 83}{87\cdots 67}a^{8}-\frac{13\cdots 30}{87\cdots 67}a^{7}-\frac{50\cdots 95}{17\cdots 34}a^{6}+\frac{77\cdots 94}{87\cdots 67}a^{5}+\frac{11\cdots 77}{17\cdots 34}a^{4}-\frac{19\cdots 08}{87\cdots 67}a^{3}-\frac{46\cdots 74}{87\cdots 67}a^{2}+\frac{19\cdots 05}{87\cdots 67}a+\frac{28\cdots 33}{87\cdots 67}$, $\frac{6911027709324}{10\cdots 69}a^{19}-\frac{116228509859823}{50\cdots 45}a^{18}-\frac{59\cdots 51}{50\cdots 45}a^{17}+\frac{36\cdots 53}{10\cdots 90}a^{16}+\frac{79\cdots 20}{10\cdots 69}a^{15}-\frac{11\cdots 72}{50\cdots 45}a^{14}-\frac{13\cdots 46}{50\cdots 45}a^{13}+\frac{36\cdots 07}{50\cdots 45}a^{12}+\frac{23\cdots 28}{50\cdots 45}a^{11}-\frac{63\cdots 07}{50\cdots 45}a^{10}-\frac{20\cdots 63}{50\cdots 45}a^{9}+\frac{11\cdots 59}{10\cdots 69}a^{8}+\frac{82\cdots 02}{50\cdots 45}a^{7}-\frac{23\cdots 56}{50\cdots 45}a^{6}-\frac{76\cdots 47}{50\cdots 45}a^{5}+\frac{76\cdots 89}{10\cdots 90}a^{4}-\frac{13\cdots 01}{50\cdots 45}a^{3}-\frac{15\cdots 23}{10\cdots 69}a^{2}+\frac{27\cdots 46}{50\cdots 45}a-\frac{22\cdots 01}{50\cdots 45}$, $\frac{27\cdots 33}{23\cdots 55}a^{19}-\frac{60\cdots 73}{46\cdots 10}a^{18}-\frac{31\cdots 11}{23\cdots 55}a^{17}+\frac{16\cdots 71}{92\cdots 02}a^{16}+\frac{24\cdots 22}{46\cdots 51}a^{15}-\frac{42\cdots 79}{46\cdots 10}a^{14}-\frac{31\cdots 63}{46\cdots 10}a^{13}+\frac{11\cdots 24}{46\cdots 51}a^{12}-\frac{10\cdots 91}{23\cdots 55}a^{11}-\frac{33\cdots 35}{92\cdots 02}a^{10}+\frac{88\cdots 81}{46\cdots 10}a^{9}+\frac{13\cdots 89}{46\cdots 10}a^{8}-\frac{80\cdots 53}{46\cdots 10}a^{7}-\frac{62\cdots 29}{46\cdots 10}a^{6}+\frac{15\cdots 27}{23\cdots 55}a^{5}+\frac{69\cdots 84}{23\cdots 55}a^{4}-\frac{63\cdots 61}{46\cdots 10}a^{3}-\frac{60\cdots 17}{23\cdots 55}a^{2}+\frac{53\cdots 47}{46\cdots 51}a+\frac{39\cdots 69}{23\cdots 55}$, $\frac{14\cdots 66}{43\cdots 35}a^{19}-\frac{51\cdots 43}{17\cdots 40}a^{18}-\frac{18\cdots 74}{43\cdots 35}a^{17}+\frac{17\cdots 58}{43\cdots 35}a^{16}+\frac{84\cdots 59}{43\cdots 35}a^{15}-\frac{17\cdots 25}{87\cdots 67}a^{14}-\frac{36\cdots 53}{87\cdots 70}a^{13}+\frac{45\cdots 31}{87\cdots 70}a^{12}+\frac{18\cdots 56}{43\cdots 35}a^{11}-\frac{64\cdots 49}{87\cdots 70}a^{10}-\frac{12\cdots 03}{87\cdots 70}a^{9}+\frac{51\cdots 33}{87\cdots 70}a^{8}-\frac{47\cdots 59}{87\cdots 70}a^{7}-\frac{44\cdots 83}{17\cdots 40}a^{6}+\frac{25\cdots 04}{43\cdots 35}a^{5}+\frac{48\cdots 41}{87\cdots 70}a^{4}-\frac{15\cdots 01}{87\cdots 70}a^{3}-\frac{41\cdots 67}{87\cdots 70}a^{2}+\frac{84\cdots 89}{43\cdots 35}a+\frac{24\cdots 53}{87\cdots 67}$, $\frac{18\cdots 97}{92\cdots 20}a^{19}-\frac{17\cdots 91}{92\cdots 20}a^{18}-\frac{21\cdots 49}{92\cdots 20}a^{17}+\frac{11\cdots 31}{46\cdots 10}a^{16}+\frac{46\cdots 51}{46\cdots 10}a^{15}-\frac{58\cdots 69}{46\cdots 10}a^{14}-\frac{85\cdots 01}{46\cdots 10}a^{13}+\frac{15\cdots 33}{46\cdots 10}a^{12}+\frac{22\cdots 43}{23\cdots 55}a^{11}-\frac{10\cdots 01}{23\cdots 55}a^{10}+\frac{24\cdots 02}{23\cdots 55}a^{9}+\frac{16\cdots 11}{46\cdots 10}a^{8}-\frac{15\cdots 27}{92\cdots 20}a^{7}-\frac{14\cdots 63}{92\cdots 20}a^{6}+\frac{74\cdots 93}{92\cdots 20}a^{5}+\frac{31\cdots 15}{92\cdots 02}a^{4}-\frac{84\cdots 97}{46\cdots 10}a^{3}-\frac{61\cdots 71}{23\cdots 55}a^{2}+\frac{74\cdots 72}{46\cdots 51}a-\frac{10\cdots 93}{46\cdots 51}$, $\frac{13\cdots 15}{18\cdots 04}a^{19}-\frac{31\cdots 81}{46\cdots 10}a^{18}-\frac{20\cdots 44}{23\cdots 55}a^{17}+\frac{82\cdots 07}{92\cdots 20}a^{16}+\frac{18\cdots 69}{46\cdots 10}a^{15}-\frac{10\cdots 68}{23\cdots 55}a^{14}-\frac{18\cdots 09}{23\cdots 55}a^{13}+\frac{27\cdots 98}{23\cdots 55}a^{12}+\frac{28\cdots 94}{46\cdots 51}a^{11}-\frac{39\cdots 66}{23\cdots 55}a^{10}+\frac{82\cdots 12}{23\cdots 55}a^{9}+\frac{31\cdots 92}{23\cdots 55}a^{8}-\frac{60\cdots 87}{18\cdots 04}a^{7}-\frac{28\cdots 67}{46\cdots 10}a^{6}+\frac{43\cdots 47}{23\cdots 55}a^{5}+\frac{24\cdots 49}{18\cdots 04}a^{4}-\frac{21\cdots 27}{46\cdots 10}a^{3}-\frac{26\cdots 17}{23\cdots 55}a^{2}+\frac{21\cdots 50}{46\cdots 51}a+\frac{15\cdots 97}{23\cdots 55}$, $\frac{42\cdots 04}{43\cdots 35}a^{19}-\frac{16\cdots 13}{17\cdots 40}a^{18}-\frac{51\cdots 89}{43\cdots 35}a^{17}+\frac{53\cdots 64}{43\cdots 35}a^{16}+\frac{23\cdots 61}{43\cdots 35}a^{15}-\frac{27\cdots 21}{43\cdots 35}a^{14}-\frac{92\cdots 63}{87\cdots 70}a^{13}+\frac{14\cdots 11}{87\cdots 70}a^{12}+\frac{35\cdots 38}{43\cdots 35}a^{11}-\frac{20\cdots 03}{87\cdots 70}a^{10}+\frac{16\cdots 29}{17\cdots 34}a^{9}+\frac{16\cdots 97}{87\cdots 70}a^{8}-\frac{42\cdots 49}{87\cdots 70}a^{7}-\frac{14\cdots 69}{17\cdots 40}a^{6}+\frac{23\cdots 59}{87\cdots 67}a^{5}+\frac{16\cdots 71}{87\cdots 70}a^{4}-\frac{11\cdots 73}{17\cdots 34}a^{3}-\frac{13\cdots 23}{87\cdots 70}a^{2}+\frac{29\cdots 49}{43\cdots 35}a+\frac{43\cdots 07}{43\cdots 35}$, $\frac{59\cdots 51}{92\cdots 20}a^{19}-\frac{14\cdots 69}{23\cdots 55}a^{18}-\frac{73\cdots 99}{92\cdots 20}a^{17}+\frac{18\cdots 27}{23\cdots 55}a^{16}+\frac{82\cdots 59}{23\cdots 55}a^{15}-\frac{38\cdots 87}{92\cdots 02}a^{14}-\frac{33\cdots 27}{46\cdots 10}a^{13}+\frac{50\cdots 93}{46\cdots 10}a^{12}+\frac{27\cdots 11}{46\cdots 10}a^{11}-\frac{72\cdots 82}{46\cdots 51}a^{10}-\frac{45\cdots 87}{46\cdots 10}a^{9}+\frac{29\cdots 78}{23\cdots 55}a^{8}-\frac{25\cdots 37}{92\cdots 20}a^{7}-\frac{25\cdots 09}{46\cdots 10}a^{6}+\frac{15\cdots 93}{92\cdots 20}a^{5}+\frac{28\cdots 17}{23\cdots 55}a^{4}-\frac{20\cdots 47}{46\cdots 10}a^{3}-\frac{48\cdots 39}{46\cdots 10}a^{2}+\frac{10\cdots 93}{23\cdots 55}a+\frac{14\cdots 11}{23\cdots 55}$, $\frac{96\cdots 33}{46\cdots 10}a^{19}-\frac{34\cdots 87}{18\cdots 04}a^{18}-\frac{24\cdots 89}{92\cdots 20}a^{17}+\frac{11\cdots 79}{46\cdots 10}a^{16}+\frac{59\cdots 09}{46\cdots 10}a^{15}-\frac{58\cdots 95}{46\cdots 51}a^{14}-\frac{13\cdots 59}{46\cdots 10}a^{13}+\frac{15\cdots 23}{46\cdots 10}a^{12}+\frac{84\cdots 27}{23\cdots 55}a^{11}-\frac{21\cdots 61}{46\cdots 10}a^{10}-\frac{11\cdots 47}{46\cdots 10}a^{9}+\frac{34\cdots 73}{92\cdots 02}a^{8}+\frac{23\cdots 99}{23\cdots 55}a^{7}-\frac{29\cdots 03}{18\cdots 04}a^{6}-\frac{23\cdots 43}{92\cdots 20}a^{5}+\frac{74\cdots 77}{23\cdots 55}a^{4}+\frac{60\cdots 81}{23\cdots 55}a^{3}-\frac{10\cdots 67}{46\cdots 10}a^{2}+\frac{15\cdots 04}{23\cdots 55}a+\frac{24\cdots 43}{23\cdots 55}$, $\frac{19\cdots 71}{46\cdots 10}a^{19}-\frac{11\cdots 41}{46\cdots 10}a^{18}-\frac{54\cdots 53}{92\cdots 20}a^{17}+\frac{14\cdots 26}{46\cdots 51}a^{16}+\frac{15\cdots 09}{46\cdots 10}a^{15}-\frac{72\cdots 79}{46\cdots 10}a^{14}-\frac{42\cdots 54}{46\cdots 51}a^{13}+\frac{86\cdots 53}{23\cdots 55}a^{12}+\frac{68\cdots 47}{46\cdots 10}a^{11}-\frac{10\cdots 14}{23\cdots 55}a^{10}-\frac{61\cdots 09}{46\cdots 10}a^{9}+\frac{14\cdots 73}{46\cdots 10}a^{8}+\frac{30\cdots 93}{46\cdots 10}a^{7}-\frac{51\cdots 07}{46\cdots 10}a^{6}-\frac{14\cdots 61}{92\cdots 20}a^{5}+\frac{45\cdots 13}{23\cdots 55}a^{4}+\frac{29\cdots 94}{23\cdots 55}a^{3}-\frac{34\cdots 18}{23\cdots 55}a^{2}+\frac{19\cdots 62}{23\cdots 55}a+\frac{21\cdots 47}{23\cdots 55}$, $\frac{50\cdots 77}{46\cdots 10}a^{19}-\frac{94\cdots 57}{92\cdots 02}a^{18}-\frac{31\cdots 23}{23\cdots 55}a^{17}+\frac{31\cdots 33}{23\cdots 55}a^{16}+\frac{29\cdots 24}{46\cdots 51}a^{15}-\frac{32\cdots 11}{46\cdots 10}a^{14}-\frac{65\cdots 75}{46\cdots 51}a^{13}+\frac{85\cdots 07}{46\cdots 10}a^{12}+\frac{70\cdots 69}{46\cdots 10}a^{11}-\frac{12\cdots 69}{46\cdots 10}a^{10}-\frac{34\cdots 58}{46\cdots 51}a^{9}+\frac{49\cdots 79}{23\cdots 55}a^{8}+\frac{42\cdots 43}{46\cdots 10}a^{7}-\frac{21\cdots 59}{23\cdots 55}a^{6}+\frac{31\cdots 97}{46\cdots 10}a^{5}+\frac{91\cdots 79}{46\cdots 10}a^{4}-\frac{10\cdots 64}{23\cdots 55}a^{3}-\frac{75\cdots 07}{46\cdots 10}a^{2}+\frac{29\cdots 47}{46\cdots 51}a+\frac{42\cdots 89}{46\cdots 51}$, $\frac{26\cdots 69}{92\cdots 20}a^{19}-\frac{32\cdots 89}{92\cdots 20}a^{18}-\frac{24\cdots 69}{92\cdots 20}a^{17}+\frac{40\cdots 19}{92\cdots 20}a^{16}+\frac{24\cdots 69}{46\cdots 10}a^{15}-\frac{49\cdots 91}{23\cdots 55}a^{14}+\frac{93\cdots 09}{46\cdots 10}a^{13}+\frac{11\cdots 28}{23\cdots 55}a^{12}-\frac{23\cdots 57}{23\cdots 55}a^{11}-\frac{27\cdots 93}{46\cdots 10}a^{10}+\frac{75\cdots 68}{46\cdots 51}a^{9}+\frac{79\cdots 07}{23\cdots 55}a^{8}-\frac{22\cdots 35}{18\cdots 04}a^{7}-\frac{62\cdots 11}{92\cdots 20}a^{6}+\frac{38\cdots 49}{92\cdots 20}a^{5}-\frac{11\cdots 21}{92\cdots 20}a^{4}-\frac{24\cdots 39}{46\cdots 10}a^{3}+\frac{21\cdots 94}{46\cdots 51}a^{2}-\frac{17\cdots 53}{23\cdots 55}a-\frac{47\cdots 57}{23\cdots 55}$, $\frac{37\cdots 27}{46\cdots 10}a^{19}-\frac{11\cdots 41}{92\cdots 02}a^{18}-\frac{11\cdots 65}{18\cdots 04}a^{17}+\frac{35\cdots 13}{23\cdots 55}a^{16}-\frac{96\cdots 49}{23\cdots 55}a^{15}-\frac{34\cdots 23}{46\cdots 10}a^{14}+\frac{33\cdots 14}{23\cdots 55}a^{13}+\frac{41\cdots 72}{23\cdots 55}a^{12}-\frac{10\cdots 58}{23\cdots 55}a^{11}-\frac{10\cdots 03}{46\cdots 10}a^{10}+\frac{15\cdots 89}{23\cdots 55}a^{9}+\frac{35\cdots 89}{23\cdots 55}a^{8}-\frac{20\cdots 77}{46\cdots 10}a^{7}-\frac{26\cdots 21}{46\cdots 10}a^{6}+\frac{13\cdots 97}{92\cdots 20}a^{5}+\frac{51\cdots 61}{46\cdots 10}a^{4}-\frac{54\cdots 47}{23\cdots 55}a^{3}-\frac{34\cdots 61}{46\cdots 10}a^{2}+\frac{60\cdots 35}{46\cdots 51}a-\frac{54\cdots 71}{23\cdots 55}$, $\frac{61\cdots 53}{92\cdots 20}a^{19}-\frac{76\cdots 01}{46\cdots 10}a^{18}-\frac{52\cdots 63}{46\cdots 10}a^{17}+\frac{10\cdots 05}{46\cdots 51}a^{16}+\frac{34\cdots 77}{46\cdots 10}a^{15}-\frac{28\cdots 24}{23\cdots 55}a^{14}-\frac{58\cdots 91}{23\cdots 55}a^{13}+\frac{16\cdots 77}{46\cdots 10}a^{12}+\frac{10\cdots 89}{23\cdots 55}a^{11}-\frac{13\cdots 97}{23\cdots 55}a^{10}-\frac{20\cdots 23}{46\cdots 10}a^{9}+\frac{25\cdots 43}{46\cdots 10}a^{8}+\frac{18\cdots 07}{92\cdots 20}a^{7}-\frac{12\cdots 91}{46\cdots 51}a^{6}-\frac{14\cdots 99}{46\cdots 10}a^{5}+\frac{16\cdots 67}{23\cdots 55}a^{4}-\frac{12\cdots 32}{46\cdots 51}a^{3}-\frac{31\cdots 67}{46\cdots 10}a^{2}+\frac{17\cdots 96}{23\cdots 55}a-\frac{50\cdots 97}{23\cdots 55}$, $\frac{28\cdots 53}{92\cdots 20}a^{19}-\frac{26\cdots 67}{92\cdots 20}a^{18}-\frac{69\cdots 63}{18\cdots 04}a^{17}+\frac{69\cdots 01}{18\cdots 04}a^{16}+\frac{16\cdots 21}{92\cdots 02}a^{15}-\frac{17\cdots 83}{92\cdots 02}a^{14}-\frac{16\cdots 71}{46\cdots 10}a^{13}+\frac{11\cdots 57}{23\cdots 55}a^{12}+\frac{77\cdots 93}{23\cdots 55}a^{11}-\frac{16\cdots 23}{23\cdots 55}a^{10}-\frac{28\cdots 47}{46\cdots 10}a^{9}+\frac{27\cdots 49}{46\cdots 10}a^{8}-\frac{87\cdots 91}{92\cdots 20}a^{7}-\frac{24\cdots 21}{92\cdots 20}a^{6}+\frac{12\cdots 67}{18\cdots 04}a^{5}+\frac{51\cdots 87}{92\cdots 20}a^{4}-\frac{81\cdots 76}{46\cdots 51}a^{3}-\frac{10\cdots 92}{23\cdots 55}a^{2}+\frac{87\cdots 12}{46\cdots 51}a+\frac{62\cdots 61}{23\cdots 55}$ 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:  \( 293289608829000000 \) (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:  \( 17 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 293289608829000000 \cdot 2}{2\cdot\sqrt{4884274415246183319344904689389832005326536704}}\cr\approx \mathstrut & 4.40044463841083 \end{aligned}\] (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 - 10*x^19 - 116*x^18 + 1313*x^17 + 4736*x^16 - 67464*x^15 - 71774*x^14 + 1744748*x^13 - 126850*x^12 - 24570618*x^11 + 14939336*x^10 + 193331394*x^9 - 163802315*x^8 - 826609978*x^7 + 787082462*x^6 + 1707884929*x^5 - 1813914636*x^4 - 1179787402*x^3 + 1644305872*x^2 - 315124556*x - 58612856) 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 - 10*x^19 - 116*x^18 + 1313*x^17 + 4736*x^16 - 67464*x^15 - 71774*x^14 + 1744748*x^13 - 126850*x^12 - 24570618*x^11 + 14939336*x^10 + 193331394*x^9 - 163802315*x^8 - 826609978*x^7 + 787082462*x^6 + 1707884929*x^5 - 1813914636*x^4 - 1179787402*x^3 + 1644305872*x^2 - 315124556*x - 58612856, 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 - 10*x^19 - 116*x^18 + 1313*x^17 + 4736*x^16 - 67464*x^15 - 71774*x^14 + 1744748*x^13 - 126850*x^12 - 24570618*x^11 + 14939336*x^10 + 193331394*x^9 - 163802315*x^8 - 826609978*x^7 + 787082462*x^6 + 1707884929*x^5 - 1813914636*x^4 - 1179787402*x^3 + 1644305872*x^2 - 315124556*x - 58612856); 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 - 10*x^19 - 116*x^18 + 1313*x^17 + 4736*x^16 - 67464*x^15 - 71774*x^14 + 1744748*x^13 - 126850*x^12 - 24570618*x^11 + 14939336*x^10 + 193331394*x^9 - 163802315*x^8 - 826609978*x^7 + 787082462*x^6 + 1707884929*x^5 - 1813914636*x^4 - 1179787402*x^3 + 1644305872*x^2 - 315124556*x - 58612856); 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

$C_{20}:C_4$ (as 20T18):

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 solvable group of order 80
The 14 conjugacy class representatives for $C_{20}:C_4$
Character table for $C_{20}:C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{51 +10 \sqrt{17}})\), 5.5.2382032.1, 10.10.8056377164681869568.1

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 sibling: 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 ${\href{/padicField/3.4.0.1}{4} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ $20$ $20$ ${\href{/padicField/13.10.0.1}{10} }^{2}$ R ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{5}$ ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{5}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ R ${\href{/padicField/59.5.0.1}{5} }^{4}$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.2.4a1.2$x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$$2$$2$$4$$C_4$$$[2]^{2}$$
2.2.2.4a1.2$x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$$2$$2$$4$$C_4$$$[2]^{2}$$
2.2.2.4a1.2$x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$$2$$2$$4$$C_4$$$[2]^{2}$$
2.2.2.4a1.2$x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$$2$$2$$4$$C_4$$$[2]^{2}$$
\(17\) Copy content Toggle raw display 17.1.4.3a1.3$x^{4} + 153$$4$$1$$3$$C_4$$$[\ ]_{4}$$
17.2.4.6a1.2$x^{8} + 64 x^{7} + 1548 x^{6} + 16960 x^{5} + 74806 x^{4} + 50880 x^{3} + 13932 x^{2} + 1728 x + 98$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
17.2.4.6a1.2$x^{8} + 64 x^{7} + 1548 x^{6} + 16960 x^{5} + 74806 x^{4} + 50880 x^{3} + 13932 x^{2} + 1728 x + 98$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(53\) Copy content Toggle raw display $\Q_{53}$$x + 51$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{53}$$x + 51$$1$$1$$0$Trivial$$[\ ]$$
53.1.2.1a1.1$x^{2} + 53$$2$$1$$1$$C_2$$$[\ ]_{2}$$
53.1.4.3a1.1$x^{4} + 53$$4$$1$$3$$C_4$$$[\ ]_{4}$$
53.1.4.3a1.1$x^{4} + 53$$4$$1$$3$$C_4$$$[\ ]_{4}$$
53.1.4.3a1.1$x^{4} + 53$$4$$1$$3$$C_4$$$[\ ]_{4}$$
53.1.4.3a1.1$x^{4} + 53$$4$$1$$3$$C_4$$$[\ ]_{4}$$

Spectrum of ring of integers

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