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Properties

Label 20.12.382...152.5
Degree $20$
Signature $[12, 4]$
Discriminant $3.825\times 10^{40}$
Root discriminant \(106.94\)
Ramified primes $2,17,37$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^9.(C_2\times F_5)$ (as 20T514)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 64*x^18 + 1244*x^16 - 6262*x^14 - 20928*x^12 + 216742*x^10 - 261295*x^8 - 616048*x^6 + 352190*x^4 + 96316*x^2 + 4913)
 
Copy content gp:K = bnfinit(y^20 - 64*y^18 + 1244*y^16 - 6262*y^14 - 20928*y^12 + 216742*y^10 - 261295*y^8 - 616048*y^6 + 352190*y^4 + 96316*y^2 + 4913, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 64*x^18 + 1244*x^16 - 6262*x^14 - 20928*x^12 + 216742*x^10 - 261295*x^8 - 616048*x^6 + 352190*x^4 + 96316*x^2 + 4913);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 64*x^18 + 1244*x^16 - 6262*x^14 - 20928*x^12 + 216742*x^10 - 261295*x^8 - 616048*x^6 + 352190*x^4 + 96316*x^2 + 4913)
 

\( x^{20} - 64 x^{18} + 1244 x^{16} - 6262 x^{14} - 20928 x^{12} + 216742 x^{10} - 261295 x^{8} + \cdots + 4913 \) 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:  $[12, 4]$
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:   \(38251599236941169123840379965550626865152\) \(\medspace = 2^{40}\cdot 17^{13}\cdot 37^{8}\) 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:  \(106.94\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(17\), \(37\) 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:  \(\Q(\sqrt{17}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}+\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{252932}a^{16}+\frac{17953}{126466}a^{14}-\frac{1}{4}a^{13}-\frac{11609}{63233}a^{12}-\frac{1}{4}a^{11}+\frac{40097}{252932}a^{10}-\frac{1}{2}a^{9}-\frac{23589}{63233}a^{8}+\frac{1}{4}a^{7}+\frac{124231}{252932}a^{6}+\frac{1}{4}a^{5}-\frac{1323}{3418}a^{4}+\frac{1}{4}a^{3}-\frac{44239}{126466}a^{2}-\frac{1}{4}a+\frac{10405}{63233}$, $\frac{1}{252932}a^{17}-\frac{27327}{252932}a^{15}-\frac{1}{4}a^{14}-\frac{11609}{63233}a^{13}+\frac{40097}{252932}a^{11}-\frac{1}{4}a^{10}+\frac{95343}{252932}a^{9}-\frac{1}{4}a^{8}-\frac{2235}{252932}a^{7}-\frac{937}{6836}a^{5}-\frac{44239}{126466}a^{3}-\frac{1}{2}a^{2}-\frac{42423}{126466}a+\frac{1}{4}$, $\frac{1}{13\cdots 88}a^{18}+\frac{11296960850977}{17\cdots 62}a^{16}-\frac{42\cdots 35}{66\cdots 94}a^{14}-\frac{1}{4}a^{13}-\frac{49\cdots 37}{33\cdots 97}a^{12}-\frac{92\cdots 09}{13\cdots 88}a^{10}-\frac{1}{2}a^{9}-\frac{41\cdots 81}{13\cdots 88}a^{8}-\frac{1}{4}a^{7}-\frac{11\cdots 85}{13\cdots 88}a^{6}-\frac{1}{2}a^{5}-\frac{51\cdots 51}{13\cdots 88}a^{4}-\frac{1}{4}a^{3}-\frac{22\cdots 47}{13\cdots 88}a^{2}+\frac{30\cdots 99}{13\cdots 88}$, $\frac{1}{22\cdots 96}a^{19}+\frac{21\cdots 85}{11\cdots 98}a^{17}-\frac{23\cdots 23}{22\cdots 96}a^{15}-\frac{1}{4}a^{14}+\frac{12\cdots 09}{56\cdots 49}a^{13}-\frac{1}{4}a^{12}-\frac{43\cdots 71}{22\cdots 96}a^{11}-\frac{1}{4}a^{10}-\frac{10\cdots 67}{11\cdots 98}a^{9}+\frac{1}{4}a^{8}-\frac{35\cdots 75}{22\cdots 96}a^{7}-\frac{1}{4}a^{6}-\frac{90\cdots 26}{56\cdots 49}a^{5}-\frac{1}{2}a^{4}+\frac{13\cdots 89}{22\cdots 96}a^{3}+\frac{1}{4}a^{2}-\frac{99\cdots 53}{22\cdots 96}a+\frac{1}{4}$ 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:  $15$
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:   $\frac{1134133401545}{19\cdots 33}a^{18}-\frac{143832130993363}{38\cdots 66}a^{16}+\frac{13\cdots 75}{19\cdots 33}a^{14}-\frac{12\cdots 35}{38\cdots 66}a^{12}-\frac{27\cdots 48}{19\cdots 33}a^{10}+\frac{23\cdots 54}{19\cdots 33}a^{8}-\frac{15\cdots 96}{19\cdots 33}a^{6}-\frac{16\cdots 45}{38\cdots 66}a^{4}-\frac{34\cdots 73}{38\cdots 66}a^{2}+\frac{14\cdots 89}{38\cdots 66}$, $\frac{37\cdots 48}{33\cdots 97}a^{18}-\frac{23\cdots 18}{33\cdots 97}a^{16}+\frac{46\cdots 32}{33\cdots 97}a^{14}-\frac{22\cdots 72}{33\cdots 97}a^{12}-\frac{78\cdots 56}{33\cdots 97}a^{10}+\frac{78\cdots 85}{33\cdots 97}a^{8}-\frac{96\cdots 08}{33\cdots 97}a^{6}-\frac{20\cdots 32}{33\cdots 97}a^{4}+\frac{14\cdots 92}{33\cdots 97}a^{2}+\frac{20\cdots 66}{33\cdots 97}$, $\frac{53764471958278}{25\cdots 69}a^{18}-\frac{31\cdots 55}{25\cdots 69}a^{16}+\frac{49\cdots 62}{25\cdots 69}a^{14}+\frac{43\cdots 78}{25\cdots 69}a^{12}-\frac{37\cdots 88}{25\cdots 69}a^{10}+\frac{69\cdots 82}{25\cdots 69}a^{8}+\frac{76\cdots 74}{25\cdots 69}a^{6}-\frac{15\cdots 07}{25\cdots 69}a^{4}-\frac{31\cdots 06}{25\cdots 69}a^{2}-\frac{21\cdots 88}{25\cdots 69}$, $\frac{61\cdots 91}{33\cdots 97}a^{18}-\frac{78\cdots 51}{66\cdots 94}a^{16}+\frac{15\cdots 51}{66\cdots 94}a^{14}-\frac{75\cdots 09}{66\cdots 94}a^{12}-\frac{35\cdots 74}{89\cdots 81}a^{10}+\frac{26\cdots 51}{66\cdots 94}a^{8}-\frac{15\cdots 21}{33\cdots 97}a^{6}-\frac{21\cdots 07}{19\cdots 33}a^{4}+\frac{50\cdots 03}{66\cdots 94}a^{2}+\frac{37\cdots 63}{66\cdots 94}$, $\frac{16\cdots 80}{33\cdots 97}a^{18}-\frac{10\cdots 69}{33\cdots 97}a^{16}+\frac{20\cdots 48}{33\cdots 97}a^{14}-\frac{10\cdots 47}{33\cdots 97}a^{12}-\frac{35\cdots 32}{33\cdots 97}a^{10}+\frac{35\cdots 69}{33\cdots 97}a^{8}-\frac{42\cdots 30}{33\cdots 97}a^{6}-\frac{95\cdots 14}{33\cdots 97}a^{4}+\frac{61\cdots 82}{33\cdots 97}a^{2}+\frac{10\cdots 55}{33\cdots 97}$, $\frac{15\cdots 26}{33\cdots 97}a^{18}-\frac{99\cdots 48}{33\cdots 97}a^{16}+\frac{19\cdots 54}{33\cdots 97}a^{14}-\frac{94\cdots 79}{33\cdots 97}a^{12}-\frac{32\cdots 16}{33\cdots 97}a^{10}+\frac{32\cdots 92}{33\cdots 97}a^{8}-\frac{41\cdots 44}{33\cdots 97}a^{6}-\frac{82\cdots 29}{33\cdots 97}a^{4}+\frac{64\cdots 24}{33\cdots 97}a^{2}+\frac{46\cdots 47}{33\cdots 97}$, $\frac{19\cdots 47}{11\cdots 98}a^{19}+\frac{1134133401545}{38\cdots 66}a^{18}-\frac{12\cdots 97}{11\cdots 98}a^{17}-\frac{143832130993363}{77\cdots 32}a^{16}+\frac{48\cdots 97}{22\cdots 96}a^{15}+\frac{13\cdots 75}{38\cdots 66}a^{14}-\frac{24\cdots 27}{22\cdots 96}a^{13}-\frac{12\cdots 35}{77\cdots 32}a^{12}-\frac{79\cdots 43}{22\cdots 96}a^{11}-\frac{13\cdots 24}{19\cdots 33}a^{10}+\frac{84\cdots 21}{22\cdots 96}a^{9}+\frac{11\cdots 27}{19\cdots 33}a^{8}-\frac{10\cdots 87}{22\cdots 96}a^{7}-\frac{76\cdots 98}{19\cdots 33}a^{6}-\frac{57\cdots 32}{56\cdots 49}a^{5}-\frac{16\cdots 45}{77\cdots 32}a^{4}+\frac{15\cdots 29}{22\cdots 96}a^{3}-\frac{34\cdots 73}{77\cdots 32}a^{2}+\frac{26\cdots 21}{22\cdots 96}a-\frac{24\cdots 77}{77\cdots 32}$, $\frac{19\cdots 47}{11\cdots 98}a^{19}-\frac{1134133401545}{38\cdots 66}a^{18}-\frac{12\cdots 97}{11\cdots 98}a^{17}+\frac{143832130993363}{77\cdots 32}a^{16}+\frac{48\cdots 97}{22\cdots 96}a^{15}-\frac{13\cdots 75}{38\cdots 66}a^{14}-\frac{24\cdots 27}{22\cdots 96}a^{13}+\frac{12\cdots 35}{77\cdots 32}a^{12}-\frac{79\cdots 43}{22\cdots 96}a^{11}+\frac{13\cdots 24}{19\cdots 33}a^{10}+\frac{84\cdots 21}{22\cdots 96}a^{9}-\frac{11\cdots 27}{19\cdots 33}a^{8}-\frac{10\cdots 87}{22\cdots 96}a^{7}+\frac{76\cdots 98}{19\cdots 33}a^{6}-\frac{57\cdots 32}{56\cdots 49}a^{5}+\frac{16\cdots 45}{77\cdots 32}a^{4}+\frac{15\cdots 29}{22\cdots 96}a^{3}+\frac{34\cdots 73}{77\cdots 32}a^{2}+\frac{26\cdots 21}{22\cdots 96}a+\frac{24\cdots 77}{77\cdots 32}$, $\frac{67\cdots 91}{11\cdots 98}a^{19}-\frac{85\cdots 15}{13\cdots 88}a^{18}-\frac{86\cdots 09}{22\cdots 96}a^{17}+\frac{57\cdots 55}{13\cdots 88}a^{16}+\frac{16\cdots 85}{22\cdots 96}a^{15}-\frac{12\cdots 29}{13\cdots 88}a^{14}-\frac{42\cdots 01}{11\cdots 98}a^{13}+\frac{41\cdots 87}{66\cdots 94}a^{12}-\frac{13\cdots 95}{11\cdots 98}a^{11}+\frac{13\cdots 77}{13\cdots 88}a^{10}+\frac{28\cdots 37}{22\cdots 96}a^{9}-\frac{13\cdots 33}{66\cdots 94}a^{8}-\frac{37\cdots 83}{22\cdots 96}a^{7}+\frac{11\cdots 65}{33\cdots 97}a^{6}-\frac{38\cdots 45}{11\cdots 98}a^{5}+\frac{76\cdots 65}{13\cdots 88}a^{4}+\frac{13\cdots 20}{56\cdots 49}a^{3}-\frac{59\cdots 65}{13\cdots 88}a^{2}+\frac{62\cdots 21}{22\cdots 96}a-\frac{25\cdots 59}{66\cdots 94}$, $\frac{67\cdots 91}{11\cdots 98}a^{19}+\frac{85\cdots 15}{13\cdots 88}a^{18}-\frac{86\cdots 09}{22\cdots 96}a^{17}-\frac{57\cdots 55}{13\cdots 88}a^{16}+\frac{16\cdots 85}{22\cdots 96}a^{15}+\frac{12\cdots 29}{13\cdots 88}a^{14}-\frac{42\cdots 01}{11\cdots 98}a^{13}-\frac{41\cdots 87}{66\cdots 94}a^{12}-\frac{13\cdots 95}{11\cdots 98}a^{11}-\frac{13\cdots 77}{13\cdots 88}a^{10}+\frac{28\cdots 37}{22\cdots 96}a^{9}+\frac{13\cdots 33}{66\cdots 94}a^{8}-\frac{37\cdots 83}{22\cdots 96}a^{7}-\frac{11\cdots 65}{33\cdots 97}a^{6}-\frac{38\cdots 45}{11\cdots 98}a^{5}-\frac{76\cdots 65}{13\cdots 88}a^{4}+\frac{13\cdots 20}{56\cdots 49}a^{3}+\frac{59\cdots 65}{13\cdots 88}a^{2}+\frac{62\cdots 21}{22\cdots 96}a+\frac{25\cdots 59}{66\cdots 94}$, $\frac{63\cdots 08}{33\cdots 97}a^{18}-\frac{40\cdots 69}{33\cdots 97}a^{16}+\frac{81\cdots 82}{33\cdots 97}a^{14}-\frac{88\cdots 25}{66\cdots 94}a^{12}-\frac{23\cdots 53}{66\cdots 94}a^{10}+\frac{14\cdots 47}{33\cdots 97}a^{8}-\frac{45\cdots 29}{66\cdots 94}a^{6}-\frac{71\cdots 87}{66\cdots 94}a^{4}+\frac{77\cdots 57}{66\cdots 94}a^{2}+\frac{64\cdots 21}{66\cdots 94}$, $\frac{10\cdots 19}{22\cdots 96}a^{19}-\frac{45\cdots 65}{13\cdots 88}a^{18}-\frac{16\cdots 42}{56\cdots 49}a^{17}+\frac{29\cdots 59}{13\cdots 88}a^{16}+\frac{13\cdots 77}{22\cdots 96}a^{15}-\frac{60\cdots 59}{13\cdots 88}a^{14}-\frac{16\cdots 51}{56\cdots 49}a^{13}+\frac{34\cdots 65}{13\cdots 88}a^{12}-\frac{53\cdots 07}{56\cdots 49}a^{11}+\frac{83\cdots 11}{13\cdots 88}a^{10}+\frac{15\cdots 71}{15\cdots 77}a^{9}-\frac{28\cdots 38}{33\cdots 97}a^{8}-\frac{29\cdots 53}{22\cdots 96}a^{7}+\frac{17\cdots 09}{13\cdots 88}a^{6}-\frac{62\cdots 87}{22\cdots 96}a^{5}+\frac{27\cdots 83}{13\cdots 88}a^{4}+\frac{42\cdots 85}{22\cdots 96}a^{3}-\frac{75\cdots 31}{33\cdots 97}a^{2}+\frac{20\cdots 71}{56\cdots 49}a-\frac{62\cdots 05}{33\cdots 97}$, $\frac{97\cdots 19}{22\cdots 96}a^{19}+\frac{16\cdots 89}{13\cdots 88}a^{18}-\frac{57\cdots 59}{22\cdots 96}a^{17}-\frac{48\cdots 23}{66\cdots 94}a^{16}+\frac{94\cdots 81}{22\cdots 96}a^{15}+\frac{81\cdots 85}{66\cdots 94}a^{14}-\frac{19\cdots 53}{22\cdots 96}a^{13}-\frac{96\cdots 24}{33\cdots 97}a^{12}-\frac{60\cdots 55}{56\cdots 49}a^{11}-\frac{20\cdots 19}{66\cdots 94}a^{10}+\frac{20\cdots 38}{56\cdots 49}a^{9}+\frac{16\cdots 11}{13\cdots 88}a^{8}-\frac{21\cdots 05}{22\cdots 96}a^{7}+\frac{17\cdots 81}{13\cdots 88}a^{6}-\frac{59\cdots 35}{11\cdots 98}a^{5}-\frac{74\cdots 93}{33\cdots 97}a^{4}+\frac{39\cdots 31}{56\cdots 49}a^{3}+\frac{13\cdots 27}{13\cdots 88}a^{2}-\frac{10\cdots 61}{22\cdots 96}a+\frac{23\cdots 77}{66\cdots 94}$, $\frac{34\cdots 81}{17\cdots 92}a^{19}+\frac{11\cdots 25}{66\cdots 94}a^{18}-\frac{54\cdots 74}{43\cdots 73}a^{17}-\frac{14\cdots 37}{13\cdots 88}a^{16}+\frac{10\cdots 13}{43\cdots 73}a^{15}+\frac{27\cdots 09}{13\cdots 88}a^{14}-\frac{22\cdots 07}{17\cdots 92}a^{13}-\frac{32\cdots 98}{33\cdots 97}a^{12}-\frac{16\cdots 93}{43\cdots 73}a^{11}-\frac{47\cdots 67}{13\cdots 88}a^{10}+\frac{76\cdots 67}{17\cdots 92}a^{9}+\frac{45\cdots 77}{13\cdots 88}a^{8}-\frac{25\cdots 28}{43\cdots 73}a^{7}-\frac{54\cdots 07}{13\cdots 88}a^{6}-\frac{50\cdots 22}{43\cdots 73}a^{5}-\frac{11\cdots 83}{13\cdots 88}a^{4}+\frac{36\cdots 48}{43\cdots 73}a^{3}+\frac{22\cdots 21}{33\cdots 97}a^{2}+\frac{24\cdots 95}{86\cdots 46}a+\frac{29\cdots 90}{33\cdots 97}$, $\frac{19\cdots 07}{11\cdots 98}a^{19}+\frac{91\cdots 71}{66\cdots 94}a^{18}-\frac{60\cdots 72}{56\cdots 49}a^{17}-\frac{29\cdots 20}{33\cdots 97}a^{16}+\frac{46\cdots 77}{22\cdots 96}a^{15}+\frac{54\cdots 62}{33\cdots 97}a^{14}-\frac{53\cdots 58}{56\cdots 49}a^{13}-\frac{98\cdots 61}{13\cdots 88}a^{12}-\frac{46\cdots 61}{11\cdots 98}a^{11}-\frac{45\cdots 21}{13\cdots 88}a^{10}+\frac{77\cdots 59}{22\cdots 96}a^{9}+\frac{90\cdots 42}{33\cdots 97}a^{8}-\frac{13\cdots 42}{56\cdots 49}a^{7}-\frac{21\cdots 05}{13\cdots 88}a^{6}-\frac{26\cdots 11}{22\cdots 96}a^{5}-\frac{12\cdots 55}{13\cdots 88}a^{4}-\frac{17\cdots 67}{11\cdots 98}a^{3}-\frac{23\cdots 97}{13\cdots 88}a^{2}+\frac{47\cdots 32}{56\cdots 49}a-\frac{82\cdots 25}{13\cdots 88}$ 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:  \( 12316530469600 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{4}\cdot 12316530469600 \cdot 2}{2\cdot\sqrt{38251599236941169123840379965550626865152}}\cr\approx \mathstrut & 0.402015694878244 \end{aligned}\] (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 - 64*x^18 + 1244*x^16 - 6262*x^14 - 20928*x^12 + 216742*x^10 - 261295*x^8 - 616048*x^6 + 352190*x^4 + 96316*x^2 + 4913) 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 - 64*x^18 + 1244*x^16 - 6262*x^14 - 20928*x^12 + 216742*x^10 - 261295*x^8 - 616048*x^6 + 352190*x^4 + 96316*x^2 + 4913, 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 - 64*x^18 + 1244*x^16 - 6262*x^14 - 20928*x^12 + 216742*x^10 - 261295*x^8 - 616048*x^6 + 352190*x^4 + 96316*x^2 + 4913); 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 = PolynomialRing(QQ); K, a = NumberField(x^20 - 64*x^18 + 1244*x^16 - 6262*x^14 - 20928*x^12 + 216742*x^10 - 261295*x^8 - 616048*x^6 + 352190*x^4 + 96316*x^2 + 4913); 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

$C_2^9.(C_2\times F_5)$ (as 20T514):

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); G, transitive_group_identification(G)
 
A solvable group of order 20480
The 74 conjugacy class representatives for $C_2^9.(C_2\times F_5)$
Character table for $C_2^9.(C_2\times F_5)$

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.6725897.1, 10.10.1482348640816627712.1

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(b)]
 
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: 20.12.38251599236941169123840379965550626865152.6

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{6}$ R ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ R ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }^{2}$ ${\href{/padicField/47.10.0.1}{10} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}$

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 2.5.4.40b1.1$x^{20} + 4 x^{17} + 4 x^{15} + 6 x^{14} + 12 x^{12} + 4 x^{11} + 8 x^{10} + 12 x^{9} + x^{8} + 16 x^{7} + 4 x^{6} + 12 x^{5} + 8 x^{4} + 12 x^{2} + 9$$4$$5$$40$20T3not computed
\(17\) Copy content Toggle raw display 17.1.2.1a1.2$x^{2} + 51$$2$$1$$1$$C_2$$$[\ ]_{2}$$
17.2.1.0a1.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
17.1.4.3a1.3$x^{4} + 153$$4$$1$$3$$C_4$$$[\ ]_{4}$$
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$$$[\ ]_{4}^{2}$$
\(37\) Copy content Toggle raw display 37.4.1.0a1.1$x^{4} + 6 x^{2} + 24 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
37.2.2.2a1.1$x^{4} + 66 x^{3} + 1093 x^{2} + 169 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
37.2.2.2a1.1$x^{4} + 66 x^{3} + 1093 x^{2} + 169 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
37.4.2.4a1.2$x^{8} + 12 x^{6} + 48 x^{5} + 40 x^{4} + 288 x^{3} + 600 x^{2} + 96 x + 41$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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