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Properties

Label 20.20.120...125.1
Degree $20$
Signature $(20, 0)$
Discriminant $1.201\times 10^{36}$
Root discriminant \(63.68\)
Ramified primes $5,7,139$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4\times D_5$ (as 20T6)

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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 - 6*x^19 - 49*x^18 + 394*x^17 + 431*x^16 - 8836*x^15 + 11137*x^14 + 75642*x^13 - 215641*x^12 - 135520*x^11 + 1183051*x^10 - 1032083*x^9 - 1892289*x^8 + 4095435*x^7 - 1533360*x^6 - 2597056*x^5 + 3317104*x^4 - 1586360*x^3 + 344740*x^2 - 27105*x - 139)
 
Copy content gp:K = bnfinit(y^20 - 6*y^19 - 49*y^18 + 394*y^17 + 431*y^16 - 8836*y^15 + 11137*y^14 + 75642*y^13 - 215641*y^12 - 135520*y^11 + 1183051*y^10 - 1032083*y^9 - 1892289*y^8 + 4095435*y^7 - 1533360*y^6 - 2597056*y^5 + 3317104*y^4 - 1586360*y^3 + 344740*y^2 - 27105*y - 139, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^19 - 49*x^18 + 394*x^17 + 431*x^16 - 8836*x^15 + 11137*x^14 + 75642*x^13 - 215641*x^12 - 135520*x^11 + 1183051*x^10 - 1032083*x^9 - 1892289*x^8 + 4095435*x^7 - 1533360*x^6 - 2597056*x^5 + 3317104*x^4 - 1586360*x^3 + 344740*x^2 - 27105*x - 139);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 6*x^19 - 49*x^18 + 394*x^17 + 431*x^16 - 8836*x^15 + 11137*x^14 + 75642*x^13 - 215641*x^12 - 135520*x^11 + 1183051*x^10 - 1032083*x^9 - 1892289*x^8 + 4095435*x^7 - 1533360*x^6 - 2597056*x^5 + 3317104*x^4 - 1586360*x^3 + 344740*x^2 - 27105*x - 139)
 

\( x^{20} - 6 x^{19} - 49 x^{18} + 394 x^{17} + 431 x^{16} - 8836 x^{15} + 11137 x^{14} + 75642 x^{13} + \cdots - 139 \) 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:   \(1201292780904647323599596221923828125\) \(\medspace = 5^{15}\cdot 7^{10}\cdot 139^{8}\) 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:  \(63.68\)
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:  $5^{3/4}7^{1/2}139^{1/2}\approx 104.29990752889705$
Ramified primes:   \(5\), \(7\), \(139\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{13}-\frac{1}{3}a^{11}-\frac{2}{9}a^{10}+\frac{4}{9}a^{9}-\frac{1}{3}a^{8}+\frac{2}{9}a^{6}+\frac{2}{9}a^{5}-\frac{4}{9}a^{4}+\frac{2}{9}a^{3}-\frac{2}{9}a^{2}+\frac{1}{3}a+\frac{1}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{14}-\frac{2}{9}a^{11}-\frac{2}{9}a^{10}-\frac{1}{3}a^{8}-\frac{4}{9}a^{7}-\frac{4}{9}a^{6}-\frac{4}{9}a^{5}+\frac{2}{9}a^{4}+\frac{1}{9}a^{3}+\frac{1}{3}a^{2}+\frac{1}{9}a-\frac{1}{3}$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{15}+\frac{1}{9}a^{12}-\frac{2}{9}a^{11}+\frac{1}{3}a^{10}+\frac{2}{9}a^{8}-\frac{1}{9}a^{7}-\frac{1}{9}a^{6}+\frac{2}{9}a^{5}+\frac{1}{9}a^{4}-\frac{1}{3}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{67\cdots 39}a^{19}-\frac{15\cdots 65}{67\cdots 39}a^{18}+\frac{49\cdots 22}{67\cdots 39}a^{17}-\frac{91\cdots 56}{22\cdots 13}a^{16}+\frac{31\cdots 46}{67\cdots 39}a^{15}-\frac{10\cdots 06}{67\cdots 39}a^{14}+\frac{12\cdots 94}{75\cdots 71}a^{13}-\frac{71\cdots 40}{67\cdots 39}a^{12}-\frac{45\cdots 71}{22\cdots 13}a^{11}+\frac{51\cdots 01}{67\cdots 39}a^{10}-\frac{30\cdots 00}{67\cdots 39}a^{9}+\frac{12\cdots 93}{67\cdots 39}a^{8}-\frac{14\cdots 68}{67\cdots 39}a^{7}+\frac{12\cdots 10}{75\cdots 71}a^{6}-\frac{12\cdots 49}{67\cdots 39}a^{5}-\frac{12\cdots 28}{15\cdots 37}a^{4}-\frac{21\cdots 02}{67\cdots 39}a^{3}+\frac{82\cdots 05}{22\cdots 13}a^{2}-\frac{94\cdots 41}{67\cdots 39}a-\frac{29\cdots 65}{67\cdots 39}$ 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:  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:  $C_{2}\times C_{2}\times C_{2}$, 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:  $19$
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:   $\frac{16\cdots 80}{34\cdots 41}a^{19}-\frac{41\cdots 68}{34\cdots 41}a^{18}-\frac{10\cdots 12}{34\cdots 41}a^{17}+\frac{31\cdots 19}{34\cdots 41}a^{16}+\frac{22\cdots 25}{34\cdots 41}a^{15}-\frac{85\cdots 52}{34\cdots 41}a^{14}-\frac{18\cdots 01}{34\cdots 41}a^{13}+\frac{33\cdots 21}{11\cdots 47}a^{12}+\frac{33\cdots 75}{34\cdots 41}a^{11}-\frac{54\cdots 92}{34\cdots 41}a^{10}+\frac{30\cdots 85}{34\cdots 41}a^{9}+\frac{42\cdots 92}{11\cdots 47}a^{8}-\frac{14\cdots 82}{34\cdots 41}a^{7}-\frac{98\cdots 77}{34\cdots 41}a^{6}+\frac{64\cdots 34}{11\cdots 47}a^{5}-\frac{83\cdots 25}{11\cdots 47}a^{4}-\frac{22\cdots 48}{11\cdots 47}a^{3}+\frac{31\cdots 31}{34\cdots 41}a^{2}-\frac{36\cdots 24}{34\cdots 41}a+\frac{95\cdots 11}{11\cdots 47}$, $\frac{26\cdots 80}{67\cdots 39}a^{19}-\frac{25\cdots 81}{22\cdots 13}a^{18}-\frac{15\cdots 26}{67\cdots 39}a^{17}+\frac{18\cdots 21}{22\cdots 13}a^{16}+\frac{10\cdots 11}{22\cdots 13}a^{15}-\frac{14\cdots 88}{67\cdots 39}a^{14}-\frac{69\cdots 95}{22\cdots 13}a^{13}+\frac{15\cdots 07}{67\cdots 39}a^{12}-\frac{40\cdots 36}{67\cdots 39}a^{11}-\frac{72\cdots 67}{67\cdots 39}a^{10}+\frac{86\cdots 65}{67\cdots 39}a^{9}+\frac{10\cdots 30}{67\cdots 39}a^{8}-\frac{85\cdots 14}{22\cdots 13}a^{7}+\frac{82\cdots 39}{67\cdots 39}a^{6}+\frac{18\cdots 61}{75\cdots 71}a^{5}-\frac{12\cdots 45}{45\cdots 11}a^{4}+\frac{74\cdots 67}{67\cdots 39}a^{3}-\frac{13\cdots 21}{67\cdots 39}a^{2}+\frac{90\cdots 39}{67\cdots 39}a+\frac{29\cdots 23}{67\cdots 39}$, $\frac{90\cdots 97}{75\cdots 71}a^{19}-\frac{14\cdots 84}{22\cdots 13}a^{18}-\frac{47\cdots 99}{75\cdots 71}a^{17}+\frac{32\cdots 95}{75\cdots 71}a^{16}+\frac{17\cdots 22}{22\cdots 13}a^{15}-\frac{75\cdots 68}{75\cdots 71}a^{14}+\frac{52\cdots 08}{75\cdots 71}a^{13}+\frac{21\cdots 42}{22\cdots 13}a^{12}-\frac{44\cdots 91}{22\cdots 13}a^{11}-\frac{20\cdots 56}{75\cdots 71}a^{10}+\frac{91\cdots 82}{75\cdots 71}a^{9}-\frac{11\cdots 02}{22\cdots 13}a^{8}-\frac{56\cdots 30}{22\cdots 13}a^{7}+\frac{75\cdots 14}{22\cdots 13}a^{6}+\frac{16\cdots 93}{22\cdots 13}a^{5}-\frac{44\cdots 19}{15\cdots 37}a^{4}+\frac{16\cdots 41}{75\cdots 71}a^{3}-\frac{14\cdots 78}{22\cdots 13}a^{2}+\frac{45\cdots 48}{75\cdots 71}a+\frac{19\cdots 14}{75\cdots 71}$, $\frac{10\cdots 41}{22\cdots 13}a^{19}-\frac{42\cdots 45}{22\cdots 13}a^{18}-\frac{62\cdots 97}{22\cdots 13}a^{17}+\frac{99\cdots 13}{75\cdots 71}a^{16}+\frac{10\cdots 32}{22\cdots 13}a^{15}-\frac{72\cdots 75}{22\cdots 13}a^{14}-\frac{10\cdots 19}{75\cdots 71}a^{13}+\frac{74\cdots 52}{22\cdots 13}a^{12}-\frac{25\cdots 37}{75\cdots 71}a^{11}-\frac{30\cdots 78}{22\cdots 13}a^{10}+\frac{64\cdots 03}{22\cdots 13}a^{9}+\frac{20\cdots 12}{22\cdots 13}a^{8}-\frac{15\cdots 82}{22\cdots 13}a^{7}+\frac{38\cdots 47}{75\cdots 71}a^{6}+\frac{65\cdots 44}{22\cdots 13}a^{5}-\frac{31\cdots 53}{50\cdots 79}a^{4}+\frac{77\cdots 81}{22\cdots 13}a^{3}-\frac{62\cdots 28}{75\cdots 71}a^{2}+\frac{16\cdots 88}{22\cdots 13}a-\frac{19\cdots 19}{22\cdots 13}$, $\frac{31\cdots 39}{22\cdots 13}a^{19}-\frac{14\cdots 55}{22\cdots 13}a^{18}-\frac{17\cdots 76}{22\cdots 13}a^{17}+\frac{33\cdots 41}{75\cdots 71}a^{16}+\frac{26\cdots 28}{22\cdots 13}a^{15}-\frac{23\cdots 59}{22\cdots 13}a^{14}+\frac{10\cdots 78}{75\cdots 71}a^{13}+\frac{23\cdots 48}{22\cdots 13}a^{12}-\frac{11\cdots 17}{75\cdots 71}a^{11}-\frac{87\cdots 12}{22\cdots 13}a^{10}+\frac{25\cdots 25}{22\cdots 13}a^{9}+\frac{49\cdots 05}{22\cdots 13}a^{8}-\frac{57\cdots 04}{22\cdots 13}a^{7}+\frac{17\cdots 21}{75\cdots 71}a^{6}+\frac{20\cdots 33}{22\cdots 13}a^{5}-\frac{12\cdots 76}{50\cdots 79}a^{4}+\frac{31\cdots 67}{22\cdots 13}a^{3}-\frac{24\cdots 14}{75\cdots 71}a^{2}+\frac{56\cdots 81}{22\cdots 13}a+\frac{54\cdots 58}{22\cdots 13}$, $\frac{70\cdots 54}{67\cdots 39}a^{19}-\frac{40\cdots 49}{67\cdots 39}a^{18}-\frac{36\cdots 27}{67\cdots 39}a^{17}+\frac{26\cdots 73}{67\cdots 39}a^{16}+\frac{40\cdots 46}{67\cdots 39}a^{15}-\frac{61\cdots 45}{67\cdots 39}a^{14}+\frac{56\cdots 22}{67\cdots 39}a^{13}+\frac{18\cdots 59}{22\cdots 13}a^{12}-\frac{44\cdots 96}{22\cdots 13}a^{11}-\frac{14\cdots 79}{67\cdots 39}a^{10}+\frac{78\cdots 87}{67\cdots 39}a^{9}-\frac{47\cdots 15}{75\cdots 71}a^{8}-\frac{15\cdots 59}{67\cdots 39}a^{7}+\frac{23\cdots 47}{67\cdots 39}a^{6}-\frac{20\cdots 95}{67\cdots 39}a^{5}-\frac{12\cdots 49}{45\cdots 11}a^{4}+\frac{16\cdots 84}{67\cdots 39}a^{3}-\frac{52\cdots 81}{67\cdots 39}a^{2}+\frac{57\cdots 32}{67\cdots 39}a+\frac{37\cdots 01}{75\cdots 71}$, $\frac{24\cdots 65}{67\cdots 39}a^{19}-\frac{91\cdots 07}{67\cdots 39}a^{18}-\frac{14\cdots 19}{67\cdots 39}a^{17}+\frac{21\cdots 55}{22\cdots 13}a^{16}+\frac{25\cdots 20}{67\cdots 39}a^{15}-\frac{15\cdots 08}{67\cdots 39}a^{14}-\frac{29\cdots 01}{22\cdots 13}a^{13}+\frac{55\cdots 38}{22\cdots 13}a^{12}-\frac{15\cdots 92}{67\cdots 39}a^{11}-\frac{68\cdots 85}{67\cdots 39}a^{10}+\frac{13\cdots 43}{67\cdots 39}a^{9}+\frac{18\cdots 81}{22\cdots 13}a^{8}-\frac{33\cdots 03}{67\cdots 39}a^{7}+\frac{23\cdots 09}{67\cdots 39}a^{6}+\frac{15\cdots 05}{67\cdots 39}a^{5}-\frac{19\cdots 87}{45\cdots 11}a^{4}+\frac{15\cdots 41}{67\cdots 39}a^{3}-\frac{31\cdots 88}{67\cdots 39}a^{2}+\frac{14\cdots 18}{67\cdots 39}a+\frac{17\cdots 27}{67\cdots 39}$, $\frac{22\cdots 50}{67\cdots 39}a^{19}-\frac{83\cdots 81}{67\cdots 39}a^{18}-\frac{13\cdots 54}{67\cdots 39}a^{17}+\frac{59\cdots 87}{67\cdots 39}a^{16}+\frac{23\cdots 39}{67\cdots 39}a^{15}-\frac{14\cdots 39}{67\cdots 39}a^{14}-\frac{86\cdots 36}{67\cdots 39}a^{13}+\frac{50\cdots 46}{22\cdots 13}a^{12}-\frac{15\cdots 75}{75\cdots 71}a^{11}-\frac{63\cdots 95}{67\cdots 39}a^{10}+\frac{41\cdots 21}{22\cdots 13}a^{9}+\frac{18\cdots 96}{22\cdots 13}a^{8}-\frac{31\cdots 32}{67\cdots 39}a^{7}+\frac{20\cdots 13}{67\cdots 39}a^{6}+\frac{50\cdots 71}{22\cdots 13}a^{5}-\frac{58\cdots 68}{15\cdots 37}a^{4}+\frac{13\cdots 96}{67\cdots 39}a^{3}-\frac{30\cdots 94}{75\cdots 71}a^{2}+\frac{13\cdots 71}{67\cdots 39}a+\frac{13\cdots 22}{67\cdots 39}$, $\frac{70\cdots 70}{67\cdots 39}a^{19}-\frac{15\cdots 97}{67\cdots 39}a^{18}-\frac{43\cdots 12}{67\cdots 39}a^{17}+\frac{11\cdots 03}{67\cdots 39}a^{16}+\frac{93\cdots 54}{67\cdots 39}a^{15}-\frac{31\cdots 30}{67\cdots 39}a^{14}-\frac{80\cdots 28}{67\cdots 39}a^{13}+\frac{12\cdots 94}{22\cdots 13}a^{12}+\frac{17\cdots 87}{67\cdots 39}a^{11}-\frac{63\cdots 69}{22\cdots 13}a^{10}+\frac{30\cdots 96}{22\cdots 13}a^{9}+\frac{13\cdots 49}{22\cdots 13}a^{8}-\frac{44\cdots 84}{67\cdots 39}a^{7}-\frac{70\cdots 84}{22\cdots 13}a^{6}+\frac{48\cdots 87}{67\cdots 39}a^{5}-\frac{10\cdots 75}{45\cdots 11}a^{4}-\frac{72\cdots 02}{67\cdots 39}a^{3}+\frac{53\cdots 11}{75\cdots 71}a^{2}-\frac{42\cdots 09}{67\cdots 39}a-\frac{76\cdots 36}{67\cdots 39}$, $\frac{65\cdots 20}{67\cdots 39}a^{19}-\frac{26\cdots 54}{67\cdots 39}a^{18}-\frac{37\cdots 29}{67\cdots 39}a^{17}+\frac{18\cdots 66}{67\cdots 39}a^{16}+\frac{67\cdots 49}{67\cdots 39}a^{15}-\frac{45\cdots 93}{67\cdots 39}a^{14}-\frac{21\cdots 73}{67\cdots 39}a^{13}+\frac{47\cdots 22}{67\cdots 39}a^{12}-\frac{15\cdots 38}{22\cdots 13}a^{11}-\frac{68\cdots 02}{22\cdots 13}a^{10}+\frac{13\cdots 48}{22\cdots 13}a^{9}+\frac{21\cdots 56}{67\cdots 39}a^{8}-\frac{10\cdots 16}{67\cdots 39}a^{7}+\frac{52\cdots 96}{67\cdots 39}a^{6}+\frac{68\cdots 17}{75\cdots 71}a^{5}-\frac{51\cdots 14}{45\cdots 11}a^{4}+\frac{28\cdots 51}{67\cdots 39}a^{3}-\frac{40\cdots 31}{67\cdots 39}a^{2}+\frac{17\cdots 96}{67\cdots 39}a-\frac{20\cdots 42}{67\cdots 39}$, $\frac{52\cdots 36}{67\cdots 39}a^{19}-\frac{11\cdots 17}{67\cdots 39}a^{18}-\frac{32\cdots 36}{67\cdots 39}a^{17}+\frac{88\cdots 37}{67\cdots 39}a^{16}+\frac{73\cdots 51}{67\cdots 39}a^{15}-\frac{24\cdots 17}{67\cdots 39}a^{14}-\frac{69\cdots 30}{67\cdots 39}a^{13}+\frac{30\cdots 83}{67\cdots 39}a^{12}+\frac{22\cdots 19}{67\cdots 39}a^{11}-\frac{17\cdots 73}{67\cdots 39}a^{10}+\frac{40\cdots 39}{67\cdots 39}a^{9}+\frac{47\cdots 33}{67\cdots 39}a^{8}-\frac{12\cdots 39}{22\cdots 13}a^{7}-\frac{17\cdots 63}{22\cdots 13}a^{6}+\frac{68\cdots 04}{67\cdots 39}a^{5}+\frac{15\cdots 05}{15\cdots 37}a^{4}-\frac{37\cdots 23}{67\cdots 39}a^{3}+\frac{15\cdots 00}{67\cdots 39}a^{2}-\frac{17\cdots 34}{67\cdots 39}a-\frac{67\cdots 86}{75\cdots 71}$, $\frac{16\cdots 60}{67\cdots 39}a^{19}-\frac{91\cdots 98}{67\cdots 39}a^{18}-\frac{88\cdots 17}{67\cdots 39}a^{17}+\frac{61\cdots 64}{67\cdots 39}a^{16}+\frac{10\cdots 15}{67\cdots 39}a^{15}-\frac{14\cdots 29}{67\cdots 39}a^{14}+\frac{10\cdots 38}{67\cdots 39}a^{13}+\frac{13\cdots 60}{67\cdots 39}a^{12}-\frac{95\cdots 59}{22\cdots 13}a^{11}-\frac{12\cdots 15}{22\cdots 13}a^{10}+\frac{58\cdots 16}{22\cdots 13}a^{9}-\frac{77\cdots 08}{67\cdots 39}a^{8}-\frac{35\cdots 76}{67\cdots 39}a^{7}+\frac{49\cdots 05}{67\cdots 39}a^{6}-\frac{16\cdots 65}{22\cdots 13}a^{5}-\frac{28\cdots 81}{45\cdots 11}a^{4}+\frac{33\cdots 99}{67\cdots 39}a^{3}-\frac{98\cdots 02}{67\cdots 39}a^{2}+\frac{97\cdots 85}{67\cdots 39}a+\frac{11\cdots 17}{67\cdots 39}$, $\frac{30\cdots 00}{75\cdots 71}a^{19}-\frac{14\cdots 48}{67\cdots 39}a^{18}-\frac{48\cdots 30}{22\cdots 13}a^{17}+\frac{10\cdots 48}{75\cdots 71}a^{16}+\frac{19\cdots 94}{67\cdots 39}a^{15}-\frac{75\cdots 88}{22\cdots 13}a^{14}+\frac{14\cdots 81}{75\cdots 71}a^{13}+\frac{21\cdots 80}{67\cdots 39}a^{12}-\frac{42\cdots 84}{67\cdots 39}a^{11}-\frac{75\cdots 57}{75\cdots 71}a^{10}+\frac{89\cdots 79}{22\cdots 13}a^{9}-\frac{84\cdots 02}{67\cdots 39}a^{8}-\frac{55\cdots 93}{67\cdots 39}a^{7}+\frac{69\cdots 65}{67\cdots 39}a^{6}+\frac{55\cdots 01}{67\cdots 39}a^{5}-\frac{42\cdots 58}{45\cdots 11}a^{4}+\frac{50\cdots 38}{75\cdots 71}a^{3}-\frac{12\cdots 17}{67\cdots 39}a^{2}+\frac{39\cdots 36}{22\cdots 13}a+\frac{52\cdots 44}{75\cdots 71}$, $\frac{20\cdots 54}{22\cdots 13}a^{19}-\frac{23\cdots 43}{67\cdots 39}a^{18}-\frac{11\cdots 14}{22\cdots 13}a^{17}+\frac{55\cdots 35}{22\cdots 13}a^{16}+\frac{60\cdots 47}{67\cdots 39}a^{15}-\frac{13\cdots 46}{22\cdots 13}a^{14}-\frac{19\cdots 59}{75\cdots 71}a^{13}+\frac{42\cdots 49}{67\cdots 39}a^{12}-\frac{43\cdots 62}{67\cdots 39}a^{11}-\frac{19\cdots 59}{75\cdots 71}a^{10}+\frac{12\cdots 17}{22\cdots 13}a^{9}+\frac{13\cdots 06}{67\cdots 39}a^{8}-\frac{88\cdots 45}{67\cdots 39}a^{7}+\frac{60\cdots 53}{67\cdots 39}a^{6}+\frac{38\cdots 67}{67\cdots 39}a^{5}-\frac{48\cdots 74}{45\cdots 11}a^{4}+\frac{46\cdots 10}{75\cdots 71}a^{3}-\frac{12\cdots 55}{67\cdots 39}a^{2}+\frac{56\cdots 60}{22\cdots 13}a-\frac{22\cdots 46}{22\cdots 13}$, $\frac{20\cdots 11}{75\cdots 71}a^{19}-\frac{35\cdots 44}{22\cdots 13}a^{18}-\frac{94\cdots 14}{67\cdots 39}a^{17}+\frac{70\cdots 56}{67\cdots 39}a^{16}+\frac{34\cdots 38}{22\cdots 13}a^{15}-\frac{16\cdots 78}{67\cdots 39}a^{14}+\frac{15\cdots 27}{67\cdots 39}a^{13}+\frac{48\cdots 88}{22\cdots 13}a^{12}-\frac{35\cdots 95}{67\cdots 39}a^{11}-\frac{12\cdots 89}{22\cdots 13}a^{10}+\frac{21\cdots 85}{67\cdots 39}a^{9}-\frac{40\cdots 30}{22\cdots 13}a^{8}-\frac{40\cdots 95}{67\cdots 39}a^{7}+\frac{21\cdots 39}{22\cdots 13}a^{6}-\frac{55\cdots 38}{75\cdots 71}a^{5}-\frac{11\cdots 57}{15\cdots 37}a^{4}+\frac{43\cdots 41}{67\cdots 39}a^{3}-\frac{13\cdots 66}{67\cdots 39}a^{2}+\frac{12\cdots 59}{67\cdots 39}a+\frac{22\cdots 54}{67\cdots 39}$, $\frac{67\cdots 79}{22\cdots 13}a^{19}-\frac{32\cdots 94}{22\cdots 13}a^{18}-\frac{10\cdots 81}{67\cdots 39}a^{17}+\frac{66\cdots 34}{67\cdots 39}a^{16}+\frac{51\cdots 92}{22\cdots 13}a^{15}-\frac{15\cdots 23}{67\cdots 39}a^{14}+\frac{59\cdots 15}{67\cdots 39}a^{13}+\frac{49\cdots 00}{22\cdots 13}a^{12}-\frac{26\cdots 66}{67\cdots 39}a^{11}-\frac{48\cdots 46}{67\cdots 39}a^{10}+\frac{17\cdots 90}{67\cdots 39}a^{9}-\frac{15\cdots 88}{22\cdots 13}a^{8}-\frac{36\cdots 68}{67\cdots 39}a^{7}+\frac{45\cdots 81}{67\cdots 39}a^{6}+\frac{31\cdots 03}{67\cdots 39}a^{5}-\frac{27\cdots 60}{45\cdots 11}a^{4}+\frac{10\cdots 41}{22\cdots 13}a^{3}-\frac{91\cdots 33}{67\cdots 39}a^{2}+\frac{90\cdots 86}{67\cdots 39}a+\frac{34\cdots 27}{67\cdots 39}$, $\frac{39\cdots 31}{67\cdots 39}a^{19}-\frac{21\cdots 34}{67\cdots 39}a^{18}-\frac{68\cdots 85}{22\cdots 13}a^{17}+\frac{14\cdots 00}{67\cdots 39}a^{16}+\frac{24\cdots 39}{67\cdots 39}a^{15}-\frac{11\cdots 06}{22\cdots 13}a^{14}+\frac{26\cdots 39}{67\cdots 39}a^{13}+\frac{10\cdots 53}{22\cdots 13}a^{12}-\frac{68\cdots 20}{67\cdots 39}a^{11}-\frac{88\cdots 17}{67\cdots 39}a^{10}+\frac{13\cdots 11}{22\cdots 13}a^{9}-\frac{21\cdots 63}{75\cdots 71}a^{8}-\frac{91\cdots 78}{75\cdots 71}a^{7}+\frac{11\cdots 57}{67\cdots 39}a^{6}-\frac{47\cdots 93}{67\cdots 39}a^{5}-\frac{67\cdots 93}{45\cdots 11}a^{4}+\frac{27\cdots 94}{22\cdots 13}a^{3}-\frac{85\cdots 30}{22\cdots 13}a^{2}+\frac{91\cdots 23}{22\cdots 13}a+\frac{18\cdots 61}{67\cdots 39}$, $\frac{23\cdots 54}{22\cdots 13}a^{19}-\frac{28\cdots 85}{67\cdots 39}a^{18}-\frac{39\cdots 07}{67\cdots 39}a^{17}+\frac{66\cdots 88}{22\cdots 13}a^{16}+\frac{68\cdots 15}{67\cdots 39}a^{15}-\frac{48\cdots 45}{67\cdots 39}a^{14}-\frac{47\cdots 95}{22\cdots 13}a^{13}+\frac{49\cdots 54}{67\cdots 39}a^{12}-\frac{56\cdots 91}{67\cdots 39}a^{11}-\frac{19\cdots 18}{67\cdots 39}a^{10}+\frac{15\cdots 16}{22\cdots 13}a^{9}+\frac{10\cdots 56}{67\cdots 39}a^{8}-\frac{10\cdots 05}{67\cdots 39}a^{7}+\frac{85\cdots 62}{67\cdots 39}a^{6}+\frac{43\cdots 86}{67\cdots 39}a^{5}-\frac{22\cdots 67}{15\cdots 37}a^{4}+\frac{54\cdots 54}{67\cdots 39}a^{3}-\frac{12\cdots 09}{67\cdots 39}a^{2}+\frac{85\cdots 85}{67\cdots 39}a-\frac{19\cdots 16}{22\cdots 13}$, $\frac{83\cdots 83}{22\cdots 13}a^{19}-\frac{15\cdots 24}{67\cdots 39}a^{18}-\frac{11\cdots 60}{67\cdots 39}a^{17}+\frac{11\cdots 70}{75\cdots 71}a^{16}+\frac{84\cdots 18}{67\cdots 39}a^{15}-\frac{22\cdots 10}{67\cdots 39}a^{14}+\frac{11\cdots 99}{22\cdots 13}a^{13}+\frac{19\cdots 34}{67\cdots 39}a^{12}-\frac{20\cdots 85}{22\cdots 13}a^{11}-\frac{29\cdots 54}{67\cdots 39}a^{10}+\frac{11\cdots 01}{22\cdots 13}a^{9}-\frac{30\cdots 21}{67\cdots 39}a^{8}-\frac{18\cdots 65}{22\cdots 13}a^{7}+\frac{39\cdots 88}{22\cdots 13}a^{6}-\frac{35\cdots 29}{75\cdots 71}a^{5}-\frac{59\cdots 29}{45\cdots 11}a^{4}+\frac{87\cdots 21}{67\cdots 39}a^{3}-\frac{29\cdots 37}{67\cdots 39}a^{2}+\frac{31\cdots 67}{67\cdots 39}a+\frac{11\cdots 08}{75\cdots 71}$ 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:  \( 386511180224 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 386511180224 \cdot 1}{2\cdot\sqrt{1201292780904647323599596221923828125}}\cr\approx \mathstrut & 0.184887497774864 \end{aligned}\] (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 - 6*x^19 - 49*x^18 + 394*x^17 + 431*x^16 - 8836*x^15 + 11137*x^14 + 75642*x^13 - 215641*x^12 - 135520*x^11 + 1183051*x^10 - 1032083*x^9 - 1892289*x^8 + 4095435*x^7 - 1533360*x^6 - 2597056*x^5 + 3317104*x^4 - 1586360*x^3 + 344740*x^2 - 27105*x - 139) 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 - 6*x^19 - 49*x^18 + 394*x^17 + 431*x^16 - 8836*x^15 + 11137*x^14 + 75642*x^13 - 215641*x^12 - 135520*x^11 + 1183051*x^10 - 1032083*x^9 - 1892289*x^8 + 4095435*x^7 - 1533360*x^6 - 2597056*x^5 + 3317104*x^4 - 1586360*x^3 + 344740*x^2 - 27105*x - 139, 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 - 6*x^19 - 49*x^18 + 394*x^17 + 431*x^16 - 8836*x^15 + 11137*x^14 + 75642*x^13 - 215641*x^12 - 135520*x^11 + 1183051*x^10 - 1032083*x^9 - 1892289*x^8 + 4095435*x^7 - 1533360*x^6 - 2597056*x^5 + 3317104*x^4 - 1586360*x^3 + 344740*x^2 - 27105*x - 139); 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 - 6*x^19 - 49*x^18 + 394*x^17 + 431*x^16 - 8836*x^15 + 11137*x^14 + 75642*x^13 - 215641*x^12 - 135520*x^11 + 1183051*x^10 - 1032083*x^9 - 1892289*x^8 + 4095435*x^7 - 1533360*x^6 - 2597056*x^5 + 3317104*x^4 - 1586360*x^3 + 344740*x^2 - 27105*x - 139); 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

$C_4\times D_5$ (as 20T6):

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 40
The 16 conjugacy class representatives for $C_4\times D_5$
Character table for $C_4\times D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{70 -14 \sqrt{5}})\), 5.5.23668225.1, 10.10.2800924373253125.1

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

Galois closure: deg 40
Degree 20 sibling: deg 20
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 $20$ ${\href{/padicField/3.4.0.1}{4} }^{5}$ R R ${\href{/padicField/11.5.0.1}{5} }^{4}$ $20$ ${\href{/padicField/17.4.0.1}{4} }^{5}$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{10}$ ${\href{/padicField/37.4.0.1}{4} }^{5}$ ${\href{/padicField/41.2.0.1}{2} }^{10}$ $20$ $20$ $20$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

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
\(5\) Copy content Toggle raw display 5.1.4.3a1.4$x^{4} + 20$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(7\) Copy content Toggle raw display 7.2.2.2a1.1$x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
7.4.2.4a1.2$x^{8} + 10 x^{6} + 8 x^{5} + 31 x^{4} + 40 x^{3} + 46 x^{2} + 24 x + 16$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
7.4.2.4a1.2$x^{8} + 10 x^{6} + 8 x^{5} + 31 x^{4} + 40 x^{3} + 46 x^{2} + 24 x + 16$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(139\) Copy content Toggle raw display $\Q_{139}$$x + 137$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{139}$$x + 137$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{139}$$x + 137$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{139}$$x + 137$$1$$1$$0$Trivial$$[\ ]$$
139.1.2.1a1.2$x^{2} + 278$$2$$1$$1$$C_2$$$[\ ]_{2}$$
139.1.2.1a1.2$x^{2} + 278$$2$$1$$1$$C_2$$$[\ ]_{2}$$
139.1.2.1a1.2$x^{2} + 278$$2$$1$$1$$C_2$$$[\ ]_{2}$$
139.1.2.1a1.2$x^{2} + 278$$2$$1$$1$$C_2$$$[\ ]_{2}$$
139.1.2.1a1.2$x^{2} + 278$$2$$1$$1$$C_2$$$[\ ]_{2}$$
139.1.2.1a1.2$x^{2} + 278$$2$$1$$1$$C_2$$$[\ ]_{2}$$
139.1.2.1a1.2$x^{2} + 278$$2$$1$$1$$C_2$$$[\ ]_{2}$$
139.1.2.1a1.2$x^{2} + 278$$2$$1$$1$$C_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)