Properties

Label 20.0.148...000.2
Degree $20$
Signature $(0, 10)$
Discriminant $1.482\times 10^{38}$
Root discriminant \(81.01\)
Ramified primes $2,3,5,173$
Class number $1968$ (GRH)
Class group [2, 2, 492] (GRH)
Galois group $D_5^2.D_4$ (as 20T157)

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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 - 8*x^19 + 66*x^18 - 232*x^17 + 967*x^16 - 2032*x^15 + 7288*x^14 - 12336*x^13 + 41090*x^12 - 56096*x^11 + 250108*x^10 - 594912*x^9 + 1890886*x^8 - 3051984*x^7 + 5344280*x^6 - 5193232*x^5 + 8220077*x^4 - 6438936*x^3 + 11123186*x^2 - 2353656*x + 1425475)
 
Copy content gp:K = bnfinit(y^20 - 8*y^19 + 66*y^18 - 232*y^17 + 967*y^16 - 2032*y^15 + 7288*y^14 - 12336*y^13 + 41090*y^12 - 56096*y^11 + 250108*y^10 - 594912*y^9 + 1890886*y^8 - 3051984*y^7 + 5344280*y^6 - 5193232*y^5 + 8220077*y^4 - 6438936*y^3 + 11123186*y^2 - 2353656*y + 1425475, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 8*x^19 + 66*x^18 - 232*x^17 + 967*x^16 - 2032*x^15 + 7288*x^14 - 12336*x^13 + 41090*x^12 - 56096*x^11 + 250108*x^10 - 594912*x^9 + 1890886*x^8 - 3051984*x^7 + 5344280*x^6 - 5193232*x^5 + 8220077*x^4 - 6438936*x^3 + 11123186*x^2 - 2353656*x + 1425475);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 8*x^19 + 66*x^18 - 232*x^17 + 967*x^16 - 2032*x^15 + 7288*x^14 - 12336*x^13 + 41090*x^12 - 56096*x^11 + 250108*x^10 - 594912*x^9 + 1890886*x^8 - 3051984*x^7 + 5344280*x^6 - 5193232*x^5 + 8220077*x^4 - 6438936*x^3 + 11123186*x^2 - 2353656*x + 1425475)
 

\( x^{20} - 8 x^{19} + 66 x^{18} - 232 x^{17} + 967 x^{16} - 2032 x^{15} + 7288 x^{14} - 12336 x^{13} + \cdots + 1425475 \) 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:  $(0, 10)$
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:   \(148184566209122179801351864839045120000\) \(\medspace = 2^{64}\cdot 3^{15}\cdot 5^{4}\cdot 173^{4}\) 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:  \(81.01\)
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:  $2^{27/8}3^{3/4}5^{1/2}173^{1/2}\approx 695.5449954322967$
Ramified primes:   \(2\), \(3\), \(5\), \(173\) 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:  \(\Q(\sqrt{3}) \)
$\Aut(K/\Q)$:   $C_2$
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 a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{3}+\frac{1}{8}a^{2}+\frac{1}{8}a+\frac{1}{8}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{8}-\frac{1}{8}a^{4}+\frac{1}{8}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{9}-\frac{1}{8}a^{5}+\frac{1}{8}a$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{12}-\frac{1}{16}a^{10}+\frac{1}{16}a^{8}-\frac{1}{16}a^{6}+\frac{1}{16}a^{4}+\frac{1}{16}a^{2}-\frac{1}{16}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{13}-\frac{1}{16}a^{11}+\frac{1}{16}a^{9}-\frac{1}{16}a^{7}+\frac{1}{16}a^{5}+\frac{1}{16}a^{3}-\frac{1}{16}a$, $\frac{1}{16}a^{16}-\frac{1}{8}a^{8}+\frac{1}{16}$, $\frac{1}{160}a^{17}-\frac{1}{32}a^{16}-\frac{1}{80}a^{15}-\frac{1}{80}a^{14}-\frac{1}{20}a^{13}+\frac{1}{20}a^{12}-\frac{3}{80}a^{11}-\frac{3}{80}a^{10}-\frac{1}{80}a^{9}+\frac{1}{16}a^{8}-\frac{11}{80}a^{7}+\frac{1}{16}a^{6}+\frac{1}{5}a^{5}-\frac{33}{80}a^{3}+\frac{31}{80}a^{2}+\frac{41}{160}a-\frac{9}{32}$, $\frac{1}{480}a^{18}-\frac{1}{480}a^{17}-\frac{1}{240}a^{16}+\frac{1}{48}a^{15}+\frac{1}{120}a^{14}-\frac{1}{120}a^{13}+\frac{13}{240}a^{12}-\frac{1}{16}a^{11}+\frac{9}{80}a^{10}-\frac{19}{240}a^{9}+\frac{19}{240}a^{8}-\frac{49}{240}a^{7}-\frac{9}{40}a^{6}-\frac{13}{120}a^{5}+\frac{47}{240}a^{4}+\frac{59}{240}a^{3}+\frac{209}{480}a^{2}-\frac{161}{480}a+\frac{5}{24}$, $\frac{1}{11\cdots 00}a^{19}+\frac{10\cdots 39}{57\cdots 00}a^{18}+\frac{24\cdots 29}{11\cdots 00}a^{17}-\frac{14\cdots 59}{57\cdots 00}a^{16}+\frac{81\cdots 67}{57\cdots 00}a^{15}-\frac{14\cdots 29}{57\cdots 00}a^{14}-\frac{38\cdots 63}{11\cdots 40}a^{13}-\frac{20\cdots 81}{19\cdots 00}a^{12}-\frac{81\cdots 99}{47\cdots 00}a^{11}+\frac{21\cdots 59}{57\cdots 00}a^{10}+\frac{71\cdots 19}{28\cdots 00}a^{9}+\frac{39\cdots 17}{57\cdots 00}a^{8}+\frac{19\cdots 63}{76\cdots 96}a^{7}-\frac{30\cdots 87}{57\cdots 00}a^{6}+\frac{11\cdots 63}{57\cdots 00}a^{5}-\frac{54\cdots 11}{25\cdots 00}a^{4}+\frac{19\cdots 31}{11\cdots 00}a^{3}-\frac{27\cdots 39}{57\cdots 20}a^{2}-\frac{46\cdots 49}{11\cdots 00}a+\frac{18\cdots 07}{12\cdots 48}$ 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:  $C_{2}\times C_{2}\times C_{492}$, which has order $1968$ (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:  $C_{2}\times C_{2}\times C_{492}$, which has order $1968$ (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);
 
Relative class number:   $1968$ (assuming GRH)

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:  $9$
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:   $\frac{24\cdots 11}{25\cdots 00}a^{19}-\frac{81\cdots 53}{16\cdots 00}a^{18}+\frac{94\cdots 87}{20\cdots 00}a^{17}-\frac{30\cdots 71}{50\cdots 00}a^{16}+\frac{11\cdots 99}{25\cdots 00}a^{15}+\frac{18\cdots 29}{83\cdots 00}a^{14}+\frac{18\cdots 59}{50\cdots 80}a^{13}+\frac{93\cdots 79}{25\cdots 00}a^{12}+\frac{18\cdots 13}{83\cdots 00}a^{11}+\frac{80\cdots 31}{31\cdots 50}a^{10}+\frac{12\cdots 87}{83\cdots 00}a^{9}-\frac{20\cdots 23}{20\cdots 00}a^{8}+\frac{70\cdots 25}{10\cdots 56}a^{7}+\frac{11\cdots 61}{25\cdots 00}a^{6}+\frac{26\cdots 87}{83\cdots 00}a^{5}+\frac{21\cdots 09}{25\cdots 00}a^{4}+\frac{16\cdots 61}{41\cdots 00}a^{3}+\frac{14\cdots 21}{33\cdots 20}a^{2}-\frac{56\cdots 89}{25\cdots 00}a+\frac{16\cdots 11}{66\cdots 12}$, $\frac{22\cdots 43}{50\cdots 00}a^{19}-\frac{27\cdots 67}{16\cdots 00}a^{18}+\frac{24\cdots 79}{16\cdots 00}a^{17}+\frac{11\cdots 11}{50\cdots 00}a^{16}-\frac{45\cdots 51}{62\cdots 00}a^{15}+\frac{88\cdots 61}{83\cdots 00}a^{14}-\frac{36\cdots 73}{50\cdots 28}a^{13}+\frac{25\cdots 51}{25\cdots 00}a^{12}-\frac{41\cdots 13}{83\cdots 00}a^{11}+\frac{83\cdots 61}{12\cdots 00}a^{10}+\frac{10\cdots 23}{83\cdots 00}a^{9}+\frac{12\cdots 81}{41\cdots 00}a^{8}-\frac{25\cdots 67}{12\cdots 20}a^{7}+\frac{66\cdots 29}{25\cdots 00}a^{6}-\frac{15\cdots 21}{41\cdots 00}a^{5}+\frac{26\cdots 61}{25\cdots 00}a^{4}-\frac{12\cdots 39}{16\cdots 00}a^{3}+\frac{53\cdots 79}{33\cdots 20}a^{2}-\frac{16\cdots 67}{50\cdots 00}a+\frac{12\cdots 41}{66\cdots 12}$, $\frac{15\cdots 37}{50\cdots 00}a^{19}+\frac{60\cdots 31}{83\cdots 00}a^{18}+\frac{61\cdots 51}{16\cdots 00}a^{17}+\frac{19\cdots 67}{25\cdots 00}a^{16}+\frac{68\cdots 49}{25\cdots 00}a^{15}+\frac{24\cdots 31}{20\cdots 00}a^{14}+\frac{86\cdots 83}{10\cdots 56}a^{13}+\frac{63\cdots 01}{62\cdots 00}a^{12}+\frac{24\cdots 09}{41\cdots 00}a^{11}+\frac{59\cdots 59}{12\cdots 00}a^{10}+\frac{46\cdots 23}{20\cdots 00}a^{9}+\frac{10\cdots 59}{41\cdots 00}a^{8}-\frac{13\cdots 41}{50\cdots 80}a^{7}+\frac{67\cdots 89}{62\cdots 00}a^{6}-\frac{12\cdots 13}{83\cdots 00}a^{5}+\frac{20\cdots 41}{62\cdots 00}a^{4}-\frac{58\cdots 11}{16\cdots 00}a^{3}+\frac{16\cdots 19}{33\cdots 52}a^{2}-\frac{56\cdots 33}{50\cdots 00}a+\frac{33\cdots 79}{33\cdots 56}$, $\frac{11\cdots 67}{50\cdots 00}a^{19}-\frac{25\cdots 13}{16\cdots 00}a^{18}+\frac{20\cdots 71}{16\cdots 00}a^{17}-\frac{16\cdots 01}{50\cdots 00}a^{16}+\frac{15\cdots 17}{12\cdots 00}a^{15}-\frac{11\cdots 01}{83\cdots 00}a^{14}+\frac{85\cdots 49}{12\cdots 20}a^{13}-\frac{10\cdots 31}{25\cdots 00}a^{12}+\frac{24\cdots 13}{83\cdots 00}a^{11}+\frac{60\cdots 79}{12\cdots 00}a^{10}+\frac{20\cdots 17}{83\cdots 00}a^{9}-\frac{33\cdots 77}{52\cdots 75}a^{8}+\frac{42\cdots 23}{25\cdots 40}a^{7}-\frac{36\cdots 69}{25\cdots 00}a^{6}-\frac{61\cdots 57}{20\cdots 00}a^{5}+\frac{23\cdots 19}{25\cdots 00}a^{4}-\frac{17\cdots 91}{16\cdots 00}a^{3}+\frac{45\cdots 81}{33\cdots 20}a^{2}-\frac{15\cdots 63}{50\cdots 00}a+\frac{39\cdots 13}{66\cdots 12}$, $\frac{50\cdots 87}{12\cdots 20}a^{19}-\frac{37\cdots 03}{83\cdots 80}a^{18}+\frac{33\cdots 02}{10\cdots 35}a^{17}-\frac{36\cdots 19}{25\cdots 64}a^{16}+\frac{26\cdots 81}{62\cdots 10}a^{15}-\frac{10\cdots 01}{83\cdots 80}a^{14}+\frac{62\cdots 52}{31\cdots 05}a^{13}-\frac{20\cdots 67}{25\cdots 40}a^{12}+\frac{55\cdots 83}{83\cdots 88}a^{11}-\frac{10\cdots 73}{25\cdots 40}a^{10}+\frac{25\cdots 81}{83\cdots 88}a^{9}-\frac{34\cdots 09}{83\cdots 80}a^{8}+\frac{18\cdots 67}{31\cdots 05}a^{7}-\frac{40\cdots 33}{25\cdots 40}a^{6}-\frac{71\cdots 51}{20\cdots 70}a^{5}-\frac{11\cdots 73}{25\cdots 40}a^{4}-\frac{20\cdots 47}{41\cdots 94}a^{3}-\frac{10\cdots 67}{41\cdots 40}a^{2}-\frac{19\cdots 17}{12\cdots 20}a-\frac{47\cdots 51}{16\cdots 28}$, $\frac{24\cdots 07}{25\cdots 00}a^{19}+\frac{62\cdots 11}{41\cdots 00}a^{18}-\frac{52\cdots 39}{83\cdots 00}a^{17}+\frac{10\cdots 67}{12\cdots 00}a^{16}-\frac{73\cdots 03}{62\cdots 00}a^{15}+\frac{43\cdots 29}{41\cdots 00}a^{14}-\frac{11\cdots 31}{62\cdots 10}a^{13}+\frac{25\cdots 21}{31\cdots 50}a^{12}+\frac{29\cdots 53}{41\cdots 00}a^{11}+\frac{51\cdots 53}{12\cdots 00}a^{10}+\frac{39\cdots 87}{41\cdots 00}a^{9}+\frac{58\cdots 29}{20\cdots 00}a^{8}-\frac{43\cdots 43}{12\cdots 20}a^{7}+\frac{15\cdots 61}{12\cdots 00}a^{6}-\frac{40\cdots 96}{52\cdots 75}a^{5}+\frac{40\cdots 43}{15\cdots 25}a^{4}-\frac{41\cdots 91}{83\cdots 00}a^{3}+\frac{17\cdots 07}{41\cdots 40}a^{2}-\frac{17\cdots 13}{25\cdots 00}a+\frac{13\cdots 19}{16\cdots 28}$, $\frac{14\cdots 96}{52\cdots 75}a^{19}-\frac{41\cdots 73}{20\cdots 00}a^{18}+\frac{16\cdots 23}{10\cdots 50}a^{17}-\frac{37\cdots 23}{83\cdots 00}a^{16}+\frac{76\cdots 97}{41\cdots 00}a^{15}-\frac{61\cdots 67}{20\cdots 00}a^{14}+\frac{22\cdots 69}{16\cdots 76}a^{13}-\frac{79\cdots 13}{41\cdots 00}a^{12}+\frac{30\cdots 37}{41\cdots 00}a^{11}-\frac{78\cdots 99}{10\cdots 50}a^{10}+\frac{21\cdots 53}{41\cdots 00}a^{9}-\frac{12\cdots 47}{10\cdots 50}a^{8}+\frac{26\cdots 87}{83\cdots 80}a^{7}-\frac{66\cdots 91}{20\cdots 00}a^{6}+\frac{27\cdots 83}{41\cdots 00}a^{5}-\frac{43\cdots 83}{41\cdots 00}a^{4}+\frac{88\cdots 63}{41\cdots 00}a^{3}-\frac{71\cdots 99}{83\cdots 88}a^{2}+\frac{14\cdots 63}{41\cdots 00}a-\frac{17\cdots 57}{11\cdots 52}$, $\frac{19\cdots 03}{16\cdots 00}a^{19}-\frac{64\cdots 83}{83\cdots 00}a^{18}+\frac{10\cdots 87}{16\cdots 00}a^{17}-\frac{59\cdots 01}{41\cdots 00}a^{16}+\frac{49\cdots 51}{83\cdots 00}a^{15}-\frac{16\cdots 31}{41\cdots 00}a^{14}+\frac{60\cdots 01}{16\cdots 60}a^{13}-\frac{14\cdots 77}{41\cdots 00}a^{12}+\frac{72\cdots 43}{41\cdots 00}a^{11}+\frac{27\cdots 63}{20\cdots 00}a^{10}+\frac{32\cdots 41}{20\cdots 00}a^{9}-\frac{95\cdots 87}{41\cdots 00}a^{8}+\frac{27\cdots 45}{33\cdots 52}a^{7}+\frac{17\cdots 07}{41\cdots 00}a^{6}-\frac{67\cdots 61}{83\cdots 00}a^{5}+\frac{18\cdots 33}{41\cdots 00}a^{4}-\frac{26\cdots 07}{16\cdots 00}a^{3}+\frac{12\cdots 51}{16\cdots 60}a^{2}-\frac{21\cdots 47}{16\cdots 00}a+\frac{25\cdots 47}{13\cdots 94}$, $\frac{26\cdots 83}{50\cdots 80}a^{19}-\frac{59\cdots 67}{33\cdots 20}a^{18}+\frac{16\cdots 83}{83\cdots 80}a^{17}+\frac{13\cdots 31}{20\cdots 12}a^{16}+\frac{95\cdots 09}{50\cdots 80}a^{15}+\frac{15\cdots 25}{33\cdots 52}a^{14}+\frac{94\cdots 61}{50\cdots 80}a^{13}+\frac{23\cdots 07}{50\cdots 80}a^{12}+\frac{18\cdots 91}{16\cdots 60}a^{11}+\frac{64\cdots 21}{25\cdots 40}a^{10}+\frac{13\cdots 57}{16\cdots 60}a^{9}+\frac{53\cdots 51}{83\cdots 80}a^{8}+\frac{69\cdots 63}{50\cdots 80}a^{7}+\frac{29\cdots 09}{50\cdots 80}a^{6}-\frac{27\cdots 55}{33\cdots 52}a^{5}+\frac{67\cdots 29}{50\cdots 80}a^{4}+\frac{37\cdots 39}{41\cdots 40}a^{3}+\frac{19\cdots 31}{66\cdots 04}a^{2}-\frac{16\cdots 41}{50\cdots 80}a+\frac{43\cdots 37}{66\cdots 12}$ 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:  \( 255930231.626 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 255930231.626 \cdot 1968}{2\cdot\sqrt{148184566209122179801351864839045120000}}\cr\approx \mathstrut & 1.9838730838 \end{aligned}\] (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 - 8*x^19 + 66*x^18 - 232*x^17 + 967*x^16 - 2032*x^15 + 7288*x^14 - 12336*x^13 + 41090*x^12 - 56096*x^11 + 250108*x^10 - 594912*x^9 + 1890886*x^8 - 3051984*x^7 + 5344280*x^6 - 5193232*x^5 + 8220077*x^4 - 6438936*x^3 + 11123186*x^2 - 2353656*x + 1425475) 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 - 8*x^19 + 66*x^18 - 232*x^17 + 967*x^16 - 2032*x^15 + 7288*x^14 - 12336*x^13 + 41090*x^12 - 56096*x^11 + 250108*x^10 - 594912*x^9 + 1890886*x^8 - 3051984*x^7 + 5344280*x^6 - 5193232*x^5 + 8220077*x^4 - 6438936*x^3 + 11123186*x^2 - 2353656*x + 1425475, 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 - 8*x^19 + 66*x^18 - 232*x^17 + 967*x^16 - 2032*x^15 + 7288*x^14 - 12336*x^13 + 41090*x^12 - 56096*x^11 + 250108*x^10 - 594912*x^9 + 1890886*x^8 - 3051984*x^7 + 5344280*x^6 - 5193232*x^5 + 8220077*x^4 - 6438936*x^3 + 11123186*x^2 - 2353656*x + 1425475); 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 - 8*x^19 + 66*x^18 - 232*x^17 + 967*x^16 - 2032*x^15 + 7288*x^14 - 12336*x^13 + 41090*x^12 - 56096*x^11 + 250108*x^10 - 594912*x^9 + 1890886*x^8 - 3051984*x^7 + 5344280*x^6 - 5193232*x^5 + 8220077*x^4 - 6438936*x^3 + 11123186*x^2 - 2353656*x + 1425475); 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

$D_5^2.D_4$ (as 20T157):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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 800
The 26 conjugacy class representatives for $D_5^2.D_4$
Character table for $D_5^2.D_4$

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{-3 + \sqrt{6}})\), 10.10.24403289466470400.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed
Arithmetically equivalent siblings: data not computed
Minimal sibling: 20.0.148184566209122179801351864839045120000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R $20$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }^{2}$ $20$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ $20$ ${\href{/padicField/37.10.0.1}{10} }^{2}$ $20$ ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

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 2.1.4.10a1.8$x^{4} + 4 x^{3} + 12 x^{2} + 8 x + 10$$4$$1$$10$$D_{4}$$$[2, 3, \frac{7}{2}]$$
2.1.16.54o1.297$x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 14$$16$$1$$54$$C_4:C_4$$$[2, 3, \frac{7}{2}, 4]$$
\(3\) Copy content Toggle raw display 3.1.4.3a1.1$x^{4} + 3$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
3.4.4.12a1.3$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 19$$4$$4$$12$$C_4:C_4$$$[\ ]_{4}^{4}$$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.4.1.0a1.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
5.2.2.2a1.1$x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
5.4.1.0a1.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
5.2.2.2a1.1$x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
\(173\) Copy content Toggle raw display $\Q_{173}$$x + 171$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{173}$$x + 171$$1$$1$$0$Trivial$$[\ ]$$
173.2.1.0a1.1$x^{2} + 169 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
173.4.1.0a1.1$x^{4} + x^{2} + 102 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
173.4.1.0a1.1$x^{4} + x^{2} + 102 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
173.2.2.2a1.1$x^{4} + 338 x^{3} + 28565 x^{2} + 849 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
173.2.2.2a1.1$x^{4} + 338 x^{3} + 28565 x^{2} + 849 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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