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Properties

Label 20.0.404...016.3
Degree $20$
Signature $[0, 10]$
Discriminant $4.043\times 10^{39}$
Root discriminant \(95.57\)
Ramified primes $2,3,41$
Class number $5$ (GRH)
Class group [5] (GRH)
Galois group $D_{10}$ (as 20T4)

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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 - 18*x^18 + 55*x^16 + 14*x^14 + 14381*x^12 - 190396*x^10 + 1133340*x^8 - 7116528*x^6 + 79490576*x^4 - 730931200*x^2 + 2621440000)
 
Copy content gp:K = bnfinit(y^20 - 18*y^18 + 55*y^16 + 14*y^14 + 14381*y^12 - 190396*y^10 + 1133340*y^8 - 7116528*y^6 + 79490576*y^4 - 730931200*y^2 + 2621440000, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 18*x^18 + 55*x^16 + 14*x^14 + 14381*x^12 - 190396*x^10 + 1133340*x^8 - 7116528*x^6 + 79490576*x^4 - 730931200*x^2 + 2621440000);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 18*x^18 + 55*x^16 + 14*x^14 + 14381*x^12 - 190396*x^10 + 1133340*x^8 - 7116528*x^6 + 79490576*x^4 - 730931200*x^2 + 2621440000)
 

\( x^{20} - 18 x^{18} + 55 x^{16} + 14 x^{14} + 14381 x^{12} - 190396 x^{10} + 1133340 x^{8} + \cdots + 2621440000 \) 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:   \(4042538127064933401151627374468337238016\) \(\medspace = 2^{30}\cdot 3^{10}\cdot 41^{16}\) 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:  \(95.57\)
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:  $2^{3/2}3^{1/2}41^{4/5}\approx 95.57244712596375$
Ramified primes:   \(2\), \(3\), \(41\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $D_{10}$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{-2}, \sqrt{-3})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{40}a^{11}-\frac{1}{8}a^{10}+\frac{1}{10}a^{9}-\frac{1}{8}a^{8}+\frac{3}{20}a^{7}-\frac{1}{8}a^{6}+\frac{9}{40}a^{5}+\frac{3}{20}a^{3}+\frac{1}{10}a$, $\frac{1}{40}a^{12}-\frac{1}{40}a^{10}-\frac{1}{8}a^{9}+\frac{1}{40}a^{8}+\frac{1}{8}a^{7}+\frac{1}{10}a^{6}+\frac{1}{8}a^{5}+\frac{3}{20}a^{4}-\frac{1}{4}a^{3}-\frac{2}{5}a^{2}-\frac{1}{2}a$, $\frac{1}{320}a^{13}-\frac{1}{8}a^{10}-\frac{1}{64}a^{9}-\frac{1}{16}a^{7}-\frac{1}{4}a^{6}-\frac{3}{64}a^{5}-\frac{1}{8}a^{4}-\frac{1}{32}a^{3}+\frac{1}{4}a^{2}-\frac{39}{80}a$, $\frac{1}{1600}a^{14}+\frac{1}{200}a^{12}-\frac{53}{1600}a^{10}-\frac{1}{8}a^{9}+\frac{47}{400}a^{8}+\frac{1}{8}a^{7}-\frac{263}{1600}a^{6}+\frac{1}{8}a^{5}-\frac{181}{800}a^{4}+\frac{1}{4}a^{3}-\frac{111}{400}a^{2}$, $\frac{1}{3200}a^{15}-\frac{1}{1600}a^{13}-\frac{13}{3200}a^{11}-\frac{1}{8}a^{10}+\frac{199}{1600}a^{9}-\frac{623}{3200}a^{7}-\frac{163}{800}a^{5}+\frac{1}{8}a^{4}-\frac{113}{400}a^{3}+\frac{3}{80}a$, $\frac{1}{956800}a^{16}-\frac{57}{239200}a^{14}-\frac{257}{73600}a^{12}-\frac{11113}{119600}a^{10}+\frac{62969}{956800}a^{8}-\frac{1}{4}a^{7}-\frac{8647}{478400}a^{6}-\frac{1}{4}a^{5}-\frac{50253}{239200}a^{4}+\frac{1}{4}a^{3}-\frac{16701}{59800}a^{2}+\frac{24}{299}$, $\frac{1}{1913600}a^{17}-\frac{57}{478400}a^{15}+\frac{203}{147200}a^{13}+\frac{847}{239200}a^{11}-\frac{62611}{1913600}a^{9}+\frac{218593}{956800}a^{7}-\frac{1}{4}a^{6}+\frac{11501}{239200}a^{5}-\frac{1}{4}a^{4}-\frac{88717}{239200}a^{3}-\frac{1}{4}a^{2}-\frac{5917}{23920}a$, $\frac{1}{19\cdots 00}a^{18}+\frac{13\cdots 31}{76\cdots 00}a^{16}+\frac{60\cdots 27}{19\cdots 00}a^{14}-\frac{25\cdots 03}{42\cdots 00}a^{12}+\frac{65\cdots 73}{19\cdots 00}a^{10}-\frac{1}{8}a^{9}-\frac{34\cdots 47}{49\cdots 00}a^{8}+\frac{1}{8}a^{7}-\frac{55\cdots 17}{49\cdots 00}a^{6}+\frac{1}{8}a^{5}-\frac{17\cdots 79}{12\cdots 00}a^{4}-\frac{1}{4}a^{3}-\frac{18\cdots 39}{12\cdots 00}a^{2}-\frac{1}{2}a-\frac{34\cdots 29}{96\cdots 99}$, $\frac{1}{15\cdots 00}a^{19}-\frac{16\cdots 09}{79\cdots 00}a^{17}+\frac{10\cdots 11}{31\cdots 00}a^{15}+\frac{14\cdots 07}{79\cdots 00}a^{13}+\frac{71\cdots 81}{15\cdots 00}a^{11}-\frac{80\cdots 99}{39\cdots 00}a^{9}-\frac{12\cdots 21}{60\cdots 00}a^{7}-\frac{1}{4}a^{6}+\frac{62\cdots 17}{98\cdots 00}a^{5}-\frac{1}{4}a^{4}-\frac{31\cdots 39}{98\cdots 00}a^{3}-\frac{1}{4}a^{2}-\frac{48\cdots 61}{33\cdots 40}a$ 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:  $C_{5}$, which has order $5$ (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:  $C_{5}$, which has order $5$ (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:  $9$
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:   \( -\frac{818741277135451}{6610186581430046924800} a^{18} + \frac{4078724557837619}{3305093290715023462400} a^{16} + \frac{4240352965344791}{1322037316286009384960} a^{14} + \frac{75549284469326723}{3305093290715023462400} a^{12} - \frac{10579215681920276991}{6610186581430046924800} a^{10} + \frac{17627696584499532789}{1652546645357511731200} a^{8} - \frac{3511536789814515269}{66101865814300469248} a^{6} + \frac{183521130919604112373}{413136661339377932800} a^{4} - \frac{2541738577728122958331}{413136661339377932800} a^{2} + \frac{1312142301878950400}{32276301667138901} \)  (order $6$) 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:   $\frac{2562571401}{35\cdots 00}a^{19}+\frac{818741277135451}{33\cdots 00}a^{18}-\frac{545632343}{773566963712000}a^{17}-\frac{40\cdots 19}{16\cdots 00}a^{16}-\frac{13434165789}{71\cdots 00}a^{15}-\frac{42\cdots 91}{66\cdots 80}a^{14}-\frac{255709940033}{17\cdots 00}a^{13}-\frac{75\cdots 23}{16\cdots 00}a^{12}+\frac{32964938455061}{35\cdots 00}a^{11}+\frac{10\cdots 91}{33\cdots 00}a^{10}-\frac{54687972297319}{88\cdots 00}a^{9}-\frac{17\cdots 89}{82\cdots 00}a^{8}+\frac{54848591725491}{17\cdots 00}a^{7}+\frac{35\cdots 69}{33\cdots 24}a^{6}-\frac{577650678019303}{22\cdots 00}a^{5}-\frac{18\cdots 73}{20\cdots 00}a^{4}+\frac{79\cdots 41}{22\cdots 00}a^{3}+\frac{25\cdots 31}{20\cdots 00}a^{2}-\frac{50637507680}{2171879903}a-\frac{25\cdots 99}{32\cdots 01}$, $\frac{71\cdots 51}{12\cdots 00}a^{19}+\frac{10\cdots 47}{98\cdots 00}a^{18}-\frac{18\cdots 73}{26\cdots 00}a^{17}-\frac{47\cdots 59}{49\cdots 00}a^{16}+\frac{35\cdots 01}{24\cdots 00}a^{15}+\frac{16\cdots 21}{98\cdots 00}a^{14}-\frac{92\cdots 03}{60\cdots 00}a^{13}+\frac{11\cdots 33}{98\cdots 60}a^{12}+\frac{96\cdots 51}{12\cdots 00}a^{11}+\frac{28\cdots 71}{39\cdots 04}a^{10}-\frac{33\cdots 29}{30\cdots 00}a^{9}-\frac{10\cdots 93}{49\cdots 80}a^{8}+\frac{15\cdots 73}{60\cdots 00}a^{7}+\frac{17\cdots 57}{24\cdots 00}a^{6}-\frac{15\cdots 13}{76\cdots 00}a^{5}-\frac{38\cdots 77}{61\cdots 00}a^{4}+\frac{39\cdots 31}{76\cdots 00}a^{3}+\frac{65\cdots 99}{61\cdots 00}a^{2}-\frac{16\cdots 80}{74\cdots 23}a-\frac{53\cdots 71}{96\cdots 99}$, $\frac{79\cdots 53}{15\cdots 00}a^{19}-\frac{14\cdots 41}{98\cdots 00}a^{18}-\frac{40\cdots 17}{79\cdots 00}a^{17}+\frac{15\cdots 53}{98\cdots 60}a^{16}-\frac{36\cdots 61}{31\cdots 00}a^{15}+\frac{19\cdots 17}{98\cdots 00}a^{14}-\frac{81\cdots 29}{79\cdots 00}a^{13}+\frac{14\cdots 09}{49\cdots 00}a^{12}+\frac{79\cdots 61}{12\cdots 00}a^{11}-\frac{18\cdots 13}{98\cdots 00}a^{10}-\frac{17\cdots 47}{39\cdots 00}a^{9}+\frac{34\cdots 47}{24\cdots 00}a^{8}+\frac{17\cdots 51}{79\cdots 00}a^{7}-\frac{15\cdots 71}{19\cdots 00}a^{6}-\frac{18\cdots 19}{98\cdots 00}a^{5}+\frac{40\cdots 03}{61\cdots 00}a^{4}+\frac{24\cdots 53}{98\cdots 00}a^{3}-\frac{92\cdots 41}{12\cdots 20}a^{2}-\frac{12\cdots 03}{77\cdots 20}a+\frac{44\cdots 69}{96\cdots 99}$, $\frac{17\cdots 57}{15\cdots 00}a^{19}-\frac{18\cdots 43}{19\cdots 00}a^{18}-\frac{26\cdots 21}{60\cdots 00}a^{17}-\frac{58\cdots 89}{98\cdots 00}a^{16}+\frac{22\cdots 99}{31\cdots 00}a^{15}+\frac{13\cdots 23}{79\cdots 08}a^{14}+\frac{50\cdots 59}{79\cdots 00}a^{13}-\frac{10\cdots 49}{98\cdots 00}a^{12}-\frac{58\cdots 03}{15\cdots 00}a^{11}+\frac{97\cdots 73}{19\cdots 00}a^{10}-\frac{14\cdots 63}{39\cdots 00}a^{9}-\frac{42\cdots 67}{49\cdots 00}a^{8}+\frac{54\cdots 43}{79\cdots 00}a^{7}+\frac{48\cdots 19}{49\cdots 00}a^{6}-\frac{30\cdots 91}{98\cdots 00}a^{5}-\frac{42\cdots 91}{24\cdots 40}a^{4}-\frac{30\cdots 03}{98\cdots 00}a^{3}+\frac{12\cdots 73}{12\cdots 00}a^{2}+\frac{92\cdots 63}{19\cdots 80}a-\frac{16\cdots 28}{96\cdots 99}$, $\frac{71\cdots 51}{12\cdots 00}a^{19}-\frac{10\cdots 47}{98\cdots 00}a^{18}-\frac{18\cdots 73}{26\cdots 00}a^{17}+\frac{47\cdots 59}{49\cdots 00}a^{16}+\frac{35\cdots 01}{24\cdots 00}a^{15}-\frac{16\cdots 21}{98\cdots 00}a^{14}-\frac{92\cdots 03}{60\cdots 00}a^{13}-\frac{11\cdots 33}{98\cdots 60}a^{12}+\frac{96\cdots 51}{12\cdots 00}a^{11}-\frac{28\cdots 71}{39\cdots 04}a^{10}-\frac{33\cdots 29}{30\cdots 00}a^{9}+\frac{10\cdots 93}{49\cdots 80}a^{8}+\frac{15\cdots 73}{60\cdots 00}a^{7}-\frac{17\cdots 57}{24\cdots 00}a^{6}-\frac{15\cdots 13}{76\cdots 00}a^{5}+\frac{38\cdots 77}{61\cdots 00}a^{4}+\frac{39\cdots 31}{76\cdots 00}a^{3}-\frac{65\cdots 99}{61\cdots 00}a^{2}-\frac{16\cdots 80}{74\cdots 23}a+\frac{53\cdots 71}{96\cdots 99}$, $\frac{74\cdots 81}{31\cdots 00}a^{19}+\frac{16\cdots 01}{24\cdots 00}a^{18}-\frac{75\cdots 53}{31\cdots 20}a^{17}-\frac{81\cdots 09}{12\cdots 00}a^{16}-\frac{71\cdots 61}{12\cdots 28}a^{15}-\frac{17\cdots 51}{10\cdots 00}a^{14}-\frac{67\cdots 41}{15\cdots 00}a^{13}-\frac{30\cdots 01}{24\cdots 40}a^{12}+\frac{95\cdots 77}{31\cdots 00}a^{11}+\frac{42\cdots 13}{49\cdots 80}a^{10}-\frac{16\cdots 83}{79\cdots 00}a^{9}-\frac{70\cdots 53}{12\cdots 20}a^{8}+\frac{83\cdots 99}{79\cdots 00}a^{7}+\frac{18\cdots 51}{61\cdots 00}a^{6}-\frac{73\cdots 69}{85\cdots 00}a^{5}-\frac{29\cdots 49}{11\cdots 00}a^{4}+\frac{23\cdots 17}{19\cdots 00}a^{3}+\frac{50\cdots 13}{15\cdots 00}a^{2}-\frac{61\cdots 39}{77\cdots 20}a-\frac{21\cdots 57}{96\cdots 99}$, $\frac{24\cdots 37}{63\cdots 40}a^{19}+\frac{14\cdots 23}{19\cdots 00}a^{18}-\frac{66\cdots 13}{15\cdots 00}a^{17}-\frac{59\cdots 39}{98\cdots 00}a^{16}-\frac{26\cdots 21}{31\cdots 00}a^{15}-\frac{45\cdots 11}{15\cdots 00}a^{14}-\frac{54\cdots 33}{15\cdots 00}a^{13}-\frac{20\cdots 27}{98\cdots 00}a^{12}+\frac{12\cdots 77}{24\cdots 00}a^{11}+\frac{17\cdots 19}{19\cdots 00}a^{10}-\frac{29\cdots 39}{79\cdots 00}a^{9}-\frac{23\cdots 81}{49\cdots 00}a^{8}+\frac{12\cdots 99}{79\cdots 00}a^{7}+\frac{12\cdots 09}{49\cdots 00}a^{6}-\frac{10\cdots 91}{79\cdots 08}a^{5}-\frac{23\cdots 61}{12\cdots 00}a^{4}+\frac{36\cdots 17}{19\cdots 00}a^{3}+\frac{39\cdots 67}{12\cdots 00}a^{2}-\frac{42\cdots 13}{38\cdots 60}a-\frac{18\cdots 68}{74\cdots 23}$, $\frac{71\cdots 51}{15\cdots 00}a^{19}+\frac{27\cdots 27}{19\cdots 00}a^{18}-\frac{24\cdots 79}{79\cdots 00}a^{17}-\frac{22\cdots 69}{42\cdots 00}a^{16}-\frac{15\cdots 11}{31\cdots 00}a^{15}-\frac{23\cdots 51}{19\cdots 00}a^{14}-\frac{29\cdots 43}{79\cdots 00}a^{13}-\frac{12\cdots 67}{98\cdots 00}a^{12}+\frac{12\cdots 31}{15\cdots 00}a^{11}+\frac{33\cdots 59}{19\cdots 00}a^{10}-\frac{18\cdots 49}{39\cdots 00}a^{9}-\frac{44\cdots 21}{49\cdots 00}a^{8}+\frac{20\cdots 09}{60\cdots 00}a^{7}+\frac{28\cdots 93}{49\cdots 00}a^{6}-\frac{25\cdots 93}{98\cdots 00}a^{5}-\frac{13\cdots 29}{24\cdots 40}a^{4}+\frac{28\cdots 71}{98\cdots 00}a^{3}+\frac{21\cdots 91}{24\cdots 40}a^{2}-\frac{11\cdots 97}{96\cdots 90}a-\frac{38\cdots 17}{96\cdots 99}$, $\frac{17\cdots 57}{15\cdots 00}a^{19}+\frac{18\cdots 43}{19\cdots 00}a^{18}-\frac{26\cdots 21}{60\cdots 00}a^{17}+\frac{58\cdots 89}{98\cdots 00}a^{16}+\frac{22\cdots 99}{31\cdots 00}a^{15}-\frac{13\cdots 23}{79\cdots 08}a^{14}+\frac{50\cdots 59}{79\cdots 00}a^{13}+\frac{10\cdots 49}{98\cdots 00}a^{12}-\frac{58\cdots 03}{15\cdots 00}a^{11}-\frac{97\cdots 73}{19\cdots 00}a^{10}-\frac{14\cdots 63}{39\cdots 00}a^{9}+\frac{42\cdots 67}{49\cdots 00}a^{8}+\frac{54\cdots 43}{79\cdots 00}a^{7}-\frac{48\cdots 19}{49\cdots 00}a^{6}-\frac{30\cdots 91}{98\cdots 00}a^{5}+\frac{42\cdots 91}{24\cdots 40}a^{4}-\frac{30\cdots 03}{98\cdots 00}a^{3}-\frac{12\cdots 73}{12\cdots 00}a^{2}+\frac{92\cdots 63}{19\cdots 80}a+\frac{16\cdots 28}{96\cdots 99}$ 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:  \( 739275400192 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 739275400192 \cdot 5}{6\cdot\sqrt{4042538127064933401151627374468337238016}}\cr\approx \mathstrut & 0.929173104237558 \end{aligned}\] (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 - 18*x^18 + 55*x^16 + 14*x^14 + 14381*x^12 - 190396*x^10 + 1133340*x^8 - 7116528*x^6 + 79490576*x^4 - 730931200*x^2 + 2621440000) 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 - 18*x^18 + 55*x^16 + 14*x^14 + 14381*x^12 - 190396*x^10 + 1133340*x^8 - 7116528*x^6 + 79490576*x^4 - 730931200*x^2 + 2621440000, 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 - 18*x^18 + 55*x^16 + 14*x^14 + 14381*x^12 - 190396*x^10 + 1133340*x^8 - 7116528*x^6 + 79490576*x^4 - 730931200*x^2 + 2621440000); 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 - 18*x^18 + 55*x^16 + 14*x^14 + 14381*x^12 - 190396*x^10 + 1133340*x^8 - 7116528*x^6 + 79490576*x^4 - 730931200*x^2 + 2621440000); 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

$D_{10}$ (as 20T4):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), 5.1.180848704.2 x5, 10.0.261650029907836928.2, 10.2.63580957267604373504.1 x5, 10.0.7947619658450546688.1 x5

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 10 siblings: 10.2.63580957267604373504.1, 10.0.7947619658450546688.1
Minimal sibling: 10.0.7947619658450546688.1

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 ${\href{/padicField/5.2.0.1}{2} }^{10}$ ${\href{/padicField/7.2.0.1}{2} }^{10}$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{10}$ ${\href{/padicField/17.2.0.1}{2} }^{10}$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{10}$ ${\href{/padicField/29.2.0.1}{2} }^{10}$ ${\href{/padicField/31.2.0.1}{2} }^{10}$ ${\href{/padicField/37.2.0.1}{2} }^{10}$ R ${\href{/padicField/43.5.0.1}{5} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{10}$ ${\href{/padicField/53.2.0.1}{2} }^{10}$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

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 2.2.2.6a1.1$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$$2$$2$$6$$C_2^2$$$[3]^{2}$$
2.2.2.6a1.1$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$$2$$2$$6$$C_2^2$$$[3]^{2}$$
2.2.2.6a1.1$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$$2$$2$$6$$C_2^2$$$[3]^{2}$$
2.2.2.6a1.1$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$$2$$2$$6$$C_2^2$$$[3]^{2}$$
2.2.2.6a1.1$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$$2$$2$$6$$C_2^2$$$[3]^{2}$$
\(3\) Copy content Toggle raw display 3.5.2.5a1.2$x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 4 x + 4$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
3.5.2.5a1.2$x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 4 x + 4$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
\(41\) Copy content Toggle raw display 41.2.5.8a1.1$x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$$5$$2$$8$$C_{10}$$$[\ ]_{5}^{2}$$
41.2.5.8a1.1$x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$$5$$2$$8$$C_{10}$$$[\ ]_{5}^{2}$$

Spectrum of ring of integers

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