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Properties

Label 20.20.176...000.1
Degree $20$
Signature $(20, 0)$
Discriminant $1.769\times 10^{43}$
Root discriminant \(145.34\)
Ramified primes $2,3,5,61$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_5^2$ (as 20T28)

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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 - 160*x^18 + 10790*x^16 - 372*x^15 - 400000*x^14 + 37080*x^13 + 8908075*x^12 - 1405800*x^11 - 122060112*x^10 + 25437000*x^9 + 1009834870*x^8 - 221247000*x^7 - 4729220200*x^6 + 784838688*x^5 + 10914530225*x^4 - 681612780*x^3 - 8806862800*x^2 + 1007795640*x + 916095316)
 
Copy content gp:K = bnfinit(y^20 - 160*y^18 + 10790*y^16 - 372*y^15 - 400000*y^14 + 37080*y^13 + 8908075*y^12 - 1405800*y^11 - 122060112*y^10 + 25437000*y^9 + 1009834870*y^8 - 221247000*y^7 - 4729220200*y^6 + 784838688*y^5 + 10914530225*y^4 - 681612780*y^3 - 8806862800*y^2 + 1007795640*y + 916095316, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 160*x^18 + 10790*x^16 - 372*x^15 - 400000*x^14 + 37080*x^13 + 8908075*x^12 - 1405800*x^11 - 122060112*x^10 + 25437000*x^9 + 1009834870*x^8 - 221247000*x^7 - 4729220200*x^6 + 784838688*x^5 + 10914530225*x^4 - 681612780*x^3 - 8806862800*x^2 + 1007795640*x + 916095316);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 160*x^18 + 10790*x^16 - 372*x^15 - 400000*x^14 + 37080*x^13 + 8908075*x^12 - 1405800*x^11 - 122060112*x^10 + 25437000*x^9 + 1009834870*x^8 - 221247000*x^7 - 4729220200*x^6 + 784838688*x^5 + 10914530225*x^4 - 681612780*x^3 - 8806862800*x^2 + 1007795640*x + 916095316)
 

\( x^{20} - 160 x^{18} + 10790 x^{16} - 372 x^{15} - 400000 x^{14} + 37080 x^{13} + 8908075 x^{12} + \cdots + 916095316 \) 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:   \(17687695508088196514062500000000000000000000\) \(\medspace = 2^{20}\cdot 3^{10}\cdot 5^{26}\cdot 61^{8}\) 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:  \(145.34\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2\cdot 3^{1/2}5^{13/10}61^{4/5}\approx 752.5155812562069$
Ramified primes:   \(2\), \(3\), \(5\), \(61\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $D_5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{10}a^{8}+\frac{1}{10}a^{7}-\frac{1}{10}a^{5}+\frac{2}{5}a^{4}-\frac{1}{10}a^{3}-\frac{1}{2}a^{2}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{10}a^{9}-\frac{1}{10}a^{7}-\frac{1}{10}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{2}{5}a^{3}+\frac{1}{10}a^{2}+\frac{2}{5}$, $\frac{1}{20}a^{10}-\frac{1}{4}a^{6}-\frac{3}{10}a^{5}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{3}{10}$, $\frac{1}{20}a^{11}-\frac{1}{4}a^{7}+\frac{1}{5}a^{6}+\frac{1}{4}a^{3}+\frac{3}{10}a$, $\frac{1}{60}a^{12}-\frac{1}{60}a^{10}-\frac{1}{20}a^{8}+\frac{1}{10}a^{7}-\frac{1}{12}a^{6}+\frac{2}{5}a^{5}+\frac{1}{20}a^{4}-\frac{1}{5}a^{3}-\frac{19}{60}a^{2}-\frac{3}{10}a-\frac{7}{30}$, $\frac{1}{60}a^{13}-\frac{1}{60}a^{11}-\frac{1}{20}a^{9}-\frac{11}{60}a^{7}-\frac{1}{10}a^{6}+\frac{3}{20}a^{5}-\frac{1}{10}a^{4}-\frac{13}{60}a^{3}-\frac{3}{10}a^{2}+\frac{1}{6}a+\frac{2}{5}$, $\frac{1}{600}a^{14}-\frac{1}{300}a^{13}+\frac{1}{200}a^{12}+\frac{1}{75}a^{11}-\frac{1}{150}a^{10}+\frac{1}{25}a^{9}-\frac{23}{600}a^{8}-\frac{1}{75}a^{7}-\frac{17}{150}a^{6}+\frac{1}{25}a^{5}-\frac{139}{600}a^{4}-\frac{29}{75}a^{3}-\frac{19}{200}a^{2}+\frac{89}{300}a-\frac{127}{300}$, $\frac{1}{36600}a^{15}-\frac{221}{36600}a^{13}+\frac{1}{150}a^{12}-\frac{17}{9150}a^{11}+\frac{29}{9150}a^{10}-\frac{151}{7320}a^{9}-\frac{41}{1525}a^{8}+\frac{877}{4575}a^{7}+\frac{337}{9150}a^{6}+\frac{10709}{36600}a^{5}+\frac{2}{5}a^{4}+\frac{239}{600}a^{3}-\frac{19}{150}a^{2}-\frac{149}{300}a+\frac{13}{150}$, $\frac{1}{36600}a^{16}+\frac{23}{36600}a^{14}-\frac{1}{150}a^{13}+\frac{9}{6100}a^{12}+\frac{119}{18300}a^{11}+\frac{709}{36600}a^{10}+\frac{101}{3050}a^{9}-\frac{71}{6100}a^{8}+\frac{2443}{18300}a^{7}+\frac{2657}{36600}a^{6}+\frac{23}{50}a^{5}+\frac{13}{600}a^{4}+\frac{83}{300}a^{3}-\frac{41}{100}a^{2}+\frac{13}{75}a-\frac{4}{25}$, $\frac{1}{36600}a^{17}-\frac{77}{12200}a^{13}+\frac{119}{18300}a^{12}-\frac{3}{2440}a^{11}+\frac{1}{4575}a^{10}+\frac{167}{7320}a^{9}-\frac{9}{6100}a^{8}-\frac{891}{12200}a^{7}-\frac{379}{1830}a^{6}+\frac{2757}{6100}a^{5}+\frac{1}{20}a^{4}-\frac{7}{200}a^{3}+\frac{11}{150}a^{2}+\frac{1}{60}a+\frac{37}{150}$, $\frac{1}{174142800}a^{18}-\frac{17}{2854800}a^{17}+\frac{83}{6697800}a^{16}-\frac{19}{2854800}a^{15}-\frac{73573}{174142800}a^{14}-\frac{496543}{87071400}a^{13}+\frac{189077}{174142800}a^{12}-\frac{135631}{19349200}a^{11}-\frac{160701}{19349200}a^{10}+\frac{1024313}{29023800}a^{9}+\frac{2877727}{58047600}a^{8}-\frac{206029}{951600}a^{7}-\frac{21571}{109800}a^{6}+\frac{325699}{2854800}a^{5}-\frac{895913}{2854800}a^{4}-\frac{40097}{142740}a^{3}+\frac{877}{2340}a^{2}+\frac{5461}{11700}a-\frac{3727}{11700}$, $\frac{1}{24\cdots 00}a^{19}-\frac{12\cdots 67}{82\cdots 00}a^{18}+\frac{56\cdots 51}{41\cdots 00}a^{17}-\frac{38\cdots 51}{24\cdots 00}a^{16}+\frac{16\cdots 87}{16\cdots 40}a^{15}+\frac{30\cdots 43}{24\cdots 60}a^{14}+\frac{19\cdots 91}{66\cdots 00}a^{13}-\frac{15\cdots 47}{24\cdots 00}a^{12}-\frac{15\cdots 59}{82\cdots 00}a^{11}-\frac{69\cdots 91}{55\cdots 48}a^{10}-\frac{33\cdots 73}{42\cdots 60}a^{9}+\frac{10\cdots 81}{27\cdots 00}a^{8}-\frac{29\cdots 69}{20\cdots 00}a^{7}+\frac{33\cdots 43}{45\cdots 00}a^{6}+\frac{26\cdots 73}{90\cdots 80}a^{5}-\frac{48\cdots 43}{50\cdots 50}a^{4}+\frac{50\cdots 67}{13\cdots 20}a^{3}-\frac{39\cdots 79}{16\cdots 00}a^{2}+\frac{20\cdots 67}{41\cdots 25}a-\frac{70\cdots 81}{16\cdots 90}$ 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:  $2$, $3$

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:  $C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $32$ (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:   $\frac{14\cdots 65}{25\cdots 08}a^{19}+\frac{41\cdots 71}{25\cdots 08}a^{18}-\frac{27\cdots 40}{31\cdots 01}a^{17}-\frac{30\cdots 63}{12\cdots 04}a^{16}+\frac{17\cdots 63}{31\cdots 10}a^{15}+\frac{18\cdots 55}{12\cdots 04}a^{14}-\frac{57\cdots 75}{31\cdots 01}a^{13}-\frac{15\cdots 20}{31\cdots 01}a^{12}+\frac{68\cdots 63}{19\cdots 16}a^{11}+\frac{17\cdots 89}{19\cdots 16}a^{10}-\frac{21\cdots 75}{52\cdots 41}a^{9}-\frac{19\cdots 25}{20\cdots 64}a^{8}+\frac{56\cdots 15}{20\cdots 64}a^{7}+\frac{11\cdots 45}{20\cdots 64}a^{6}-\frac{58\cdots 33}{63\cdots 02}a^{5}-\frac{16\cdots 75}{10\cdots 82}a^{4}+\frac{51\cdots 27}{41\cdots 28}a^{3}+\frac{57\cdots 75}{41\cdots 28}a^{2}-\frac{49\cdots 51}{80\cdots 14}a-\frac{95\cdots 39}{10\cdots 20}$, $\frac{19\cdots 75}{10\cdots 89}a^{19}-\frac{15\cdots 95}{54\cdots 08}a^{18}-\frac{51\cdots 25}{18\cdots 36}a^{17}-\frac{39\cdots 75}{20\cdots 78}a^{16}+\frac{30\cdots 55}{18\cdots 36}a^{15}+\frac{78\cdots 95}{16\cdots 24}a^{14}-\frac{10\cdots 11}{20\cdots 78}a^{13}-\frac{45\cdots 85}{16\cdots 24}a^{12}+\frac{51\cdots 11}{54\cdots 08}a^{11}+\frac{21\cdots 01}{27\cdots 40}a^{10}-\frac{16\cdots 05}{18\cdots 61}a^{9}-\frac{10\cdots 15}{89\cdots 28}a^{8}+\frac{10\cdots 45}{26\cdots 84}a^{7}+\frac{62\cdots 99}{74\cdots 44}a^{6}+\frac{28\cdots 19}{27\cdots 40}a^{5}-\frac{52\cdots 25}{26\cdots 84}a^{4}-\frac{51\cdots 63}{11\cdots 76}a^{3}-\frac{16\cdots 05}{51\cdots 92}a^{2}+\frac{32\cdots 40}{16\cdots 49}a+\frac{30\cdots 27}{33\cdots 80}$, $\frac{33\cdots 65}{16\cdots 24}a^{19}-\frac{32\cdots 49}{16\cdots 24}a^{18}-\frac{12\cdots 55}{40\cdots 56}a^{17}+\frac{36\cdots 59}{16\cdots 24}a^{16}+\frac{15\cdots 67}{81\cdots 20}a^{15}-\frac{19\cdots 65}{20\cdots 78}a^{14}-\frac{10\cdots 73}{16\cdots 24}a^{13}+\frac{93\cdots 35}{54\cdots 08}a^{12}+\frac{65\cdots 39}{54\cdots 08}a^{11}-\frac{80\cdots 31}{11\cdots 10}a^{10}-\frac{41\cdots 05}{29\cdots 76}a^{9}-\frac{16\cdots 45}{89\cdots 28}a^{8}+\frac{60\cdots 25}{66\cdots 96}a^{7}+\frac{56\cdots 09}{26\cdots 84}a^{6}-\frac{24\cdots 13}{81\cdots 20}a^{5}-\frac{27\cdots 25}{13\cdots 92}a^{4}+\frac{39\cdots 95}{10\cdots 84}a^{3}-\frac{41\cdots 70}{12\cdots 73}a^{2}-\frac{23\cdots 39}{33\cdots 98}a+\frac{93\cdots 31}{11\cdots 66}$, $\frac{12\cdots 67}{24\cdots 00}a^{19}+\frac{43\cdots 41}{82\cdots 00}a^{18}-\frac{29\cdots 41}{41\cdots 00}a^{17}-\frac{19\cdots 63}{24\cdots 00}a^{16}+\frac{32\cdots 77}{82\cdots 00}a^{15}+\frac{58\cdots 91}{12\cdots 00}a^{14}-\frac{26\cdots 47}{24\cdots 00}a^{13}-\frac{37\cdots 43}{24\cdots 00}a^{12}+\frac{48\cdots 15}{33\cdots 88}a^{11}+\frac{11\cdots 23}{41\cdots 00}a^{10}-\frac{64\cdots 29}{82\cdots 00}a^{9}-\frac{85\cdots 67}{27\cdots 00}a^{8}-\frac{63\cdots 41}{20\cdots 00}a^{7}+\frac{50\cdots 97}{27\cdots 40}a^{6}+\frac{73\cdots 89}{13\cdots 00}a^{5}-\frac{54\cdots 29}{10\cdots 00}a^{4}-\frac{29\cdots 21}{17\cdots 00}a^{3}+\frac{60\cdots 43}{12\cdots 00}a^{2}-\frac{10\cdots 33}{41\cdots 25}a-\frac{38\cdots 07}{83\cdots 50}$, $\frac{33\cdots 77}{15\cdots 25}a^{19}+\frac{14\cdots 09}{49\cdots 20}a^{18}-\frac{74\cdots 17}{24\cdots 00}a^{17}-\frac{52\cdots 11}{12\cdots 00}a^{16}+\frac{81\cdots 29}{49\cdots 20}a^{15}+\frac{61\cdots 33}{24\cdots 00}a^{14}-\frac{58\cdots 09}{12\cdots 00}a^{13}-\frac{16\cdots 13}{21\cdots 00}a^{12}+\frac{61\cdots 39}{82\cdots 00}a^{11}+\frac{11\cdots 41}{82\cdots 00}a^{10}-\frac{90\cdots 81}{13\cdots 00}a^{9}-\frac{39\cdots 33}{27\cdots 00}a^{8}+\frac{11\cdots 21}{40\cdots 00}a^{7}+\frac{16\cdots 39}{20\cdots 00}a^{6}-\frac{79\cdots 53}{40\cdots 00}a^{5}-\frac{18\cdots 99}{81\cdots 20}a^{4}-\frac{11\cdots 57}{10\cdots 00}a^{3}+\frac{74\cdots 61}{33\cdots 00}a^{2}-\frac{57\cdots 23}{16\cdots 90}a-\frac{19\cdots 43}{27\cdots 50}$, $\frac{21\cdots 21}{24\cdots 60}a^{19}+\frac{62\cdots 77}{12\cdots 00}a^{18}-\frac{19\cdots 98}{15\cdots 25}a^{17}-\frac{37\cdots 05}{49\cdots 32}a^{16}+\frac{91\cdots 11}{12\cdots 00}a^{15}+\frac{58\cdots 77}{12\cdots 00}a^{14}-\frac{28\cdots 19}{12\cdots 00}a^{13}-\frac{80\cdots 04}{51\cdots 75}a^{12}+\frac{33\cdots 57}{82\cdots 20}a^{11}+\frac{61\cdots 93}{20\cdots 00}a^{10}-\frac{16\cdots 19}{41\cdots 00}a^{9}-\frac{68\cdots 21}{20\cdots 00}a^{8}+\frac{19\cdots 23}{10\cdots 00}a^{7}+\frac{16\cdots 45}{81\cdots 12}a^{6}-\frac{39\cdots 33}{20\cdots 00}a^{5}-\frac{12\cdots 93}{20\cdots 00}a^{4}-\frac{48\cdots 33}{50\cdots 50}a^{3}+\frac{41\cdots 13}{66\cdots 60}a^{2}+\frac{79\cdots 41}{33\cdots 80}a-\frac{78\cdots 57}{18\cdots 00}$, $\frac{65\cdots 73}{24\cdots 00}a^{19}+\frac{17\cdots 43}{24\cdots 00}a^{18}-\frac{12\cdots 09}{31\cdots 50}a^{17}-\frac{26\cdots 73}{24\cdots 00}a^{16}+\frac{62\cdots 43}{24\cdots 00}a^{15}+\frac{40\cdots 43}{62\cdots 00}a^{14}-\frac{20\cdots 53}{24\cdots 00}a^{13}-\frac{34\cdots 37}{16\cdots 40}a^{12}+\frac{45\cdots 73}{27\cdots 00}a^{11}+\frac{80\cdots 03}{20\cdots 00}a^{10}-\frac{53\cdots 73}{27\cdots 00}a^{9}-\frac{34\cdots 09}{82\cdots 00}a^{8}+\frac{25\cdots 73}{20\cdots 80}a^{7}+\frac{99\cdots 73}{40\cdots 00}a^{6}-\frac{16\cdots 21}{40\cdots 00}a^{5}-\frac{55\cdots 25}{81\cdots 12}a^{4}+\frac{19\cdots 61}{40\cdots 60}a^{3}+\frac{22\cdots 58}{41\cdots 25}a^{2}-\frac{10\cdots 86}{41\cdots 25}a-\frac{36\cdots 79}{27\cdots 50}$, $\frac{14\cdots 21}{41\cdots 00}a^{19}+\frac{16\cdots 79}{12\cdots 80}a^{18}-\frac{26\cdots 61}{55\cdots 80}a^{17}-\frac{12\cdots 07}{69\cdots 00}a^{16}+\frac{17\cdots 33}{63\cdots 00}a^{15}+\frac{29\cdots 83}{27\cdots 00}a^{14}-\frac{11\cdots 47}{13\cdots 62}a^{13}-\frac{25\cdots 51}{82\cdots 00}a^{12}+\frac{15\cdots 43}{11\cdots 96}a^{11}+\frac{43\cdots 07}{82\cdots 00}a^{10}-\frac{15\cdots 43}{10\cdots 50}a^{9}-\frac{27\cdots 41}{55\cdots 80}a^{8}+\frac{11\cdots 31}{13\cdots 00}a^{7}+\frac{17\cdots 93}{67\cdots 60}a^{6}-\frac{31\cdots 97}{13\cdots 00}a^{5}-\frac{29\cdots 93}{45\cdots 00}a^{4}+\frac{14\cdots 43}{67\cdots 26}a^{3}+\frac{56\cdots 37}{11\cdots 00}a^{2}-\frac{10\cdots 59}{13\cdots 75}a-\frac{15\cdots 23}{27\cdots 50}$, $\frac{41\cdots 59}{24\cdots 00}a^{19}+\frac{39\cdots 97}{24\cdots 00}a^{18}-\frac{14\cdots 41}{62\cdots 90}a^{17}-\frac{41\cdots 11}{16\cdots 40}a^{16}+\frac{33\cdots 17}{24\cdots 00}a^{15}+\frac{16\cdots 07}{10\cdots 50}a^{14}-\frac{32\cdots 17}{82\cdots 00}a^{13}-\frac{27\cdots 11}{49\cdots 20}a^{12}+\frac{13\cdots 31}{21\cdots 00}a^{11}+\frac{35\cdots 11}{31\cdots 00}a^{10}-\frac{40\cdots 01}{82\cdots 00}a^{9}-\frac{36\cdots 51}{27\cdots 00}a^{8}+\frac{27\cdots 83}{20\cdots 80}a^{7}+\frac{36\cdots 73}{40\cdots 00}a^{6}+\frac{17\cdots 57}{40\cdots 00}a^{5}-\frac{19\cdots 93}{67\cdots 00}a^{4}-\frac{45\cdots 27}{20\cdots 00}a^{3}+\frac{62\cdots 43}{22\cdots 20}a^{2}-\frac{15\cdots 73}{42\cdots 00}a-\frac{78\cdots 17}{33\cdots 80}$, $\frac{13\cdots 83}{62\cdots 00}a^{19}-\frac{17\cdots 91}{12\cdots 80}a^{18}-\frac{18\cdots 17}{62\cdots 90}a^{17}+\frac{14\cdots 01}{12\cdots 00}a^{16}+\frac{20\cdots 67}{12\cdots 00}a^{15}-\frac{28\cdots 39}{12\cdots 00}a^{14}-\frac{59\cdots 67}{12\cdots 00}a^{13}-\frac{92\cdots 13}{10\cdots 50}a^{12}+\frac{40\cdots 16}{51\cdots 75}a^{11}+\frac{21\cdots 51}{41\cdots 00}a^{10}-\frac{94\cdots 49}{13\cdots 00}a^{9}-\frac{10\cdots 49}{10\cdots 50}a^{8}+\frac{28\cdots 93}{10\cdots 00}a^{7}+\frac{18\cdots 71}{20\cdots 00}a^{6}-\frac{57\cdots 93}{20\cdots 00}a^{5}-\frac{72\cdots 13}{20\cdots 00}a^{4}-\frac{16\cdots 27}{20\cdots 00}a^{3}+\frac{14\cdots 38}{41\cdots 25}a^{2}+\frac{82\cdots 73}{33\cdots 80}a-\frac{12\cdots 44}{55\cdots 83}$, $\frac{96\cdots 11}{62\cdots 00}a^{19}-\frac{44\cdots 79}{62\cdots 00}a^{18}-\frac{30\cdots 73}{12\cdots 00}a^{17}+\frac{53\cdots 61}{51\cdots 75}a^{16}+\frac{51\cdots 72}{31\cdots 45}a^{15}-\frac{84\cdots 43}{13\cdots 00}a^{14}-\frac{25\cdots 51}{41\cdots 00}a^{13}+\frac{24\cdots 47}{12\cdots 00}a^{12}+\frac{52\cdots 91}{41\cdots 00}a^{11}-\frac{36\cdots 81}{10\cdots 50}a^{10}-\frac{62\cdots 71}{41\cdots 00}a^{9}+\frac{15\cdots 71}{41\cdots 00}a^{8}+\frac{20\cdots 23}{20\cdots 00}a^{7}-\frac{21\cdots 79}{10\cdots 00}a^{6}-\frac{20\cdots 27}{50\cdots 50}a^{5}+\frac{53\cdots 13}{90\cdots 68}a^{4}+\frac{16\cdots 69}{20\cdots 00}a^{3}-\frac{63\cdots 07}{11\cdots 00}a^{2}+\frac{80\cdots 93}{13\cdots 75}a+\frac{91\cdots 19}{16\cdots 00}$, $\frac{56\cdots 79}{41\cdots 60}a^{19}+\frac{10\cdots 61}{24\cdots 00}a^{18}-\frac{50\cdots 97}{24\cdots 00}a^{17}-\frac{20\cdots 71}{31\cdots 50}a^{16}+\frac{31\cdots 53}{24\cdots 00}a^{15}+\frac{20\cdots 31}{49\cdots 20}a^{14}-\frac{50\cdots 63}{12\cdots 00}a^{13}-\frac{33\cdots 81}{24\cdots 00}a^{12}+\frac{83\cdots 37}{11\cdots 96}a^{11}+\frac{83\cdots 31}{33\cdots 88}a^{10}-\frac{16\cdots 59}{21\cdots 80}a^{9}-\frac{22\cdots 43}{82\cdots 00}a^{8}+\frac{50\cdots 47}{13\cdots 00}a^{7}+\frac{16\cdots 07}{10\cdots 90}a^{6}-\frac{25\cdots 57}{81\cdots 20}a^{5}-\frac{31\cdots 89}{81\cdots 20}a^{4}-\frac{11\cdots 59}{50\cdots 45}a^{3}+\frac{35\cdots 31}{33\cdots 00}a^{2}+\frac{50\cdots 59}{83\cdots 50}a-\frac{11\cdots 79}{83\cdots 50}$, $\frac{56\cdots 11}{62\cdots 00}a^{19}+\frac{37\cdots 21}{24\cdots 00}a^{18}-\frac{36\cdots 89}{24\cdots 00}a^{17}-\frac{13\cdots 41}{62\cdots 00}a^{16}+\frac{24\cdots 21}{24\cdots 00}a^{15}+\frac{32\cdots 67}{24\cdots 00}a^{14}-\frac{45\cdots 99}{12\cdots 00}a^{13}-\frac{34\cdots 19}{82\cdots 00}a^{12}+\frac{34\cdots 73}{42\cdots 60}a^{11}+\frac{46\cdots 91}{63\cdots 00}a^{10}-\frac{45\cdots 17}{41\cdots 00}a^{9}-\frac{20\cdots 73}{27\cdots 00}a^{8}+\frac{36\cdots 17}{40\cdots 00}a^{7}+\frac{46\cdots 19}{10\cdots 89}a^{6}-\frac{16\cdots 89}{40\cdots 00}a^{5}-\frac{64\cdots 49}{40\cdots 00}a^{4}+\frac{22\cdots 53}{25\cdots 25}a^{3}+\frac{94\cdots 67}{33\cdots 00}a^{2}-\frac{40\cdots 61}{64\cdots 50}a-\frac{16\cdots 13}{18\cdots 10}$, $\frac{87\cdots 21}{12\cdots 00}a^{19}-\frac{87\cdots 87}{24\cdots 00}a^{18}-\frac{24\cdots 07}{24\cdots 00}a^{17}+\frac{61\cdots 21}{12\cdots 00}a^{16}+\frac{14\cdots 13}{24\cdots 00}a^{15}-\frac{71\cdots 87}{24\cdots 00}a^{14}-\frac{20\cdots 29}{12\cdots 58}a^{13}+\frac{11\cdots 37}{12\cdots 80}a^{12}+\frac{23\cdots 39}{82\cdots 00}a^{11}-\frac{12\cdots 39}{82\cdots 00}a^{10}-\frac{13\cdots 31}{51\cdots 75}a^{9}+\frac{84\cdots 07}{55\cdots 80}a^{8}+\frac{96\cdots 11}{81\cdots 20}a^{7}-\frac{16\cdots 21}{20\cdots 00}a^{6}-\frac{10\cdots 73}{40\cdots 00}a^{5}+\frac{32\cdots 73}{16\cdots 24}a^{4}+\frac{23\cdots 27}{10\cdots 00}a^{3}-\frac{56\cdots 11}{33\cdots 00}a^{2}+\frac{12\cdots 83}{83\cdots 50}a+\frac{16\cdots 13}{92\cdots 50}$, $\frac{24\cdots 83}{49\cdots 20}a^{19}+\frac{27\cdots 31}{49\cdots 20}a^{18}-\frac{98\cdots 07}{12\cdots 00}a^{17}-\frac{36\cdots 17}{49\cdots 20}a^{16}+\frac{13\cdots 43}{24\cdots 00}a^{15}+\frac{47\cdots 69}{12\cdots 00}a^{14}-\frac{19\cdots 99}{99\cdots 64}a^{13}-\frac{79\cdots 67}{82\cdots 00}a^{12}+\frac{13\cdots 55}{33\cdots 88}a^{11}+\frac{54\cdots 43}{51\cdots 75}a^{10}-\frac{43\cdots 21}{82\cdots 00}a^{9}-\frac{17\cdots 91}{16\cdots 40}a^{8}+\frac{76\cdots 13}{20\cdots 00}a^{7}-\frac{59\cdots 79}{81\cdots 20}a^{6}-\frac{54\cdots 49}{40\cdots 00}a^{5}+\frac{25\cdots 71}{50\cdots 50}a^{4}+\frac{67\cdots 21}{40\cdots 60}a^{3}-\frac{63\cdots 61}{66\cdots 60}a^{2}-\frac{34\cdots 47}{33\cdots 80}a-\frac{70\cdots 39}{55\cdots 00}$, $\frac{10\cdots 91}{15\cdots 25}a^{19}-\frac{21\cdots 09}{10\cdots 50}a^{18}-\frac{14\cdots 33}{13\cdots 00}a^{17}+\frac{39\cdots 29}{12\cdots 00}a^{16}+\frac{14\cdots 33}{20\cdots 00}a^{15}-\frac{63\cdots 81}{31\cdots 50}a^{14}-\frac{32\cdots 03}{12\cdots 00}a^{13}+\frac{89\cdots 99}{12\cdots 00}a^{12}+\frac{22\cdots 17}{41\cdots 00}a^{11}-\frac{63\cdots 93}{41\cdots 00}a^{10}-\frac{28\cdots 57}{41\cdots 00}a^{9}+\frac{79\cdots 31}{41\cdots 00}a^{8}+\frac{96\cdots 31}{20\cdots 00}a^{7}-\frac{30\cdots 19}{22\cdots 00}a^{6}-\frac{53\cdots 61}{33\cdots 00}a^{5}+\frac{12\cdots 13}{25\cdots 25}a^{4}+\frac{78\cdots 51}{52\cdots 00}a^{3}-\frac{14\cdots 91}{25\cdots 00}a^{2}+\frac{71\cdots 49}{16\cdots 90}a+\frac{98\cdots 07}{16\cdots 00}$, $\frac{16\cdots 33}{15\cdots 25}a^{19}-\frac{15\cdots 33}{12\cdots 00}a^{18}-\frac{25\cdots 04}{15\cdots 25}a^{17}+\frac{24\cdots 87}{12\cdots 00}a^{16}+\frac{13\cdots 53}{12\cdots 00}a^{15}-\frac{34\cdots 41}{24\cdots 60}a^{14}-\frac{95\cdots 21}{24\cdots 60}a^{13}+\frac{10\cdots 59}{20\cdots 00}a^{12}+\frac{27\cdots 53}{33\cdots 00}a^{11}-\frac{62\cdots 12}{51\cdots 75}a^{10}-\frac{82\cdots 23}{82\cdots 20}a^{9}+\frac{13\cdots 58}{79\cdots 55}a^{8}+\frac{71\cdots 83}{10\cdots 00}a^{7}-\frac{25\cdots 43}{20\cdots 00}a^{6}-\frac{49\cdots 57}{20\cdots 00}a^{5}+\frac{89\cdots 49}{20\cdots 00}a^{4}+\frac{56\cdots 17}{20\cdots 00}a^{3}-\frac{17\cdots 43}{33\cdots 00}a^{2}+\frac{26\cdots 68}{83\cdots 45}a+\frac{67\cdots 51}{11\cdots 60}$, $\frac{30\cdots 39}{82\cdots 00}a^{19}-\frac{45\cdots 01}{24\cdots 00}a^{18}-\frac{35\cdots 51}{62\cdots 00}a^{17}+\frac{67\cdots 21}{24\cdots 00}a^{16}+\frac{18\cdots 87}{49\cdots 20}a^{15}-\frac{16\cdots 91}{95\cdots 60}a^{14}-\frac{47\cdots 73}{38\cdots 40}a^{13}+\frac{13\cdots 91}{24\cdots 00}a^{12}+\frac{68\cdots 01}{27\cdots 00}a^{11}-\frac{43\cdots 29}{41\cdots 00}a^{10}-\frac{92\cdots 27}{33\cdots 88}a^{9}+\frac{38\cdots 21}{33\cdots 88}a^{8}+\frac{58\cdots 47}{33\cdots 00}a^{7}-\frac{26\cdots 49}{40\cdots 00}a^{6}-\frac{21\cdots 41}{40\cdots 00}a^{5}+\frac{24\cdots 47}{15\cdots 00}a^{4}+\frac{18\cdots 27}{20\cdots 00}a^{3}-\frac{32\cdots 89}{33\cdots 00}a^{2}-\frac{17\cdots 13}{33\cdots 80}a-\frac{97\cdots 11}{16\cdots 00}$, $\frac{33\cdots 27}{24\cdots 00}a^{19}+\frac{22\cdots 39}{15\cdots 25}a^{18}-\frac{97\cdots 21}{49\cdots 20}a^{17}-\frac{20\cdots 81}{82\cdots 00}a^{16}+\frac{35\cdots 63}{31\cdots 50}a^{15}+\frac{13\cdots 57}{82\cdots 00}a^{14}-\frac{29\cdots 01}{82\cdots 00}a^{13}-\frac{72\cdots 29}{12\cdots 00}a^{12}+\frac{33\cdots 63}{51\cdots 75}a^{11}+\frac{98\cdots 13}{82\cdots 00}a^{10}-\frac{18\cdots 99}{27\cdots 00}a^{9}-\frac{58\cdots 57}{41\cdots 00}a^{8}+\frac{15\cdots 93}{40\cdots 00}a^{7}+\frac{37\cdots 11}{40\cdots 00}a^{6}-\frac{19\cdots 37}{20\cdots 00}a^{5}-\frac{39\cdots 49}{13\cdots 00}a^{4}+\frac{17\cdots 71}{25\cdots 25}a^{3}+\frac{14\cdots 21}{55\cdots 00}a^{2}-\frac{17\cdots 91}{46\cdots 25}a-\frac{48\cdots 43}{16\cdots 00}$ 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:  \( 9814449721720000 \) (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:  \( 15 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 9814449721720000 \cdot 1}{2\cdot\sqrt{17687695508088196514062500000000000000000000}}\cr\approx \mathstrut & 1.22348949690707 \end{aligned}\] (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 - 160*x^18 + 10790*x^16 - 372*x^15 - 400000*x^14 + 37080*x^13 + 8908075*x^12 - 1405800*x^11 - 122060112*x^10 + 25437000*x^9 + 1009834870*x^8 - 221247000*x^7 - 4729220200*x^6 + 784838688*x^5 + 10914530225*x^4 - 681612780*x^3 - 8806862800*x^2 + 1007795640*x + 916095316) 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 - 160*x^18 + 10790*x^16 - 372*x^15 - 400000*x^14 + 37080*x^13 + 8908075*x^12 - 1405800*x^11 - 122060112*x^10 + 25437000*x^9 + 1009834870*x^8 - 221247000*x^7 - 4729220200*x^6 + 784838688*x^5 + 10914530225*x^4 - 681612780*x^3 - 8806862800*x^2 + 1007795640*x + 916095316, 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 - 160*x^18 + 10790*x^16 - 372*x^15 - 400000*x^14 + 37080*x^13 + 8908075*x^12 - 1405800*x^11 - 122060112*x^10 + 25437000*x^9 + 1009834870*x^8 - 221247000*x^7 - 4729220200*x^6 + 784838688*x^5 + 10914530225*x^4 - 681612780*x^3 - 8806862800*x^2 + 1007795640*x + 916095316); 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 - 160*x^18 + 10790*x^16 - 372*x^15 - 400000*x^14 + 37080*x^13 + 8908075*x^12 - 1405800*x^11 - 122060112*x^10 + 25437000*x^9 + 1009834870*x^8 - 221247000*x^7 - 4729220200*x^6 + 784838688*x^5 + 10914530225*x^4 - 681612780*x^3 - 8806862800*x^2 + 1007795640*x + 916095316); 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

$D_5^2$ (as 20T28):

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 100
The 16 conjugacy class representatives for $D_5^2$
Character table for $D_5^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}, \sqrt{5})\), 10.10.841134840750000000000.1 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: data not computed
Degree 20 sibling: data not computed
Degree 25 sibling: data not computed
Minimal sibling: 10.10.841134840750000000000.1

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 ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.5.0.1}{5} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{10}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{10}$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{10}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{10}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.2.4a1.1$x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 5$$2$$2$$4$$C_2^2$$$[2]^{2}$$
2.2.2.4a1.1$x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 5$$2$$2$$4$$C_2^2$$$[2]^{2}$$
2.2.2.4a1.1$x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 5$$2$$2$$4$$C_2^2$$$[2]^{2}$$
2.2.2.4a1.1$x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 5$$2$$2$$4$$C_2^2$$$[2]^{2}$$
2.2.2.4a1.1$x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 5$$2$$2$$4$$C_2^2$$$[2]^{2}$$
\(3\) Copy content Toggle raw display 3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(5\) Copy content Toggle raw display 5.2.10.26a3.1$x^{20} + 40 x^{19} + 740 x^{18} + 8400 x^{17} + 65460 x^{16} + 371328 x^{15} + 1586880 x^{14} + 5218560 x^{13} + 13381920 x^{12} + 26970880 x^{11} + 42904960 x^{10} + 53941760 x^{9} + 53527695 x^{8} + 41748720 x^{7} + 25391640 x^{6} + 11887776 x^{5} + 4199400 x^{4} + 1085760 x^{3} + 195680 x^{2} + 22400 x + 1269$$10$$2$$26$20T4not computed
\(61\) Copy content Toggle raw display 61.5.1.0a1.1$x^{5} + 12 x + 59$$1$$5$$0$$C_5$$$[\ ]^{5}$$
61.5.1.0a1.1$x^{5} + 12 x + 59$$1$$5$$0$$C_5$$$[\ ]^{5}$$
61.1.5.4a1.1$x^{5} + 61$$5$$1$$4$$C_5$$$[\ ]_{5}$$
61.1.5.4a1.1$x^{5} + 61$$5$$1$$4$$C_5$$$[\ ]_{5}$$

Spectrum of ring of integers

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