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Properties

Label 20.20.516...944.1
Degree $20$
Signature $(20, 0)$
Discriminant $5.169\times 10^{38}$
Root discriminant \(86.23\)
Ramified primes $2,3,7,11,13$
Class number $2$ (GRH)
Class group [2] (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 - 83*x^18 - 108*x^17 + 2380*x^16 + 5760*x^15 - 27524*x^14 - 103968*x^13 + 77348*x^12 + 735960*x^11 + 616108*x^10 - 1615050*x^9 - 3200232*x^8 - 714960*x^7 + 2411685*x^6 + 1666554*x^5 - 423490*x^4 - 635688*x^3 - 65073*x^2 + 67764*x + 15313)
 
Copy content gp:K = bnfinit(y^20 - 83*y^18 - 108*y^17 + 2380*y^16 + 5760*y^15 - 27524*y^14 - 103968*y^13 + 77348*y^12 + 735960*y^11 + 616108*y^10 - 1615050*y^9 - 3200232*y^8 - 714960*y^7 + 2411685*y^6 + 1666554*y^5 - 423490*y^4 - 635688*y^3 - 65073*y^2 + 67764*y + 15313, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 83*x^18 - 108*x^17 + 2380*x^16 + 5760*x^15 - 27524*x^14 - 103968*x^13 + 77348*x^12 + 735960*x^11 + 616108*x^10 - 1615050*x^9 - 3200232*x^8 - 714960*x^7 + 2411685*x^6 + 1666554*x^5 - 423490*x^4 - 635688*x^3 - 65073*x^2 + 67764*x + 15313);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 83*x^18 - 108*x^17 + 2380*x^16 + 5760*x^15 - 27524*x^14 - 103968*x^13 + 77348*x^12 + 735960*x^11 + 616108*x^10 - 1615050*x^9 - 3200232*x^8 - 714960*x^7 + 2411685*x^6 + 1666554*x^5 - 423490*x^4 - 635688*x^3 - 65073*x^2 + 67764*x + 15313)
 

\( x^{20} - 83 x^{18} - 108 x^{17} + 2380 x^{16} + 5760 x^{15} - 27524 x^{14} - 103968 x^{13} + \cdots + 15313 \) 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:   \(516853980088694624404163683037236690944\) \(\medspace = 2^{20}\cdot 3^{10}\cdot 7^{10}\cdot 11^{8}\cdot 13^{10}\) 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:  \(86.23\)
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:  $2\cdot 3^{1/2}7^{1/2}11^{4/5}13^{1/2}\approx 225.02225229230086$
Ramified primes:   \(2\), \(3\), \(7\), \(11\), \(13\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13}a^{16}+\frac{4}{13}a^{14}+\frac{5}{13}a^{13}-\frac{1}{13}a^{12}+\frac{4}{13}a^{11}-\frac{2}{13}a^{10}-\frac{1}{13}a^{9}+\frac{2}{13}a^{8}-\frac{6}{13}a^{7}+\frac{6}{13}a^{6}-\frac{6}{13}a^{5}+\frac{6}{13}a^{4}-\frac{5}{13}a^{3}-\frac{4}{13}a^{2}-\frac{4}{13}a+\frac{1}{13}$, $\frac{1}{13}a^{17}+\frac{4}{13}a^{15}+\frac{5}{13}a^{14}-\frac{1}{13}a^{13}+\frac{4}{13}a^{12}-\frac{2}{13}a^{11}-\frac{1}{13}a^{10}+\frac{2}{13}a^{9}-\frac{6}{13}a^{8}+\frac{6}{13}a^{7}-\frac{6}{13}a^{6}+\frac{6}{13}a^{5}-\frac{5}{13}a^{4}-\frac{4}{13}a^{3}-\frac{4}{13}a^{2}+\frac{1}{13}a$, $\frac{1}{46657}a^{18}-\frac{1465}{46657}a^{17}+\frac{1280}{46657}a^{16}-\frac{13122}{46657}a^{15}+\frac{10739}{46657}a^{14}-\frac{4878}{46657}a^{13}+\frac{3158}{46657}a^{12}+\frac{1845}{46657}a^{11}+\frac{18363}{46657}a^{10}+\frac{434}{3589}a^{9}+\frac{155}{46657}a^{8}+\frac{16815}{46657}a^{7}-\frac{2463}{46657}a^{6}-\frac{10874}{46657}a^{5}-\frac{13194}{46657}a^{4}-\frac{16202}{46657}a^{3}+\frac{3045}{46657}a^{2}-\frac{9260}{46657}a+\frac{12859}{46657}$, $\frac{1}{49\cdots 61}a^{19}+\frac{34\cdots 47}{49\cdots 61}a^{18}+\frac{16\cdots 02}{49\cdots 61}a^{17}-\frac{10\cdots 01}{49\cdots 61}a^{16}+\frac{14\cdots 87}{49\cdots 61}a^{15}+\frac{45\cdots 93}{49\cdots 61}a^{14}-\frac{41\cdots 41}{49\cdots 61}a^{13}+\frac{12\cdots 32}{49\cdots 61}a^{12}-\frac{75\cdots 25}{49\cdots 61}a^{11}+\frac{13\cdots 44}{49\cdots 61}a^{10}-\frac{53\cdots 88}{49\cdots 61}a^{9}-\frac{19\cdots 69}{49\cdots 61}a^{8}-\frac{10\cdots 53}{49\cdots 61}a^{7}+\frac{40\cdots 87}{37\cdots 97}a^{6}+\frac{15\cdots 56}{49\cdots 61}a^{5}+\frac{30\cdots 03}{49\cdots 61}a^{4}+\frac{11\cdots 76}{49\cdots 61}a^{3}+\frac{90\cdots 31}{49\cdots 61}a^{2}+\frac{24\cdots 38}{49\cdots 61}a-\frac{12\cdots 62}{49\cdots 61}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  $C_{2}$, which has order $2$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $19$
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:   $\frac{29\cdots 49}{49\cdots 61}a^{19}-\frac{82\cdots 16}{49\cdots 61}a^{18}-\frac{22\cdots 05}{49\cdots 61}a^{17}+\frac{31\cdots 51}{49\cdots 61}a^{16}+\frac{62\cdots 23}{49\cdots 61}a^{15}-\frac{94\cdots 57}{49\cdots 61}a^{14}-\frac{63\cdots 80}{37\cdots 97}a^{13}-\frac{71\cdots 92}{49\cdots 61}a^{12}+\frac{48\cdots 90}{49\cdots 61}a^{11}+\frac{82\cdots 90}{49\cdots 61}a^{10}-\frac{85\cdots 28}{49\cdots 61}a^{9}-\frac{27\cdots 26}{49\cdots 61}a^{8}-\frac{92\cdots 42}{49\cdots 61}a^{7}+\frac{19\cdots 56}{49\cdots 61}a^{6}+\frac{13\cdots 08}{49\cdots 61}a^{5}-\frac{41\cdots 59}{49\cdots 61}a^{4}-\frac{48\cdots 64}{49\cdots 61}a^{3}-\frac{13\cdots 30}{49\cdots 61}a^{2}+\frac{51\cdots 86}{49\cdots 61}a+\frac{78\cdots 50}{49\cdots 61}$, $\frac{24\cdots 46}{49\cdots 61}a^{19}-\frac{47\cdots 13}{49\cdots 61}a^{18}-\frac{15\cdots 11}{37\cdots 97}a^{17}+\frac{12\cdots 06}{49\cdots 61}a^{16}+\frac{58\cdots 75}{49\cdots 61}a^{15}+\frac{28\cdots 02}{49\cdots 61}a^{14}-\frac{79\cdots 98}{49\cdots 61}a^{13}-\frac{10\cdots 24}{49\cdots 61}a^{12}+\frac{47\cdots 99}{49\cdots 61}a^{11}+\frac{10\cdots 72}{49\cdots 61}a^{10}-\frac{87\cdots 87}{49\cdots 61}a^{9}-\frac{33\cdots 37}{49\cdots 61}a^{8}-\frac{82\cdots 19}{49\cdots 61}a^{7}+\frac{30\cdots 46}{49\cdots 61}a^{6}+\frac{14\cdots 99}{49\cdots 61}a^{5}-\frac{11\cdots 19}{49\cdots 61}a^{4}-\frac{41\cdots 11}{37\cdots 97}a^{3}+\frac{19\cdots 58}{49\cdots 61}a^{2}+\frac{52\cdots 06}{49\cdots 61}a-\frac{87\cdots 53}{49\cdots 61}$, $\frac{12\cdots 56}{20\cdots 99}a^{19}-\frac{14\cdots 65}{20\cdots 99}a^{18}-\frac{98\cdots 27}{20\cdots 99}a^{17}-\frac{11\cdots 17}{20\cdots 99}a^{16}+\frac{28\cdots 07}{20\cdots 99}a^{15}+\frac{34\cdots 30}{20\cdots 99}a^{14}-\frac{37\cdots 48}{20\cdots 99}a^{13}-\frac{80\cdots 14}{20\cdots 99}a^{12}+\frac{19\cdots 25}{20\cdots 99}a^{11}+\frac{50\cdots 89}{15\cdots 23}a^{10}-\frac{52\cdots 99}{20\cdots 99}a^{9}-\frac{18\cdots 01}{20\cdots 99}a^{8}-\frac{15\cdots 85}{20\cdots 99}a^{7}+\frac{10\cdots 01}{20\cdots 99}a^{6}+\frac{16\cdots 94}{20\cdots 99}a^{5}+\frac{64\cdots 17}{20\cdots 99}a^{4}-\frac{14\cdots 53}{55\cdots 27}a^{3}-\frac{96\cdots 23}{15\cdots 23}a^{2}+\frac{54\cdots 39}{20\cdots 99}a+\frac{17\cdots 50}{20\cdots 99}$, $\frac{13\cdots 27}{49\cdots 61}a^{19}-\frac{13\cdots 02}{49\cdots 61}a^{18}-\frac{10\cdots 48}{49\cdots 61}a^{17}-\frac{30\cdots 88}{49\cdots 61}a^{16}+\frac{24\cdots 46}{37\cdots 97}a^{15}+\frac{43\cdots 07}{49\cdots 61}a^{14}-\frac{41\cdots 98}{49\cdots 61}a^{13}-\frac{94\cdots 39}{49\cdots 61}a^{12}+\frac{20\cdots 71}{49\cdots 61}a^{11}+\frac{76\cdots 69}{49\cdots 61}a^{10}-\frac{11\cdots 87}{49\cdots 61}a^{9}-\frac{21\cdots 47}{49\cdots 61}a^{8}-\frac{14\cdots 86}{37\cdots 97}a^{7}+\frac{10\cdots 00}{49\cdots 61}a^{6}+\frac{19\cdots 95}{49\cdots 61}a^{5}+\frac{13\cdots 72}{49\cdots 61}a^{4}-\frac{62\cdots 99}{49\cdots 61}a^{3}-\frac{15\cdots 18}{49\cdots 61}a^{2}+\frac{60\cdots 34}{49\cdots 61}a+\frac{19\cdots 31}{49\cdots 61}$, $\frac{28\cdots 77}{49\cdots 61}a^{19}-\frac{11\cdots 59}{49\cdots 61}a^{18}-\frac{20\cdots 34}{49\cdots 61}a^{17}+\frac{59\cdots 86}{49\cdots 61}a^{16}+\frac{55\cdots 82}{49\cdots 61}a^{15}-\frac{90\cdots 04}{49\cdots 61}a^{14}-\frac{75\cdots 42}{49\cdots 61}a^{13}+\frac{39\cdots 74}{49\cdots 61}a^{12}+\frac{52\cdots 20}{49\cdots 61}a^{11}+\frac{13\cdots 43}{49\cdots 61}a^{10}-\frac{16\cdots 64}{49\cdots 61}a^{9}-\frac{11\cdots 06}{49\cdots 61}a^{8}+\frac{22\cdots 59}{49\cdots 61}a^{7}+\frac{19\cdots 77}{49\cdots 61}a^{6}-\frac{15\cdots 97}{49\cdots 61}a^{5}-\frac{14\cdots 25}{49\cdots 61}a^{4}+\frac{49\cdots 57}{49\cdots 61}a^{3}+\frac{46\cdots 13}{49\cdots 61}a^{2}-\frac{54\cdots 75}{49\cdots 61}a-\frac{46\cdots 65}{49\cdots 61}$, $\frac{23\cdots 38}{49\cdots 61}a^{19}-\frac{55\cdots 45}{49\cdots 61}a^{18}-\frac{18\cdots 36}{49\cdots 61}a^{17}+\frac{16\cdots 23}{49\cdots 61}a^{16}+\frac{39\cdots 16}{37\cdots 97}a^{15}+\frac{16\cdots 54}{49\cdots 61}a^{14}-\frac{64\cdots 49}{49\cdots 61}a^{13}-\frac{88\cdots 75}{49\cdots 61}a^{12}+\frac{33\cdots 46}{49\cdots 61}a^{11}+\frac{64\cdots 27}{37\cdots 97}a^{10}-\frac{25\cdots 00}{49\cdots 61}a^{9}-\frac{18\cdots 98}{37\cdots 97}a^{8}-\frac{19\cdots 83}{49\cdots 61}a^{7}+\frac{74\cdots 14}{49\cdots 61}a^{6}+\frac{16\cdots 56}{49\cdots 61}a^{5}+\frac{36\cdots 28}{49\cdots 61}a^{4}-\frac{40\cdots 76}{49\cdots 61}a^{3}-\frac{15\cdots 38}{49\cdots 61}a^{2}+\frac{30\cdots 68}{49\cdots 61}a+\frac{14\cdots 42}{49\cdots 61}$, $\frac{61\cdots 47}{49\cdots 61}a^{19}-\frac{41\cdots 66}{49\cdots 61}a^{18}-\frac{50\cdots 73}{49\cdots 61}a^{17}-\frac{33\cdots 80}{49\cdots 61}a^{16}+\frac{14\cdots 82}{49\cdots 61}a^{15}+\frac{25\cdots 90}{49\cdots 61}a^{14}-\frac{17\cdots 25}{49\cdots 61}a^{13}-\frac{51\cdots 29}{49\cdots 61}a^{12}+\frac{73\cdots 67}{49\cdots 61}a^{11}+\frac{39\cdots 10}{49\cdots 61}a^{10}+\frac{16\cdots 29}{49\cdots 61}a^{9}-\frac{10\cdots 83}{49\cdots 61}a^{8}-\frac{13\cdots 47}{49\cdots 61}a^{7}+\frac{19\cdots 06}{49\cdots 61}a^{6}+\frac{12\cdots 05}{49\cdots 61}a^{5}+\frac{39\cdots 32}{49\cdots 61}a^{4}-\frac{39\cdots 53}{49\cdots 61}a^{3}-\frac{18\cdots 55}{49\cdots 61}a^{2}+\frac{39\cdots 23}{49\cdots 61}a+\frac{16\cdots 81}{37\cdots 97}$, $\frac{89\cdots 17}{49\cdots 61}a^{19}-\frac{21\cdots 86}{49\cdots 61}a^{18}-\frac{69\cdots 23}{49\cdots 61}a^{17}+\frac{66\cdots 06}{49\cdots 61}a^{16}+\frac{19\cdots 80}{49\cdots 61}a^{15}+\frac{52\cdots 98}{49\cdots 61}a^{14}-\frac{25\cdots 97}{49\cdots 61}a^{13}-\frac{32\cdots 85}{49\cdots 61}a^{12}+\frac{14\cdots 19}{49\cdots 61}a^{11}+\frac{24\cdots 94}{37\cdots 97}a^{10}-\frac{19\cdots 04}{49\cdots 61}a^{9}-\frac{99\cdots 71}{49\cdots 61}a^{8}-\frac{54\cdots 38}{49\cdots 61}a^{7}+\frac{50\cdots 52}{37\cdots 97}a^{6}+\frac{67\cdots 59}{49\cdots 61}a^{5}-\frac{68\cdots 81}{49\cdots 61}a^{4}-\frac{23\cdots 84}{49\cdots 61}a^{3}-\frac{35\cdots 12}{49\cdots 61}a^{2}+\frac{18\cdots 01}{37\cdots 97}a+\frac{59\cdots 41}{49\cdots 61}$, $\frac{10\cdots 51}{49\cdots 61}a^{19}-\frac{23\cdots 59}{37\cdots 97}a^{18}-\frac{81\cdots 71}{49\cdots 61}a^{17}+\frac{11\cdots 03}{49\cdots 61}a^{16}+\frac{23\cdots 78}{49\cdots 61}a^{15}-\frac{53\cdots 60}{49\cdots 61}a^{14}-\frac{31\cdots 09}{49\cdots 61}a^{13}-\frac{23\cdots 55}{49\cdots 61}a^{12}+\frac{53\cdots 72}{13\cdots 53}a^{11}+\frac{30\cdots 73}{49\cdots 61}a^{10}-\frac{45\cdots 74}{49\cdots 61}a^{9}-\frac{11\cdots 69}{49\cdots 61}a^{8}+\frac{15\cdots 91}{37\cdots 97}a^{7}+\frac{12\cdots 05}{49\cdots 61}a^{6}+\frac{41\cdots 82}{49\cdots 61}a^{5}-\frac{16\cdots 61}{13\cdots 53}a^{4}-\frac{21\cdots 94}{49\cdots 61}a^{3}+\frac{12\cdots 36}{49\cdots 61}a^{2}+\frac{18\cdots 64}{37\cdots 97}a-\frac{88\cdots 96}{49\cdots 61}$, $\frac{36\cdots 79}{49\cdots 61}a^{19}-\frac{10\cdots 30}{49\cdots 61}a^{18}-\frac{27\cdots 32}{49\cdots 61}a^{17}+\frac{38\cdots 45}{49\cdots 61}a^{16}+\frac{77\cdots 96}{49\cdots 61}a^{15}-\frac{90\cdots 22}{49\cdots 61}a^{14}-\frac{10\cdots 97}{49\cdots 61}a^{13}-\frac{70\cdots 81}{37\cdots 97}a^{12}+\frac{44\cdots 29}{37\cdots 97}a^{11}+\frac{10\cdots 15}{49\cdots 61}a^{10}-\frac{70\cdots 36}{37\cdots 97}a^{9}-\frac{32\cdots 26}{49\cdots 61}a^{8}-\frac{14\cdots 94}{49\cdots 61}a^{7}+\frac{18\cdots 08}{49\cdots 61}a^{6}+\frac{16\cdots 66}{49\cdots 61}a^{5}-\frac{22\cdots 81}{49\cdots 61}a^{4}-\frac{48\cdots 40}{49\cdots 61}a^{3}-\frac{52\cdots 30}{49\cdots 61}a^{2}+\frac{44\cdots 20}{49\cdots 61}a+\frac{93\cdots 00}{49\cdots 61}$, $\frac{13\cdots 28}{49\cdots 61}a^{19}-\frac{25\cdots 91}{49\cdots 61}a^{18}-\frac{11\cdots 94}{49\cdots 61}a^{17}+\frac{57\cdots 32}{49\cdots 61}a^{16}+\frac{25\cdots 98}{37\cdots 97}a^{15}+\frac{18\cdots 15}{49\cdots 61}a^{14}-\frac{45\cdots 18}{49\cdots 61}a^{13}-\frac{64\cdots 62}{49\cdots 61}a^{12}+\frac{26\cdots 36}{49\cdots 61}a^{11}+\frac{59\cdots 48}{49\cdots 61}a^{10}-\frac{49\cdots 23}{49\cdots 61}a^{9}-\frac{19\cdots 81}{49\cdots 61}a^{8}-\frac{44\cdots 17}{49\cdots 61}a^{7}+\frac{17\cdots 49}{49\cdots 61}a^{6}+\frac{66\cdots 38}{49\cdots 61}a^{5}-\frac{79\cdots 16}{49\cdots 61}a^{4}-\frac{17\cdots 25}{37\cdots 97}a^{3}+\frac{18\cdots 63}{49\cdots 61}a^{2}+\frac{17\cdots 18}{49\cdots 61}a-\frac{15\cdots 68}{49\cdots 61}$, $\frac{11\cdots 12}{20\cdots 99}a^{19}+\frac{15\cdots 52}{20\cdots 99}a^{18}-\frac{10\cdots 56}{20\cdots 99}a^{17}-\frac{23\cdots 23}{20\cdots 99}a^{16}+\frac{32\cdots 50}{20\cdots 99}a^{15}+\frac{94\cdots 44}{20\cdots 99}a^{14}-\frac{39\cdots 49}{20\cdots 99}a^{13}-\frac{15\cdots 63}{20\cdots 99}a^{12}+\frac{14\cdots 62}{20\cdots 99}a^{11}+\frac{11\cdots 11}{20\cdots 99}a^{10}+\frac{54\cdots 67}{20\cdots 99}a^{9}-\frac{28\cdots 65}{20\cdots 99}a^{8}-\frac{37\cdots 32}{20\cdots 99}a^{7}+\frac{75\cdots 12}{20\cdots 99}a^{6}+\frac{33\cdots 11}{20\cdots 99}a^{5}+\frac{96\cdots 63}{20\cdots 99}a^{4}-\frac{26\cdots 95}{55\cdots 27}a^{3}-\frac{55\cdots 42}{20\cdots 99}a^{2}+\frac{10\cdots 74}{20\cdots 99}a+\frac{67\cdots 94}{20\cdots 99}$, $\frac{54\cdots 92}{49\cdots 61}a^{19}-\frac{23\cdots 60}{49\cdots 61}a^{18}-\frac{44\cdots 89}{49\cdots 61}a^{17}-\frac{41\cdots 52}{49\cdots 61}a^{16}+\frac{12\cdots 99}{49\cdots 61}a^{15}+\frac{26\cdots 74}{49\cdots 61}a^{14}-\frac{14\cdots 14}{49\cdots 61}a^{13}-\frac{50\cdots 50}{49\cdots 61}a^{12}+\frac{39\cdots 05}{49\cdots 61}a^{11}+\frac{35\cdots 28}{49\cdots 61}a^{10}+\frac{30\cdots 20}{49\cdots 61}a^{9}-\frac{73\cdots 89}{49\cdots 61}a^{8}-\frac{15\cdots 86}{49\cdots 61}a^{7}-\frac{47\cdots 53}{49\cdots 61}a^{6}+\frac{86\cdots 02}{49\cdots 61}a^{5}+\frac{59\cdots 94}{49\cdots 61}a^{4}-\frac{10\cdots 24}{49\cdots 61}a^{3}-\frac{13\cdots 79}{49\cdots 61}a^{2}-\frac{96\cdots 86}{49\cdots 61}a+\frac{77\cdots 31}{49\cdots 61}$, $\frac{21\cdots 11}{49\cdots 61}a^{19}-\frac{95\cdots 01}{49\cdots 61}a^{18}-\frac{14\cdots 01}{49\cdots 61}a^{17}+\frac{48\cdots 08}{49\cdots 61}a^{16}+\frac{39\cdots 31}{49\cdots 61}a^{15}-\frac{21\cdots 68}{13\cdots 53}a^{14}-\frac{41\cdots 10}{37\cdots 97}a^{13}+\frac{43\cdots 01}{49\cdots 61}a^{12}+\frac{36\cdots 56}{49\cdots 61}a^{11}+\frac{79\cdots 48}{37\cdots 97}a^{10}-\frac{92\cdots 35}{37\cdots 97}a^{9}-\frac{16\cdots 90}{13\cdots 53}a^{8}+\frac{16\cdots 04}{49\cdots 61}a^{7}+\frac{11\cdots 90}{49\cdots 61}a^{6}-\frac{27\cdots 76}{13\cdots 53}a^{5}-\frac{74\cdots 74}{49\cdots 61}a^{4}+\frac{30\cdots 74}{49\cdots 61}a^{3}+\frac{21\cdots 33}{49\cdots 61}a^{2}-\frac{32\cdots 31}{49\cdots 61}a-\frac{21\cdots 00}{49\cdots 61}$, $\frac{51\cdots 14}{49\cdots 61}a^{19}+\frac{23\cdots 34}{49\cdots 61}a^{18}-\frac{48\cdots 49}{49\cdots 61}a^{17}-\frac{18\cdots 29}{37\cdots 97}a^{16}+\frac{14\cdots 57}{49\cdots 61}a^{15}+\frac{81\cdots 52}{49\cdots 61}a^{14}-\frac{14\cdots 80}{49\cdots 61}a^{13}-\frac{12\cdots 30}{49\cdots 61}a^{12}-\frac{62\cdots 85}{13\cdots 53}a^{11}+\frac{81\cdots 49}{49\cdots 61}a^{10}+\frac{10\cdots 76}{49\cdots 61}a^{9}-\frac{18\cdots 18}{49\cdots 61}a^{8}-\frac{41\cdots 97}{49\cdots 61}a^{7}-\frac{28\cdots 59}{37\cdots 97}a^{6}+\frac{35\cdots 51}{49\cdots 61}a^{5}+\frac{37\cdots 29}{13\cdots 53}a^{4}-\frac{11\cdots 07}{49\cdots 61}a^{3}-\frac{53\cdots 19}{49\cdots 61}a^{2}+\frac{11\cdots 63}{49\cdots 61}a+\frac{56\cdots 35}{49\cdots 61}$, $\frac{28\cdots 81}{49\cdots 61}a^{19}-\frac{18\cdots 53}{49\cdots 61}a^{18}-\frac{23\cdots 67}{49\cdots 61}a^{17}-\frac{14\cdots 36}{49\cdots 61}a^{16}+\frac{14\cdots 58}{10\cdots 81}a^{15}+\frac{11\cdots 99}{49\cdots 61}a^{14}-\frac{86\cdots 88}{49\cdots 61}a^{13}-\frac{23\cdots 01}{49\cdots 61}a^{12}+\frac{29\cdots 08}{37\cdots 97}a^{11}+\frac{18\cdots 41}{49\cdots 61}a^{10}+\frac{45\cdots 30}{49\cdots 61}a^{9}-\frac{49\cdots 22}{49\cdots 61}a^{8}-\frac{56\cdots 96}{49\cdots 61}a^{7}+\frac{19\cdots 15}{49\cdots 61}a^{6}+\frac{58\cdots 74}{49\cdots 61}a^{5}+\frac{11\cdots 21}{49\cdots 61}a^{4}-\frac{19\cdots 16}{49\cdots 61}a^{3}-\frac{73\cdots 96}{49\cdots 61}a^{2}+\frac{21\cdots 05}{49\cdots 61}a+\frac{90\cdots 23}{49\cdots 61}$, $\frac{53\cdots 22}{49\cdots 61}a^{19}-\frac{25\cdots 72}{49\cdots 61}a^{18}-\frac{44\cdots 33}{49\cdots 61}a^{17}-\frac{36\cdots 65}{49\cdots 61}a^{16}+\frac{12\cdots 41}{49\cdots 61}a^{15}+\frac{24\cdots 46}{49\cdots 61}a^{14}-\frac{15\cdots 82}{49\cdots 61}a^{13}-\frac{48\cdots 65}{49\cdots 61}a^{12}+\frac{61\cdots 72}{49\cdots 61}a^{11}+\frac{36\cdots 78}{49\cdots 61}a^{10}+\frac{17\cdots 52}{49\cdots 61}a^{9}-\frac{92\cdots 92}{49\cdots 61}a^{8}-\frac{13\cdots 19}{49\cdots 61}a^{7}+\frac{15\cdots 86}{49\cdots 61}a^{6}+\frac{12\cdots 08}{49\cdots 61}a^{5}+\frac{38\cdots 51}{49\cdots 61}a^{4}-\frac{38\cdots 87}{49\cdots 61}a^{3}-\frac{17\cdots 07}{49\cdots 61}a^{2}+\frac{39\cdots 01}{49\cdots 61}a+\frac{19\cdots 44}{49\cdots 61}$, $\frac{31\cdots 69}{57\cdots 91}a^{19}-\frac{76\cdots 72}{57\cdots 91}a^{18}-\frac{24\cdots 84}{57\cdots 91}a^{17}+\frac{25\cdots 64}{57\cdots 91}a^{16}+\frac{71\cdots 64}{57\cdots 91}a^{15}+\frac{10\cdots 07}{57\cdots 91}a^{14}-\frac{95\cdots 11}{57\cdots 91}a^{13}-\frac{10\cdots 36}{57\cdots 91}a^{12}+\frac{57\cdots 85}{57\cdots 91}a^{11}+\frac{10\cdots 13}{57\cdots 91}a^{10}-\frac{10\cdots 82}{57\cdots 91}a^{9}-\frac{36\cdots 59}{57\cdots 91}a^{8}-\frac{72\cdots 30}{44\cdots 07}a^{7}+\frac{30\cdots 43}{57\cdots 91}a^{6}+\frac{16\cdots 51}{57\cdots 91}a^{5}-\frac{10\cdots 89}{57\cdots 91}a^{4}-\frac{59\cdots 46}{57\cdots 91}a^{3}+\frac{14\cdots 04}{57\cdots 91}a^{2}+\frac{59\cdots 87}{57\cdots 91}a-\frac{51\cdots 54}{57\cdots 91}$, $\frac{67\cdots 85}{49\cdots 61}a^{19}-\frac{76\cdots 00}{49\cdots 61}a^{18}-\frac{54\cdots 65}{49\cdots 61}a^{17}-\frac{10\cdots 77}{49\cdots 61}a^{16}+\frac{16\cdots 21}{49\cdots 61}a^{15}+\frac{15\cdots 16}{37\cdots 97}a^{14}-\frac{20\cdots 83}{49\cdots 61}a^{13}-\frac{46\cdots 61}{49\cdots 61}a^{12}+\frac{10\cdots 06}{49\cdots 61}a^{11}+\frac{37\cdots 96}{49\cdots 61}a^{10}+\frac{16\cdots 55}{49\cdots 61}a^{9}-\frac{10\cdots 31}{49\cdots 61}a^{8}-\frac{97\cdots 83}{49\cdots 61}a^{7}+\frac{53\cdots 67}{49\cdots 61}a^{6}+\frac{10\cdots 64}{49\cdots 61}a^{5}+\frac{77\cdots 80}{49\cdots 61}a^{4}-\frac{33\cdots 16}{49\cdots 61}a^{3}-\frac{23\cdots 67}{13\cdots 53}a^{2}+\frac{36\cdots 92}{49\cdots 61}a+\frac{12\cdots 48}{49\cdots 61}$ 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:  \( 8904025444480 \) (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:  \( 18 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 8904025444480 \cdot 2}{2\cdot\sqrt{516853980088694624404163683037236690944}}\cr\approx \mathstrut & 0.410678884009933 \end{aligned}\] (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 - 83*x^18 - 108*x^17 + 2380*x^16 + 5760*x^15 - 27524*x^14 - 103968*x^13 + 77348*x^12 + 735960*x^11 + 616108*x^10 - 1615050*x^9 - 3200232*x^8 - 714960*x^7 + 2411685*x^6 + 1666554*x^5 - 423490*x^4 - 635688*x^3 - 65073*x^2 + 67764*x + 15313) 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 - 83*x^18 - 108*x^17 + 2380*x^16 + 5760*x^15 - 27524*x^14 - 103968*x^13 + 77348*x^12 + 735960*x^11 + 616108*x^10 - 1615050*x^9 - 3200232*x^8 - 714960*x^7 + 2411685*x^6 + 1666554*x^5 - 423490*x^4 - 635688*x^3 - 65073*x^2 + 67764*x + 15313, 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 - 83*x^18 - 108*x^17 + 2380*x^16 + 5760*x^15 - 27524*x^14 - 103968*x^13 + 77348*x^12 + 735960*x^11 + 616108*x^10 - 1615050*x^9 - 3200232*x^8 - 714960*x^7 + 2411685*x^6 + 1666554*x^5 - 423490*x^4 - 635688*x^3 - 65073*x^2 + 67764*x + 15313); 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 - 83*x^18 - 108*x^17 + 2380*x^16 + 5760*x^15 - 27524*x^14 - 103968*x^13 + 77348*x^12 + 735960*x^11 + 616108*x^10 - 1615050*x^9 - 3200232*x^8 - 714960*x^7 + 2411685*x^6 + 1666554*x^5 - 423490*x^4 - 635688*x^3 - 65073*x^2 + 67764*x + 15313); 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

\(\Q(\sqrt{91}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{273}) \), \(\Q(\sqrt{3}, \sqrt{91})\), 10.10.249828821987576832.1 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: 10.10.249828821987576832.1

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 ${\href{/padicField/5.10.0.1}{10} }^{2}$ R R R ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{10}$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{10}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{10}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

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 2.5.2.10a1.2$x^{10} + 2 x^{7} + 4 x^{5} + x^{4} + 4 x^{2} + 9$$2$$5$$10$$C_{10}$$$[2]^{5}$$
2.5.2.10a1.2$x^{10} + 2 x^{7} + 4 x^{5} + x^{4} + 4 x^{2} + 9$$2$$5$$10$$C_{10}$$$[2]^{5}$$
\(3\) Copy content Toggle raw display 3.5.2.5a1.1$x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 7 x + 1$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
3.5.2.5a1.1$x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 7 x + 1$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
\(7\) Copy content Toggle raw display 7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(11\) Copy content Toggle raw display 11.5.1.0a1.1$x^{5} + 10 x^{2} + 9$$1$$5$$0$$C_5$$$[\ ]^{5}$$
11.1.5.4a1.3$x^{5} + 44$$5$$1$$4$$C_5$$$[\ ]_{5}$$
11.5.1.0a1.1$x^{5} + 10 x^{2} + 9$$1$$5$$0$$C_5$$$[\ ]^{5}$$
11.1.5.4a1.3$x^{5} + 44$$5$$1$$4$$C_5$$$[\ ]_{5}$$
\(13\) Copy content Toggle raw display 13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$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)