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Properties

Label 20.4.838...125.2
Degree $20$
Signature $[4, 8]$
Discriminant $8.387\times 10^{55}$
Root discriminant \(625.43\)
Ramified primes $5,7,41$
Class number $5$ (GRH)
Class group [5] (GRH)
Galois group $C_2\times F_5$ (as 20T9)

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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 - 5*x^19 - 35*x^18 + 215*x^17 + 420*x^16 + 39885*x^15 - 185975*x^14 + 1641450*x^13 - 9551200*x^12 + 14329275*x^11 + 696150660*x^10 - 2107342900*x^9 + 41741808575*x^8 - 105758201050*x^7 + 911902235600*x^6 + 3715280644050*x^5 - 5344988560625*x^4 + 221169857291125*x^3 - 60583973026250*x^2 + 691307088931500*x + 13445341298628775)
 
Copy content gp:K = bnfinit(y^20 - 5*y^19 - 35*y^18 + 215*y^17 + 420*y^16 + 39885*y^15 - 185975*y^14 + 1641450*y^13 - 9551200*y^12 + 14329275*y^11 + 696150660*y^10 - 2107342900*y^9 + 41741808575*y^8 - 105758201050*y^7 + 911902235600*y^6 + 3715280644050*y^5 - 5344988560625*y^4 + 221169857291125*y^3 - 60583973026250*y^2 + 691307088931500*y + 13445341298628775, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 - 35*x^18 + 215*x^17 + 420*x^16 + 39885*x^15 - 185975*x^14 + 1641450*x^13 - 9551200*x^12 + 14329275*x^11 + 696150660*x^10 - 2107342900*x^9 + 41741808575*x^8 - 105758201050*x^7 + 911902235600*x^6 + 3715280644050*x^5 - 5344988560625*x^4 + 221169857291125*x^3 - 60583973026250*x^2 + 691307088931500*x + 13445341298628775);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 5*x^19 - 35*x^18 + 215*x^17 + 420*x^16 + 39885*x^15 - 185975*x^14 + 1641450*x^13 - 9551200*x^12 + 14329275*x^11 + 696150660*x^10 - 2107342900*x^9 + 41741808575*x^8 - 105758201050*x^7 + 911902235600*x^6 + 3715280644050*x^5 - 5344988560625*x^4 + 221169857291125*x^3 - 60583973026250*x^2 + 691307088931500*x + 13445341298628775)
 

\( x^{20} - 5 x^{19} - 35 x^{18} + 215 x^{17} + 420 x^{16} + 39885 x^{15} - 185975 x^{14} + \cdots + 13\!\cdots\!75 \) 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:  $[4, 8]$
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:   \(83867218967598487041513866410602857358753681182861328125\) \(\medspace = 5^{31}\cdot 7^{10}\cdot 41^{16}\) 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:  \(625.43\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $5^{31/20}7^{1/2}41^{4/5}\approx 625.4313002647299$
Ramified primes:   \(5\), \(7\), \(41\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_4$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{205}a^{10}+\frac{9}{41}a^{9}-\frac{1}{41}a^{8}+\frac{10}{41}a^{7}-\frac{5}{41}a^{6}-\frac{8}{41}a^{5}+\frac{8}{41}a^{4}+\frac{20}{41}a^{2}+\frac{5}{41}a+\frac{15}{41}$, $\frac{1}{205}a^{11}+\frac{4}{41}a^{9}+\frac{14}{41}a^{8}-\frac{4}{41}a^{7}+\frac{12}{41}a^{6}-\frac{1}{41}a^{5}+\frac{9}{41}a^{4}+\frac{20}{41}a^{3}+\frac{7}{41}a^{2}-\frac{5}{41}a-\frac{19}{41}$, $\frac{1}{410}a^{12}+\frac{39}{82}a^{9}+\frac{8}{41}a^{8}-\frac{12}{41}a^{7}+\frac{17}{82}a^{6}-\frac{18}{41}a^{5}+\frac{12}{41}a^{4}+\frac{7}{82}a^{3}-\frac{18}{41}a^{2}+\frac{2}{41}a-\frac{13}{82}$, $\frac{1}{2050}a^{13}-\frac{1}{410}a^{10}+\frac{51}{205}a^{9}+\frac{88}{205}a^{8}-\frac{3}{82}a^{7}-\frac{2}{41}a^{6}-\frac{18}{41}a^{5}-\frac{7}{82}a^{4}+\frac{23}{205}a^{3}-\frac{6}{41}a^{2}+\frac{27}{82}a-\frac{13}{41}$, $\frac{1}{2050}a^{14}-\frac{1}{410}a^{11}+\frac{48}{205}a^{9}+\frac{17}{82}a^{8}-\frac{20}{41}a^{7}-\frac{9}{41}a^{6}-\frac{11}{82}a^{5}+\frac{33}{205}a^{4}-\frac{6}{41}a^{3}+\frac{37}{82}a^{2}+\frac{19}{41}a+\frac{14}{41}$, $\frac{1}{2050}a^{15}+\frac{6}{41}a^{9}-\frac{5}{41}a^{8}-\frac{9}{41}a^{7}-\frac{3}{41}a^{6}+\frac{18}{205}a^{5}-\frac{9}{41}a^{4}-\frac{19}{41}a^{3}-\frac{16}{41}a^{2}-\frac{19}{41}a+\frac{23}{82}$, $\frac{1}{2050}a^{16}+\frac{12}{41}a^{9}-\frac{20}{41}a^{8}-\frac{16}{41}a^{7}-\frac{52}{205}a^{6}-\frac{15}{41}a^{5}-\frac{13}{41}a^{4}-\frac{16}{41}a^{3}-\frac{4}{41}a^{2}-\frac{31}{82}a+\frac{1}{41}$, $\frac{1}{2050}a^{17}+\frac{14}{41}a^{9}+\frac{3}{41}a^{8}+\frac{23}{205}a^{7}-\frac{2}{41}a^{6}+\frac{16}{41}a^{5}-\frac{4}{41}a^{4}-\frac{4}{41}a^{3}+\frac{29}{82}a^{2}-\frac{12}{41}a+\frac{2}{41}$, $\frac{1}{84050}a^{18}+\frac{1}{42025}a^{17}+\frac{11}{84050}a^{16}-\frac{13}{84050}a^{15}-\frac{8}{42025}a^{14}-\frac{3}{84050}a^{13}-\frac{4}{8405}a^{12}+\frac{9}{8405}a^{11}+\frac{5}{3362}a^{10}-\frac{2271}{8405}a^{9}-\frac{1621}{8405}a^{8}-\frac{203}{16810}a^{7}-\frac{832}{8405}a^{6}+\frac{611}{8405}a^{5}+\frac{8249}{16810}a^{4}-\frac{133}{410}a^{3}-\frac{224}{1681}a^{2}+\frac{132}{1681}a-\frac{1113}{3362}$, $\frac{1}{11\cdots 50}a^{19}+\frac{22\cdots 19}{11\cdots 50}a^{18}+\frac{44\cdots 07}{11\cdots 50}a^{17}-\frac{13\cdots 22}{57\cdots 25}a^{16}+\frac{17\cdots 89}{11\cdots 50}a^{15}+\frac{21\cdots 19}{57\cdots 25}a^{14}+\frac{71\cdots 69}{11\cdots 05}a^{13}+\frac{59\cdots 59}{28\cdots 05}a^{12}+\frac{18\cdots 94}{11\cdots 05}a^{11}+\frac{12\cdots 62}{23\cdots 01}a^{10}-\frac{49\cdots 28}{11\cdots 05}a^{9}+\frac{13\cdots 12}{76\cdots 55}a^{8}-\frac{65\cdots 24}{11\cdots 05}a^{7}+\frac{18\cdots 03}{11\cdots 05}a^{6}-\frac{19\cdots 03}{11\cdots 05}a^{5}-\frac{48\cdots 27}{23\cdots 10}a^{4}+\frac{12\cdots 09}{46\cdots 02}a^{3}+\frac{11\cdots 69}{46\cdots 02}a^{2}+\frac{20\cdots 08}{23\cdots 01}a+\frac{13\cdots 15}{46\cdots 02}$ 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:  $C_{10}$, which has order $10$ (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:  $11$
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:   $\frac{51\cdots 82}{75\cdots 29}a^{19}-\frac{64\cdots 01}{15\cdots 58}a^{18}-\frac{13\cdots 71}{75\cdots 29}a^{17}+\frac{83\cdots 25}{15\cdots 58}a^{16}+\frac{28\cdots 92}{37\cdots 45}a^{15}+\frac{24\cdots 51}{75\cdots 29}a^{14}-\frac{29\cdots 01}{15\cdots 58}a^{13}+\frac{80\cdots 62}{75\cdots 29}a^{12}-\frac{94\cdots 84}{75\cdots 29}a^{11}+\frac{84\cdots 21}{15\cdots 58}a^{10}+\frac{23\cdots 30}{75\cdots 29}a^{9}-\frac{56\cdots 35}{49\cdots 79}a^{8}+\frac{39\cdots 95}{15\cdots 58}a^{7}-\frac{13\cdots 70}{75\cdots 29}a^{6}+\frac{71\cdots 26}{75\cdots 29}a^{5}-\frac{32\cdots 15}{15\cdots 58}a^{4}+\frac{16\cdots 45}{15\cdots 58}a^{3}+\frac{45\cdots 85}{75\cdots 29}a^{2}-\frac{28\cdots 20}{75\cdots 29}a+\frac{15\cdots 17}{75\cdots 29}$, $\frac{31\cdots 33}{68\cdots 50}a^{19}+\frac{58\cdots 09}{68\cdots 50}a^{18}-\frac{69\cdots 59}{34\cdots 25}a^{17}+\frac{29\cdots 43}{34\cdots 25}a^{16}+\frac{14\cdots 16}{34\cdots 25}a^{15}-\frac{15\cdots 71}{68\cdots 05}a^{14}+\frac{46\cdots 99}{13\cdots 10}a^{13}-\frac{33\cdots 21}{68\cdots 05}a^{12}+\frac{39\cdots 34}{68\cdots 05}a^{11}-\frac{69\cdots 99}{13\cdots 10}a^{10}+\frac{18\cdots 64}{68\cdots 05}a^{9}-\frac{47\cdots 98}{45\cdots 55}a^{8}-\frac{60\cdots 73}{13\cdots 10}a^{7}+\frac{42\cdots 83}{68\cdots 05}a^{6}-\frac{40\cdots 94}{68\cdots 05}a^{5}+\frac{65\cdots 98}{13\cdots 21}a^{4}-\frac{50\cdots 31}{27\cdots 42}a^{3}+\frac{53\cdots 78}{13\cdots 21}a^{2}+\frac{99\cdots 73}{27\cdots 42}a-\frac{45\cdots 55}{13\cdots 21}$, $\frac{42\cdots 02}{34\cdots 25}a^{19}-\frac{16\cdots 94}{34\cdots 25}a^{18}-\frac{66\cdots 92}{34\cdots 25}a^{17}-\frac{82\cdots 67}{68\cdots 50}a^{16}+\frac{60\cdots 41}{68\cdots 50}a^{15}+\frac{44\cdots 63}{68\cdots 05}a^{14}-\frac{16\cdots 71}{68\cdots 05}a^{13}+\frac{17\cdots 86}{68\cdots 05}a^{12}-\frac{14\cdots 46}{68\cdots 05}a^{11}+\frac{80\cdots 43}{68\cdots 05}a^{10}+\frac{20\cdots 02}{68\cdots 05}a^{9}-\frac{31\cdots 89}{45\cdots 55}a^{8}+\frac{40\cdots 08}{68\cdots 05}a^{7}-\frac{17\cdots 51}{68\cdots 05}a^{6}+\frac{18\cdots 83}{68\cdots 05}a^{5}-\frac{51\cdots 20}{13\cdots 21}a^{4}+\frac{58\cdots 89}{13\cdots 21}a^{3}+\frac{18\cdots 63}{13\cdots 21}a^{2}-\frac{13\cdots 47}{27\cdots 42}a+\frac{21\cdots 81}{27\cdots 42}$, $\frac{86\cdots 79}{46\cdots 02}a^{19}-\frac{88\cdots 04}{57\cdots 25}a^{18}-\frac{77\cdots 59}{57\cdots 25}a^{17}+\frac{25\cdots 77}{23\cdots 10}a^{16}+\frac{17\cdots 81}{23\cdots 10}a^{15}+\frac{74\cdots 99}{23\cdots 10}a^{14}-\frac{22\cdots 47}{23\cdots 10}a^{13}+\frac{35\cdots 22}{23\cdots 01}a^{12}+\frac{17\cdots 61}{23\cdots 10}a^{11}+\frac{19\cdots 31}{23\cdots 10}a^{10}+\frac{18\cdots 24}{23\cdots 01}a^{9}-\frac{10\cdots 63}{15\cdots 10}a^{8}+\frac{14\cdots 37}{56\cdots 10}a^{7}-\frac{59\cdots 16}{23\cdots 01}a^{6}+\frac{58\cdots 97}{46\cdots 02}a^{5}+\frac{20\cdots 99}{23\cdots 01}a^{4}+\frac{28\cdots 29}{23\cdots 01}a^{3}+\frac{57\cdots 93}{46\cdots 02}a^{2}+\frac{18\cdots 48}{23\cdots 01}a+\frac{24\cdots 93}{46\cdots 02}$, $\frac{39\cdots 03}{46\cdots 02}a^{19}-\frac{53\cdots 31}{11\cdots 50}a^{18}-\frac{29\cdots 73}{57\cdots 25}a^{17}+\frac{35\cdots 49}{11\cdots 05}a^{16}+\frac{80\cdots 09}{46\cdots 02}a^{15}+\frac{62\cdots 03}{23\cdots 10}a^{14}-\frac{27\cdots 79}{11\cdots 05}a^{13}+\frac{18\cdots 72}{23\cdots 01}a^{12}-\frac{82\cdots 29}{46\cdots 02}a^{11}+\frac{27\cdots 03}{11\cdots 05}a^{10}+\frac{13\cdots 30}{23\cdots 01}a^{9}-\frac{34\cdots 41}{15\cdots 10}a^{8}+\frac{66\cdots 07}{28\cdots 05}a^{7}-\frac{11\cdots 10}{23\cdots 01}a^{6}+\frac{15\cdots 71}{46\cdots 02}a^{5}+\frac{20\cdots 81}{46\cdots 02}a^{4}-\frac{75\cdots 83}{46\cdots 02}a^{3}+\frac{32\cdots 01}{46\cdots 02}a^{2}+\frac{87\cdots 05}{23\cdots 01}a-\frac{25\cdots 87}{46\cdots 02}$, $\frac{90\cdots 03}{57\cdots 25}a^{19}-\frac{78\cdots 07}{57\cdots 25}a^{18}-\frac{40\cdots 86}{57\cdots 25}a^{17}+\frac{15\cdots 38}{57\cdots 25}a^{16}-\frac{19\cdots 47}{11\cdots 50}a^{15}+\frac{76\cdots 71}{11\cdots 50}a^{14}-\frac{12\cdots 19}{23\cdots 10}a^{13}+\frac{10\cdots 23}{23\cdots 10}a^{12}-\frac{81\cdots 91}{23\cdots 10}a^{11}+\frac{34\cdots 29}{23\cdots 10}a^{10}+\frac{86\cdots 97}{23\cdots 10}a^{9}-\frac{59\cdots 89}{15\cdots 10}a^{8}+\frac{18\cdots 93}{23\cdots 10}a^{7}-\frac{11\cdots 39}{23\cdots 10}a^{6}+\frac{71\cdots 03}{23\cdots 10}a^{5}-\frac{23\cdots 79}{23\cdots 10}a^{4}+\frac{10\cdots 75}{46\cdots 02}a^{3}+\frac{60\cdots 57}{46\cdots 02}a^{2}-\frac{40\cdots 47}{46\cdots 02}a+\frac{73\cdots 26}{23\cdots 01}$, $\frac{17\cdots 72}{57\cdots 25}a^{19}+\frac{14\cdots 91}{57\cdots 25}a^{18}-\frac{23\cdots 19}{11\cdots 50}a^{17}-\frac{82\cdots 09}{57\cdots 25}a^{16}+\frac{27\cdots 67}{11\cdots 50}a^{15}+\frac{10\cdots 34}{57\cdots 25}a^{14}+\frac{28\cdots 79}{23\cdots 10}a^{13}+\frac{90\cdots 11}{46\cdots 02}a^{12}+\frac{13\cdots 24}{23\cdots 01}a^{11}-\frac{77\cdots 81}{23\cdots 10}a^{10}+\frac{43\cdots 67}{23\cdots 10}a^{9}+\frac{14\cdots 41}{76\cdots 55}a^{8}+\frac{39\cdots 01}{23\cdots 10}a^{7}+\frac{22\cdots 37}{23\cdots 10}a^{6}+\frac{18\cdots 91}{11\cdots 05}a^{5}+\frac{53\cdots 03}{23\cdots 10}a^{4}+\frac{55\cdots 77}{46\cdots 02}a^{3}+\frac{37\cdots 05}{46\cdots 02}a^{2}+\frac{27\cdots 67}{46\cdots 02}a+\frac{29\cdots 27}{23\cdots 01}$, $\frac{10\cdots 17}{57\cdots 75}a^{19}-\frac{11\cdots 53}{11\cdots 50}a^{18}-\frac{73\cdots 54}{57\cdots 75}a^{17}+\frac{82\cdots 53}{11\cdots 50}a^{16}+\frac{59\cdots 03}{11\cdots 95}a^{15}+\frac{63\cdots 77}{11\cdots 95}a^{14}-\frac{13\cdots 27}{23\cdots 90}a^{13}+\frac{26\cdots 93}{23\cdots 90}a^{12}+\frac{28\cdots 24}{11\cdots 95}a^{11}+\frac{70\cdots 63}{23\cdots 90}a^{10}+\frac{29\cdots 69}{23\cdots 90}a^{9}-\frac{34\cdots 89}{76\cdots 45}a^{8}+\frac{21\cdots 57}{56\cdots 90}a^{7}-\frac{22\cdots 47}{23\cdots 90}a^{6}+\frac{10\cdots 16}{23\cdots 79}a^{5}+\frac{46\cdots 11}{46\cdots 58}a^{4}-\frac{66\cdots 98}{23\cdots 79}a^{3}+\frac{13\cdots 43}{23\cdots 79}a^{2}+\frac{16\cdots 80}{23\cdots 79}a-\frac{20\cdots 19}{46\cdots 58}$, $\frac{33\cdots 91}{11\cdots 50}a^{19}-\frac{60\cdots 69}{11\cdots 50}a^{18}-\frac{17\cdots 17}{11\cdots 50}a^{17}+\frac{11\cdots 29}{11\cdots 50}a^{16}+\frac{75\cdots 57}{23\cdots 10}a^{15}+\frac{15\cdots 43}{11\cdots 05}a^{14}-\frac{15\cdots 16}{11\cdots 05}a^{13}+\frac{30\cdots 12}{11\cdots 05}a^{12}-\frac{29\cdots 76}{11\cdots 05}a^{11}-\frac{21\cdots 00}{23\cdots 01}a^{10}+\frac{21\cdots 48}{11\cdots 05}a^{9}+\frac{12\cdots 13}{76\cdots 55}a^{8}+\frac{31\cdots 74}{28\cdots 05}a^{7}-\frac{70\cdots 38}{11\cdots 05}a^{6}+\frac{82\cdots 84}{23\cdots 01}a^{5}+\frac{47\cdots 89}{46\cdots 02}a^{4}-\frac{35\cdots 77}{46\cdots 02}a^{3}+\frac{17\cdots 61}{46\cdots 02}a^{2}-\frac{48\cdots 31}{46\cdots 02}a-\frac{32\cdots 83}{46\cdots 02}$, $\frac{16\cdots 94}{57\cdots 25}a^{19}+\frac{71\cdots 47}{11\cdots 50}a^{18}-\frac{55\cdots 08}{57\cdots 25}a^{17}-\frac{50\cdots 17}{11\cdots 50}a^{16}-\frac{20\cdots 91}{57\cdots 25}a^{15}+\frac{98\cdots 57}{11\cdots 50}a^{14}+\frac{74\cdots 39}{11\cdots 50}a^{13}+\frac{94\cdots 29}{23\cdots 10}a^{12}-\frac{18\cdots 73}{23\cdots 10}a^{11}-\frac{33\cdots 67}{23\cdots 10}a^{10}+\frac{35\cdots 53}{23\cdots 10}a^{9}-\frac{14\cdots 19}{15\cdots 10}a^{8}+\frac{35\cdots 29}{23\cdots 10}a^{7}-\frac{81\cdots 57}{23\cdots 10}a^{6}-\frac{28\cdots 57}{23\cdots 10}a^{5}-\frac{86\cdots 83}{23\cdots 10}a^{4}-\frac{81\cdots 38}{11\cdots 05}a^{3}-\frac{83\cdots 85}{46\cdots 02}a^{2}-\frac{14\cdots 01}{23\cdots 01}a-\frac{20\cdots 91}{46\cdots 02}$, $\frac{40\cdots 49}{11\cdots 50}a^{19}-\frac{24\cdots 51}{11\cdots 50}a^{18}-\frac{11\cdots 69}{57\cdots 25}a^{17}+\frac{53\cdots 07}{57\cdots 25}a^{16}+\frac{84\cdots 63}{14\cdots 25}a^{15}+\frac{16\cdots 21}{11\cdots 05}a^{14}-\frac{10\cdots 27}{11\cdots 50}a^{13}+\frac{23\cdots 71}{11\cdots 05}a^{12}-\frac{44\cdots 59}{11\cdots 05}a^{11}+\frac{31\cdots 71}{23\cdots 10}a^{10}+\frac{70\cdots 41}{23\cdots 01}a^{9}-\frac{53\cdots 59}{76\cdots 55}a^{8}+\frac{20\cdots 57}{23\cdots 10}a^{7}-\frac{74\cdots 18}{11\cdots 05}a^{6}+\frac{24\cdots 53}{11\cdots 05}a^{5}+\frac{20\cdots 20}{23\cdots 01}a^{4}-\frac{13\cdots 07}{23\cdots 10}a^{3}+\frac{94\cdots 31}{23\cdots 01}a^{2}-\frac{72\cdots 87}{46\cdots 02}a-\frac{30\cdots 68}{23\cdots 01}$ 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:  \( 31758563963704467000 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 31758563963704467000 \cdot 5}{2\cdot\sqrt{83867218967598487041513866410602857358753681182861328125}}\cr\approx \mathstrut & 0.336948613568146 \end{aligned}\] (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 - 5*x^19 - 35*x^18 + 215*x^17 + 420*x^16 + 39885*x^15 - 185975*x^14 + 1641450*x^13 - 9551200*x^12 + 14329275*x^11 + 696150660*x^10 - 2107342900*x^9 + 41741808575*x^8 - 105758201050*x^7 + 911902235600*x^6 + 3715280644050*x^5 - 5344988560625*x^4 + 221169857291125*x^3 - 60583973026250*x^2 + 691307088931500*x + 13445341298628775) 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 - 5*x^19 - 35*x^18 + 215*x^17 + 420*x^16 + 39885*x^15 - 185975*x^14 + 1641450*x^13 - 9551200*x^12 + 14329275*x^11 + 696150660*x^10 - 2107342900*x^9 + 41741808575*x^8 - 105758201050*x^7 + 911902235600*x^6 + 3715280644050*x^5 - 5344988560625*x^4 + 221169857291125*x^3 - 60583973026250*x^2 + 691307088931500*x + 13445341298628775, 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 - 5*x^19 - 35*x^18 + 215*x^17 + 420*x^16 + 39885*x^15 - 185975*x^14 + 1641450*x^13 - 9551200*x^12 + 14329275*x^11 + 696150660*x^10 - 2107342900*x^9 + 41741808575*x^8 - 105758201050*x^7 + 911902235600*x^6 + 3715280644050*x^5 - 5344988560625*x^4 + 221169857291125*x^3 - 60583973026250*x^2 + 691307088931500*x + 13445341298628775); 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 - 5*x^19 - 35*x^18 + 215*x^17 + 420*x^16 + 39885*x^15 - 185975*x^14 + 1641450*x^13 - 9551200*x^12 + 14329275*x^11 + 696150660*x^10 - 2107342900*x^9 + 41741808575*x^8 - 105758201050*x^7 + 911902235600*x^6 + 3715280644050*x^5 - 5344988560625*x^4 + 221169857291125*x^3 - 60583973026250*x^2 + 691307088931500*x + 13445341298628775); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2\times F_5$ (as 20T9):

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 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.6125.1, 5.1.220762578125.3, 10.2.243680579501983642578125.4

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

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

Sibling fields

Galois closure: deg 40
Degree 10 siblings: deg 10, 10.0.819107899937967816162109375.1
Degree 20 sibling: deg 20
Minimal sibling: 10.0.819107899937967816162109375.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{5}$ ${\href{/padicField/3.4.0.1}{4} }^{5}$ R R ${\href{/padicField/11.5.0.1}{5} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{5}$ ${\href{/padicField/17.4.0.1}{4} }^{5}$ ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{5}$ ${\href{/padicField/29.2.0.1}{2} }^{10}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{5}$ R ${\href{/padicField/43.4.0.1}{4} }^{5}$ ${\href{/padicField/47.4.0.1}{4} }^{5}$ ${\href{/padicField/53.4.0.1}{4} }^{5}$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display 5.1.20.31a4.25$x^{20} + 10 x^{12} + 20$$20$$1$$31$not computednot computed
\(7\) Copy content Toggle raw display 7.2.2.2a1.1$x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
7.4.2.4a1.2$x^{8} + 10 x^{6} + 8 x^{5} + 31 x^{4} + 40 x^{3} + 46 x^{2} + 24 x + 16$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
7.4.2.4a1.2$x^{8} + 10 x^{6} + 8 x^{5} + 31 x^{4} + 40 x^{3} + 46 x^{2} + 24 x + 16$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(41\) 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)