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Properties

Label 20.0.213...125.1
Degree $20$
Signature $(0, 10)$
Discriminant $2.132\times 10^{33}$
Root discriminant \(46.39\)
Ramified primes $3,5,239$
Class number $116$ (GRH)
Class group [2, 58] (GRH)
Galois group $C_4\times D_5$ (as 20T6)

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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 - 5*x^19 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*x + 1)
 
Copy content gp:K = bnfinit(y^20 - 5*y^19 + 13*y^18 - 17*y^17 + 94*y^16 - 151*y^15 + 406*y^14 - 633*y^13 + 3653*y^12 - 772*y^11 + 13426*y^10 + 5719*y^9 + 24764*y^8 + 15388*y^7 + 21892*y^6 + 15269*y^5 + 9379*y^4 + 3157*y^3 + 732*y^2 - 45*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 5*x^19 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*x + 1)
 

\( x^{20} - 5 x^{19} + 13 x^{18} - 17 x^{17} + 94 x^{16} - 151 x^{15} + 406 x^{14} - 633 x^{13} + 3653 x^{12} + \cdots + 1 \) 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:   \(2131588214553606414985382080078125\) \(\medspace = 3^{8}\cdot 5^{15}\cdot 239^{8}\) 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:  \(46.39\)
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:  $3^{1/2}5^{3/4}239^{1/2}\approx 89.53381316204927$
Ramified primes:   \(3\), \(5\), \(239\) 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:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19}a^{16}-\frac{3}{19}a^{15}+\frac{2}{19}a^{14}-\frac{6}{19}a^{13}-\frac{6}{19}a^{12}+\frac{5}{19}a^{11}+\frac{5}{19}a^{10}-\frac{7}{19}a^{9}+\frac{6}{19}a^{8}-\frac{2}{19}a^{7}-\frac{1}{19}a^{6}+\frac{9}{19}a^{5}+\frac{4}{19}a^{4}+\frac{2}{19}a^{3}-\frac{1}{19}a^{2}-\frac{8}{19}$, $\frac{1}{19}a^{17}-\frac{7}{19}a^{15}-\frac{5}{19}a^{13}+\frac{6}{19}a^{12}+\frac{1}{19}a^{11}+\frac{8}{19}a^{10}+\frac{4}{19}a^{9}-\frac{3}{19}a^{8}-\frac{7}{19}a^{7}+\frac{6}{19}a^{6}-\frac{7}{19}a^{5}-\frac{5}{19}a^{4}+\frac{5}{19}a^{3}-\frac{3}{19}a^{2}-\frac{8}{19}a-\frac{5}{19}$, $\frac{1}{19}a^{18}-\frac{2}{19}a^{15}+\frac{9}{19}a^{14}+\frac{2}{19}a^{13}-\frac{3}{19}a^{12}+\frac{5}{19}a^{11}+\frac{1}{19}a^{10}+\frac{5}{19}a^{9}-\frac{3}{19}a^{8}-\frac{8}{19}a^{7}+\frac{5}{19}a^{6}+\frac{1}{19}a^{5}-\frac{5}{19}a^{4}-\frac{8}{19}a^{3}+\frac{4}{19}a^{2}-\frac{5}{19}a+\frac{1}{19}$, $\frac{1}{15\cdots 79}a^{19}-\frac{32\cdots 16}{15\cdots 79}a^{18}+\frac{22\cdots 40}{15\cdots 79}a^{17}+\frac{98\cdots 67}{81\cdots 41}a^{16}-\frac{62\cdots 36}{15\cdots 79}a^{15}-\frac{12\cdots 24}{15\cdots 79}a^{14}-\frac{14\cdots 14}{15\cdots 79}a^{13}+\frac{33\cdots 07}{15\cdots 79}a^{12}-\frac{46\cdots 01}{15\cdots 79}a^{11}-\frac{68\cdots 07}{15\cdots 79}a^{10}-\frac{11\cdots 37}{15\cdots 79}a^{9}-\frac{30\cdots 46}{15\cdots 79}a^{8}-\frac{48\cdots 58}{15\cdots 79}a^{7}+\frac{78\cdots 00}{15\cdots 79}a^{6}-\frac{99\cdots 62}{15\cdots 79}a^{5}-\frac{88\cdots 79}{15\cdots 79}a^{4}+\frac{22\cdots 57}{15\cdots 79}a^{3}+\frac{46\cdots 05}{15\cdots 79}a^{2}-\frac{33\cdots 14}{15\cdots 79}a+\frac{27\cdots 55}{15\cdots 79}$ Copy content Toggle raw display

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

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

Class group and class number

Ideal class group:  $C_{2}\times C_{58}$, which has order $116$ (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:  $C_{2}\times C_{58}$, which has order $116$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 
Relative class number:   $116$ (assuming GRH)

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $9$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( \frac{462818769889456179651310650579408872}{15512369205008039820354470756707694779} a^{19} - \frac{2326769500830031245644024226778428991}{15512369205008039820354470756707694779} a^{18} + \frac{6085575599032599985406446477638831728}{15512369205008039820354470756707694779} a^{17} - \frac{8053672957771882543603366298796728394}{15512369205008039820354470756707694779} a^{16} + \frac{43756386035050661857706352506913597896}{15512369205008039820354470756707694779} a^{15} - \frac{71071874341284803146102106521707118592}{15512369205008039820354470756707694779} a^{14} + \frac{190193241180812691445636467388065236429}{15512369205008039820354470756707694779} a^{13} - \frac{298259079478417551045537111755855720027}{15512369205008039820354470756707694779} a^{12} + \frac{1699680945567312614727430986045645406878}{15512369205008039820354470756707694779} a^{11} - \frac{403976620459062232554254939005395478672}{15512369205008039820354470756707694779} a^{10} + \frac{6238463018903085141089317736446009784682}{15512369205008039820354470756707694779} a^{9} + \frac{2499055278537939215613511471752905341233}{15512369205008039820354470756707694779} a^{8} + \frac{11448874579390192561509181938382218220782}{15512369205008039820354470756707694779} a^{7} + \frac{6933857183413525225748035473749834037042}{15512369205008039820354470756707694779} a^{6} + \frac{10074706929359956358303595959856066709952}{15512369205008039820354470756707694779} a^{5} + \frac{7034096685754392810557671000997154146244}{15512369205008039820354470756707694779} a^{4} + \frac{4294624621290086941447127430527707227167}{15512369205008039820354470756707694779} a^{3} + \frac{1532784301452226441723579802687424978039}{15512369205008039820354470756707694779} a^{2} + \frac{380233125383883954419361157061624425244}{15512369205008039820354470756707694779} a + \frac{1934096680512204750531796611047313906}{15512369205008039820354470756707694779} \)  (order $10$) 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:   $\frac{25\cdots 72}{15\cdots 79}a^{19}-\frac{12\cdots 44}{15\cdots 79}a^{18}+\frac{16\cdots 82}{81\cdots 41}a^{17}-\frac{38\cdots 06}{15\cdots 79}a^{16}+\frac{23\cdots 04}{15\cdots 79}a^{15}-\frac{35\cdots 06}{15\cdots 79}a^{14}+\frac{98\cdots 92}{15\cdots 79}a^{13}-\frac{14\cdots 14}{15\cdots 79}a^{12}+\frac{91\cdots 64}{15\cdots 79}a^{11}-\frac{64\cdots 62}{15\cdots 79}a^{10}+\frac{34\cdots 98}{15\cdots 79}a^{9}+\frac{19\cdots 22}{15\cdots 79}a^{8}+\frac{65\cdots 60}{15\cdots 79}a^{7}+\frac{47\cdots 87}{15\cdots 79}a^{6}+\frac{61\cdots 98}{15\cdots 79}a^{5}+\frac{45\cdots 72}{15\cdots 79}a^{4}+\frac{28\cdots 94}{15\cdots 79}a^{3}+\frac{10\cdots 12}{15\cdots 79}a^{2}+\frac{22\cdots 11}{15\cdots 79}a+\frac{13\cdots 14}{15\cdots 79}$, $\frac{43\cdots 03}{15\cdots 79}a^{19}-\frac{21\cdots 66}{15\cdots 79}a^{18}+\frac{55\cdots 11}{15\cdots 79}a^{17}-\frac{71\cdots 06}{15\cdots 79}a^{16}+\frac{40\cdots 50}{15\cdots 79}a^{15}-\frac{63\cdots 86}{15\cdots 79}a^{14}+\frac{17\cdots 29}{15\cdots 79}a^{13}-\frac{26\cdots 98}{15\cdots 79}a^{12}+\frac{15\cdots 92}{15\cdots 79}a^{11}-\frac{27\cdots 69}{15\cdots 79}a^{10}+\frac{58\cdots 65}{15\cdots 79}a^{9}+\frac{27\cdots 44}{15\cdots 79}a^{8}+\frac{10\cdots 97}{15\cdots 79}a^{7}+\frac{71\cdots 44}{15\cdots 79}a^{6}+\frac{97\cdots 84}{15\cdots 79}a^{5}+\frac{70\cdots 95}{15\cdots 79}a^{4}+\frac{43\cdots 12}{15\cdots 79}a^{3}+\frac{15\cdots 50}{15\cdots 79}a^{2}+\frac{41\cdots 48}{15\cdots 79}a+\frac{19\cdots 01}{15\cdots 79}$, $\frac{33\cdots 88}{15\cdots 79}a^{19}-\frac{16\cdots 64}{15\cdots 79}a^{18}+\frac{41\cdots 60}{15\cdots 79}a^{17}-\frac{53\cdots 94}{15\cdots 79}a^{16}+\frac{30\cdots 84}{15\cdots 79}a^{15}-\frac{47\cdots 57}{15\cdots 79}a^{14}+\frac{13\cdots 94}{15\cdots 79}a^{13}-\frac{20\cdots 75}{15\cdots 79}a^{12}+\frac{11\cdots 64}{15\cdots 79}a^{11}-\frac{17\cdots 86}{15\cdots 79}a^{10}+\frac{44\cdots 13}{15\cdots 79}a^{9}+\frac{21\cdots 79}{15\cdots 79}a^{8}+\frac{84\cdots 64}{15\cdots 79}a^{7}+\frac{55\cdots 34}{15\cdots 79}a^{6}+\frac{77\cdots 04}{15\cdots 79}a^{5}+\frac{54\cdots 42}{15\cdots 79}a^{4}+\frac{35\cdots 12}{15\cdots 79}a^{3}+\frac{12\cdots 24}{15\cdots 79}a^{2}+\frac{33\cdots 26}{15\cdots 79}a-\frac{66\cdots 94}{15\cdots 79}$, $\frac{40\cdots 09}{15\cdots 79}a^{19}-\frac{21\cdots 88}{15\cdots 79}a^{18}+\frac{60\cdots 50}{15\cdots 79}a^{17}-\frac{89\cdots 60}{15\cdots 79}a^{16}+\frac{41\cdots 72}{15\cdots 79}a^{15}-\frac{76\cdots 03}{15\cdots 79}a^{14}+\frac{19\cdots 66}{15\cdots 79}a^{13}-\frac{32\cdots 66}{15\cdots 79}a^{12}+\frac{15\cdots 04}{15\cdots 79}a^{11}-\frac{87\cdots 86}{15\cdots 79}a^{10}+\frac{56\cdots 53}{15\cdots 79}a^{9}+\frac{29\cdots 36}{15\cdots 79}a^{8}+\frac{95\cdots 24}{15\cdots 79}a^{7}+\frac{26\cdots 40}{15\cdots 79}a^{6}+\frac{72\cdots 00}{15\cdots 79}a^{5}+\frac{33\cdots 75}{15\cdots 79}a^{4}+\frac{20\cdots 96}{15\cdots 79}a^{3}+\frac{31\cdots 06}{15\cdots 79}a^{2}+\frac{13\cdots 00}{15\cdots 79}a-\frac{78\cdots 32}{15\cdots 79}$, $\frac{38\cdots 35}{81\cdots 41}a^{19}-\frac{47\cdots 51}{15\cdots 79}a^{18}+\frac{15\cdots 87}{15\cdots 79}a^{17}-\frac{27\cdots 54}{15\cdots 79}a^{16}+\frac{89\cdots 71}{15\cdots 79}a^{15}-\frac{21\cdots 54}{15\cdots 79}a^{14}+\frac{47\cdots 77}{15\cdots 79}a^{13}-\frac{92\cdots 87}{15\cdots 79}a^{12}+\frac{34\cdots 24}{15\cdots 79}a^{11}-\frac{46\cdots 97}{15\cdots 79}a^{10}+\frac{11\cdots 17}{15\cdots 79}a^{9}-\frac{10\cdots 80}{15\cdots 79}a^{8}+\frac{13\cdots 22}{15\cdots 79}a^{7}-\frac{13\cdots 24}{15\cdots 79}a^{6}+\frac{21\cdots 91}{15\cdots 79}a^{5}-\frac{74\cdots 49}{15\cdots 79}a^{4}-\frac{74\cdots 02}{15\cdots 79}a^{3}-\frac{34\cdots 11}{15\cdots 79}a^{2}-\frac{17\cdots 16}{15\cdots 79}a-\frac{22\cdots 97}{81\cdots 41}$, $\frac{91\cdots 48}{15\cdots 79}a^{19}-\frac{39\cdots 44}{15\cdots 79}a^{18}+\frac{87\cdots 22}{15\cdots 79}a^{17}-\frac{69\cdots 93}{15\cdots 79}a^{16}+\frac{74\cdots 45}{15\cdots 79}a^{15}-\frac{79\cdots 82}{15\cdots 79}a^{14}+\frac{26\cdots 94}{15\cdots 79}a^{13}-\frac{31\cdots 42}{15\cdots 79}a^{12}+\frac{29\cdots 23}{15\cdots 79}a^{11}+\frac{15\cdots 72}{15\cdots 79}a^{10}+\frac{11\cdots 32}{15\cdots 79}a^{9}+\frac{13\cdots 22}{15\cdots 79}a^{8}+\frac{25\cdots 58}{15\cdots 79}a^{7}+\frac{15\cdots 79}{81\cdots 41}a^{6}+\frac{28\cdots 15}{15\cdots 79}a^{5}+\frac{25\cdots 42}{15\cdots 79}a^{4}+\frac{17\cdots 57}{15\cdots 79}a^{3}+\frac{69\cdots 54}{15\cdots 79}a^{2}+\frac{22\cdots 48}{15\cdots 79}a+\frac{86\cdots 10}{15\cdots 79}$, $\frac{96\cdots 61}{15\cdots 79}a^{19}-\frac{51\cdots 09}{15\cdots 79}a^{18}+\frac{14\cdots 07}{15\cdots 79}a^{17}-\frac{20\cdots 46}{15\cdots 79}a^{16}+\frac{96\cdots 88}{15\cdots 79}a^{15}-\frac{17\cdots 18}{15\cdots 79}a^{14}+\frac{44\cdots 01}{15\cdots 79}a^{13}-\frac{75\cdots 59}{15\cdots 79}a^{12}+\frac{37\cdots 46}{15\cdots 79}a^{11}-\frac{10\cdots 89}{81\cdots 41}a^{10}+\frac{13\cdots 41}{15\cdots 79}a^{9}+\frac{10\cdots 24}{15\cdots 79}a^{8}+\frac{22\cdots 00}{15\cdots 79}a^{7}+\frac{70\cdots 08}{15\cdots 79}a^{6}+\frac{17\cdots 46}{15\cdots 79}a^{5}+\frac{83\cdots 65}{15\cdots 79}a^{4}+\frac{50\cdots 99}{15\cdots 79}a^{3}+\frac{79\cdots 00}{15\cdots 79}a^{2}-\frac{51\cdots 31}{15\cdots 79}a-\frac{15\cdots 55}{15\cdots 79}$, $\frac{26\cdots 10}{15\cdots 79}a^{19}-\frac{13\cdots 30}{15\cdots 79}a^{18}+\frac{37\cdots 80}{15\cdots 79}a^{17}-\frac{54\cdots 94}{15\cdots 79}a^{16}+\frac{13\cdots 20}{81\cdots 41}a^{15}-\frac{46\cdots 74}{15\cdots 79}a^{14}+\frac{11\cdots 01}{15\cdots 79}a^{13}-\frac{19\cdots 56}{15\cdots 79}a^{12}+\frac{99\cdots 66}{15\cdots 79}a^{11}-\frac{48\cdots 80}{15\cdots 79}a^{10}+\frac{35\cdots 95}{15\cdots 79}a^{9}+\frac{47\cdots 74}{15\cdots 79}a^{8}+\frac{58\cdots 18}{15\cdots 79}a^{7}+\frac{22\cdots 31}{15\cdots 79}a^{6}+\frac{44\cdots 04}{15\cdots 79}a^{5}+\frac{25\cdots 13}{15\cdots 79}a^{4}+\frac{13\cdots 05}{15\cdots 79}a^{3}+\frac{28\cdots 20}{15\cdots 79}a^{2}+\frac{95\cdots 82}{15\cdots 79}a-\frac{56\cdots 45}{15\cdots 79}$, $\frac{76\cdots 51}{15\cdots 79}a^{19}-\frac{38\cdots 23}{15\cdots 79}a^{18}+\frac{10\cdots 29}{15\cdots 79}a^{17}-\frac{13\cdots 47}{15\cdots 79}a^{16}+\frac{71\cdots 69}{15\cdots 79}a^{15}-\frac{11\cdots 37}{15\cdots 79}a^{14}+\frac{30\cdots 46}{15\cdots 79}a^{13}-\frac{48\cdots 53}{15\cdots 79}a^{12}+\frac{27\cdots 79}{15\cdots 79}a^{11}-\frac{73\cdots 51}{15\cdots 79}a^{10}+\frac{98\cdots 54}{15\cdots 79}a^{9}+\frac{38\cdots 98}{15\cdots 79}a^{8}+\frac{17\cdots 78}{15\cdots 79}a^{7}+\frac{10\cdots 82}{15\cdots 79}a^{6}+\frac{13\cdots 70}{15\cdots 79}a^{5}+\frac{97\cdots 03}{15\cdots 79}a^{4}+\frac{47\cdots 69}{15\cdots 79}a^{3}+\frac{10\cdots 34}{15\cdots 79}a^{2}+\frac{18\cdots 07}{15\cdots 79}a-\frac{37\cdots 28}{15\cdots 79}$ 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:  \( 10123076.9997 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 10123076.9997 \cdot 116}{10\cdot\sqrt{2131588214553606414985382080078125}}\cr\approx \mathstrut & 0.243903233471 \end{aligned}\] (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 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*x + 1) 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 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*x + 1, 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 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*x + 1); 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 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*x + 1); 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

$C_4\times D_5$ (as 20T6):

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 40
The 16 conjugacy class representatives for $C_4\times D_5$
Character table for $C_4\times D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.12852225.1, 10.10.825898437253125.1

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

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

Sibling fields

Galois closure: deg 40
Degree 20 sibling: deg 20
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $20$ R R $20$ ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/padicField/19.2.0.1}{2} }^{10}$ ${\href{/padicField/23.4.0.1}{4} }^{5}$ ${\href{/padicField/29.2.0.1}{2} }^{10}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/padicField/47.4.0.1}{4} }^{5}$ ${\href{/padicField/53.4.0.1}{4} }^{5}$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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

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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
\(3\) Copy content Toggle raw display 3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(5\) Copy content Toggle raw display 5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(239\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$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)