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Properties

Label 20.4.513...000.2
Degree $20$
Signature $[4, 8]$
Discriminant $5.136\times 10^{33}$
Root discriminant \(48.48\)
Ramified primes $2,5,29$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $C_2^9.(C_2\times F_5)$ (as 20T514)

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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^16 - 185*x^14 - 1560*x^12 - 8180*x^10 - 2445*x^8 - 19895*x^6 + 11180*x^4 - 1575*x^2 + 20)
 
Copy content gp:K = bnfinit(y^20 - 5*y^16 - 185*y^14 - 1560*y^12 - 8180*y^10 - 2445*y^8 - 19895*y^6 + 11180*y^4 - 1575*y^2 + 20, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^16 - 185*x^14 - 1560*x^12 - 8180*x^10 - 2445*x^8 - 19895*x^6 + 11180*x^4 - 1575*x^2 + 20);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 5*x^16 - 185*x^14 - 1560*x^12 - 8180*x^10 - 2445*x^8 - 19895*x^6 + 11180*x^4 - 1575*x^2 + 20)
 

\( x^{20} - 5x^{16} - 185x^{14} - 1560x^{12} - 8180x^{10} - 2445x^{8} - 19895x^{6} + 11180x^{4} - 1575x^{2} + 20 \) 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:   \(5135586343996594238281250000000000\) \(\medspace = 2^{10}\cdot 5^{23}\cdot 29^{10}\) 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:  \(48.48\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(5\), \(29\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{17\cdots 02}a^{18}-\frac{12\cdots 32}{86\cdots 51}a^{16}+\frac{15\cdots 23}{86\cdots 51}a^{14}+\frac{36\cdots 19}{17\cdots 02}a^{12}-\frac{1}{2}a^{11}-\frac{19\cdots 67}{86\cdots 51}a^{10}+\frac{63\cdots 11}{86\cdots 51}a^{8}-\frac{1}{2}a^{7}-\frac{58\cdots 07}{15\cdots 82}a^{6}+\frac{11\cdots 04}{86\cdots 51}a^{4}-\frac{1}{2}a^{3}+\frac{36\cdots 49}{15\cdots 82}a^{2}-\frac{1}{2}a+\frac{25\cdots 40}{86\cdots 51}$, $\frac{1}{17\cdots 02}a^{19}-\frac{12\cdots 32}{86\cdots 51}a^{17}+\frac{15\cdots 23}{86\cdots 51}a^{15}+\frac{36\cdots 19}{17\cdots 02}a^{13}-\frac{1}{2}a^{12}-\frac{19\cdots 67}{86\cdots 51}a^{11}+\frac{63\cdots 11}{86\cdots 51}a^{9}-\frac{1}{2}a^{8}-\frac{58\cdots 07}{15\cdots 82}a^{7}+\frac{11\cdots 04}{86\cdots 51}a^{5}-\frac{1}{2}a^{4}+\frac{36\cdots 49}{15\cdots 82}a^{3}-\frac{1}{2}a^{2}+\frac{25\cdots 40}{86\cdots 51}a$ Copy content Toggle raw display

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

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

Class group and class number

Ideal class group:  $C_{4}$, which has order $4$ (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:  $C_{2}\times C_{4}$, 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:  $11$
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:   $\frac{75\cdots 47}{86\cdots 51}a^{19}+\frac{36\cdots 78}{86\cdots 51}a^{18}+\frac{37\cdots 17}{17\cdots 02}a^{17}+\frac{18\cdots 17}{17\cdots 02}a^{16}-\frac{37\cdots 24}{86\cdots 51}a^{15}-\frac{18\cdots 37}{86\cdots 51}a^{14}-\frac{14\cdots 43}{86\cdots 51}a^{13}-\frac{13\cdots 77}{17\cdots 02}a^{12}-\frac{12\cdots 14}{86\cdots 51}a^{11}-\frac{11\cdots 77}{17\cdots 02}a^{10}-\frac{13\cdots 33}{17\cdots 02}a^{9}-\frac{31\cdots 38}{86\cdots 51}a^{8}-\frac{31\cdots 09}{78\cdots 41}a^{7}-\frac{30\cdots 11}{15\cdots 82}a^{6}-\frac{31\cdots 03}{17\cdots 02}a^{5}-\frac{77\cdots 50}{86\cdots 51}a^{4}+\frac{81\cdots 11}{15\cdots 82}a^{3}+\frac{39\cdots 77}{15\cdots 82}a^{2}-\frac{11\cdots 63}{17\cdots 02}a-\frac{31\cdots 76}{86\cdots 51}$, $\frac{60\cdots 83}{17\cdots 02}a^{19}-\frac{60\cdots 40}{86\cdots 51}a^{18}-\frac{15\cdots 42}{86\cdots 51}a^{17}-\frac{17\cdots 91}{17\cdots 02}a^{16}-\frac{15\cdots 79}{86\cdots 51}a^{15}+\frac{59\cdots 23}{17\cdots 02}a^{14}-\frac{10\cdots 07}{17\cdots 02}a^{13}+\frac{11\cdots 57}{86\cdots 51}a^{12}-\frac{43\cdots 66}{86\cdots 51}a^{11}+\frac{95\cdots 87}{86\cdots 51}a^{10}-\frac{44\cdots 31}{17\cdots 02}a^{9}+\frac{10\cdots 93}{17\cdots 02}a^{8}+\frac{58\cdots 72}{78\cdots 41}a^{7}+\frac{20\cdots 41}{78\cdots 41}a^{6}-\frac{51\cdots 34}{86\cdots 51}a^{5}+\frac{12\cdots 06}{86\cdots 51}a^{4}+\frac{12\cdots 33}{15\cdots 82}a^{3}-\frac{44\cdots 38}{78\cdots 41}a^{2}-\frac{23\cdots 57}{17\cdots 02}a+\frac{49\cdots 26}{86\cdots 51}$, $\frac{23\cdots 41}{17\cdots 02}a^{19}-\frac{30\cdots 21}{17\cdots 02}a^{18}+\frac{37\cdots 23}{17\cdots 02}a^{17}-\frac{20\cdots 83}{17\cdots 02}a^{16}-\frac{11\cdots 31}{17\cdots 02}a^{15}+\frac{74\cdots 48}{86\cdots 51}a^{14}-\frac{43\cdots 79}{17\cdots 02}a^{13}+\frac{28\cdots 94}{86\cdots 51}a^{12}-\frac{37\cdots 07}{17\cdots 02}a^{11}+\frac{50\cdots 97}{17\cdots 02}a^{10}-\frac{19\cdots 83}{17\cdots 02}a^{9}+\frac{13\cdots 27}{86\cdots 51}a^{8}-\frac{40\cdots 50}{78\cdots 41}a^{7}+\frac{10\cdots 76}{78\cdots 41}a^{6}-\frac{24\cdots 28}{86\cdots 51}a^{5}+\frac{32\cdots 57}{86\cdots 51}a^{4}+\frac{84\cdots 66}{78\cdots 41}a^{3}+\frac{13\cdots 61}{78\cdots 41}a^{2}-\frac{16\cdots 93}{17\cdots 02}a-\frac{21\cdots 38}{86\cdots 51}$, $\frac{23\cdots 41}{17\cdots 02}a^{19}+\frac{30\cdots 21}{17\cdots 02}a^{18}+\frac{37\cdots 23}{17\cdots 02}a^{17}+\frac{20\cdots 83}{17\cdots 02}a^{16}-\frac{11\cdots 31}{17\cdots 02}a^{15}-\frac{74\cdots 48}{86\cdots 51}a^{14}-\frac{43\cdots 79}{17\cdots 02}a^{13}-\frac{28\cdots 94}{86\cdots 51}a^{12}-\frac{37\cdots 07}{17\cdots 02}a^{11}-\frac{50\cdots 97}{17\cdots 02}a^{10}-\frac{19\cdots 83}{17\cdots 02}a^{9}-\frac{13\cdots 27}{86\cdots 51}a^{8}-\frac{40\cdots 50}{78\cdots 41}a^{7}-\frac{10\cdots 76}{78\cdots 41}a^{6}-\frac{24\cdots 28}{86\cdots 51}a^{5}-\frac{32\cdots 57}{86\cdots 51}a^{4}+\frac{84\cdots 66}{78\cdots 41}a^{3}-\frac{13\cdots 61}{78\cdots 41}a^{2}-\frac{16\cdots 93}{17\cdots 02}a+\frac{21\cdots 38}{86\cdots 51}$, $\frac{23\cdots 47}{86\cdots 51}a^{18}+\frac{47\cdots 95}{86\cdots 51}a^{16}-\frac{11\cdots 05}{86\cdots 51}a^{14}-\frac{43\cdots 25}{86\cdots 51}a^{12}-\frac{37\cdots 51}{86\cdots 51}a^{10}-\frac{19\cdots 15}{86\cdots 51}a^{8}-\frac{88\cdots 13}{78\cdots 41}a^{6}-\frac{48\cdots 40}{86\cdots 51}a^{4}+\frac{14\cdots 19}{78\cdots 41}a^{2}-\frac{44\cdots 49}{86\cdots 51}$, $\frac{27\cdots 96}{86\cdots 51}a^{19}-\frac{19\cdots 75}{78\cdots 41}a^{18}+\frac{17\cdots 79}{17\cdots 02}a^{17}-\frac{481377547657675}{15\cdots 82}a^{16}-\frac{27\cdots 23}{17\cdots 02}a^{15}+\frac{19\cdots 31}{15\cdots 82}a^{14}-\frac{10\cdots 27}{17\cdots 02}a^{13}+\frac{73\cdots 91}{15\cdots 82}a^{12}-\frac{43\cdots 43}{86\cdots 51}a^{11}+\frac{61\cdots 31}{15\cdots 82}a^{10}-\frac{45\cdots 07}{17\cdots 02}a^{9}+\frac{32\cdots 03}{15\cdots 82}a^{8}-\frac{13\cdots 99}{15\cdots 82}a^{7}+\frac{47\cdots 23}{78\cdots 41}a^{6}-\frac{11\cdots 05}{17\cdots 02}a^{5}+\frac{77\cdots 89}{15\cdots 82}a^{4}+\frac{53\cdots 07}{15\cdots 82}a^{3}-\frac{22\cdots 23}{78\cdots 41}a^{2}-\frac{74\cdots 85}{17\cdots 02}a+\frac{30\cdots 96}{78\cdots 41}$, $\frac{74\cdots 87}{17\cdots 02}a^{19}-\frac{45\cdots 87}{17\cdots 02}a^{18}-\frac{23\cdots 62}{86\cdots 51}a^{17}-\frac{12\cdots 65}{86\cdots 51}a^{16}-\frac{19\cdots 25}{86\cdots 51}a^{15}+\frac{21\cdots 79}{17\cdots 02}a^{14}-\frac{68\cdots 86}{86\cdots 51}a^{13}+\frac{85\cdots 05}{17\cdots 02}a^{12}-\frac{10\cdots 11}{17\cdots 02}a^{11}+\frac{37\cdots 84}{86\cdots 51}a^{10}-\frac{26\cdots 49}{86\cdots 51}a^{9}+\frac{41\cdots 79}{17\cdots 02}a^{8}+\frac{20\cdots 47}{15\cdots 82}a^{7}+\frac{15\cdots 75}{78\cdots 41}a^{6}-\frac{12\cdots 11}{17\cdots 02}a^{5}+\frac{10\cdots 47}{17\cdots 02}a^{4}+\frac{16\cdots 23}{15\cdots 82}a^{3}+\frac{16\cdots 87}{78\cdots 41}a^{2}-\frac{15\cdots 58}{86\cdots 51}a-\frac{19\cdots 29}{86\cdots 51}$, $\frac{40\cdots 43}{17\cdots 02}a^{19}-\frac{58\cdots 20}{86\cdots 51}a^{18}+\frac{15\cdots 66}{86\cdots 51}a^{17}-\frac{51\cdots 43}{86\cdots 51}a^{16}-\frac{19\cdots 61}{17\cdots 02}a^{15}+\frac{58\cdots 27}{17\cdots 02}a^{14}-\frac{37\cdots 80}{86\cdots 51}a^{13}+\frac{21\cdots 41}{17\cdots 02}a^{12}-\frac{31\cdots 66}{86\cdots 51}a^{11}+\frac{92\cdots 24}{86\cdots 51}a^{10}-\frac{16\cdots 72}{86\cdots 51}a^{9}+\frac{48\cdots 87}{86\cdots 51}a^{8}-\frac{57\cdots 04}{78\cdots 41}a^{7}+\frac{33\cdots 49}{15\cdots 82}a^{6}-\frac{40\cdots 92}{86\cdots 51}a^{5}+\frac{11\cdots 58}{86\cdots 51}a^{4}+\frac{34\cdots 67}{15\cdots 82}a^{3}-\frac{50\cdots 38}{78\cdots 41}a^{2}-\frac{30\cdots 86}{86\cdots 51}a-\frac{12\cdots 61}{86\cdots 51}$, $\frac{68\cdots 41}{17\cdots 02}a^{19}-\frac{13\cdots 65}{17\cdots 02}a^{18}+\frac{34\cdots 09}{17\cdots 02}a^{17}-\frac{11\cdots 54}{86\cdots 51}a^{16}-\frac{17\cdots 03}{86\cdots 51}a^{15}+\frac{34\cdots 83}{86\cdots 51}a^{14}-\frac{63\cdots 23}{86\cdots 51}a^{13}+\frac{25\cdots 93}{17\cdots 02}a^{12}-\frac{10\cdots 69}{17\cdots 02}a^{11}+\frac{22\cdots 87}{17\cdots 02}a^{10}-\frac{28\cdots 66}{86\cdots 51}a^{9}+\frac{11\cdots 31}{17\cdots 02}a^{8}-\frac{89\cdots 20}{78\cdots 41}a^{7}+\frac{48\cdots 41}{15\cdots 82}a^{6}-\frac{69\cdots 39}{86\cdots 51}a^{5}+\frac{14\cdots 58}{86\cdots 51}a^{4}+\frac{31\cdots 77}{78\cdots 41}a^{3}-\frac{48\cdots 50}{78\cdots 41}a^{2}-\frac{40\cdots 68}{86\cdots 51}a+\frac{33\cdots 03}{86\cdots 51}$, $\frac{84\cdots 87}{17\cdots 02}a^{19}+\frac{81\cdots 95}{15\cdots 82}a^{18}+\frac{13\cdots 97}{17\cdots 02}a^{17}+\frac{88\cdots 45}{78\cdots 41}a^{16}-\frac{41\cdots 35}{17\cdots 02}a^{15}-\frac{20\cdots 01}{78\cdots 41}a^{14}-\frac{78\cdots 20}{86\cdots 51}a^{13}-\frac{75\cdots 97}{78\cdots 41}a^{12}-\frac{13\cdots 71}{17\cdots 02}a^{11}-\frac{13\cdots 73}{15\cdots 82}a^{10}-\frac{35\cdots 11}{86\cdots 51}a^{9}-\frac{34\cdots 29}{78\cdots 41}a^{8}-\frac{14\cdots 61}{78\cdots 41}a^{7}-\frac{34\cdots 77}{15\cdots 82}a^{6}-\frac{17\cdots 11}{17\cdots 02}a^{5}-\frac{16\cdots 85}{15\cdots 82}a^{4}+\frac{59\cdots 65}{15\cdots 82}a^{3}+\frac{55\cdots 91}{15\cdots 82}a^{2}-\frac{52\cdots 07}{17\cdots 02}a+\frac{20\cdots 48}{78\cdots 41}$, $\frac{10\cdots 41}{17\cdots 02}a^{19}+\frac{10\cdots 87}{17\cdots 02}a^{18}+\frac{51\cdots 85}{17\cdots 02}a^{17}-\frac{22\cdots 35}{17\cdots 02}a^{16}-\frac{26\cdots 82}{86\cdots 51}a^{15}-\frac{25\cdots 91}{86\cdots 51}a^{14}-\frac{19\cdots 71}{17\cdots 02}a^{13}-\frac{18\cdots 31}{17\cdots 02}a^{12}-\frac{16\cdots 61}{17\cdots 02}a^{11}-\frac{15\cdots 63}{17\cdots 02}a^{10}-\frac{88\cdots 33}{17\cdots 02}a^{9}-\frac{83\cdots 97}{17\cdots 02}a^{8}-\frac{13\cdots 78}{78\cdots 41}a^{7}-\frac{10\cdots 70}{78\cdots 41}a^{6}-\frac{21\cdots 27}{17\cdots 02}a^{5}-\frac{20\cdots 89}{17\cdots 02}a^{4}+\frac{99\cdots 29}{15\cdots 82}a^{3}+\frac{10\cdots 89}{15\cdots 82}a^{2}-\frac{67\cdots 53}{86\cdots 51}a-\frac{78\cdots 47}{86\cdots 51}$ 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:  \( 1863737091.17 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 1863737091.17 \cdot 4}{2\cdot\sqrt{5135586343996594238281250000000000}}\cr\approx \mathstrut & 2.02152239027 \end{aligned}\] (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^16 - 185*x^14 - 1560*x^12 - 8180*x^10 - 2445*x^8 - 19895*x^6 + 11180*x^4 - 1575*x^2 + 20) 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^16 - 185*x^14 - 1560*x^12 - 8180*x^10 - 2445*x^8 - 19895*x^6 + 11180*x^4 - 1575*x^2 + 20, 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^16 - 185*x^14 - 1560*x^12 - 8180*x^10 - 2445*x^8 - 19895*x^6 + 11180*x^4 - 1575*x^2 + 20); 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 = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^16 - 185*x^14 - 1560*x^12 - 8180*x^10 - 2445*x^8 - 19895*x^6 + 11180*x^4 - 1575*x^2 + 20); 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

$C_2^9.(C_2\times F_5)$ (as 20T514):

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); G, transitive_group_identification(G)
 
A solvable group of order 20480
The 74 conjugacy class representatives for $C_2^9.(C_2\times F_5)$
Character table for $C_2^9.(C_2\times F_5)$

Intermediate fields

\(\Q(\sqrt{145}) \), 5.1.2628125.1, 10.2.1001520947265625.1

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed
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 R ${\href{/padicField/3.8.0.1}{8} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{6}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ R ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(2\) Copy content Toggle raw display 2.1.2.2a1.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$$[2]$$
2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.8.1.0a1.1$x^{8} + x^{4} + x^{3} + x^{2} + 1$$1$$8$$0$$C_8$$$[\ ]^{8}$$
2.4.2.8a2.2$x^{8} + 4 x^{5} + 2 x^{4} + 3 x^{2} + 4 x + 7$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$$[2, 2, 2, 2]^{4}$$
\(5\) Copy content Toggle raw display 5.1.20.23a4.4$x^{20} + 5 x^{4} + 20$$20$$1$$23$not computednot computed
\(29\) Copy content Toggle raw display 29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
29.4.2.4a1.2$x^{8} + 4 x^{6} + 30 x^{5} + 8 x^{4} + 60 x^{3} + 233 x^{2} + 60 x + 33$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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