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Properties

Label 20.10.443...544.2
Degree $20$
Signature $[10, 5]$
Discriminant $-4.432\times 10^{30}$
Root discriminant \(34.07\)
Ramified primes $2,89$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^9.(C_2\times S_5)$ (as 20T797)

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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 + 8*x^18 + 34*x^16 - 42*x^14 - 724*x^12 - 902*x^10 + 837*x^8 + 2882*x^6 - 1527*x^4 + 102*x^2 - 1)
 
Copy content gp:K = bnfinit(y^20 + 8*y^18 + 34*y^16 - 42*y^14 - 724*y^12 - 902*y^10 + 837*y^8 + 2882*y^6 - 1527*y^4 + 102*y^2 - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 8*x^18 + 34*x^16 - 42*x^14 - 724*x^12 - 902*x^10 + 837*x^8 + 2882*x^6 - 1527*x^4 + 102*x^2 - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 + 8*x^18 + 34*x^16 - 42*x^14 - 724*x^12 - 902*x^10 + 837*x^8 + 2882*x^6 - 1527*x^4 + 102*x^2 - 1)
 

\( x^{20} + 8 x^{18} + 34 x^{16} - 42 x^{14} - 724 x^{12} - 902 x^{10} + 837 x^{8} + 2882 x^{6} - 1527 x^{4} + \cdots - 1 \) 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:  $[10, 5]$
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:   \(-4432204969617689467720296300544\) \(\medspace = -\,2^{50}\cdot 89^{8}\) 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:  \(34.07\)
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:  not computed
Ramified primes:   \(2\), \(89\) 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:  \(\Q(\sqrt{-1}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{12}a^{16}+\frac{1}{3}a^{14}+\frac{1}{12}a^{12}-\frac{1}{2}a^{10}+\frac{1}{4}a^{8}+\frac{1}{3}a^{6}+\frac{1}{6}a^{4}-\frac{1}{6}a^{2}-\frac{5}{12}$, $\frac{1}{12}a^{17}+\frac{1}{3}a^{15}+\frac{1}{12}a^{13}-\frac{1}{2}a^{11}+\frac{1}{4}a^{9}+\frac{1}{3}a^{7}+\frac{1}{6}a^{5}-\frac{1}{6}a^{3}-\frac{5}{12}a$, $\frac{1}{71\cdots 32}a^{18}+\frac{51602553474089}{71\cdots 32}a^{16}-\frac{1}{2}a^{15}-\frac{20\cdots 79}{71\cdots 32}a^{14}-\frac{1}{2}a^{13}+\frac{217033327318747}{71\cdots 32}a^{12}-\frac{296127161199989}{23\cdots 44}a^{10}-\frac{32\cdots 29}{71\cdots 32}a^{8}-\frac{1}{2}a^{7}+\frac{127193463524871}{11\cdots 72}a^{6}-\frac{1}{2}a^{5}+\frac{203080345916521}{593438642823386}a^{4}-\frac{1}{2}a^{3}-\frac{27\cdots 19}{71\cdots 32}a^{2}-\frac{1}{2}a-\frac{16\cdots 17}{71\cdots 32}$, $\frac{1}{71\cdots 32}a^{19}+\frac{51602553474089}{71\cdots 32}a^{17}-\frac{20\cdots 79}{71\cdots 32}a^{15}-\frac{1}{2}a^{14}+\frac{217033327318747}{71\cdots 32}a^{13}-\frac{1}{2}a^{12}-\frac{296127161199989}{23\cdots 44}a^{11}-\frac{32\cdots 29}{71\cdots 32}a^{9}+\frac{127193463524871}{11\cdots 72}a^{7}-\frac{1}{2}a^{6}+\frac{203080345916521}{593438642823386}a^{5}-\frac{1}{2}a^{4}-\frac{27\cdots 19}{71\cdots 32}a^{3}-\frac{1}{2}a^{2}-\frac{16\cdots 17}{71\cdots 32}a-\frac{1}{2}$ 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:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (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:  $14$
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:   $\frac{47172086387053}{17\cdots 58}a^{18}+\frac{190518188097896}{890157964235079}a^{16}+\frac{16\cdots 15}{17\cdots 58}a^{14}-\frac{309251812024881}{296719321411693}a^{12}-\frac{11\cdots 59}{593438642823386}a^{10}-\frac{22\cdots 49}{890157964235079}a^{8}+\frac{18\cdots 06}{890157964235079}a^{6}+\frac{68\cdots 20}{890157964235079}a^{4}-\frac{63\cdots 99}{17\cdots 58}a^{2}+\frac{262311694307768}{296719321411693}$, $\frac{242545213853977}{11\cdots 72}a^{19}+\frac{19\cdots 01}{11\cdots 72}a^{17}+\frac{82\cdots 85}{11\cdots 72}a^{15}-\frac{10\cdots 49}{11\cdots 72}a^{13}-\frac{17\cdots 27}{11\cdots 72}a^{11}-\frac{22\cdots 41}{11\cdots 72}a^{9}+\frac{10\cdots 67}{593438642823386}a^{7}+\frac{17\cdots 50}{296719321411693}a^{5}-\frac{35\cdots 95}{11\cdots 72}a^{3}+\frac{15\cdots 95}{11\cdots 72}a$, $\frac{4778393269823}{593438642823386}a^{18}+\frac{18533870868986}{296719321411693}a^{16}+\frac{152831205627453}{593438642823386}a^{14}-\frac{121826203622076}{296719321411693}a^{12}-\frac{34\cdots 51}{593438642823386}a^{10}-\frac{17\cdots 48}{296719321411693}a^{8}+\frac{26\cdots 56}{296719321411693}a^{6}+\frac{67\cdots 83}{296719321411693}a^{4}-\frac{10\cdots 57}{593438642823386}a^{2}+\frac{146574718384281}{296719321411693}$, $\frac{33998276105207}{17\cdots 58}a^{18}+\frac{135761594638285}{890157964235079}a^{16}+\frac{11\cdots 65}{17\cdots 58}a^{14}-\frac{241849899353051}{296719321411693}a^{12}-\frac{82\cdots 85}{593438642823386}a^{10}-\frac{15\cdots 26}{890157964235079}a^{8}+\frac{14\cdots 98}{890157964235079}a^{6}+\frac{48\cdots 26}{890157964235079}a^{4}-\frac{53\cdots 41}{17\cdots 58}a^{2}+\frac{444616560271088}{296719321411693}$, $\frac{56805463}{106341482452}a^{18}+\frac{474877661}{106341482452}a^{16}+\frac{2203137379}{106341482452}a^{14}-\frac{769008441}{106341482452}a^{12}-\frac{37869924953}{106341482452}a^{10}-\frac{68635498877}{106341482452}a^{8}-\frac{25170718295}{53170741226}a^{6}+\frac{11079596590}{26585370613}a^{4}-\frac{3114025461}{106341482452}a^{2}+\frac{44078875719}{106341482452}$, $\frac{90676386669625}{35\cdots 16}a^{18}+\frac{120578256785653}{593438642823386}a^{16}+\frac{10\cdots 39}{11\cdots 72}a^{14}-\frac{973210383861022}{890157964235079}a^{12}-\frac{21\cdots 55}{11\cdots 72}a^{10}-\frac{40\cdots 09}{17\cdots 58}a^{8}+\frac{39\cdots 83}{17\cdots 58}a^{6}+\frac{13\cdots 53}{17\cdots 58}a^{4}-\frac{47\cdots 75}{11\cdots 72}a^{2}+\frac{42\cdots 41}{17\cdots 58}$, $\frac{667413837885523}{71\cdots 32}a^{19}-\frac{187336758632801}{71\cdots 32}a^{18}+\frac{17\cdots 13}{23\cdots 44}a^{17}-\frac{506398830414261}{23\cdots 44}a^{16}+\frac{76\cdots 53}{23\cdots 44}a^{15}-\frac{21\cdots 43}{23\cdots 44}a^{14}-\frac{26\cdots 71}{71\cdots 32}a^{13}+\frac{71\cdots 07}{71\cdots 32}a^{12}-\frac{16\cdots 55}{23\cdots 44}a^{11}+\frac{45\cdots 09}{23\cdots 44}a^{10}-\frac{62\cdots 55}{71\cdots 32}a^{9}+\frac{18\cdots 91}{71\cdots 32}a^{8}+\frac{26\cdots 45}{35\cdots 16}a^{7}-\frac{67\cdots 51}{35\cdots 16}a^{6}+\frac{24\cdots 96}{890157964235079}a^{5}-\frac{13\cdots 13}{17\cdots 58}a^{4}-\frac{30\cdots 47}{23\cdots 44}a^{3}+\frac{72\cdots 93}{23\cdots 44}a^{2}+\frac{27\cdots 33}{71\cdots 32}a+\frac{22\cdots 59}{71\cdots 32}$, $\frac{667413837885523}{71\cdots 32}a^{19}+\frac{187336758632801}{71\cdots 32}a^{18}+\frac{17\cdots 13}{23\cdots 44}a^{17}+\frac{506398830414261}{23\cdots 44}a^{16}+\frac{76\cdots 53}{23\cdots 44}a^{15}+\frac{21\cdots 43}{23\cdots 44}a^{14}-\frac{26\cdots 71}{71\cdots 32}a^{13}-\frac{71\cdots 07}{71\cdots 32}a^{12}-\frac{16\cdots 55}{23\cdots 44}a^{11}-\frac{45\cdots 09}{23\cdots 44}a^{10}-\frac{62\cdots 55}{71\cdots 32}a^{9}-\frac{18\cdots 91}{71\cdots 32}a^{8}+\frac{26\cdots 45}{35\cdots 16}a^{7}+\frac{67\cdots 51}{35\cdots 16}a^{6}+\frac{24\cdots 96}{890157964235079}a^{5}+\frac{13\cdots 13}{17\cdots 58}a^{4}-\frac{30\cdots 47}{23\cdots 44}a^{3}-\frac{72\cdots 93}{23\cdots 44}a^{2}+\frac{27\cdots 33}{71\cdots 32}a-\frac{22\cdots 59}{71\cdots 32}$, $\frac{21\cdots 13}{71\cdots 32}a^{19}-\frac{112486476152279}{35\cdots 16}a^{18}+\frac{56\cdots 77}{23\cdots 44}a^{17}-\frac{448560023462023}{17\cdots 58}a^{16}+\frac{23\cdots 11}{23\cdots 44}a^{15}-\frac{38\cdots 73}{35\cdots 16}a^{14}-\frac{86\cdots 15}{71\cdots 32}a^{13}+\frac{804148904996351}{593438642823386}a^{12}-\frac{50\cdots 25}{23\cdots 44}a^{11}+\frac{27\cdots 45}{11\cdots 72}a^{10}-\frac{19\cdots 87}{71\cdots 32}a^{9}+\frac{49\cdots 37}{17\cdots 58}a^{8}+\frac{85\cdots 53}{35\cdots 16}a^{7}-\frac{24\cdots 93}{890157964235079}a^{6}+\frac{76\cdots 17}{890157964235079}a^{5}-\frac{81\cdots 70}{890157964235079}a^{4}-\frac{10\cdots 13}{23\cdots 44}a^{3}+\frac{17\cdots 41}{35\cdots 16}a^{2}+\frac{14\cdots 41}{71\cdots 32}a-\frac{819416505653929}{296719321411693}$, $\frac{31\cdots 71}{71\cdots 32}a^{19}-\frac{419913916342219}{71\cdots 32}a^{18}+\frac{25\cdots 33}{71\cdots 32}a^{17}-\frac{11\cdots 11}{23\cdots 44}a^{16}+\frac{10\cdots 67}{71\cdots 32}a^{15}-\frac{47\cdots 37}{23\cdots 44}a^{14}-\frac{13\cdots 01}{71\cdots 32}a^{13}+\frac{17\cdots 17}{71\cdots 32}a^{12}-\frac{76\cdots 99}{23\cdots 44}a^{11}+\frac{10\cdots 43}{23\cdots 44}a^{10}-\frac{28\cdots 09}{71\cdots 32}a^{9}+\frac{37\cdots 17}{71\cdots 32}a^{8}+\frac{43\cdots 51}{11\cdots 72}a^{7}-\frac{18\cdots 33}{35\cdots 16}a^{6}+\frac{76\cdots 69}{593438642823386}a^{5}-\frac{30\cdots 75}{17\cdots 58}a^{4}-\frac{47\cdots 29}{71\cdots 32}a^{3}+\frac{22\cdots 59}{23\cdots 44}a^{2}+\frac{28\cdots 95}{71\cdots 32}a-\frac{34\cdots 91}{71\cdots 32}$, $\frac{13\cdots 45}{23\cdots 44}a^{19}+\frac{8762287453913}{35\cdots 16}a^{18}+\frac{33\cdots 49}{71\cdots 32}a^{17}+\frac{64656272940967}{35\cdots 16}a^{16}+\frac{14\cdots 23}{71\cdots 32}a^{15}+\frac{254288554834219}{35\cdots 16}a^{14}-\frac{16\cdots 49}{71\cdots 32}a^{13}-\frac{555611258560897}{35\cdots 16}a^{12}-\frac{10\cdots 83}{23\cdots 44}a^{11}-\frac{20\cdots 45}{11\cdots 72}a^{10}-\frac{12\cdots 67}{23\cdots 44}a^{9}-\frac{40\cdots 11}{35\cdots 16}a^{8}+\frac{16\cdots 11}{35\cdots 16}a^{7}+\frac{10\cdots 58}{296719321411693}a^{6}+\frac{15\cdots 23}{890157964235079}a^{5}+\frac{35\cdots 17}{593438642823386}a^{4}-\frac{59\cdots 33}{71\cdots 32}a^{3}-\frac{27\cdots 89}{35\cdots 16}a^{2}+\frac{25\cdots 71}{71\cdots 32}a+\frac{90\cdots 15}{35\cdots 16}$, $\frac{368310919289897}{11\cdots 72}a^{19}+\frac{12755202317913}{23\cdots 44}a^{18}+\frac{44\cdots 55}{17\cdots 58}a^{17}+\frac{352429018265933}{71\cdots 32}a^{16}+\frac{37\cdots 89}{35\cdots 16}a^{15}+\frac{16\cdots 47}{71\cdots 32}a^{14}-\frac{23\cdots 43}{17\cdots 58}a^{13}+\frac{122086041136907}{71\cdots 32}a^{12}-\frac{26\cdots 37}{11\cdots 72}a^{11}-\frac{96\cdots 11}{23\cdots 44}a^{10}-\frac{16\cdots 95}{593438642823386}a^{9}-\frac{22\cdots 55}{23\cdots 44}a^{8}+\frac{23\cdots 94}{890157964235079}a^{7}-\frac{11\cdots 73}{35\cdots 16}a^{6}+\frac{80\cdots 49}{890157964235079}a^{5}+\frac{16\cdots 18}{890157964235079}a^{4}-\frac{16\cdots 21}{35\cdots 16}a^{3}+\frac{84\cdots 39}{71\cdots 32}a^{2}+\frac{13\cdots 80}{890157964235079}a-\frac{70\cdots 81}{71\cdots 32}$, $\frac{256046693280037}{17\cdots 58}a^{19}-\frac{381707088771299}{71\cdots 32}a^{18}+\frac{10\cdots 88}{890157964235079}a^{17}-\frac{10\cdots 57}{23\cdots 44}a^{16}+\frac{44\cdots 82}{890157964235079}a^{15}-\frac{44\cdots 05}{23\cdots 44}a^{14}-\frac{99\cdots 67}{17\cdots 58}a^{13}+\frac{14\cdots 35}{71\cdots 32}a^{12}-\frac{62\cdots 95}{593438642823386}a^{11}+\frac{92\cdots 51}{23\cdots 44}a^{10}-\frac{12\cdots 47}{890157964235079}a^{9}+\frac{38\cdots 99}{71\cdots 32}a^{8}+\frac{64\cdots 15}{593438642823386}a^{7}-\frac{13\cdots 95}{35\cdots 16}a^{6}+\frac{25\cdots 09}{593438642823386}a^{5}-\frac{28\cdots 77}{17\cdots 58}a^{4}-\frac{16\cdots 67}{890157964235079}a^{3}+\frac{14\cdots 43}{23\cdots 44}a^{2}-\frac{10\cdots 67}{17\cdots 58}a+\frac{19\cdots 35}{71\cdots 32}$, $\frac{470877565040395}{23\cdots 44}a^{19}+\frac{40609220342977}{890157964235079}a^{18}+\frac{11\cdots 17}{71\cdots 32}a^{17}+\frac{657084090003097}{17\cdots 58}a^{16}+\frac{47\cdots 29}{71\cdots 32}a^{15}+\frac{28\cdots 57}{17\cdots 58}a^{14}-\frac{59\cdots 41}{71\cdots 32}a^{13}-\frac{15\cdots 24}{890157964235079}a^{12}-\frac{34\cdots 41}{23\cdots 44}a^{11}-\frac{98\cdots 94}{296719321411693}a^{10}-\frac{42\cdots 07}{23\cdots 44}a^{9}-\frac{78\cdots 73}{17\cdots 58}a^{8}+\frac{59\cdots 37}{35\cdots 16}a^{7}+\frac{19\cdots 45}{593438642823386}a^{6}+\frac{10\cdots 47}{17\cdots 58}a^{5}+\frac{78\cdots 63}{593438642823386}a^{4}-\frac{21\cdots 63}{71\cdots 32}a^{3}-\frac{10\cdots 33}{17\cdots 58}a^{2}+\frac{14\cdots 27}{71\cdots 32}a+\frac{19\cdots 57}{890157964235079}$ 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:  \( 179247197.076 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{5}\cdot 179247197.076 \cdot 1}{2\cdot\sqrt{4432204969617689467720296300544}}\cr\approx \mathstrut & 0.426885848391 \end{aligned}\] (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 + 8*x^18 + 34*x^16 - 42*x^14 - 724*x^12 - 902*x^10 + 837*x^8 + 2882*x^6 - 1527*x^4 + 102*x^2 - 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 + 8*x^18 + 34*x^16 - 42*x^14 - 724*x^12 - 902*x^10 + 837*x^8 + 2882*x^6 - 1527*x^4 + 102*x^2 - 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 + 8*x^18 + 34*x^16 - 42*x^14 - 724*x^12 - 902*x^10 + 837*x^8 + 2882*x^6 - 1527*x^4 + 102*x^2 - 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 + 8*x^18 + 34*x^16 - 42*x^14 - 724*x^12 - 902*x^10 + 837*x^8 + 2882*x^6 - 1527*x^4 + 102*x^2 - 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

$C_2^9.(C_2\times S_5)$ (as 20T797):

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 non-solvable group of order 122880
The 108 conjugacy class representatives for $C_2^9.(C_2\times S_5)$
Character table for $C_2^9.(C_2\times S_5)$

Intermediate fields

\(\Q(\sqrt{2}) \), 5.3.31684.1, 10.6.8223751012352.1

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

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 20.8.34626601325138198966564814848.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{5}$ ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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 2.1.8.20d1.1$x^{8} + 4 x^{5} + 2 x^{4} + 2$$8$$1$$20$$Q_8:C_2$$$[2, 3, 3]^{2}$$
2.3.4.30a2.83$x^{12} + 8 x^{10} + 4 x^{9} + 22 x^{8} + 24 x^{7} + 34 x^{6} + 52 x^{5} + 41 x^{4} + 44 x^{3} + 34 x^{2} + 16 x + 7$$4$$3$$30$12T134$$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{3}$$
\(89\) Copy content Toggle raw display 89.2.1.0a1.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
89.2.1.0a1.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
89.1.3.2a1.1$x^{3} + 89$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
89.1.3.2a1.1$x^{3} + 89$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
89.4.1.0a1.1$x^{4} + 4 x^{2} + 72 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
89.2.3.4a1.2$x^{6} + 246 x^{5} + 20181 x^{4} + 552844 x^{3} + 60543 x^{2} + 2214 x + 116$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$

Spectrum of ring of integers

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