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Properties

Label 22.20.412...904.1
Degree $22$
Signature $[20, 1]$
Discriminant $-4.129\times 10^{45}$
Root discriminant \(118.43\)
Ramified primes $2,74843$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{11}.\PSL(2,11)$ (as 22T42)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^22 - 68*x^20 + 1749*x^18 - 21878*x^16 + 146579*x^14 - 538577*x^12 + 1028049*x^10 - 796919*x^8 - 100770*x^6 + 299644*x^4 - 34200*x^2 + 729)
 
Copy content gp:K = bnfinit(y^22 - 68*y^20 + 1749*y^18 - 21878*y^16 + 146579*y^14 - 538577*y^12 + 1028049*y^10 - 796919*y^8 - 100770*y^6 + 299644*y^4 - 34200*y^2 + 729, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 68*x^20 + 1749*x^18 - 21878*x^16 + 146579*x^14 - 538577*x^12 + 1028049*x^10 - 796919*x^8 - 100770*x^6 + 299644*x^4 - 34200*x^2 + 729);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 68*x^20 + 1749*x^18 - 21878*x^16 + 146579*x^14 - 538577*x^12 + 1028049*x^10 - 796919*x^8 - 100770*x^6 + 299644*x^4 - 34200*x^2 + 729)
 

\( x^{22} - 68 x^{20} + 1749 x^{18} - 21878 x^{16} + 146579 x^{14} - 538577 x^{12} + 1028049 x^{10} + \cdots + 729 \) 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:  $22$
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:  $[20, 1]$
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:   \(-4129233136056857981979443884256982828952059904\) \(\medspace = -\,2^{22}\cdot 74843^{8}\) 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:  \(118.43\)
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:  not computed
Ramified primes:   \(2\), \(74843\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{17}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{18}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{19}+\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{3}a^{5}$, $\frac{1}{12\cdots 89}a^{20}-\frac{16\cdots 85}{12\cdots 89}a^{18}+\frac{22\cdots 05}{41\cdots 63}a^{16}+\frac{29\cdots 65}{12\cdots 89}a^{14}+\frac{14\cdots 25}{12\cdots 89}a^{12}+\frac{52\cdots 51}{12\cdots 89}a^{10}+\frac{21\cdots 13}{13\cdots 21}a^{8}-\frac{14\cdots 81}{12\cdots 89}a^{6}-\frac{15\cdots 41}{41\cdots 63}a^{4}+\frac{49\cdots 91}{12\cdots 89}a^{2}-\frac{44\cdots 69}{13\cdots 21}$, $\frac{1}{11\cdots 01}a^{21}-\frac{85\cdots 11}{11\cdots 01}a^{19}+\frac{30\cdots 47}{37\cdots 67}a^{17}+\frac{44\cdots 28}{11\cdots 01}a^{15}+\frac{18\cdots 77}{11\cdots 01}a^{13}-\frac{28\cdots 90}{11\cdots 01}a^{11}+\frac{11\cdots 07}{37\cdots 67}a^{9}-\frac{39\cdots 48}{11\cdots 01}a^{7}+\frac{26\cdots 22}{37\cdots 67}a^{5}-\frac{36\cdots 39}{11\cdots 01}a^{3}+\frac{55\cdots 15}{12\cdots 89}a$ 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:  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:  $C_{2}$, which has order $2$ (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:  $20$
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:   $\frac{49\cdots 02}{41\cdots 63}a^{20}-\frac{10\cdots 63}{13\cdots 21}a^{18}+\frac{78\cdots 17}{41\cdots 63}a^{16}-\frac{30\cdots 79}{13\cdots 21}a^{14}+\frac{52\cdots 48}{41\cdots 63}a^{12}-\frac{16\cdots 95}{41\cdots 63}a^{10}+\frac{24\cdots 09}{41\cdots 63}a^{8}-\frac{40\cdots 27}{13\cdots 21}a^{6}-\frac{16\cdots 89}{13\cdots 21}a^{4}+\frac{44\cdots 15}{41\cdots 63}a^{2}-\frac{54\cdots 85}{13\cdots 21}$, $\frac{31\cdots 83}{12\cdots 89}a^{20}-\frac{21\cdots 07}{12\cdots 89}a^{18}+\frac{18\cdots 96}{41\cdots 63}a^{16}-\frac{67\cdots 28}{12\cdots 89}a^{14}+\frac{45\cdots 80}{12\cdots 89}a^{12}-\frac{16\cdots 66}{12\cdots 89}a^{10}+\frac{34\cdots 21}{13\cdots 21}a^{8}-\frac{22\cdots 70}{12\cdots 89}a^{6}-\frac{14\cdots 82}{41\cdots 63}a^{4}+\frac{84\cdots 29}{12\cdots 89}a^{2}-\frac{62\cdots 93}{13\cdots 21}$, $\frac{28\cdots 86}{41\cdots 63}a^{20}-\frac{19\cdots 86}{41\cdots 63}a^{18}+\frac{17\cdots 62}{13\cdots 21}a^{16}-\frac{66\cdots 31}{41\cdots 63}a^{14}+\frac{46\cdots 56}{41\cdots 63}a^{12}-\frac{17\cdots 44}{41\cdots 63}a^{10}+\frac{12\cdots 32}{13\cdots 21}a^{8}-\frac{29\cdots 54}{41\cdots 63}a^{6}-\frac{14\cdots 19}{13\cdots 21}a^{4}+\frac{11\cdots 44}{41\cdots 63}a^{2}-\frac{39\cdots 33}{13\cdots 21}$, $\frac{77\cdots 88}{41\cdots 63}a^{20}-\frac{52\cdots 75}{41\cdots 63}a^{18}+\frac{13\cdots 03}{41\cdots 63}a^{16}-\frac{15\cdots 68}{41\cdots 63}a^{14}+\frac{33\cdots 68}{13\cdots 21}a^{12}-\frac{11\cdots 13}{13\cdots 21}a^{10}+\frac{60\cdots 05}{41\cdots 63}a^{8}-\frac{41\cdots 35}{41\cdots 63}a^{6}-\frac{30\cdots 08}{13\cdots 21}a^{4}+\frac{15\cdots 59}{41\cdots 63}a^{2}-\frac{44\cdots 18}{13\cdots 21}$, $\frac{12\cdots 39}{11\cdots 01}a^{21}-\frac{84\cdots 02}{11\cdots 01}a^{19}+\frac{72\cdots 11}{37\cdots 67}a^{17}-\frac{27\cdots 22}{11\cdots 01}a^{15}+\frac{18\cdots 70}{11\cdots 01}a^{13}-\frac{65\cdots 55}{11\cdots 01}a^{11}+\frac{41\cdots 49}{37\cdots 67}a^{9}-\frac{91\cdots 04}{11\cdots 01}a^{7}-\frac{55\cdots 26}{37\cdots 67}a^{5}+\frac{35\cdots 62}{11\cdots 01}a^{3}-\frac{27\cdots 48}{12\cdots 89}a$, $\frac{10\cdots 93}{11\cdots 01}a^{21}-\frac{69\cdots 59}{11\cdots 01}a^{19}+\frac{59\cdots 89}{37\cdots 67}a^{17}-\frac{22\cdots 60}{11\cdots 01}a^{15}+\frac{14\cdots 56}{11\cdots 01}a^{13}-\frac{54\cdots 93}{11\cdots 01}a^{11}+\frac{34\cdots 79}{37\cdots 67}a^{9}-\frac{73\cdots 95}{11\cdots 01}a^{7}-\frac{74\cdots 88}{37\cdots 67}a^{5}+\frac{32\cdots 59}{11\cdots 01}a^{3}+\frac{24\cdots 66}{12\cdots 89}a$, $\frac{26\cdots 14}{41\cdots 63}a^{20}-\frac{60\cdots 36}{13\cdots 21}a^{18}+\frac{15\cdots 71}{13\cdots 21}a^{16}-\frac{58\cdots 68}{41\cdots 63}a^{14}+\frac{12\cdots 03}{13\cdots 21}a^{12}-\frac{14\cdots 78}{41\cdots 63}a^{10}+\frac{25\cdots 10}{41\cdots 63}a^{8}-\frac{60\cdots 19}{13\cdots 21}a^{6}-\frac{43\cdots 82}{41\cdots 63}a^{4}+\frac{73\cdots 20}{41\cdots 63}a^{2}-\frac{86\cdots 89}{13\cdots 21}$, $\frac{17\cdots 41}{11\cdots 01}a^{21}-\frac{11\cdots 79}{11\cdots 01}a^{19}+\frac{10\cdots 55}{37\cdots 67}a^{17}-\frac{38\cdots 36}{11\cdots 01}a^{15}+\frac{25\cdots 09}{11\cdots 01}a^{13}-\frac{93\cdots 81}{11\cdots 01}a^{11}+\frac{58\cdots 72}{37\cdots 67}a^{9}-\frac{13\cdots 62}{11\cdots 01}a^{7}-\frac{72\cdots 54}{37\cdots 67}a^{5}+\frac{51\cdots 65}{11\cdots 01}a^{3}-\frac{49\cdots 73}{12\cdots 89}a$, $\frac{11\cdots 32}{12\cdots 89}a^{20}-\frac{73\cdots 13}{12\cdots 89}a^{18}+\frac{19\cdots 78}{13\cdots 21}a^{16}-\frac{18\cdots 44}{12\cdots 89}a^{14}+\frac{10\cdots 74}{12\cdots 89}a^{12}-\frac{29\cdots 36}{12\cdots 89}a^{10}+\frac{11\cdots 18}{41\cdots 63}a^{8}+\frac{24\cdots 21}{12\cdots 89}a^{6}-\frac{27\cdots 34}{41\cdots 63}a^{4}-\frac{28\cdots 30}{12\cdots 89}a^{2}+\frac{33\cdots 48}{13\cdots 21}$, $\frac{17\cdots 08}{12\cdots 89}a^{20}-\frac{11\cdots 78}{12\cdots 89}a^{18}+\frac{10\cdots 97}{41\cdots 63}a^{16}-\frac{37\cdots 45}{12\cdots 89}a^{14}+\frac{24\cdots 55}{12\cdots 89}a^{12}-\frac{81\cdots 92}{12\cdots 89}a^{10}+\frac{39\cdots 10}{41\cdots 63}a^{8}+\frac{12\cdots 05}{12\cdots 89}a^{6}-\frac{64\cdots 79}{41\cdots 63}a^{4}+\frac{11\cdots 45}{12\cdots 89}a^{2}-\frac{65\cdots 49}{13\cdots 21}$, $\frac{25\cdots 95}{37\cdots 67}a^{21}+\frac{36\cdots 81}{12\cdots 89}a^{20}-\frac{17\cdots 97}{37\cdots 67}a^{19}-\frac{24\cdots 51}{12\cdots 89}a^{18}+\frac{14\cdots 72}{12\cdots 89}a^{17}+\frac{20\cdots 10}{41\cdots 63}a^{16}-\frac{55\cdots 38}{37\cdots 67}a^{15}-\frac{77\cdots 48}{12\cdots 89}a^{14}+\frac{37\cdots 88}{37\cdots 67}a^{13}+\frac{50\cdots 51}{12\cdots 89}a^{12}-\frac{13\cdots 94}{37\cdots 67}a^{11}-\frac{18\cdots 84}{12\cdots 89}a^{10}+\frac{87\cdots 28}{12\cdots 89}a^{9}+\frac{37\cdots 08}{13\cdots 21}a^{8}-\frac{20\cdots 27}{37\cdots 67}a^{7}-\frac{24\cdots 37}{12\cdots 89}a^{6}-\frac{92\cdots 95}{12\cdots 89}a^{5}-\frac{14\cdots 04}{41\cdots 63}a^{4}+\frac{76\cdots 50}{37\cdots 67}a^{3}+\frac{96\cdots 74}{12\cdots 89}a^{2}-\frac{88\cdots 48}{41\cdots 63}a-\frac{91\cdots 85}{13\cdots 21}$, $\frac{51\cdots 59}{37\cdots 67}a^{21}+\frac{10\cdots 27}{12\cdots 89}a^{20}-\frac{34\cdots 64}{37\cdots 67}a^{19}-\frac{71\cdots 36}{12\cdots 89}a^{18}+\frac{29\cdots 22}{12\cdots 89}a^{17}+\frac{61\cdots 02}{41\cdots 63}a^{16}-\frac{11\cdots 12}{37\cdots 67}a^{15}-\frac{23\cdots 09}{12\cdots 89}a^{14}+\frac{76\cdots 76}{37\cdots 67}a^{13}+\frac{15\cdots 91}{12\cdots 89}a^{12}-\frac{28\cdots 04}{37\cdots 67}a^{11}-\frac{57\cdots 52}{12\cdots 89}a^{10}+\frac{18\cdots 36}{12\cdots 89}a^{9}+\frac{36\cdots 69}{41\cdots 63}a^{8}-\frac{44\cdots 37}{37\cdots 67}a^{7}-\frac{84\cdots 45}{12\cdots 89}a^{6}-\frac{13\cdots 95}{12\cdots 89}a^{5}-\frac{14\cdots 67}{13\cdots 21}a^{4}+\frac{17\cdots 25}{37\cdots 67}a^{3}+\frac{33\cdots 20}{12\cdots 89}a^{2}-\frac{87\cdots 21}{13\cdots 21}a-\frac{22\cdots 91}{13\cdots 21}$, $\frac{46\cdots 26}{11\cdots 01}a^{21}-\frac{97\cdots 97}{13\cdots 21}a^{20}-\frac{31\cdots 70}{11\cdots 01}a^{19}+\frac{19\cdots 60}{41\cdots 63}a^{18}+\frac{26\cdots 35}{37\cdots 67}a^{17}-\frac{51\cdots 08}{41\cdots 63}a^{16}-\frac{10\cdots 58}{11\cdots 01}a^{15}+\frac{64\cdots 41}{41\cdots 63}a^{14}+\frac{67\cdots 61}{11\cdots 01}a^{13}-\frac{42\cdots 91}{41\cdots 63}a^{12}-\frac{24\cdots 16}{11\cdots 01}a^{11}+\frac{52\cdots 26}{13\cdots 21}a^{10}+\frac{15\cdots 26}{37\cdots 67}a^{9}-\frac{29\cdots 59}{41\cdots 63}a^{8}-\frac{34\cdots 53}{11\cdots 01}a^{7}+\frac{76\cdots 63}{13\cdots 21}a^{6}-\frac{20\cdots 57}{37\cdots 67}a^{5}+\frac{29\cdots 34}{41\cdots 63}a^{4}+\frac{13\cdots 17}{11\cdots 01}a^{3}-\frac{82\cdots 41}{41\cdots 63}a^{2}-\frac{11\cdots 24}{12\cdots 89}a+\frac{20\cdots 45}{13\cdots 21}$, $\frac{27\cdots 26}{11\cdots 01}a^{21}+\frac{11\cdots 45}{12\cdots 89}a^{20}-\frac{18\cdots 63}{11\cdots 01}a^{19}-\frac{78\cdots 47}{12\cdots 89}a^{18}+\frac{16\cdots 64}{37\cdots 67}a^{17}+\frac{63\cdots 00}{41\cdots 63}a^{16}-\frac{62\cdots 53}{11\cdots 01}a^{15}-\frac{21\cdots 21}{12\cdots 89}a^{14}+\frac{43\cdots 14}{11\cdots 01}a^{13}+\frac{12\cdots 91}{12\cdots 89}a^{12}-\frac{16\cdots 58}{11\cdots 01}a^{11}-\frac{34\cdots 85}{12\cdots 89}a^{10}+\frac{11\cdots 26}{37\cdots 67}a^{9}+\frac{11\cdots 69}{41\cdots 63}a^{8}-\frac{29\cdots 36}{11\cdots 01}a^{7}+\frac{98\cdots 07}{12\cdots 89}a^{6}-\frac{11\cdots 15}{37\cdots 67}a^{5}-\frac{12\cdots 75}{13\cdots 21}a^{4}+\frac{11\cdots 10}{11\cdots 01}a^{3}-\frac{76\cdots 38}{12\cdots 89}a^{2}-\frac{81\cdots 95}{12\cdots 89}a-\frac{81\cdots 39}{13\cdots 21}$, $\frac{87\cdots 83}{11\cdots 01}a^{21}-\frac{30\cdots 68}{13\cdots 21}a^{20}-\frac{59\cdots 01}{11\cdots 01}a^{19}+\frac{61\cdots 01}{41\cdots 63}a^{18}+\frac{50\cdots 91}{37\cdots 67}a^{17}-\frac{15\cdots 98}{41\cdots 63}a^{16}-\frac{19\cdots 62}{11\cdots 01}a^{15}+\frac{66\cdots 58}{13\cdots 21}a^{14}+\frac{12\cdots 54}{11\cdots 01}a^{13}-\frac{44\cdots 52}{13\cdots 21}a^{12}-\frac{46\cdots 06}{11\cdots 01}a^{11}+\frac{49\cdots 80}{41\cdots 63}a^{10}+\frac{28\cdots 37}{37\cdots 67}a^{9}-\frac{30\cdots 50}{13\cdots 21}a^{8}-\frac{62\cdots 02}{11\cdots 01}a^{7}+\frac{68\cdots 53}{41\cdots 63}a^{6}-\frac{48\cdots 83}{37\cdots 67}a^{5}+\frac{15\cdots 86}{41\cdots 63}a^{4}+\frac{25\cdots 06}{11\cdots 01}a^{3}-\frac{27\cdots 54}{41\cdots 63}a^{2}-\frac{86\cdots 62}{12\cdots 89}a+\frac{27\cdots 72}{13\cdots 21}$, $\frac{12\cdots 85}{11\cdots 01}a^{21}+\frac{11\cdots 59}{12\cdots 89}a^{20}-\frac{81\cdots 12}{11\cdots 01}a^{19}-\frac{78\cdots 40}{12\cdots 89}a^{18}+\frac{70\cdots 88}{37\cdots 67}a^{17}+\frac{22\cdots 39}{13\cdots 21}a^{16}-\frac{26\cdots 11}{11\cdots 01}a^{15}-\frac{24\cdots 58}{12\cdots 89}a^{14}+\frac{17\cdots 39}{11\cdots 01}a^{13}+\frac{16\cdots 15}{12\cdots 89}a^{12}-\frac{64\cdots 82}{11\cdots 01}a^{11}-\frac{57\cdots 40}{12\cdots 89}a^{10}+\frac{40\cdots 47}{37\cdots 67}a^{9}+\frac{35\cdots 91}{41\cdots 63}a^{8}-\frac{89\cdots 87}{11\cdots 01}a^{7}-\frac{76\cdots 90}{12\cdots 89}a^{6}-\frac{58\cdots 25}{37\cdots 67}a^{5}-\frac{63\cdots 74}{41\cdots 63}a^{4}+\frac{34\cdots 23}{11\cdots 01}a^{3}+\frac{32\cdots 28}{12\cdots 89}a^{2}-\frac{23\cdots 86}{12\cdots 89}a+\frac{28\cdots 41}{13\cdots 21}$, $\frac{11\cdots 65}{11\cdots 01}a^{21}-\frac{46\cdots 93}{12\cdots 89}a^{20}-\frac{81\cdots 21}{11\cdots 01}a^{19}+\frac{31\cdots 27}{12\cdots 89}a^{18}+\frac{69\cdots 47}{37\cdots 67}a^{17}-\frac{89\cdots 17}{13\cdots 21}a^{16}-\frac{26\cdots 52}{11\cdots 01}a^{15}+\frac{10\cdots 66}{12\cdots 89}a^{14}+\frac{17\cdots 52}{11\cdots 01}a^{13}-\frac{67\cdots 83}{12\cdots 89}a^{12}-\frac{62\cdots 34}{11\cdots 01}a^{11}+\frac{24\cdots 82}{12\cdots 89}a^{10}+\frac{39\cdots 29}{37\cdots 67}a^{9}-\frac{15\cdots 80}{41\cdots 63}a^{8}-\frac{85\cdots 33}{11\cdots 01}a^{7}+\frac{38\cdots 47}{12\cdots 89}a^{6}-\frac{62\cdots 08}{37\cdots 67}a^{5}-\frac{28\cdots 05}{41\cdots 63}a^{4}+\frac{35\cdots 36}{11\cdots 01}a^{3}-\frac{76\cdots 46}{12\cdots 89}a^{2}-\frac{38\cdots 46}{12\cdots 89}a+\frac{78\cdots 63}{13\cdots 21}$, $\frac{11\cdots 51}{11\cdots 01}a^{21}-\frac{42\cdots 98}{13\cdots 21}a^{20}-\frac{80\cdots 81}{11\cdots 01}a^{19}+\frac{85\cdots 15}{41\cdots 63}a^{18}+\frac{69\cdots 35}{37\cdots 67}a^{17}-\frac{22\cdots 68}{41\cdots 63}a^{16}-\frac{25\cdots 80}{11\cdots 01}a^{15}+\frac{27\cdots 17}{41\cdots 63}a^{14}+\frac{17\cdots 62}{11\cdots 01}a^{13}-\frac{18\cdots 37}{41\cdots 63}a^{12}-\frac{62\cdots 07}{11\cdots 01}a^{11}+\frac{22\cdots 48}{13\cdots 21}a^{10}+\frac{38\cdots 72}{37\cdots 67}a^{9}-\frac{41\cdots 72}{13\cdots 21}a^{8}-\frac{84\cdots 28}{11\cdots 01}a^{7}+\frac{88\cdots 10}{41\cdots 63}a^{6}-\frac{65\cdots 29}{37\cdots 67}a^{5}+\frac{21\cdots 00}{41\cdots 63}a^{4}+\frac{33\cdots 45}{11\cdots 01}a^{3}-\frac{12\cdots 49}{13\cdots 21}a^{2}-\frac{11\cdots 93}{12\cdots 89}a+\frac{31\cdots 38}{13\cdots 21}$, $\frac{21\cdots 37}{11\cdots 01}a^{21}-\frac{74\cdots 21}{12\cdots 89}a^{20}-\frac{14\cdots 13}{11\cdots 01}a^{19}+\frac{49\cdots 22}{12\cdots 89}a^{18}+\frac{12\cdots 17}{37\cdots 67}a^{17}-\frac{42\cdots 95}{41\cdots 63}a^{16}-\frac{46\cdots 86}{11\cdots 01}a^{15}+\frac{15\cdots 32}{12\cdots 89}a^{14}+\frac{30\cdots 08}{11\cdots 01}a^{13}-\frac{99\cdots 86}{12\cdots 89}a^{12}-\frac{11\cdots 59}{11\cdots 01}a^{11}+\frac{35\cdots 74}{12\cdots 89}a^{10}+\frac{70\cdots 86}{37\cdots 67}a^{9}-\frac{70\cdots 95}{13\cdots 21}a^{8}-\frac{15\cdots 44}{11\cdots 01}a^{7}+\frac{46\cdots 50}{12\cdots 89}a^{6}-\frac{10\cdots 25}{37\cdots 67}a^{5}+\frac{31\cdots 68}{41\cdots 63}a^{4}+\frac{63\cdots 04}{11\cdots 01}a^{3}-\frac{19\cdots 45}{12\cdots 89}a^{2}-\frac{23\cdots 90}{12\cdots 89}a+\frac{59\cdots 56}{13\cdots 21}$, $\frac{73\cdots 77}{11\cdots 01}a^{21}+\frac{31\cdots 47}{12\cdots 89}a^{20}-\frac{49\cdots 42}{11\cdots 01}a^{19}-\frac{20\cdots 22}{12\cdots 89}a^{18}+\frac{42\cdots 84}{37\cdots 67}a^{17}+\frac{57\cdots 36}{13\cdots 21}a^{16}-\frac{16\cdots 49}{11\cdots 01}a^{15}-\frac{62\cdots 02}{12\cdots 89}a^{14}+\frac{10\cdots 84}{11\cdots 01}a^{13}+\frac{39\cdots 13}{12\cdots 89}a^{12}-\frac{38\cdots 64}{11\cdots 01}a^{11}-\frac{13\cdots 33}{12\cdots 89}a^{10}+\frac{24\cdots 03}{37\cdots 67}a^{9}+\frac{24\cdots 35}{13\cdots 21}a^{8}-\frac{52\cdots 55}{11\cdots 01}a^{7}-\frac{14\cdots 12}{12\cdots 89}a^{6}-\frac{37\cdots 39}{37\cdots 67}a^{5}-\frac{31\cdots 99}{13\cdots 21}a^{4}+\frac{21\cdots 23}{11\cdots 01}a^{3}+\frac{58\cdots 45}{12\cdots 89}a^{2}-\frac{21\cdots 88}{12\cdots 89}a-\frac{54\cdots 92}{13\cdots 21}$ 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:  \( 7215216869150000 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{1}\cdot 7215216869150000 \cdot 1}{2\cdot\sqrt{4129233136056857981979443884256982828952059904}}\cr\approx \mathstrut & 0.369883086860875 \end{aligned}\] (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^22 - 68*x^20 + 1749*x^18 - 21878*x^16 + 146579*x^14 - 538577*x^12 + 1028049*x^10 - 796919*x^8 - 100770*x^6 + 299644*x^4 - 34200*x^2 + 729) 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^22 - 68*x^20 + 1749*x^18 - 21878*x^16 + 146579*x^14 - 538577*x^12 + 1028049*x^10 - 796919*x^8 - 100770*x^6 + 299644*x^4 - 34200*x^2 + 729, 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^22 - 68*x^20 + 1749*x^18 - 21878*x^16 + 146579*x^14 - 538577*x^12 + 1028049*x^10 - 796919*x^8 - 100770*x^6 + 299644*x^4 - 34200*x^2 + 729); 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^22 - 68*x^20 + 1749*x^18 - 21878*x^16 + 146579*x^14 - 538577*x^12 + 1028049*x^10 - 796919*x^8 - 100770*x^6 + 299644*x^4 - 34200*x^2 + 729); 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

$C_2^{11}.\PSL(2,11)$ (as 22T42):

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 1351680
The 112 conjugacy class representatives for $C_2^{11}.\PSL(2,11)$
Character table for $C_2^{11}.\PSL(2,11)$

Intermediate fields

11.11.31376518243389673201.2

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 22 sibling: data not computed
Degree 44 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 ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.4.0.1}{4} }$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ $22$ ${\href{/padicField/13.5.0.1}{5} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.11.0.1}{11} }^{2}$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ $22$ ${\href{/padicField/29.5.0.1}{5} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ $22$ ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.11.0.1}{11} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.11.0.1}{11} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 2.11.2.22a72.1$x^{22} + 2 x^{20} + 2 x^{18} + 2 x^{17} + 2 x^{15} + 2 x^{13} + 4 x^{11} + 4 x^{9} + 2 x^{8} + 2 x^{7} + 4 x^{6} + 3 x^{4} + 2 x^{2} + 3$$2$$11$$22$not computednot computed
\(74843\) Copy content Toggle raw display $\Q_{74843}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{74843}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $4$$2$$2$$2$
Deg $8$$2$$4$$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)