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Properties

Label 22.14.226...241.3
Degree $22$
Signature $[14, 4]$
Discriminant $2.266\times 10^{37}$
Root discriminant \(49.88\)
Ramified prime $1297$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.D_{11}$ (as 22T30)

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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 - 7*x^21 - 9*x^20 + 171*x^19 - 276*x^18 - 1022*x^17 + 3668*x^16 - 1134*x^15 - 10988*x^14 + 19765*x^13 - 17789*x^12 + 6*x^11 + 111532*x^10 - 268518*x^9 - 21573*x^8 + 707388*x^7 - 501170*x^6 - 789327*x^5 + 795886*x^4 + 400629*x^3 - 428800*x^2 - 69431*x + 68279)
 
Copy content gp:K = bnfinit(y^22 - 7*y^21 - 9*y^20 + 171*y^19 - 276*y^18 - 1022*y^17 + 3668*y^16 - 1134*y^15 - 10988*y^14 + 19765*y^13 - 17789*y^12 + 6*y^11 + 111532*y^10 - 268518*y^9 - 21573*y^8 + 707388*y^7 - 501170*y^6 - 789327*y^5 + 795886*y^4 + 400629*y^3 - 428800*y^2 - 69431*y + 68279, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 7*x^21 - 9*x^20 + 171*x^19 - 276*x^18 - 1022*x^17 + 3668*x^16 - 1134*x^15 - 10988*x^14 + 19765*x^13 - 17789*x^12 + 6*x^11 + 111532*x^10 - 268518*x^9 - 21573*x^8 + 707388*x^7 - 501170*x^6 - 789327*x^5 + 795886*x^4 + 400629*x^3 - 428800*x^2 - 69431*x + 68279);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 7*x^21 - 9*x^20 + 171*x^19 - 276*x^18 - 1022*x^17 + 3668*x^16 - 1134*x^15 - 10988*x^14 + 19765*x^13 - 17789*x^12 + 6*x^11 + 111532*x^10 - 268518*x^9 - 21573*x^8 + 707388*x^7 - 501170*x^6 - 789327*x^5 + 795886*x^4 + 400629*x^3 - 428800*x^2 - 69431*x + 68279)
 

\( x^{22} - 7 x^{21} - 9 x^{20} + 171 x^{19} - 276 x^{18} - 1022 x^{17} + 3668 x^{16} - 1134 x^{15} + \cdots + 68279 \) 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:  $[14, 4]$
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:   \(22661033510180079603495293971842498241\) \(\medspace = 1297^{12}\) 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:  \(49.88\)
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:  $1297^{3/4}\approx 216.12498794754728$
Ramified primes:   \(1297\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{25}a^{20}+\frac{6}{25}a^{19}-\frac{1}{5}a^{18}+\frac{12}{25}a^{17}-\frac{2}{5}a^{15}-\frac{12}{25}a^{14}-\frac{2}{5}a^{11}+\frac{6}{25}a^{10}-\frac{1}{25}a^{9}+\frac{6}{25}a^{7}+\frac{1}{5}a^{6}+\frac{9}{25}a^{5}+\frac{2}{25}a^{4}+\frac{8}{25}a^{3}-\frac{8}{25}a^{2}+\frac{8}{25}a-\frac{4}{25}$, $\frac{1}{31\cdots 75}a^{21}-\frac{17\cdots 41}{31\cdots 75}a^{20}+\frac{38\cdots 63}{31\cdots 75}a^{19}+\frac{11\cdots 47}{31\cdots 75}a^{18}-\frac{83\cdots 14}{31\cdots 75}a^{17}+\frac{27\cdots 93}{63\cdots 35}a^{16}-\frac{12\cdots 17}{31\cdots 75}a^{15}+\frac{12\cdots 14}{31\cdots 75}a^{14}-\frac{12\cdots 49}{12\cdots 87}a^{13}-\frac{89\cdots 12}{63\cdots 35}a^{12}-\frac{85\cdots 74}{31\cdots 75}a^{11}-\frac{89\cdots 58}{31\cdots 75}a^{10}+\frac{62\cdots 47}{31\cdots 75}a^{9}-\frac{21\cdots 19}{31\cdots 75}a^{8}-\frac{79\cdots 02}{31\cdots 75}a^{7}+\frac{23\cdots 99}{31\cdots 75}a^{6}+\frac{51\cdots 29}{31\cdots 75}a^{5}+\frac{11\cdots 64}{31\cdots 75}a^{4}+\frac{53\cdots 91}{31\cdots 75}a^{3}-\frac{56\cdots 91}{31\cdots 75}a^{2}-\frac{24\cdots 46}{63\cdots 35}a-\frac{99\cdots 62}{31\cdots 75}$ 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:  $17$
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:   $\frac{23\cdots 06}{31\cdots 75}a^{21}-\frac{36\cdots 33}{63\cdots 35}a^{20}-\frac{48\cdots 61}{31\cdots 75}a^{19}+\frac{39\cdots 27}{31\cdots 75}a^{18}-\frac{97\cdots 87}{31\cdots 75}a^{17}-\frac{29\cdots 82}{63\cdots 35}a^{16}+\frac{95\cdots 13}{31\cdots 75}a^{15}-\frac{10\cdots 88}{31\cdots 75}a^{14}-\frac{57\cdots 85}{12\cdots 87}a^{13}+\frac{10\cdots 43}{63\cdots 35}a^{12}-\frac{89\cdots 79}{31\cdots 75}a^{11}+\frac{88\cdots 63}{31\cdots 75}a^{10}+\frac{16\cdots 51}{31\cdots 75}a^{9}-\frac{73\cdots 14}{31\cdots 75}a^{8}+\frac{59\cdots 24}{31\cdots 75}a^{7}+\frac{98\cdots 24}{31\cdots 75}a^{6}-\frac{17\cdots 97}{31\cdots 75}a^{5}-\frac{13\cdots 54}{31\cdots 75}a^{4}+\frac{14\cdots 94}{31\cdots 75}a^{3}-\frac{22\cdots 19}{31\cdots 75}a^{2}-\frac{29\cdots 57}{31\cdots 75}a+\frac{76\cdots 29}{31\cdots 75}$, $\frac{61\cdots 51}{31\cdots 75}a^{21}-\frac{47\cdots 54}{31\cdots 75}a^{20}-\frac{47\cdots 48}{63\cdots 35}a^{19}+\frac{10\cdots 62}{31\cdots 75}a^{18}-\frac{47\cdots 89}{63\cdots 35}a^{17}-\frac{87\cdots 47}{63\cdots 35}a^{16}+\frac{24\cdots 13}{31\cdots 75}a^{15}-\frac{48\cdots 41}{63\cdots 35}a^{14}-\frac{16\cdots 08}{12\cdots 87}a^{13}+\frac{27\cdots 88}{63\cdots 35}a^{12}-\frac{21\cdots 94}{31\cdots 75}a^{11}+\frac{19\cdots 64}{31\cdots 75}a^{10}+\frac{95\cdots 62}{63\cdots 35}a^{9}-\frac{18\cdots 19}{31\cdots 75}a^{8}+\frac{25\cdots 69}{63\cdots 35}a^{7}+\frac{27\cdots 84}{31\cdots 75}a^{6}-\frac{43\cdots 38}{31\cdots 75}a^{5}-\frac{80\cdots 12}{31\cdots 75}a^{4}+\frac{36\cdots 87}{31\cdots 75}a^{3}-\frac{55\cdots 37}{31\cdots 75}a^{2}-\frac{80\cdots 34}{31\cdots 75}a+\frac{42\cdots 78}{63\cdots 35}$, $\frac{14\cdots 28}{63\cdots 35}a^{21}-\frac{13\cdots 82}{31\cdots 75}a^{20}-\frac{34\cdots 82}{31\cdots 75}a^{19}+\frac{19\cdots 68}{63\cdots 35}a^{18}+\frac{43\cdots 86}{31\cdots 75}a^{17}-\frac{77\cdots 03}{12\cdots 87}a^{16}-\frac{10\cdots 87}{63\cdots 35}a^{15}+\frac{12\cdots 39}{31\cdots 75}a^{14}-\frac{67\cdots 35}{12\cdots 87}a^{13}-\frac{61\cdots 18}{12\cdots 87}a^{12}+\frac{11\cdots 97}{63\cdots 35}a^{11}-\frac{10\cdots 22}{31\cdots 75}a^{10}+\frac{15\cdots 72}{31\cdots 75}a^{9}+\frac{26\cdots 83}{63\cdots 35}a^{8}-\frac{10\cdots 57}{31\cdots 75}a^{7}+\frac{32\cdots 24}{12\cdots 87}a^{6}+\frac{15\cdots 32}{31\cdots 75}a^{5}-\frac{24\cdots 49}{31\cdots 75}a^{4}-\frac{82\cdots 71}{31\cdots 75}a^{3}+\frac{19\cdots 21}{31\cdots 75}a^{2}+\frac{89\cdots 14}{31\cdots 75}a-\frac{44\cdots 62}{31\cdots 75}$, $\frac{14\cdots 74}{31\cdots 75}a^{21}-\frac{21\cdots 69}{63\cdots 35}a^{20}-\frac{64\cdots 79}{31\cdots 75}a^{19}+\frac{23\cdots 08}{31\cdots 75}a^{18}-\frac{51\cdots 93}{31\cdots 75}a^{17}-\frac{20\cdots 43}{63\cdots 35}a^{16}+\frac{53\cdots 77}{31\cdots 75}a^{15}-\frac{50\cdots 82}{31\cdots 75}a^{14}-\frac{38\cdots 12}{12\cdots 87}a^{13}+\frac{59\cdots 22}{63\cdots 35}a^{12}-\frac{46\cdots 41}{31\cdots 75}a^{11}+\frac{41\cdots 67}{31\cdots 75}a^{10}+\frac{10\cdots 39}{31\cdots 75}a^{9}-\frac{40\cdots 81}{31\cdots 75}a^{8}+\frac{25\cdots 11}{31\cdots 75}a^{7}+\frac{61\cdots 46}{31\cdots 75}a^{6}-\frac{89\cdots 78}{31\cdots 75}a^{5}-\frac{22\cdots 86}{31\cdots 75}a^{4}+\frac{72\cdots 46}{31\cdots 75}a^{3}-\frac{61\cdots 21}{31\cdots 75}a^{2}-\frac{14\cdots 58}{31\cdots 75}a+\frac{19\cdots 56}{31\cdots 75}$, $\frac{21\cdots 96}{31\cdots 75}a^{21}-\frac{19\cdots 82}{31\cdots 75}a^{20}+\frac{21\cdots 22}{31\cdots 75}a^{19}+\frac{36\cdots 92}{31\cdots 75}a^{18}-\frac{14\cdots 21}{31\cdots 75}a^{17}+\frac{68\cdots 93}{63\cdots 35}a^{16}+\frac{10\cdots 28}{31\cdots 75}a^{15}-\frac{23\cdots 79}{31\cdots 75}a^{14}+\frac{82\cdots 09}{12\cdots 87}a^{13}+\frac{14\cdots 73}{63\cdots 35}a^{12}-\frac{16\cdots 69}{31\cdots 75}a^{11}+\frac{20\cdots 56}{31\cdots 75}a^{10}+\frac{28\cdots 33}{31\cdots 75}a^{9}-\frac{91\cdots 74}{31\cdots 75}a^{8}+\frac{15\cdots 32}{31\cdots 75}a^{7}+\frac{66\cdots 74}{31\cdots 75}a^{6}-\frac{61\cdots 41}{63\cdots 35}a^{5}+\frac{24\cdots 52}{31\cdots 75}a^{4}+\frac{16\cdots 43}{31\cdots 75}a^{3}-\frac{24\cdots 93}{31\cdots 75}a^{2}+\frac{18\cdots 77}{31\cdots 75}a+\frac{54\cdots 07}{31\cdots 75}$, $\frac{23\cdots 06}{31\cdots 75}a^{21}-\frac{36\cdots 33}{63\cdots 35}a^{20}-\frac{48\cdots 61}{31\cdots 75}a^{19}+\frac{39\cdots 27}{31\cdots 75}a^{18}-\frac{97\cdots 87}{31\cdots 75}a^{17}-\frac{29\cdots 82}{63\cdots 35}a^{16}+\frac{95\cdots 13}{31\cdots 75}a^{15}-\frac{10\cdots 88}{31\cdots 75}a^{14}-\frac{57\cdots 85}{12\cdots 87}a^{13}+\frac{10\cdots 43}{63\cdots 35}a^{12}-\frac{89\cdots 79}{31\cdots 75}a^{11}+\frac{88\cdots 63}{31\cdots 75}a^{10}+\frac{16\cdots 51}{31\cdots 75}a^{9}-\frac{73\cdots 14}{31\cdots 75}a^{8}+\frac{59\cdots 24}{31\cdots 75}a^{7}+\frac{98\cdots 24}{31\cdots 75}a^{6}-\frac{17\cdots 97}{31\cdots 75}a^{5}-\frac{13\cdots 54}{31\cdots 75}a^{4}+\frac{14\cdots 94}{31\cdots 75}a^{3}-\frac{22\cdots 19}{31\cdots 75}a^{2}-\frac{29\cdots 57}{31\cdots 75}a+\frac{39\cdots 04}{31\cdots 75}$, $\frac{16\cdots 96}{31\cdots 75}a^{21}-\frac{10\cdots 23}{31\cdots 75}a^{20}-\frac{16\cdots 24}{31\cdots 75}a^{19}+\frac{25\cdots 47}{31\cdots 75}a^{18}-\frac{37\cdots 88}{31\cdots 75}a^{17}-\frac{30\cdots 92}{63\cdots 35}a^{16}+\frac{48\cdots 13}{31\cdots 75}a^{15}-\frac{14\cdots 12}{31\cdots 75}a^{14}-\frac{49\cdots 96}{12\cdots 87}a^{13}+\frac{45\cdots 03}{63\cdots 35}a^{12}-\frac{31\cdots 84}{31\cdots 75}a^{11}+\frac{42\cdots 17}{63\cdots 35}a^{10}+\frac{13\cdots 24}{31\cdots 75}a^{9}-\frac{34\cdots 99}{31\cdots 75}a^{8}-\frac{25\cdots 14}{31\cdots 75}a^{7}+\frac{69\cdots 44}{31\cdots 75}a^{6}-\frac{33\cdots 24}{31\cdots 75}a^{5}-\frac{15\cdots 76}{63\cdots 35}a^{4}+\frac{30\cdots 43}{63\cdots 35}a^{3}+\frac{97\cdots 67}{63\cdots 35}a^{2}+\frac{72\cdots 49}{31\cdots 75}a-\frac{14\cdots 04}{31\cdots 75}$, $\frac{15\cdots 48}{31\cdots 75}a^{21}-\frac{12\cdots 16}{31\cdots 75}a^{20}-\frac{23\cdots 89}{31\cdots 75}a^{19}+\frac{27\cdots 21}{31\cdots 75}a^{18}-\frac{68\cdots 23}{31\cdots 75}a^{17}-\frac{18\cdots 81}{63\cdots 35}a^{16}+\frac{65\cdots 89}{31\cdots 75}a^{15}-\frac{80\cdots 52}{31\cdots 75}a^{14}-\frac{36\cdots 57}{12\cdots 87}a^{13}+\frac{77\cdots 64}{63\cdots 35}a^{12}-\frac{65\cdots 72}{31\cdots 75}a^{11}+\frac{64\cdots 53}{31\cdots 75}a^{10}+\frac{10\cdots 29}{31\cdots 75}a^{9}-\frac{51\cdots 87}{31\cdots 75}a^{8}+\frac{46\cdots 41}{31\cdots 75}a^{7}+\frac{63\cdots 87}{31\cdots 75}a^{6}-\frac{26\cdots 83}{63\cdots 35}a^{5}+\frac{76\cdots 01}{31\cdots 75}a^{4}+\frac{10\cdots 34}{31\cdots 75}a^{3}-\frac{38\cdots 09}{31\cdots 75}a^{2}-\frac{22\cdots 24}{31\cdots 75}a+\frac{94\cdots 41}{31\cdots 75}$, $\frac{49\cdots 57}{31\cdots 75}a^{21}-\frac{78\cdots 31}{63\cdots 35}a^{20}-\frac{95\cdots 17}{31\cdots 75}a^{19}+\frac{85\cdots 19}{31\cdots 75}a^{18}-\frac{21\cdots 89}{31\cdots 75}a^{17}-\frac{61\cdots 14}{63\cdots 35}a^{16}+\frac{20\cdots 11}{31\cdots 75}a^{15}-\frac{24\cdots 61}{31\cdots 75}a^{14}-\frac{12\cdots 98}{12\cdots 87}a^{13}+\frac{24\cdots 66}{63\cdots 35}a^{12}-\frac{20\cdots 13}{31\cdots 75}a^{11}+\frac{19\cdots 36}{31\cdots 75}a^{10}+\frac{35\cdots 72}{31\cdots 75}a^{9}-\frac{16\cdots 33}{31\cdots 75}a^{8}+\frac{13\cdots 53}{31\cdots 75}a^{7}+\frac{20\cdots 03}{31\cdots 75}a^{6}-\frac{41\cdots 84}{31\cdots 75}a^{5}+\frac{41\cdots 12}{31\cdots 75}a^{4}+\frac{33\cdots 93}{31\cdots 75}a^{3}-\frac{10\cdots 43}{31\cdots 75}a^{2}-\frac{74\cdots 29}{31\cdots 75}a+\frac{22\cdots 63}{31\cdots 75}$, $\frac{28\cdots 36}{31\cdots 75}a^{21}-\frac{17\cdots 57}{31\cdots 75}a^{20}-\frac{39\cdots 68}{31\cdots 75}a^{19}+\frac{45\cdots 72}{31\cdots 75}a^{18}-\frac{43\cdots 26}{31\cdots 75}a^{17}-\frac{66\cdots 77}{63\cdots 35}a^{16}+\frac{81\cdots 73}{31\cdots 75}a^{15}+\frac{31\cdots 26}{31\cdots 75}a^{14}-\frac{12\cdots 35}{12\cdots 87}a^{13}+\frac{72\cdots 83}{63\cdots 35}a^{12}-\frac{20\cdots 29}{31\cdots 75}a^{11}-\frac{29\cdots 24}{31\cdots 75}a^{10}+\frac{31\cdots 73}{31\cdots 75}a^{9}-\frac{55\cdots 59}{31\cdots 75}a^{8}-\frac{49\cdots 58}{31\cdots 75}a^{7}+\frac{18\cdots 09}{31\cdots 75}a^{6}-\frac{50\cdots 82}{63\cdots 35}a^{5}-\frac{26\cdots 83}{31\cdots 75}a^{4}+\frac{75\cdots 53}{31\cdots 75}a^{3}+\frac{16\cdots 47}{31\cdots 75}a^{2}-\frac{32\cdots 53}{31\cdots 75}a-\frac{35\cdots 33}{31\cdots 75}$, $\frac{97\cdots 94}{31\cdots 75}a^{21}-\frac{72\cdots 02}{31\cdots 75}a^{20}-\frac{51\cdots 16}{31\cdots 75}a^{19}+\frac{16\cdots 33}{31\cdots 75}a^{18}-\frac{34\cdots 17}{31\cdots 75}a^{17}-\frac{15\cdots 73}{63\cdots 35}a^{16}+\frac{37\cdots 57}{31\cdots 75}a^{15}-\frac{31\cdots 08}{31\cdots 75}a^{14}-\frac{29\cdots 01}{12\cdots 87}a^{13}+\frac{41\cdots 92}{63\cdots 35}a^{12}-\frac{30\cdots 76}{31\cdots 75}a^{11}+\frac{48\cdots 12}{63\cdots 35}a^{10}+\frac{82\cdots 66}{31\cdots 75}a^{9}-\frac{28\cdots 36}{31\cdots 75}a^{8}+\frac{14\cdots 49}{31\cdots 75}a^{7}+\frac{46\cdots 66}{31\cdots 75}a^{6}-\frac{61\cdots 56}{31\cdots 75}a^{5}-\frac{46\cdots 56}{63\cdots 35}a^{4}+\frac{11\cdots 99}{63\cdots 35}a^{3}-\frac{11\cdots 59}{63\cdots 35}a^{2}-\frac{12\cdots 04}{31\cdots 75}a+\frac{79\cdots 39}{31\cdots 75}$, $\frac{91\cdots 89}{31\cdots 75}a^{21}-\frac{60\cdots 63}{31\cdots 75}a^{20}-\frac{10\cdots 52}{31\cdots 75}a^{19}+\frac{14\cdots 53}{31\cdots 75}a^{18}-\frac{19\cdots 14}{31\cdots 75}a^{17}-\frac{18\cdots 93}{63\cdots 35}a^{16}+\frac{28\cdots 27}{31\cdots 75}a^{15}-\frac{39\cdots 86}{31\cdots 75}a^{14}-\frac{33\cdots 71}{12\cdots 87}a^{13}+\frac{27\cdots 62}{63\cdots 35}a^{12}-\frac{15\cdots 96}{31\cdots 75}a^{11}+\frac{44\cdots 54}{31\cdots 75}a^{10}+\frac{89\cdots 47}{31\cdots 75}a^{9}-\frac{20\cdots 91}{31\cdots 75}a^{8}-\frac{50\cdots 87}{31\cdots 75}a^{7}+\frac{48\cdots 16}{31\cdots 75}a^{6}-\frac{44\cdots 99}{63\cdots 35}a^{5}-\frac{55\cdots 57}{31\cdots 75}a^{4}+\frac{24\cdots 87}{31\cdots 75}a^{3}+\frac{28\cdots 88}{31\cdots 75}a^{2}-\frac{44\cdots 32}{31\cdots 75}a-\frac{42\cdots 37}{31\cdots 75}$, $\frac{53\cdots 89}{31\cdots 75}a^{21}-\frac{76\cdots 47}{63\cdots 35}a^{20}-\frac{41\cdots 59}{31\cdots 75}a^{19}+\frac{91\cdots 38}{31\cdots 75}a^{18}-\frac{16\cdots 78}{31\cdots 75}a^{17}-\frac{99\cdots 78}{63\cdots 35}a^{16}+\frac{19\cdots 72}{31\cdots 75}a^{15}-\frac{10\cdots 97}{31\cdots 75}a^{14}-\frac{20\cdots 69}{12\cdots 87}a^{13}+\frac{22\cdots 27}{63\cdots 35}a^{12}-\frac{12\cdots 01}{31\cdots 75}a^{11}+\frac{54\cdots 22}{31\cdots 75}a^{10}+\frac{55\cdots 44}{31\cdots 75}a^{9}-\frac{15\cdots 41}{31\cdots 75}a^{8}+\frac{29\cdots 56}{31\cdots 75}a^{7}+\frac{32\cdots 31}{31\cdots 75}a^{6}-\frac{31\cdots 68}{31\cdots 75}a^{5}-\frac{26\cdots 76}{31\cdots 75}a^{4}+\frac{35\cdots 61}{31\cdots 75}a^{3}+\frac{73\cdots 39}{31\cdots 75}a^{2}-\frac{11\cdots 83}{31\cdots 75}a-\frac{17\cdots 24}{31\cdots 75}$, $\frac{30\cdots 87}{31\cdots 75}a^{21}-\frac{24\cdots 02}{31\cdots 75}a^{20}+\frac{98\cdots 21}{31\cdots 75}a^{19}+\frac{50\cdots 14}{31\cdots 75}a^{18}-\frac{13\cdots 63}{31\cdots 75}a^{17}-\frac{25\cdots 44}{63\cdots 35}a^{16}+\frac{12\cdots 46}{31\cdots 75}a^{15}-\frac{18\cdots 12}{31\cdots 75}a^{14}-\frac{31\cdots 28}{12\cdots 87}a^{13}+\frac{13\cdots 51}{63\cdots 35}a^{12}-\frac{13\cdots 63}{31\cdots 75}a^{11}+\frac{17\cdots 69}{31\cdots 75}a^{10}+\frac{10\cdots 49}{31\cdots 75}a^{9}-\frac{89\cdots 03}{31\cdots 75}a^{8}+\frac{10\cdots 66}{31\cdots 75}a^{7}+\frac{61\cdots 38}{31\cdots 75}a^{6}-\frac{21\cdots 42}{31\cdots 75}a^{5}+\frac{68\cdots 98}{31\cdots 75}a^{4}+\frac{12\cdots 62}{31\cdots 75}a^{3}-\frac{85\cdots 12}{31\cdots 75}a^{2}-\frac{30\cdots 08}{63\cdots 35}a+\frac{21\cdots 46}{31\cdots 75}$, $\frac{94\cdots 52}{31\cdots 75}a^{21}-\frac{76\cdots 67}{31\cdots 75}a^{20}-\frac{22\cdots 09}{31\cdots 75}a^{19}+\frac{16\cdots 69}{31\cdots 75}a^{18}-\frac{43\cdots 98}{31\cdots 75}a^{17}-\frac{98\cdots 69}{63\cdots 35}a^{16}+\frac{40\cdots 66}{31\cdots 75}a^{15}-\frac{54\cdots 02}{31\cdots 75}a^{14}-\frac{18\cdots 92}{12\cdots 87}a^{13}+\frac{47\cdots 81}{63\cdots 35}a^{12}-\frac{42\cdots 98}{31\cdots 75}a^{11}+\frac{45\cdots 74}{31\cdots 75}a^{10}+\frac{57\cdots 79}{31\cdots 75}a^{9}-\frac{31\cdots 63}{31\cdots 75}a^{8}+\frac{32\cdots 61}{31\cdots 75}a^{7}+\frac{32\cdots 98}{31\cdots 75}a^{6}-\frac{84\cdots 07}{31\cdots 75}a^{5}+\frac{16\cdots 83}{31\cdots 75}a^{4}+\frac{61\cdots 27}{31\cdots 75}a^{3}-\frac{31\cdots 77}{31\cdots 75}a^{2}-\frac{23\cdots 98}{63\cdots 35}a+\frac{78\cdots 16}{31\cdots 75}$, $\frac{24\cdots 14}{31\cdots 75}a^{21}-\frac{17\cdots 06}{31\cdots 75}a^{20}-\frac{43\cdots 82}{63\cdots 35}a^{19}+\frac{40\cdots 18}{31\cdots 75}a^{18}-\frac{13\cdots 46}{63\cdots 35}a^{17}-\frac{44\cdots 93}{63\cdots 35}a^{16}+\frac{81\cdots 82}{31\cdots 75}a^{15}-\frac{74\cdots 94}{63\cdots 35}a^{14}-\frac{79\cdots 20}{12\cdots 87}a^{13}+\frac{81\cdots 87}{63\cdots 35}a^{12}-\frac{54\cdots 91}{31\cdots 75}a^{11}+\frac{34\cdots 21}{31\cdots 75}a^{10}+\frac{43\cdots 48}{63\cdots 35}a^{9}-\frac{58\cdots 91}{31\cdots 75}a^{8}+\frac{11\cdots 46}{63\cdots 35}a^{7}+\frac{11\cdots 76}{31\cdots 75}a^{6}-\frac{89\cdots 82}{31\cdots 75}a^{5}-\frac{10\cdots 93}{31\cdots 75}a^{4}+\frac{80\cdots 68}{31\cdots 75}a^{3}+\frac{43\cdots 82}{31\cdots 75}a^{2}-\frac{14\cdots 01}{31\cdots 75}a-\frac{10\cdots 08}{63\cdots 35}$, $\frac{20\cdots 53}{31\cdots 75}a^{21}-\frac{75\cdots 57}{31\cdots 75}a^{20}+\frac{86\cdots 67}{63\cdots 35}a^{19}+\frac{66\cdots 36}{31\cdots 75}a^{18}-\frac{42\cdots 29}{12\cdots 87}a^{17}+\frac{38\cdots 99}{63\cdots 35}a^{16}+\frac{51\cdots 89}{31\cdots 75}a^{15}-\frac{90\cdots 95}{12\cdots 87}a^{14}+\frac{71\cdots 39}{12\cdots 87}a^{13}+\frac{86\cdots 49}{63\cdots 35}a^{12}-\frac{12\cdots 82}{31\cdots 75}a^{11}+\frac{19\cdots 22}{31\cdots 75}a^{10}-\frac{63\cdots 35}{12\cdots 87}a^{9}-\frac{49\cdots 32}{31\cdots 75}a^{8}+\frac{33\cdots 58}{63\cdots 35}a^{7}-\frac{83\cdots 98}{31\cdots 75}a^{6}-\frac{25\cdots 19}{31\cdots 75}a^{5}+\frac{32\cdots 99}{31\cdots 75}a^{4}+\frac{12\cdots 76}{31\cdots 75}a^{3}-\frac{24\cdots 26}{31\cdots 75}a^{2}-\frac{11\cdots 12}{31\cdots 75}a+\frac{20\cdots 66}{12\cdots 87}$ 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:  \( 62984644695.5 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 62984644695.5 \cdot 1}{2\cdot\sqrt{22661033510180079603495293971842498241}}\cr\approx \mathstrut & 0.168929012895 \end{aligned}\] (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 - 7*x^21 - 9*x^20 + 171*x^19 - 276*x^18 - 1022*x^17 + 3668*x^16 - 1134*x^15 - 10988*x^14 + 19765*x^13 - 17789*x^12 + 6*x^11 + 111532*x^10 - 268518*x^9 - 21573*x^8 + 707388*x^7 - 501170*x^6 - 789327*x^5 + 795886*x^4 + 400629*x^3 - 428800*x^2 - 69431*x + 68279) 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 - 7*x^21 - 9*x^20 + 171*x^19 - 276*x^18 - 1022*x^17 + 3668*x^16 - 1134*x^15 - 10988*x^14 + 19765*x^13 - 17789*x^12 + 6*x^11 + 111532*x^10 - 268518*x^9 - 21573*x^8 + 707388*x^7 - 501170*x^6 - 789327*x^5 + 795886*x^4 + 400629*x^3 - 428800*x^2 - 69431*x + 68279, 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 - 7*x^21 - 9*x^20 + 171*x^19 - 276*x^18 - 1022*x^17 + 3668*x^16 - 1134*x^15 - 10988*x^14 + 19765*x^13 - 17789*x^12 + 6*x^11 + 111532*x^10 - 268518*x^9 - 21573*x^8 + 707388*x^7 - 501170*x^6 - 789327*x^5 + 795886*x^4 + 400629*x^3 - 428800*x^2 - 69431*x + 68279); 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^22 - 7*x^21 - 9*x^20 + 171*x^19 - 276*x^18 - 1022*x^17 + 3668*x^16 - 1134*x^15 - 10988*x^14 + 19765*x^13 - 17789*x^12 + 6*x^11 + 111532*x^10 - 268518*x^9 - 21573*x^8 + 707388*x^7 - 501170*x^6 - 789327*x^5 + 795886*x^4 + 400629*x^3 - 428800*x^2 - 69431*x + 68279); 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

$C_2^{10}.D_{11}$ (as 22T30):

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 22528
The 100 conjugacy class representatives for $C_2^{10}.D_{11}$
Character table for $C_2^{10}.D_{11}$

Intermediate fields

11.11.3670285774226257.1

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 22 siblings: data not computed
Degree 44 siblings: data not computed
Minimal sibling: 22.14.17471883970840462300304775614373553.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.11.0.1}{11} }^{2}$ ${\href{/padicField/3.11.0.1}{11} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }^{5}$ ${\href{/padicField/7.11.0.1}{11} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{5}$ ${\href{/padicField/13.11.0.1}{11} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{5}$ ${\href{/padicField/19.11.0.1}{11} }^{2}$ ${\href{/padicField/23.11.0.1}{11} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{9}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ ${\href{/padicField/47.11.0.1}{11} }^{2}$ ${\href{/padicField/53.11.0.1}{11} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}$

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
\(1297\) Copy content Toggle raw display $\Q_{1297}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1297}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$4$$1$$3$
Deg $4$$4$$1$$3$

Spectrum of ring of integers

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