↑

Properties

Label 22.22.904...448.1
Degree $22$
Signature $[22, 0]$
Discriminant $9.046\times 10^{46}$
Root discriminant \(136.27\)
Ramified primes $2,3,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.C_{11}:C_{10}$ (as 22T36)

Related objects

Downloads

Learn more

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^22 - 99*x^20 + 3663*x^18 - 62997*x^16 + 515625*x^14 - 1891296*x^12 + 2785728*x^10 - 1631619*x^8 + 296538*x^6 - 17523*x^4 + 396*x^2 - 3)
 
Copy content gp:K = bnfinit(y^22 - 99*y^20 + 3663*y^18 - 62997*y^16 + 515625*y^14 - 1891296*y^12 + 2785728*y^10 - 1631619*y^8 + 296538*y^6 - 17523*y^4 + 396*y^2 - 3, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 99*x^20 + 3663*x^18 - 62997*x^16 + 515625*x^14 - 1891296*x^12 + 2785728*x^10 - 1631619*x^8 + 296538*x^6 - 17523*x^4 + 396*x^2 - 3);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 99*x^20 + 3663*x^18 - 62997*x^16 + 515625*x^14 - 1891296*x^12 + 2785728*x^10 - 1631619*x^8 + 296538*x^6 - 17523*x^4 + 396*x^2 - 3)
 

\( x^{22} - 99 x^{20} + 3663 x^{18} - 62997 x^{16} + 515625 x^{14} - 1891296 x^{12} + 2785728 x^{10} + \cdots - 3 \) 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:  $[22, 0]$
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:   \(90463257479444681369926681096111744478113640448\) \(\medspace = 2^{12}\cdot 3^{21}\cdot 11^{32}\) 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:  \(136.27\)
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:   \(2\), \(3\), \(11\) 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:  \(\Q(\sqrt{3}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{14\cdots 06}a^{20}-\frac{20\cdots 09}{14\cdots 06}a^{18}+\frac{14\cdots 79}{14\cdots 06}a^{16}-\frac{1}{2}a^{15}+\frac{24\cdots 41}{14\cdots 06}a^{14}-\frac{1}{2}a^{13}+\frac{10\cdots 41}{72\cdots 03}a^{12}-\frac{1}{2}a^{11}-\frac{41\cdots 91}{72\cdots 03}a^{10}-\frac{32\cdots 53}{72\cdots 03}a^{8}-\frac{21\cdots 31}{72\cdots 03}a^{6}-\frac{34\cdots 30}{72\cdots 03}a^{4}-\frac{1}{2}a^{3}+\frac{11\cdots 89}{14\cdots 06}a^{2}-\frac{56\cdots 29}{14\cdots 06}$, $\frac{1}{14\cdots 06}a^{21}-\frac{20\cdots 09}{14\cdots 06}a^{19}+\frac{14\cdots 79}{14\cdots 06}a^{17}-\frac{1}{2}a^{16}+\frac{24\cdots 41}{14\cdots 06}a^{15}-\frac{1}{2}a^{14}+\frac{10\cdots 41}{72\cdots 03}a^{13}-\frac{1}{2}a^{12}-\frac{41\cdots 91}{72\cdots 03}a^{11}-\frac{32\cdots 53}{72\cdots 03}a^{9}-\frac{21\cdots 31}{72\cdots 03}a^{7}-\frac{34\cdots 30}{72\cdots 03}a^{5}-\frac{1}{2}a^{4}+\frac{11\cdots 89}{14\cdots 06}a^{3}-\frac{56\cdots 29}{14\cdots 06}a$ 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:  $C_{2}\times C_{2}\times C_{2}$, 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:  $21$
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:   $\frac{30\cdots 71}{72\cdots 03}a^{20}-\frac{29\cdots 43}{72\cdots 03}a^{18}+\frac{11\cdots 00}{72\cdots 03}a^{16}-\frac{19\cdots 16}{72\cdots 03}a^{14}+\frac{15\cdots 58}{72\cdots 03}a^{12}-\frac{56\cdots 46}{72\cdots 03}a^{10}+\frac{83\cdots 40}{72\cdots 03}a^{8}-\frac{47\cdots 08}{72\cdots 03}a^{6}+\frac{81\cdots 16}{72\cdots 03}a^{4}-\frac{39\cdots 29}{72\cdots 03}a^{2}+\frac{52\cdots 74}{72\cdots 03}$, $\frac{33\cdots 05}{14\cdots 06}a^{21}-\frac{88\cdots 21}{14\cdots 06}a^{20}-\frac{16\cdots 63}{72\cdots 03}a^{19}+\frac{43\cdots 65}{72\cdots 03}a^{18}+\frac{62\cdots 71}{72\cdots 03}a^{17}-\frac{16\cdots 62}{72\cdots 03}a^{16}-\frac{10\cdots 66}{72\cdots 03}a^{15}+\frac{55\cdots 39}{14\cdots 06}a^{14}+\frac{17\cdots 81}{14\cdots 06}a^{13}-\frac{45\cdots 13}{14\cdots 06}a^{12}-\frac{63\cdots 85}{14\cdots 06}a^{11}+\frac{16\cdots 49}{14\cdots 06}a^{10}+\frac{90\cdots 27}{14\cdots 06}a^{9}-\frac{11\cdots 71}{72\cdots 03}a^{8}-\frac{50\cdots 07}{14\cdots 06}a^{7}+\frac{13\cdots 77}{14\cdots 06}a^{6}+\frac{35\cdots 56}{72\cdots 03}a^{5}-\frac{96\cdots 46}{72\cdots 03}a^{4}-\frac{14\cdots 77}{14\cdots 06}a^{3}+\frac{25\cdots 95}{72\cdots 03}a^{2}+\frac{29\cdots 90}{72\cdots 03}a-\frac{33\cdots 59}{14\cdots 06}$, $\frac{67\cdots 28}{72\cdots 03}a^{20}-\frac{66\cdots 83}{72\cdots 03}a^{18}+\frac{24\cdots 77}{72\cdots 03}a^{16}-\frac{42\cdots 17}{72\cdots 03}a^{14}+\frac{34\cdots 22}{72\cdots 03}a^{12}-\frac{12\cdots 08}{72\cdots 03}a^{10}+\frac{18\cdots 69}{72\cdots 03}a^{8}-\frac{10\cdots 46}{72\cdots 03}a^{6}+\frac{18\cdots 68}{72\cdots 03}a^{4}-\frac{84\cdots 61}{72\cdots 03}a^{2}+\frac{95\cdots 06}{72\cdots 03}$, $\frac{13\cdots 11}{72\cdots 03}a^{20}-\frac{13\cdots 68}{72\cdots 03}a^{18}+\frac{51\cdots 83}{72\cdots 03}a^{16}-\frac{87\cdots 89}{72\cdots 03}a^{14}+\frac{72\cdots 78}{72\cdots 03}a^{12}-\frac{26\cdots 52}{72\cdots 03}a^{10}+\frac{39\cdots 95}{72\cdots 03}a^{8}-\frac{23\cdots 65}{72\cdots 03}a^{6}+\frac{39\cdots 07}{72\cdots 03}a^{4}-\frac{17\cdots 27}{72\cdots 03}a^{2}+\frac{19\cdots 88}{72\cdots 03}$, $\frac{52\cdots 76}{72\cdots 03}a^{21}-\frac{53\cdots 47}{72\cdots 03}a^{20}-\frac{10\cdots 91}{14\cdots 06}a^{19}+\frac{52\cdots 33}{72\cdots 03}a^{18}+\frac{38\cdots 01}{14\cdots 06}a^{17}-\frac{39\cdots 85}{14\cdots 06}a^{16}-\frac{65\cdots 87}{14\cdots 06}a^{15}+\frac{33\cdots 98}{72\cdots 03}a^{14}+\frac{26\cdots 12}{72\cdots 03}a^{13}-\frac{27\cdots 30}{72\cdots 03}a^{12}-\frac{97\cdots 05}{72\cdots 03}a^{11}+\frac{20\cdots 71}{14\cdots 06}a^{10}+\frac{14\cdots 66}{72\cdots 03}a^{9}-\frac{14\cdots 18}{72\cdots 03}a^{8}-\frac{16\cdots 95}{14\cdots 06}a^{7}+\frac{84\cdots 10}{72\cdots 03}a^{6}+\frac{13\cdots 28}{72\cdots 03}a^{5}-\frac{28\cdots 91}{14\cdots 06}a^{4}-\frac{55\cdots 03}{72\cdots 03}a^{3}+\frac{13\cdots 63}{14\cdots 06}a^{2}+\frac{59\cdots 53}{72\cdots 03}a-\frac{72\cdots 58}{72\cdots 03}$, $\frac{25\cdots 77}{14\cdots 06}a^{21}-\frac{16\cdots 54}{72\cdots 03}a^{20}-\frac{24\cdots 93}{14\cdots 06}a^{19}+\frac{16\cdots 44}{72\cdots 03}a^{18}+\frac{45\cdots 14}{72\cdots 03}a^{17}-\frac{12\cdots 45}{14\cdots 06}a^{16}-\frac{78\cdots 23}{72\cdots 03}a^{15}+\frac{10\cdots 27}{72\cdots 03}a^{14}+\frac{12\cdots 89}{14\cdots 06}a^{13}-\frac{17\cdots 03}{14\cdots 06}a^{12}-\frac{23\cdots 10}{72\cdots 03}a^{11}+\frac{31\cdots 88}{72\cdots 03}a^{10}+\frac{32\cdots 49}{72\cdots 03}a^{9}-\frac{92\cdots 31}{14\cdots 06}a^{8}-\frac{17\cdots 50}{72\cdots 03}a^{7}+\frac{26\cdots 78}{72\cdots 03}a^{6}+\frac{38\cdots 97}{14\cdots 06}a^{5}-\frac{89\cdots 07}{14\cdots 06}a^{4}+\frac{86\cdots 21}{14\cdots 06}a^{3}+\frac{43\cdots 25}{14\cdots 06}a^{2}-\frac{83\cdots 87}{14\cdots 06}a-\frac{59\cdots 05}{14\cdots 06}$, $\frac{36\cdots 45}{72\cdots 03}a^{20}-\frac{35\cdots 25}{72\cdots 03}a^{18}+\frac{13\cdots 67}{72\cdots 03}a^{16}-\frac{22\cdots 13}{72\cdots 03}a^{14}+\frac{18\cdots 88}{72\cdots 03}a^{12}-\frac{67\cdots 92}{72\cdots 03}a^{10}+\frac{96\cdots 10}{72\cdots 03}a^{8}-\frac{53\cdots 08}{72\cdots 03}a^{6}+\frac{80\cdots 90}{72\cdots 03}a^{4}-\frac{24\cdots 45}{72\cdots 03}a^{2}+\frac{20\cdots 30}{72\cdots 03}$, $\frac{30\cdots 99}{72\cdots 03}a^{20}-\frac{30\cdots 96}{72\cdots 03}a^{18}+\frac{11\cdots 94}{72\cdots 03}a^{16}-\frac{19\cdots 00}{72\cdots 03}a^{14}+\frac{15\cdots 81}{72\cdots 03}a^{12}-\frac{56\cdots 93}{72\cdots 03}a^{10}+\frac{83\cdots 46}{72\cdots 03}a^{8}-\frac{47\cdots 46}{72\cdots 03}a^{6}+\frac{80\cdots 79}{72\cdots 03}a^{4}-\frac{36\cdots 75}{72\cdots 03}a^{2}+\frac{43\cdots 38}{72\cdots 03}$, $\frac{16\cdots 68}{72\cdots 03}a^{21}+\frac{10\cdots 53}{14\cdots 06}a^{20}-\frac{16\cdots 72}{72\cdots 03}a^{19}-\frac{10\cdots 63}{14\cdots 06}a^{18}+\frac{12\cdots 61}{14\cdots 06}a^{17}+\frac{37\cdots 23}{14\cdots 06}a^{16}-\frac{10\cdots 68}{72\cdots 03}a^{15}-\frac{32\cdots 61}{72\cdots 03}a^{14}+\frac{86\cdots 44}{72\cdots 03}a^{13}+\frac{26\cdots 92}{72\cdots 03}a^{12}-\frac{63\cdots 85}{14\cdots 06}a^{11}-\frac{97\cdots 73}{72\cdots 03}a^{10}+\frac{46\cdots 51}{72\cdots 03}a^{9}+\frac{28\cdots 97}{14\cdots 06}a^{8}-\frac{26\cdots 09}{72\cdots 03}a^{7}-\frac{79\cdots 20}{72\cdots 03}a^{6}+\frac{88\cdots 23}{14\cdots 06}a^{5}+\frac{12\cdots 95}{72\cdots 03}a^{4}-\frac{39\cdots 13}{14\cdots 06}a^{3}-\frac{50\cdots 08}{72\cdots 03}a^{2}+\frac{24\cdots 57}{72\cdots 03}a+\frac{77\cdots 92}{72\cdots 03}$, $\frac{37\cdots 10}{72\cdots 03}a^{21}+\frac{64\cdots 62}{72\cdots 03}a^{20}-\frac{36\cdots 96}{72\cdots 03}a^{19}-\frac{64\cdots 15}{72\cdots 03}a^{18}+\frac{27\cdots 19}{14\cdots 06}a^{17}+\frac{23\cdots 47}{72\cdots 03}a^{16}-\frac{46\cdots 99}{14\cdots 06}a^{15}-\frac{81\cdots 19}{14\cdots 06}a^{14}+\frac{38\cdots 79}{14\cdots 06}a^{13}+\frac{33\cdots 64}{72\cdots 03}a^{12}-\frac{70\cdots 33}{72\cdots 03}a^{11}-\frac{12\cdots 22}{72\cdots 03}a^{10}+\frac{10\cdots 26}{72\cdots 03}a^{9}+\frac{35\cdots 61}{14\cdots 06}a^{8}-\frac{58\cdots 39}{72\cdots 03}a^{7}-\frac{10\cdots 22}{72\cdots 03}a^{6}+\frac{19\cdots 83}{14\cdots 06}a^{5}+\frac{16\cdots 45}{72\cdots 03}a^{4}-\frac{41\cdots 26}{72\cdots 03}a^{3}-\frac{14\cdots 71}{14\cdots 06}a^{2}+\frac{45\cdots 20}{72\cdots 03}a+\frac{16\cdots 65}{14\cdots 06}$, $\frac{15\cdots 03}{14\cdots 06}a^{21}+\frac{51\cdots 51}{14\cdots 06}a^{20}-\frac{75\cdots 83}{72\cdots 03}a^{19}-\frac{50\cdots 57}{14\cdots 06}a^{18}+\frac{27\cdots 63}{72\cdots 03}a^{17}+\frac{18\cdots 33}{14\cdots 06}a^{16}-\frac{95\cdots 75}{14\cdots 06}a^{15}-\frac{16\cdots 56}{72\cdots 03}a^{14}+\frac{77\cdots 85}{14\cdots 06}a^{13}+\frac{26\cdots 27}{14\cdots 06}a^{12}-\frac{28\cdots 99}{14\cdots 06}a^{11}-\frac{97\cdots 99}{14\cdots 06}a^{10}+\frac{20\cdots 21}{72\cdots 03}a^{9}+\frac{72\cdots 79}{72\cdots 03}a^{8}-\frac{23\cdots 35}{14\cdots 06}a^{7}-\frac{42\cdots 96}{72\cdots 03}a^{6}+\frac{18\cdots 03}{72\cdots 03}a^{5}+\frac{82\cdots 64}{72\cdots 03}a^{4}-\frac{76\cdots 94}{72\cdots 03}a^{3}-\frac{50\cdots 11}{72\cdots 03}a^{2}+\frac{16\cdots 55}{14\cdots 06}a+\frac{13\cdots 75}{14\cdots 06}$, $\frac{23\cdots 58}{72\cdots 03}a^{21}-\frac{75\cdots 08}{72\cdots 03}a^{20}-\frac{46\cdots 81}{14\cdots 06}a^{19}+\frac{15\cdots 15}{14\cdots 06}a^{18}+\frac{86\cdots 50}{72\cdots 03}a^{17}-\frac{27\cdots 24}{72\cdots 03}a^{16}-\frac{29\cdots 37}{14\cdots 06}a^{15}+\frac{47\cdots 57}{72\cdots 03}a^{14}+\frac{24\cdots 03}{14\cdots 06}a^{13}-\frac{39\cdots 15}{72\cdots 03}a^{12}-\frac{44\cdots 00}{72\cdots 03}a^{11}+\frac{28\cdots 89}{14\cdots 06}a^{10}+\frac{12\cdots 57}{14\cdots 06}a^{9}-\frac{41\cdots 99}{14\cdots 06}a^{8}-\frac{73\cdots 71}{14\cdots 06}a^{7}+\frac{23\cdots 81}{14\cdots 06}a^{6}+\frac{12\cdots 55}{14\cdots 06}a^{5}-\frac{39\cdots 05}{14\cdots 06}a^{4}-\frac{53\cdots 85}{14\cdots 06}a^{3}+\frac{86\cdots 58}{72\cdots 03}a^{2}+\frac{59\cdots 23}{14\cdots 06}a-\frac{19\cdots 05}{14\cdots 06}$, $\frac{31\cdots 10}{72\cdots 03}a^{21}+\frac{26\cdots 93}{14\cdots 06}a^{20}-\frac{63\cdots 91}{14\cdots 06}a^{19}-\frac{26\cdots 27}{14\cdots 06}a^{18}+\frac{11\cdots 35}{72\cdots 03}a^{17}+\frac{49\cdots 91}{72\cdots 03}a^{16}-\frac{40\cdots 39}{14\cdots 06}a^{15}-\frac{86\cdots 88}{72\cdots 03}a^{14}+\frac{16\cdots 73}{72\cdots 03}a^{13}+\frac{72\cdots 04}{72\cdots 03}a^{12}-\frac{12\cdots 13}{14\cdots 06}a^{11}-\frac{56\cdots 91}{14\cdots 06}a^{10}+\frac{88\cdots 34}{72\cdots 03}a^{9}+\frac{93\cdots 91}{14\cdots 06}a^{8}-\frac{10\cdots 37}{14\cdots 06}a^{7}-\frac{30\cdots 04}{72\cdots 03}a^{6}+\frac{18\cdots 33}{14\cdots 06}a^{5}+\frac{10\cdots 87}{14\cdots 06}a^{4}-\frac{10\cdots 49}{14\cdots 06}a^{3}-\frac{54\cdots 67}{14\cdots 06}a^{2}+\frac{85\cdots 66}{72\cdots 03}a+\frac{61\cdots 37}{72\cdots 03}$, $\frac{84\cdots 95}{14\cdots 06}a^{21}-\frac{81\cdots 20}{72\cdots 03}a^{20}-\frac{42\cdots 94}{72\cdots 03}a^{19}+\frac{80\cdots 97}{72\cdots 03}a^{18}+\frac{15\cdots 83}{72\cdots 03}a^{17}-\frac{29\cdots 63}{72\cdots 03}a^{16}-\frac{26\cdots 51}{72\cdots 03}a^{15}+\frac{10\cdots 67}{14\cdots 06}a^{14}+\frac{21\cdots 34}{72\cdots 03}a^{13}-\frac{84\cdots 65}{14\cdots 06}a^{12}-\frac{80\cdots 83}{72\cdots 03}a^{11}+\frac{30\cdots 01}{14\cdots 06}a^{10}+\frac{11\cdots 94}{72\cdots 03}a^{9}-\frac{22\cdots 52}{72\cdots 03}a^{8}-\frac{13\cdots 13}{14\cdots 06}a^{7}+\frac{13\cdots 29}{72\cdots 03}a^{6}+\frac{12\cdots 95}{72\cdots 03}a^{5}-\frac{25\cdots 62}{72\cdots 03}a^{4}-\frac{13\cdots 27}{14\cdots 06}a^{3}+\frac{32\cdots 35}{14\cdots 06}a^{2}+\frac{22\cdots 85}{14\cdots 06}a-\frac{23\cdots 74}{72\cdots 03}$, $\frac{89\cdots 45}{72\cdots 03}a^{21}-\frac{36\cdots 75}{72\cdots 03}a^{20}-\frac{17\cdots 99}{14\cdots 06}a^{19}+\frac{36\cdots 97}{72\cdots 03}a^{18}+\frac{32\cdots 31}{72\cdots 03}a^{17}-\frac{26\cdots 41}{14\cdots 06}a^{16}-\frac{56\cdots 11}{72\cdots 03}a^{15}+\frac{46\cdots 49}{14\cdots 06}a^{14}+\frac{92\cdots 13}{14\cdots 06}a^{13}-\frac{18\cdots 26}{72\cdots 03}a^{12}-\frac{16\cdots 71}{72\cdots 03}a^{11}+\frac{13\cdots 61}{14\cdots 06}a^{10}+\frac{24\cdots 29}{72\cdots 03}a^{9}-\frac{20\cdots 05}{14\cdots 06}a^{8}-\frac{28\cdots 05}{14\cdots 06}a^{7}+\frac{57\cdots 01}{72\cdots 03}a^{6}+\frac{50\cdots 35}{14\cdots 06}a^{5}-\frac{19\cdots 09}{14\cdots 06}a^{4}-\frac{13\cdots 75}{72\cdots 03}a^{3}+\frac{45\cdots 53}{72\cdots 03}a^{2}+\frac{20\cdots 60}{72\cdots 03}a-\frac{11\cdots 13}{14\cdots 06}$, $\frac{25\cdots 77}{14\cdots 06}a^{21}+\frac{16\cdots 54}{72\cdots 03}a^{20}-\frac{24\cdots 93}{14\cdots 06}a^{19}-\frac{16\cdots 44}{72\cdots 03}a^{18}+\frac{45\cdots 14}{72\cdots 03}a^{17}+\frac{12\cdots 45}{14\cdots 06}a^{16}-\frac{78\cdots 23}{72\cdots 03}a^{15}-\frac{10\cdots 27}{72\cdots 03}a^{14}+\frac{12\cdots 89}{14\cdots 06}a^{13}+\frac{17\cdots 03}{14\cdots 06}a^{12}-\frac{23\cdots 10}{72\cdots 03}a^{11}-\frac{31\cdots 88}{72\cdots 03}a^{10}+\frac{32\cdots 49}{72\cdots 03}a^{9}+\frac{92\cdots 31}{14\cdots 06}a^{8}-\frac{17\cdots 50}{72\cdots 03}a^{7}-\frac{26\cdots 78}{72\cdots 03}a^{6}+\frac{38\cdots 97}{14\cdots 06}a^{5}+\frac{89\cdots 07}{14\cdots 06}a^{4}+\frac{86\cdots 21}{14\cdots 06}a^{3}-\frac{43\cdots 25}{14\cdots 06}a^{2}-\frac{83\cdots 87}{14\cdots 06}a+\frac{59\cdots 05}{14\cdots 06}$, $\frac{61\cdots 02}{72\cdots 03}a^{21}-\frac{10\cdots 39}{72\cdots 03}a^{20}-\frac{12\cdots 69}{14\cdots 06}a^{19}+\frac{10\cdots 56}{72\cdots 03}a^{18}+\frac{44\cdots 95}{14\cdots 06}a^{17}-\frac{80\cdots 77}{14\cdots 06}a^{16}-\frac{76\cdots 99}{14\cdots 06}a^{15}+\frac{68\cdots 39}{72\cdots 03}a^{14}+\frac{31\cdots 25}{72\cdots 03}a^{13}-\frac{56\cdots 86}{72\cdots 03}a^{12}-\frac{11\cdots 36}{72\cdots 03}a^{11}+\frac{41\cdots 67}{14\cdots 06}a^{10}+\frac{16\cdots 59}{72\cdots 03}a^{9}-\frac{30\cdots 67}{72\cdots 03}a^{8}-\frac{19\cdots 49}{14\cdots 06}a^{7}+\frac{17\cdots 49}{72\cdots 03}a^{6}+\frac{15\cdots 15}{72\cdots 03}a^{5}-\frac{57\cdots 37}{14\cdots 06}a^{4}-\frac{68\cdots 16}{72\cdots 03}a^{3}+\frac{25\cdots 05}{14\cdots 06}a^{2}+\frac{76\cdots 85}{72\cdots 03}a-\frac{14\cdots 40}{72\cdots 03}$, $\frac{11\cdots 14}{72\cdots 03}a^{21}-\frac{13\cdots 51}{14\cdots 06}a^{20}-\frac{22\cdots 99}{14\cdots 06}a^{19}+\frac{64\cdots 13}{72\cdots 03}a^{18}+\frac{81\cdots 95}{14\cdots 06}a^{17}-\frac{47\cdots 37}{14\cdots 06}a^{16}-\frac{14\cdots 63}{14\cdots 06}a^{15}+\frac{41\cdots 39}{72\cdots 03}a^{14}+\frac{11\cdots 77}{14\cdots 06}a^{13}-\frac{33\cdots 28}{72\cdots 03}a^{12}-\frac{41\cdots 57}{14\cdots 06}a^{11}+\frac{24\cdots 67}{14\cdots 06}a^{10}+\frac{61\cdots 35}{14\cdots 06}a^{9}-\frac{18\cdots 70}{72\cdots 03}a^{8}-\frac{34\cdots 37}{14\cdots 06}a^{7}+\frac{21\cdots 37}{14\cdots 06}a^{6}+\frac{28\cdots 71}{72\cdots 03}a^{5}-\frac{38\cdots 27}{14\cdots 06}a^{4}-\frac{11\cdots 26}{72\cdots 03}a^{3}+\frac{10\cdots 90}{72\cdots 03}a^{2}+\frac{26\cdots 25}{14\cdots 06}a-\frac{25\cdots 29}{14\cdots 06}$, $\frac{43\cdots 91}{14\cdots 06}a^{21}-\frac{60\cdots 55}{14\cdots 06}a^{20}-\frac{43\cdots 71}{14\cdots 06}a^{19}+\frac{30\cdots 45}{72\cdots 03}a^{18}+\frac{15\cdots 63}{14\cdots 06}a^{17}-\frac{22\cdots 25}{14\cdots 06}a^{16}-\frac{27\cdots 27}{14\cdots 06}a^{15}+\frac{38\cdots 75}{14\cdots 06}a^{14}+\frac{22\cdots 39}{14\cdots 06}a^{13}-\frac{31\cdots 13}{14\cdots 06}a^{12}-\frac{82\cdots 21}{14\cdots 06}a^{11}+\frac{57\cdots 87}{72\cdots 03}a^{10}+\frac{12\cdots 49}{14\cdots 06}a^{9}-\frac{83\cdots 78}{72\cdots 03}a^{8}-\frac{34\cdots 80}{72\cdots 03}a^{7}+\frac{95\cdots 19}{14\cdots 06}a^{6}+\frac{58\cdots 27}{72\cdots 03}a^{5}-\frac{16\cdots 51}{14\cdots 06}a^{4}-\frac{56\cdots 15}{14\cdots 06}a^{3}+\frac{77\cdots 09}{14\cdots 06}a^{2}+\frac{38\cdots 16}{72\cdots 03}a-\frac{10\cdots 09}{14\cdots 06}$, $\frac{38\cdots 17}{72\cdots 03}a^{21}-\frac{86\cdots 47}{14\cdots 06}a^{20}-\frac{76\cdots 83}{14\cdots 06}a^{19}+\frac{85\cdots 11}{14\cdots 06}a^{18}+\frac{14\cdots 28}{72\cdots 03}a^{17}-\frac{15\cdots 06}{72\cdots 03}a^{16}-\frac{48\cdots 79}{14\cdots 06}a^{15}+\frac{27\cdots 56}{72\cdots 03}a^{14}+\frac{19\cdots 52}{72\cdots 03}a^{13}-\frac{22\cdots 96}{72\cdots 03}a^{12}-\frac{14\cdots 23}{14\cdots 06}a^{11}+\frac{16\cdots 71}{14\cdots 06}a^{10}+\frac{10\cdots 10}{72\cdots 03}a^{9}-\frac{24\cdots 07}{14\cdots 06}a^{8}-\frac{12\cdots 25}{14\cdots 06}a^{7}+\frac{70\cdots 96}{72\cdots 03}a^{6}+\frac{21\cdots 07}{14\cdots 06}a^{5}-\frac{24\cdots 71}{14\cdots 06}a^{4}-\frac{10\cdots 43}{14\cdots 06}a^{3}+\frac{13\cdots 51}{14\cdots 06}a^{2}+\frac{75\cdots 69}{72\cdots 03}a-\frac{95\cdots 24}{72\cdots 03}$, $\frac{95\cdots 01}{14\cdots 06}a^{21}-\frac{60\cdots 44}{72\cdots 03}a^{20}-\frac{47\cdots 69}{72\cdots 03}a^{19}+\frac{11\cdots 39}{14\cdots 06}a^{18}+\frac{17\cdots 23}{72\cdots 03}a^{17}-\frac{44\cdots 51}{14\cdots 06}a^{16}-\frac{60\cdots 59}{14\cdots 06}a^{15}+\frac{38\cdots 34}{72\cdots 03}a^{14}+\frac{24\cdots 70}{72\cdots 03}a^{13}-\frac{62\cdots 29}{14\cdots 06}a^{12}-\frac{89\cdots 07}{72\cdots 03}a^{11}+\frac{22\cdots 83}{14\cdots 06}a^{10}+\frac{26\cdots 67}{14\cdots 06}a^{9}-\frac{16\cdots 64}{72\cdots 03}a^{8}-\frac{14\cdots 57}{14\cdots 06}a^{7}+\frac{19\cdots 89}{14\cdots 06}a^{6}+\frac{12\cdots 92}{72\cdots 03}a^{5}-\frac{15\cdots 10}{72\cdots 03}a^{4}-\frac{53\cdots 49}{72\cdots 03}a^{3}+\frac{14\cdots 83}{14\cdots 06}a^{2}+\frac{58\cdots 78}{72\cdots 03}a-\frac{81\cdots 80}{72\cdots 03}$ 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:  \( 165654542977000000 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{22}\cdot(2\pi)^{0}\cdot 165654542977000000 \cdot 1}{2\cdot\sqrt{90463257479444681369926681096111744478113640448}}\cr\approx \mathstrut & 1.15504032992262 \end{aligned}\] (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 - 99*x^20 + 3663*x^18 - 62997*x^16 + 515625*x^14 - 1891296*x^12 + 2785728*x^10 - 1631619*x^8 + 296538*x^6 - 17523*x^4 + 396*x^2 - 3) 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 - 99*x^20 + 3663*x^18 - 62997*x^16 + 515625*x^14 - 1891296*x^12 + 2785728*x^10 - 1631619*x^8 + 296538*x^6 - 17523*x^4 + 396*x^2 - 3, 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 - 99*x^20 + 3663*x^18 - 62997*x^16 + 515625*x^14 - 1891296*x^12 + 2785728*x^10 - 1631619*x^8 + 296538*x^6 - 17523*x^4 + 396*x^2 - 3); 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 - 99*x^20 + 3663*x^18 - 62997*x^16 + 515625*x^14 - 1891296*x^12 + 2785728*x^10 - 1631619*x^8 + 296538*x^6 - 17523*x^4 + 396*x^2 - 3); 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

$C_2^{10}.C_{11}:C_{10}$ (as 22T36):

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 solvable group of order 112640
The 80 conjugacy class representatives for $C_2^{10}.C_{11}:C_{10}$
Character table for $C_2^{10}.C_{11}:C_{10}$

Intermediate fields

11.11.2713285598714072534889.1

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 22 siblings: 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 R ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ R ${\href{/padicField/13.5.0.1}{5} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.5.0.1}{5} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.11.0.1}{11} }^{2}$ ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ $22$ ${\href{/padicField/47.5.0.1}{5} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.1.2.2a1.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$$[2]$$
2.5.1.0a1.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$$[\ ]^{5}$$
2.5.1.0a1.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$$[\ ]^{5}$$
2.5.2.10a2.2$x^{10} + 2 x^{7} + 2 x^{6} + 2 x^{5} + x^{4} + 2 x^{3} + 6 x^{2} + 2 x + 3$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$$[2, 2, 2, 2]^{10}$$
\(3\) Copy content Toggle raw display 3.1.22.21a1.2$x^{22} + 6$$22$$1$$21$22T5$$[\ ]_{22}^{5}$$
\(11\) Copy content Toggle raw display 11.1.11.16a2.1$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$$[\frac{8}{5}]_{5}$$
11.1.11.16a2.1$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$$[\frac{8}{5}]_{5}$$

Spectrum of ring of integers

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