↑

Properties

Label 22.14.617...624.1
Degree $22$
Signature $[14, 4]$
Discriminant $6.172\times 10^{39}$
Root discriminant \(64.37\)
Ramified primes $2,7,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.F_{11}$ (as 22T34)

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 - 33*x^20 - 44*x^19 + 253*x^18 + 660*x^17 + 1177*x^16 + 2992*x^15 - 2266*x^14 - 27214*x^13 - 75812*x^12 - 258772*x^11 - 808192*x^10 - 1679018*x^9 - 2491038*x^8 - 3007114*x^7 - 3012009*x^6 - 2244528*x^5 - 1081113*x^4 - 281710*x^3 - 22011*x^2 + 4268*x + 561)
 
Copy content gp:K = bnfinit(y^22 - 33*y^20 - 44*y^19 + 253*y^18 + 660*y^17 + 1177*y^16 + 2992*y^15 - 2266*y^14 - 27214*y^13 - 75812*y^12 - 258772*y^11 - 808192*y^10 - 1679018*y^9 - 2491038*y^8 - 3007114*y^7 - 3012009*y^6 - 2244528*y^5 - 1081113*y^4 - 281710*y^3 - 22011*y^2 + 4268*y + 561, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 33*x^20 - 44*x^19 + 253*x^18 + 660*x^17 + 1177*x^16 + 2992*x^15 - 2266*x^14 - 27214*x^13 - 75812*x^12 - 258772*x^11 - 808192*x^10 - 1679018*x^9 - 2491038*x^8 - 3007114*x^7 - 3012009*x^6 - 2244528*x^5 - 1081113*x^4 - 281710*x^3 - 22011*x^2 + 4268*x + 561);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 33*x^20 - 44*x^19 + 253*x^18 + 660*x^17 + 1177*x^16 + 2992*x^15 - 2266*x^14 - 27214*x^13 - 75812*x^12 - 258772*x^11 - 808192*x^10 - 1679018*x^9 - 2491038*x^8 - 3007114*x^7 - 3012009*x^6 - 2244528*x^5 - 1081113*x^4 - 281710*x^3 - 22011*x^2 + 4268*x + 561)
 

\( x^{22} - 33 x^{20} - 44 x^{19} + 253 x^{18} + 660 x^{17} + 1177 x^{16} + 2992 x^{15} - 2266 x^{14} + \cdots + 561 \) 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:   \(6172475179135960522642404800603172634624\) \(\medspace = 2^{28}\cdot 7^{10}\cdot 11^{22}\) 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:  \(64.37\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(7\), \(11\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{55\cdots 63}a^{21}+\frac{17\cdots 46}{55\cdots 63}a^{20}-\frac{24\cdots 08}{55\cdots 63}a^{19}-\frac{17\cdots 05}{42\cdots 51}a^{18}-\frac{22\cdots 36}{55\cdots 63}a^{17}-\frac{12\cdots 12}{55\cdots 63}a^{16}-\frac{57\cdots 15}{55\cdots 63}a^{15}+\frac{17\cdots 65}{55\cdots 63}a^{14}+\frac{23\cdots 51}{55\cdots 63}a^{13}-\frac{49\cdots 60}{55\cdots 63}a^{12}+\frac{18\cdots 61}{55\cdots 63}a^{11}+\frac{23\cdots 03}{55\cdots 63}a^{10}-\frac{45\cdots 92}{55\cdots 63}a^{9}-\frac{27\cdots 32}{55\cdots 63}a^{8}-\frac{10\cdots 39}{55\cdots 63}a^{7}+\frac{20\cdots 33}{55\cdots 63}a^{6}+\frac{36\cdots 45}{55\cdots 63}a^{5}-\frac{18\cdots 42}{55\cdots 63}a^{4}+\frac{13\cdots 97}{55\cdots 63}a^{3}-\frac{23\cdots 93}{55\cdots 63}a^{2}+\frac{14\cdots 10}{42\cdots 51}a+\frac{12\cdots 00}{55\cdots 63}$ 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:   $\frac{23\cdots 60}{55\cdots 63}a^{21}-\frac{17\cdots 82}{55\cdots 63}a^{20}-\frac{76\cdots 18}{55\cdots 63}a^{19}-\frac{35\cdots 61}{42\cdots 51}a^{18}+\frac{63\cdots 63}{55\cdots 63}a^{17}+\frac{10\cdots 75}{55\cdots 63}a^{16}+\frac{19\cdots 46}{55\cdots 63}a^{15}+\frac{55\cdots 19}{55\cdots 63}a^{14}-\frac{96\cdots 80}{55\cdots 63}a^{13}-\frac{57\cdots 91}{55\cdots 63}a^{12}-\frac{13\cdots 58}{55\cdots 63}a^{11}-\frac{50\cdots 71}{55\cdots 63}a^{10}-\frac{15\cdots 85}{55\cdots 63}a^{9}-\frac{28\cdots 46}{55\cdots 63}a^{8}-\frac{37\cdots 96}{55\cdots 63}a^{7}-\frac{42\cdots 23}{55\cdots 63}a^{6}-\frac{38\cdots 02}{55\cdots 63}a^{5}-\frac{23\cdots 85}{55\cdots 63}a^{4}-\frac{75\cdots 48}{55\cdots 63}a^{3}-\frac{78\cdots 95}{55\cdots 63}a^{2}+\frac{87\cdots 96}{42\cdots 51}a+\frac{16\cdots 93}{55\cdots 63}$, $\frac{11\cdots 23}{55\cdots 63}a^{21}-\frac{92\cdots 40}{55\cdots 63}a^{20}-\frac{36\cdots 96}{55\cdots 63}a^{19}-\frac{15\cdots 48}{42\cdots 51}a^{18}+\frac{29\cdots 47}{55\cdots 63}a^{17}+\frac{49\cdots 96}{55\cdots 63}a^{16}+\frac{92\cdots 54}{55\cdots 63}a^{15}+\frac{26\cdots 29}{55\cdots 63}a^{14}-\frac{46\cdots 35}{55\cdots 63}a^{13}-\frac{26\cdots 21}{55\cdots 63}a^{12}-\frac{63\cdots 90}{55\cdots 63}a^{11}-\frac{23\cdots 32}{55\cdots 63}a^{10}-\frac{71\cdots 62}{55\cdots 63}a^{9}-\frac{13\cdots 49}{55\cdots 63}a^{8}-\frac{17\cdots 12}{55\cdots 63}a^{7}-\frac{19\cdots 20}{55\cdots 63}a^{6}-\frac{17\cdots 84}{55\cdots 63}a^{5}-\frac{10\cdots 89}{55\cdots 63}a^{4}-\frac{34\cdots 26}{55\cdots 63}a^{3}-\frac{34\cdots 54}{55\cdots 63}a^{2}+\frac{48\cdots 85}{42\cdots 51}a+\frac{92\cdots 65}{55\cdots 63}$, $\frac{26\cdots 67}{55\cdots 63}a^{21}-\frac{30\cdots 36}{55\cdots 63}a^{20}-\frac{82\cdots 41}{55\cdots 63}a^{19}-\frac{13\cdots 26}{42\cdots 51}a^{18}+\frac{68\cdots 20}{55\cdots 63}a^{17}+\frac{91\cdots 18}{55\cdots 63}a^{16}+\frac{20\cdots 27}{55\cdots 63}a^{15}+\frac{54\cdots 92}{55\cdots 63}a^{14}-\frac{12\cdots 14}{55\cdots 63}a^{13}-\frac{56\cdots 63}{55\cdots 63}a^{12}-\frac{13\cdots 67}{55\cdots 63}a^{11}-\frac{52\cdots 58}{55\cdots 63}a^{10}-\frac{14\cdots 34}{55\cdots 63}a^{9}-\frac{26\cdots 92}{55\cdots 63}a^{8}-\frac{34\cdots 11}{55\cdots 63}a^{7}-\frac{38\cdots 81}{55\cdots 63}a^{6}-\frac{33\cdots 09}{55\cdots 63}a^{5}-\frac{19\cdots 60}{55\cdots 63}a^{4}-\frac{60\cdots 36}{55\cdots 63}a^{3}-\frac{65\cdots 27}{55\cdots 63}a^{2}+\frac{60\cdots 48}{42\cdots 51}a+\frac{13\cdots 62}{55\cdots 63}$, $\frac{55\cdots 61}{10\cdots 11}a^{21}-\frac{48\cdots 45}{10\cdots 11}a^{20}-\frac{17\cdots 24}{10\cdots 11}a^{19}-\frac{89\cdots 49}{10\cdots 11}a^{18}+\frac{14\cdots 05}{10\cdots 11}a^{17}+\frac{23\cdots 50}{10\cdots 11}a^{16}+\frac{45\cdots 91}{10\cdots 11}a^{15}+\frac{12\cdots 36}{10\cdots 11}a^{14}-\frac{23\cdots 57}{10\cdots 11}a^{13}-\frac{12\cdots 99}{10\cdots 11}a^{12}-\frac{30\cdots 01}{10\cdots 11}a^{11}-\frac{11\cdots 94}{10\cdots 11}a^{10}-\frac{34\cdots 44}{10\cdots 11}a^{9}-\frac{63\cdots 67}{10\cdots 11}a^{8}-\frac{84\cdots 11}{10\cdots 11}a^{7}-\frac{95\cdots 90}{10\cdots 11}a^{6}-\frac{86\cdots 12}{10\cdots 11}a^{5}-\frac{51\cdots 64}{10\cdots 11}a^{4}-\frac{16\cdots 19}{10\cdots 11}a^{3}-\frac{17\cdots 18}{10\cdots 11}a^{2}+\frac{25\cdots 83}{10\cdots 11}a+\frac{37\cdots 59}{10\cdots 11}$, $\frac{12\cdots 87}{10\cdots 11}a^{21}-\frac{11\cdots 93}{10\cdots 11}a^{20}-\frac{39\cdots 97}{10\cdots 11}a^{19}-\frac{16\cdots 42}{10\cdots 11}a^{18}+\frac{32\cdots 37}{10\cdots 11}a^{17}+\frac{50\cdots 39}{10\cdots 11}a^{16}+\frac{96\cdots 33}{10\cdots 11}a^{15}+\frac{27\cdots 16}{10\cdots 11}a^{14}-\frac{53\cdots 94}{10\cdots 11}a^{13}-\frac{28\cdots 39}{10\cdots 11}a^{12}-\frac{66\cdots 91}{10\cdots 11}a^{11}-\frac{25\cdots 91}{10\cdots 11}a^{10}-\frac{74\cdots 77}{10\cdots 11}a^{9}-\frac{13\cdots 63}{10\cdots 11}a^{8}-\frac{17\cdots 51}{10\cdots 11}a^{7}-\frac{20\cdots 29}{10\cdots 11}a^{6}-\frac{17\cdots 01}{10\cdots 11}a^{5}-\frac{10\cdots 31}{10\cdots 11}a^{4}-\frac{33\cdots 54}{10\cdots 11}a^{3}-\frac{34\cdots 11}{10\cdots 11}a^{2}+\frac{47\cdots 94}{10\cdots 11}a+\frac{76\cdots 47}{10\cdots 11}$, $\frac{37\cdots 27}{55\cdots 63}a^{21}-\frac{51\cdots 60}{55\cdots 63}a^{20}-\frac{11\cdots 26}{55\cdots 63}a^{19}-\frac{88\cdots 51}{42\cdots 51}a^{18}+\frac{96\cdots 58}{55\cdots 63}a^{17}+\frac{11\cdots 86}{55\cdots 63}a^{16}+\frac{26\cdots 18}{55\cdots 63}a^{15}+\frac{73\cdots 18}{55\cdots 63}a^{14}-\frac{19\cdots 48}{55\cdots 63}a^{13}-\frac{76\cdots 03}{55\cdots 63}a^{12}-\frac{17\cdots 26}{55\cdots 63}a^{11}-\frac{71\cdots 70}{55\cdots 63}a^{10}-\frac{19\cdots 17}{55\cdots 63}a^{9}-\frac{33\cdots 60}{55\cdots 63}a^{8}-\frac{42\cdots 22}{55\cdots 63}a^{7}-\frac{47\cdots 80}{55\cdots 63}a^{6}-\frac{40\cdots 07}{55\cdots 63}a^{5}-\frac{21\cdots 47}{55\cdots 63}a^{4}-\frac{57\cdots 54}{55\cdots 63}a^{3}-\frac{48\cdots 78}{55\cdots 63}a^{2}+\frac{56\cdots 63}{42\cdots 51}a+\frac{11\cdots 46}{55\cdots 63}$, $\frac{28\cdots 47}{55\cdots 63}a^{21}-\frac{22\cdots 42}{55\cdots 63}a^{20}-\frac{93\cdots 34}{55\cdots 63}a^{19}-\frac{41\cdots 88}{42\cdots 51}a^{18}+\frac{77\cdots 96}{55\cdots 63}a^{17}+\frac{12\cdots 23}{55\cdots 63}a^{16}+\frac{23\cdots 27}{55\cdots 63}a^{15}+\frac{67\cdots 16}{55\cdots 63}a^{14}-\frac{11\cdots 16}{55\cdots 63}a^{13}-\frac{69\cdots 52}{55\cdots 63}a^{12}-\frac{16\cdots 87}{55\cdots 63}a^{11}-\frac{61\cdots 25}{55\cdots 63}a^{10}-\frac{18\cdots 06}{55\cdots 63}a^{9}-\frac{33\cdots 86}{55\cdots 63}a^{8}-\frac{44\cdots 81}{55\cdots 63}a^{7}-\frac{50\cdots 51}{55\cdots 63}a^{6}-\frac{45\cdots 19}{55\cdots 63}a^{5}-\frac{26\cdots 58}{55\cdots 63}a^{4}-\frac{83\cdots 37}{55\cdots 63}a^{3}-\frac{80\cdots 18}{55\cdots 63}a^{2}+\frac{93\cdots 14}{42\cdots 51}a+\frac{17\cdots 72}{55\cdots 63}$, $\frac{33\cdots 46}{55\cdots 63}a^{21}-\frac{33\cdots 99}{55\cdots 63}a^{20}-\frac{10\cdots 93}{55\cdots 63}a^{19}-\frac{30\cdots 54}{42\cdots 51}a^{18}+\frac{90\cdots 54}{55\cdots 63}a^{17}+\frac{13\cdots 70}{55\cdots 63}a^{16}+\frac{25\cdots 77}{55\cdots 63}a^{15}+\frac{73\cdots 35}{55\cdots 63}a^{14}-\frac{15\cdots 80}{55\cdots 63}a^{13}-\frac{76\cdots 59}{55\cdots 63}a^{12}-\frac{17\cdots 45}{55\cdots 63}a^{11}-\frac{68\cdots 12}{55\cdots 63}a^{10}-\frac{20\cdots 80}{55\cdots 63}a^{9}-\frac{35\cdots 07}{55\cdots 63}a^{8}-\frac{45\cdots 31}{55\cdots 63}a^{7}-\frac{51\cdots 86}{55\cdots 63}a^{6}-\frac{45\cdots 52}{55\cdots 63}a^{5}-\frac{25\cdots 61}{55\cdots 63}a^{4}-\frac{70\cdots 30}{55\cdots 63}a^{3}-\frac{53\cdots 97}{55\cdots 63}a^{2}+\frac{78\cdots 42}{42\cdots 51}a+\frac{12\cdots 13}{55\cdots 63}$, $\frac{52\cdots 55}{55\cdots 63}a^{21}-\frac{45\cdots 36}{55\cdots 63}a^{20}-\frac{17\cdots 73}{55\cdots 63}a^{19}-\frac{65\cdots 34}{42\cdots 51}a^{18}+\frac{14\cdots 43}{55\cdots 63}a^{17}+\frac{22\cdots 21}{55\cdots 63}a^{16}+\frac{42\cdots 84}{55\cdots 63}a^{15}+\frac{12\cdots 36}{55\cdots 63}a^{14}-\frac{22\cdots 01}{55\cdots 63}a^{13}-\frac{12\cdots 18}{55\cdots 63}a^{12}-\frac{29\cdots 72}{55\cdots 63}a^{11}-\frac{11\cdots 80}{55\cdots 63}a^{10}-\frac{32\cdots 81}{55\cdots 63}a^{9}-\frac{59\cdots 30}{55\cdots 63}a^{8}-\frac{79\cdots 94}{55\cdots 63}a^{7}-\frac{89\cdots 43}{55\cdots 63}a^{6}-\frac{80\cdots 47}{55\cdots 63}a^{5}-\frac{47\cdots 68}{55\cdots 63}a^{4}-\frac{14\cdots 35}{55\cdots 63}a^{3}-\frac{13\cdots 80}{55\cdots 63}a^{2}+\frac{18\cdots 46}{42\cdots 51}a+\frac{28\cdots 24}{55\cdots 63}$, $\frac{34\cdots 89}{55\cdots 63}a^{21}-\frac{30\cdots 36}{55\cdots 63}a^{20}-\frac{11\cdots 49}{55\cdots 63}a^{19}-\frac{42\cdots 65}{42\cdots 51}a^{18}+\frac{92\cdots 73}{55\cdots 63}a^{17}+\frac{14\cdots 00}{55\cdots 63}a^{16}+\frac{28\cdots 90}{55\cdots 63}a^{15}+\frac{79\cdots 33}{55\cdots 63}a^{14}-\frac{14\cdots 27}{55\cdots 63}a^{13}-\frac{81\cdots 48}{55\cdots 63}a^{12}-\frac{19\cdots 83}{55\cdots 63}a^{11}-\frac{73\cdots 43}{55\cdots 63}a^{10}-\frac{21\cdots 92}{55\cdots 63}a^{9}-\frac{39\cdots 39}{55\cdots 63}a^{8}-\frac{52\cdots 41}{55\cdots 63}a^{7}-\frac{59\cdots 62}{55\cdots 63}a^{6}-\frac{53\cdots 70}{55\cdots 63}a^{5}-\frac{31\cdots 56}{55\cdots 63}a^{4}-\frac{10\cdots 14}{55\cdots 63}a^{3}-\frac{11\cdots 69}{55\cdots 63}a^{2}+\frac{10\cdots 69}{42\cdots 51}a+\frac{23\cdots 03}{55\cdots 63}$, $\frac{56\cdots 87}{55\cdots 63}a^{21}-\frac{53\cdots 32}{55\cdots 63}a^{20}-\frac{18\cdots 59}{55\cdots 63}a^{19}-\frac{59\cdots 09}{42\cdots 51}a^{18}+\frac{14\cdots 20}{55\cdots 63}a^{17}+\frac{22\cdots 53}{55\cdots 63}a^{16}+\frac{44\cdots 64}{55\cdots 63}a^{15}+\frac{12\cdots 55}{55\cdots 63}a^{14}-\frac{24\cdots 93}{55\cdots 63}a^{13}-\frac{12\cdots 85}{55\cdots 63}a^{12}-\frac{30\cdots 97}{55\cdots 63}a^{11}-\frac{11\cdots 30}{55\cdots 63}a^{10}-\frac{34\cdots 10}{55\cdots 63}a^{9}-\frac{61\cdots 70}{55\cdots 63}a^{8}-\frac{81\cdots 77}{55\cdots 63}a^{7}-\frac{92\cdots 30}{55\cdots 63}a^{6}-\frac{82\cdots 02}{55\cdots 63}a^{5}-\frac{48\cdots 57}{55\cdots 63}a^{4}-\frac{15\cdots 02}{55\cdots 63}a^{3}-\frac{15\cdots 26}{55\cdots 63}a^{2}+\frac{16\cdots 02}{42\cdots 51}a+\frac{33\cdots 47}{55\cdots 63}$, $\frac{12\cdots 63}{55\cdots 63}a^{21}-\frac{11\cdots 08}{55\cdots 63}a^{20}-\frac{39\cdots 61}{55\cdots 63}a^{19}-\frac{13\cdots 03}{42\cdots 51}a^{18}+\frac{32\cdots 16}{55\cdots 63}a^{17}+\frac{50\cdots 93}{55\cdots 63}a^{16}+\frac{96\cdots 31}{55\cdots 63}a^{15}+\frac{27\cdots 27}{55\cdots 63}a^{14}-\frac{53\cdots 69}{55\cdots 63}a^{13}-\frac{28\cdots 21}{55\cdots 63}a^{12}-\frac{66\cdots 15}{55\cdots 63}a^{11}-\frac{25\cdots 25}{55\cdots 63}a^{10}-\frac{75\cdots 67}{55\cdots 63}a^{9}-\frac{13\cdots 75}{55\cdots 63}a^{8}-\frac{17\cdots 47}{55\cdots 63}a^{7}-\frac{20\cdots 67}{55\cdots 63}a^{6}-\frac{18\cdots 20}{55\cdots 63}a^{5}-\frac{10\cdots 82}{55\cdots 63}a^{4}-\frac{32\cdots 46}{55\cdots 63}a^{3}-\frac{32\cdots 02}{55\cdots 63}a^{2}+\frac{35\cdots 35}{42\cdots 51}a+\frac{69\cdots 84}{55\cdots 63}$, $\frac{71\cdots 23}{55\cdots 63}a^{21}-\frac{57\cdots 77}{55\cdots 63}a^{20}-\frac{23\cdots 38}{55\cdots 63}a^{19}-\frac{10\cdots 46}{42\cdots 51}a^{18}+\frac{19\cdots 71}{55\cdots 63}a^{17}+\frac{32\cdots 79}{55\cdots 63}a^{16}+\frac{59\cdots 93}{55\cdots 63}a^{15}+\frac{16\cdots 03}{55\cdots 63}a^{14}-\frac{29\cdots 55}{55\cdots 63}a^{13}-\frac{17\cdots 91}{55\cdots 63}a^{12}-\frac{40\cdots 54}{55\cdots 63}a^{11}-\frac{15\cdots 94}{55\cdots 63}a^{10}-\frac{45\cdots 19}{55\cdots 63}a^{9}-\frac{84\cdots 81}{55\cdots 63}a^{8}-\frac{11\cdots 29}{55\cdots 63}a^{7}-\frac{12\cdots 87}{55\cdots 63}a^{6}-\frac{11\cdots 62}{55\cdots 63}a^{5}-\frac{70\cdots 37}{55\cdots 63}a^{4}-\frac{23\cdots 35}{55\cdots 63}a^{3}-\frac{25\cdots 50}{55\cdots 63}a^{2}+\frac{25\cdots 67}{42\cdots 51}a+\frac{53\cdots 50}{55\cdots 63}$, $\frac{17\cdots 25}{42\cdots 51}a^{21}-\frac{29\cdots 77}{42\cdots 51}a^{20}-\frac{55\cdots 99}{42\cdots 51}a^{19}+\frac{14\cdots 53}{42\cdots 51}a^{18}+\frac{47\cdots 78}{42\cdots 51}a^{17}+\frac{39\cdots 54}{42\cdots 51}a^{16}+\frac{10\cdots 40}{42\cdots 51}a^{15}+\frac{31\cdots 99}{42\cdots 51}a^{14}-\frac{10\cdots 68}{42\cdots 51}a^{13}-\frac{34\cdots 91}{42\cdots 51}a^{12}-\frac{70\cdots 84}{42\cdots 51}a^{11}-\frac{31\cdots 84}{42\cdots 51}a^{10}-\frac{85\cdots 12}{42\cdots 51}a^{9}-\frac{13\cdots 90}{42\cdots 51}a^{8}-\frac{15\cdots 58}{42\cdots 51}a^{7}-\frac{16\cdots 15}{42\cdots 51}a^{6}-\frac{11\cdots 21}{42\cdots 51}a^{5}-\frac{36\cdots 93}{42\cdots 51}a^{4}+\frac{55\cdots 20}{42\cdots 51}a^{3}+\frac{43\cdots 59}{42\cdots 51}a^{2}-\frac{18\cdots 51}{42\cdots 51}a-\frac{58\cdots 18}{42\cdots 51}$, $\frac{34\cdots 66}{55\cdots 63}a^{21}-\frac{31\cdots 41}{55\cdots 63}a^{20}-\frac{11\cdots 94}{55\cdots 63}a^{19}-\frac{39\cdots 91}{42\cdots 51}a^{18}+\frac{91\cdots 81}{55\cdots 63}a^{17}+\frac{14\cdots 61}{55\cdots 63}a^{16}+\frac{27\cdots 02}{55\cdots 63}a^{15}+\frac{78\cdots 80}{55\cdots 63}a^{14}-\frac{14\cdots 13}{55\cdots 63}a^{13}-\frac{80\cdots 03}{55\cdots 63}a^{12}-\frac{18\cdots 38}{55\cdots 63}a^{11}-\frac{72\cdots 89}{55\cdots 63}a^{10}-\frac{21\cdots 69}{55\cdots 63}a^{9}-\frac{38\cdots 12}{55\cdots 63}a^{8}-\frac{51\cdots 65}{55\cdots 63}a^{7}-\frac{57\cdots 79}{55\cdots 63}a^{6}-\frac{51\cdots 16}{55\cdots 63}a^{5}-\frac{30\cdots 44}{55\cdots 63}a^{4}-\frac{97\cdots 86}{55\cdots 63}a^{3}-\frac{10\cdots 29}{55\cdots 63}a^{2}+\frac{10\cdots 92}{42\cdots 51}a+\frac{21\cdots 58}{55\cdots 63}$, $\frac{60\cdots 84}{55\cdots 63}a^{21}-\frac{12\cdots 89}{55\cdots 63}a^{20}-\frac{20\cdots 19}{55\cdots 63}a^{19}-\frac{17\cdots 35}{42\cdots 51}a^{18}+\frac{16\cdots 19}{55\cdots 63}a^{17}+\frac{36\cdots 41}{55\cdots 63}a^{16}+\frac{61\cdots 52}{55\cdots 63}a^{15}+\frac{16\cdots 43}{55\cdots 63}a^{14}-\frac{18\cdots 42}{55\cdots 63}a^{13}-\frac{16\cdots 18}{55\cdots 63}a^{12}-\frac{42\cdots 94}{55\cdots 63}a^{11}-\frac{14\cdots 69}{55\cdots 63}a^{10}-\frac{45\cdots 57}{55\cdots 63}a^{9}-\frac{90\cdots 57}{55\cdots 63}a^{8}-\frac{12\cdots 10}{55\cdots 63}a^{7}-\frac{14\cdots 26}{55\cdots 63}a^{6}-\frac{14\cdots 67}{55\cdots 63}a^{5}-\frac{94\cdots 82}{55\cdots 63}a^{4}-\frac{35\cdots 56}{55\cdots 63}a^{3}-\frac{50\cdots 97}{55\cdots 63}a^{2}+\frac{38\cdots 30}{42\cdots 51}a+\frac{96\cdots 46}{55\cdots 63}$, $\frac{17\cdots 69}{55\cdots 63}a^{21}-\frac{94\cdots 40}{55\cdots 63}a^{20}-\frac{57\cdots 79}{55\cdots 63}a^{19}-\frac{35\cdots 99}{42\cdots 51}a^{18}+\frac{46\cdots 30}{55\cdots 63}a^{17}+\frac{90\cdots 17}{55\cdots 63}a^{16}+\frac{15\cdots 19}{55\cdots 63}a^{15}+\frac{44\cdots 12}{55\cdots 63}a^{14}-\frac{63\cdots 64}{55\cdots 63}a^{13}-\frac{44\cdots 41}{55\cdots 63}a^{12}-\frac{10\cdots 08}{55\cdots 63}a^{11}-\frac{39\cdots 80}{55\cdots 63}a^{10}-\frac{12\cdots 63}{55\cdots 63}a^{9}-\frac{23\cdots 02}{55\cdots 63}a^{8}-\frac{31\cdots 14}{55\cdots 63}a^{7}-\frac{36\cdots 04}{55\cdots 63}a^{6}-\frac{33\cdots 77}{55\cdots 63}a^{5}-\frac{21\cdots 12}{55\cdots 63}a^{4}-\frac{75\cdots 28}{55\cdots 63}a^{3}-\frac{81\cdots 36}{55\cdots 63}a^{2}+\frac{13\cdots 07}{42\cdots 51}a+\frac{27\cdots 04}{55\cdots 63}$ 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:  \( 10182928874900 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 10182928874900 \cdot 1}{2\cdot\sqrt{6172475179135960522642404800603172634624}}\cr\approx \mathstrut & 1.65482644030311 \end{aligned}\] (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 - 33*x^20 - 44*x^19 + 253*x^18 + 660*x^17 + 1177*x^16 + 2992*x^15 - 2266*x^14 - 27214*x^13 - 75812*x^12 - 258772*x^11 - 808192*x^10 - 1679018*x^9 - 2491038*x^8 - 3007114*x^7 - 3012009*x^6 - 2244528*x^5 - 1081113*x^4 - 281710*x^3 - 22011*x^2 + 4268*x + 561) 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 - 33*x^20 - 44*x^19 + 253*x^18 + 660*x^17 + 1177*x^16 + 2992*x^15 - 2266*x^14 - 27214*x^13 - 75812*x^12 - 258772*x^11 - 808192*x^10 - 1679018*x^9 - 2491038*x^8 - 3007114*x^7 - 3012009*x^6 - 2244528*x^5 - 1081113*x^4 - 281710*x^3 - 22011*x^2 + 4268*x + 561, 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 - 33*x^20 - 44*x^19 + 253*x^18 + 660*x^17 + 1177*x^16 + 2992*x^15 - 2266*x^14 - 27214*x^13 - 75812*x^12 - 258772*x^11 - 808192*x^10 - 1679018*x^9 - 2491038*x^8 - 3007114*x^7 - 3012009*x^6 - 2244528*x^5 - 1081113*x^4 - 281710*x^3 - 22011*x^2 + 4268*x + 561); 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 - 33*x^20 - 44*x^19 + 253*x^18 + 660*x^17 + 1177*x^16 + 2992*x^15 - 2266*x^14 - 27214*x^13 - 75812*x^12 - 258772*x^11 - 808192*x^10 - 1679018*x^9 - 2491038*x^8 - 3007114*x^7 - 3012009*x^6 - 2244528*x^5 - 1081113*x^4 - 281710*x^3 - 22011*x^2 + 4268*x + 561); 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

$C_2^{10}.F_{11}$ (as 22T34):

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 112640
The 44 conjugacy class representatives for $C_2^{10}.F_{11}$
Character table for $C_2^{10}.F_{11}$

Intermediate fields

11.11.4910318845910094848.1

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 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 ${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ $20{,}\,{\href{/padicField/5.2.0.1}{2} }$ R R ${\href{/padicField/13.5.0.1}{5} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.5.0.1}{5} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.11.0.1}{11} }^{2}$ ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.5.0.1}{5} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ $20{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.1.22.28a1.1$x^{22} + 2 x^{7} + 2$$22$$1$$28$22T34not computed
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$$[\ ]$$
7.5.2.5a1.1$x^{10} + 2 x^{6} + 8 x^{5} + x^{2} + 15 x + 16$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
7.5.2.5a1.1$x^{10} + 2 x^{6} + 8 x^{5} + x^{2} + 15 x + 16$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
\(11\) Copy content Toggle raw display 11.1.11.11a1.1$x^{11} + 11 x + 11$$11$$1$$11$$F_{11}$$$[\frac{11}{10}]_{10}$$
11.1.11.11a1.1$x^{11} + 11 x + 11$$11$$1$$11$$F_{11}$$$[\frac{11}{10}]_{10}$$

Spectrum of ring of integers

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