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Properties

Label 22.12.611...500.1
Degree $22$
Signature $[12, 5]$
Discriminant $-6.112\times 10^{40}$
Root discriminant \(71.44\)
Ramified primes $2,3,5,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.C_{11}:C_{10}$ (as 22T36)

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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 - x^21 - 30*x^20 + 35*x^19 + 265*x^18 - 687*x^17 + 357*x^16 + 7305*x^15 - 18675*x^14 - 35840*x^13 + 103856*x^12 + 58224*x^11 - 211240*x^10 + 163795*x^9 - 64050*x^8 - 651165*x^7 + 823710*x^6 + 466155*x^5 - 852830*x^4 + 161150*x^3 + 139095*x^2 - 61765*x + 6985)
 
Copy content gp:K = bnfinit(y^22 - y^21 - 30*y^20 + 35*y^19 + 265*y^18 - 687*y^17 + 357*y^16 + 7305*y^15 - 18675*y^14 - 35840*y^13 + 103856*y^12 + 58224*y^11 - 211240*y^10 + 163795*y^9 - 64050*y^8 - 651165*y^7 + 823710*y^6 + 466155*y^5 - 852830*y^4 + 161150*y^3 + 139095*y^2 - 61765*y + 6985, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^21 - 30*x^20 + 35*x^19 + 265*x^18 - 687*x^17 + 357*x^16 + 7305*x^15 - 18675*x^14 - 35840*x^13 + 103856*x^12 + 58224*x^11 - 211240*x^10 + 163795*x^9 - 64050*x^8 - 651165*x^7 + 823710*x^6 + 466155*x^5 - 852830*x^4 + 161150*x^3 + 139095*x^2 - 61765*x + 6985);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - x^21 - 30*x^20 + 35*x^19 + 265*x^18 - 687*x^17 + 357*x^16 + 7305*x^15 - 18675*x^14 - 35840*x^13 + 103856*x^12 + 58224*x^11 - 211240*x^10 + 163795*x^9 - 64050*x^8 - 651165*x^7 + 823710*x^6 + 466155*x^5 - 852830*x^4 + 161150*x^3 + 139095*x^2 - 61765*x + 6985)
 

\( x^{22} - x^{21} - 30 x^{20} + 35 x^{19} + 265 x^{18} - 687 x^{17} + 357 x^{16} + 7305 x^{15} + \cdots + 6985 \) 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:  $[12, 5]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-61117859389674098630012726211547851562500\) \(\medspace = -\,2^{2}\cdot 3^{20}\cdot 5^{20}\cdot 11^{16}\) 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:  \(71.44\)
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:  $2\cdot 3^{10/11}5^{10/11}11^{4/5}\approx 159.70481706565957$
Ramified primes:   \(2\), \(3\), \(5\), \(11\) 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:  \(\Q(\sqrt{-1}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{6}a^{12}-\frac{1}{6}a^{11}-\frac{1}{6}a^{10}-\frac{1}{3}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}+\frac{1}{3}a^{3}+\frac{1}{6}a^{2}+\frac{1}{6}a-\frac{1}{6}$, $\frac{1}{6}a^{13}-\frac{1}{6}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{6}a^{14}-\frac{1}{6}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{6}a^{5}-\frac{1}{2}a^{4}-\frac{1}{3}a^{2}-\frac{1}{2}a$, $\frac{1}{6}a^{15}+\frac{1}{3}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{6}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a+\frac{1}{6}$, $\frac{1}{6}a^{16}+\frac{1}{3}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{6}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}+\frac{1}{6}a$, $\frac{1}{660}a^{17}+\frac{7}{330}a^{16}+\frac{7}{132}a^{15}-\frac{3}{44}a^{14}+\frac{7}{132}a^{13}-\frac{31}{660}a^{12}+\frac{19}{165}a^{11}-\frac{5}{66}a^{10}-\frac{25}{132}a^{9}+\frac{17}{66}a^{8}-\frac{4}{33}a^{7}+\frac{17}{132}a^{6}-\frac{13}{44}a^{5}-\frac{1}{6}a^{4}-\frac{1}{6}a^{3}+\frac{1}{6}a^{2}-\frac{1}{12}a-\frac{1}{12}$, $\frac{1}{660}a^{18}-\frac{17}{220}a^{16}+\frac{1}{44}a^{15}+\frac{1}{132}a^{14}+\frac{29}{660}a^{13}-\frac{2}{33}a^{12}+\frac{8}{55}a^{11}+\frac{9}{44}a^{10}+\frac{8}{33}a^{9}+\frac{3}{11}a^{8}+\frac{7}{44}a^{7}-\frac{19}{44}a^{6}+\frac{31}{66}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{3}$, $\frac{1}{660}a^{19}-\frac{41}{660}a^{16}+\frac{1}{22}a^{15}+\frac{1}{15}a^{14}-\frac{1}{44}a^{13}-\frac{1}{12}a^{12}+\frac{17}{220}a^{11}-\frac{19}{66}a^{10}-\frac{17}{44}a^{9}+\frac{13}{44}a^{8}-\frac{59}{132}a^{7}-\frac{13}{44}a^{6}+\frac{35}{132}a^{5}-\frac{1}{2}a^{4}+\frac{1}{12}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{660}a^{20}+\frac{9}{110}a^{16}+\frac{49}{660}a^{15}+\frac{1}{66}a^{14}-\frac{5}{66}a^{13}-\frac{1}{66}a^{12}+\frac{1}{10}a^{11}+\frac{1}{132}a^{10}+\frac{4}{11}a^{9}+\frac{5}{44}a^{8}+\frac{3}{44}a^{7}-\frac{19}{66}a^{6}+\frac{7}{132}a^{5}-\frac{1}{12}a^{4}+\frac{1}{12}a^{3}+\frac{5}{12}a^{2}-\frac{1}{6}a-\frac{1}{4}$, $\frac{1}{20\cdots 20}a^{21}-\frac{10\cdots 11}{23\cdots 80}a^{20}+\frac{42\cdots 29}{69\cdots 40}a^{19}+\frac{95\cdots 89}{20\cdots 20}a^{18}+\frac{37\cdots 79}{34\cdots 70}a^{17}-\frac{53\cdots 01}{69\cdots 40}a^{16}+\frac{16\cdots 49}{34\cdots 70}a^{15}+\frac{38\cdots 31}{69\cdots 40}a^{14}+\frac{37\cdots 03}{17\cdots 85}a^{13}-\frac{11\cdots 79}{20\cdots 20}a^{12}+\frac{43\cdots 97}{34\cdots 70}a^{11}-\frac{12\cdots 56}{34\cdots 77}a^{10}-\frac{28\cdots 95}{20\cdots 62}a^{9}+\frac{44\cdots 91}{13\cdots 08}a^{8}+\frac{49\cdots 71}{13\cdots 08}a^{7}-\frac{26\cdots 23}{12\cdots 28}a^{6}-\frac{31\cdots 27}{12\cdots 28}a^{5}-\frac{25\cdots 69}{63\cdots 14}a^{4}+\frac{14\cdots 65}{38\cdots 84}a^{3}+\frac{13\cdots 09}{12\cdots 28}a^{2}-\frac{12\cdots 33}{42\cdots 76}a+\frac{37\cdots 01}{19\cdots 42}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $16$
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:   $\frac{46\cdots 51}{52\cdots 55}a^{21}-\frac{21\cdots 31}{34\cdots 77}a^{20}-\frac{18\cdots 67}{69\cdots 40}a^{19}+\frac{24\cdots 69}{10\cdots 10}a^{18}+\frac{14\cdots 46}{58\cdots 95}a^{17}-\frac{12\cdots 09}{23\cdots 80}a^{16}+\frac{36\cdots 53}{23\cdots 18}a^{15}+\frac{22\cdots 91}{34\cdots 70}a^{14}-\frac{10\cdots 31}{69\cdots 40}a^{13}-\frac{75\cdots 41}{20\cdots 20}a^{12}+\frac{57\cdots 39}{69\cdots 40}a^{11}+\frac{53\cdots 15}{69\cdots 54}a^{10}-\frac{69\cdots 95}{41\cdots 24}a^{9}+\frac{45\cdots 57}{46\cdots 36}a^{8}-\frac{13\cdots 09}{46\cdots 36}a^{7}-\frac{27\cdots 17}{46\cdots 36}a^{6}+\frac{78\cdots 19}{13\cdots 08}a^{5}+\frac{36\cdots 07}{63\cdots 14}a^{4}-\frac{22\cdots 35}{38\cdots 84}a^{3}-\frac{12\cdots 35}{42\cdots 76}a^{2}+\frac{48\cdots 29}{42\cdots 76}a-\frac{81\cdots 87}{38\cdots 84}$, $\frac{11\cdots 09}{20\cdots 20}a^{21}-\frac{25\cdots 27}{69\cdots 40}a^{20}-\frac{10\cdots 63}{63\cdots 14}a^{19}+\frac{14\cdots 13}{10\cdots 10}a^{18}+\frac{78\cdots 67}{52\cdots 45}a^{17}-\frac{15\cdots 95}{46\cdots 36}a^{16}+\frac{62\cdots 37}{69\cdots 40}a^{15}+\frac{46\cdots 30}{11\cdots 59}a^{14}-\frac{51\cdots 11}{58\cdots 95}a^{13}-\frac{10\cdots 83}{47\cdots 05}a^{12}+\frac{34\cdots 87}{69\cdots 40}a^{11}+\frac{59\cdots 55}{12\cdots 28}a^{10}-\frac{37\cdots 23}{38\cdots 84}a^{9}+\frac{13\cdots 17}{23\cdots 18}a^{8}-\frac{78\cdots 11}{46\cdots 36}a^{7}-\frac{49\cdots 31}{13\cdots 08}a^{6}+\frac{38\cdots 67}{11\cdots 59}a^{5}+\frac{75\cdots 81}{21\cdots 38}a^{4}-\frac{32\cdots 77}{95\cdots 21}a^{3}-\frac{87\cdots 93}{42\cdots 76}a^{2}+\frac{85\cdots 99}{12\cdots 28}a-\frac{46\cdots 61}{38\cdots 84}$, $\frac{10\cdots 37}{69\cdots 40}a^{21}-\frac{47\cdots 77}{69\cdots 54}a^{20}-\frac{81\cdots 21}{17\cdots 85}a^{19}+\frac{19\cdots 39}{69\cdots 40}a^{18}+\frac{90\cdots 23}{21\cdots 80}a^{17}-\frac{28\cdots 49}{34\cdots 70}a^{16}+\frac{31\cdots 04}{34\cdots 77}a^{15}+\frac{39\cdots 37}{34\cdots 70}a^{14}-\frac{39\cdots 86}{17\cdots 85}a^{13}-\frac{15\cdots 63}{23\cdots 80}a^{12}+\frac{85\cdots 91}{69\cdots 40}a^{11}+\frac{20\cdots 97}{12\cdots 28}a^{10}-\frac{25\cdots 74}{10\cdots 69}a^{9}+\frac{55\cdots 67}{46\cdots 36}a^{8}-\frac{43\cdots 01}{13\cdots 08}a^{7}-\frac{14\cdots 81}{13\cdots 08}a^{6}+\frac{24\cdots 01}{34\cdots 77}a^{5}+\frac{14\cdots 23}{12\cdots 28}a^{4}-\frac{88\cdots 47}{12\cdots 28}a^{3}-\frac{62\cdots 59}{42\cdots 76}a^{2}+\frac{57\cdots 09}{42\cdots 76}a-\frac{23\cdots 55}{12\cdots 28}$, $\frac{31\cdots 47}{20\cdots 20}a^{21}-\frac{13\cdots 97}{13\cdots 08}a^{20}-\frac{10\cdots 27}{23\cdots 80}a^{19}+\frac{14\cdots 07}{38\cdots 84}a^{18}+\frac{56\cdots 49}{13\cdots 08}a^{17}-\frac{61\cdots 87}{69\cdots 40}a^{16}+\frac{32\cdots 71}{13\cdots 08}a^{15}+\frac{38\cdots 63}{34\cdots 70}a^{14}-\frac{10\cdots 47}{42\cdots 76}a^{13}-\frac{12\cdots 23}{20\cdots 62}a^{12}+\frac{15\cdots 43}{11\cdots 90}a^{11}+\frac{45\cdots 49}{34\cdots 77}a^{10}-\frac{11\cdots 31}{41\cdots 24}a^{9}+\frac{21\cdots 93}{13\cdots 08}a^{8}-\frac{58\cdots 55}{13\cdots 08}a^{7}-\frac{34\cdots 95}{34\cdots 77}a^{6}+\frac{62\cdots 39}{69\cdots 54}a^{5}+\frac{10\cdots 06}{10\cdots 69}a^{4}-\frac{36\cdots 51}{38\cdots 84}a^{3}-\frac{73\cdots 97}{12\cdots 28}a^{2}+\frac{11\cdots 47}{63\cdots 14}a-\frac{12\cdots 05}{38\cdots 84}$, $\frac{33\cdots 51}{11\cdots 90}a^{21}-\frac{14\cdots 77}{69\cdots 40}a^{20}-\frac{15\cdots 97}{17\cdots 85}a^{19}+\frac{48\cdots 81}{63\cdots 40}a^{18}+\frac{54\cdots 09}{69\cdots 40}a^{17}-\frac{12\cdots 17}{69\cdots 40}a^{16}+\frac{36\cdots 37}{69\cdots 40}a^{15}+\frac{66\cdots 39}{31\cdots 70}a^{14}-\frac{16\cdots 83}{34\cdots 70}a^{13}-\frac{73\cdots 09}{63\cdots 40}a^{12}+\frac{41\cdots 54}{15\cdots 35}a^{11}+\frac{84\cdots 77}{34\cdots 77}a^{10}-\frac{74\cdots 81}{13\cdots 08}a^{9}+\frac{44\cdots 61}{13\cdots 08}a^{8}-\frac{65\cdots 31}{69\cdots 54}a^{7}-\frac{59\cdots 05}{31\cdots 07}a^{6}+\frac{12\cdots 21}{69\cdots 54}a^{5}+\frac{23\cdots 77}{12\cdots 28}a^{4}-\frac{24\cdots 91}{12\cdots 28}a^{3}-\frac{16\cdots 75}{21\cdots 38}a^{2}+\frac{39\cdots 60}{10\cdots 69}a-\frac{74\cdots 58}{10\cdots 69}$, $\frac{10\cdots 46}{52\cdots 55}a^{21}-\frac{81\cdots 19}{63\cdots 40}a^{20}-\frac{13\cdots 13}{23\cdots 80}a^{19}+\frac{10\cdots 69}{20\cdots 20}a^{18}+\frac{93\cdots 99}{17\cdots 85}a^{17}-\frac{20\cdots 48}{17\cdots 85}a^{16}+\frac{53\cdots 06}{17\cdots 85}a^{15}+\frac{30\cdots 03}{21\cdots 80}a^{14}-\frac{55\cdots 92}{17\cdots 85}a^{13}-\frac{16\cdots 53}{20\cdots 20}a^{12}+\frac{12\cdots 09}{69\cdots 40}a^{11}+\frac{54\cdots 89}{31\cdots 07}a^{10}-\frac{14\cdots 65}{41\cdots 24}a^{9}+\frac{14\cdots 05}{69\cdots 54}a^{8}-\frac{81\cdots 11}{13\cdots 08}a^{7}-\frac{45\cdots 34}{34\cdots 77}a^{6}+\frac{13\cdots 64}{11\cdots 59}a^{5}+\frac{16\cdots 95}{12\cdots 28}a^{4}-\frac{23\cdots 33}{19\cdots 42}a^{3}-\frac{36\cdots 91}{42\cdots 76}a^{2}+\frac{75\cdots 48}{31\cdots 07}a-\frac{40\cdots 07}{95\cdots 21}$, $\frac{13\cdots 97}{52\cdots 55}a^{21}-\frac{42\cdots 99}{23\cdots 80}a^{20}-\frac{16\cdots 13}{21\cdots 80}a^{19}+\frac{71\cdots 37}{10\cdots 10}a^{18}+\frac{32\cdots 41}{46\cdots 36}a^{17}-\frac{10\cdots 39}{69\cdots 40}a^{16}+\frac{82\cdots 53}{17\cdots 85}a^{15}+\frac{13\cdots 69}{69\cdots 40}a^{14}-\frac{22\cdots 19}{52\cdots 45}a^{13}-\frac{21\cdots 41}{20\cdots 62}a^{12}+\frac{55\cdots 91}{23\cdots 80}a^{11}+\frac{30\cdots 75}{13\cdots 08}a^{10}-\frac{91\cdots 79}{19\cdots 42}a^{9}+\frac{33\cdots 51}{11\cdots 59}a^{8}-\frac{58\cdots 47}{69\cdots 54}a^{7}-\frac{59\cdots 05}{34\cdots 77}a^{6}+\frac{22\cdots 83}{13\cdots 08}a^{5}+\frac{70\cdots 09}{42\cdots 76}a^{4}-\frac{16\cdots 23}{95\cdots 21}a^{3}-\frac{15\cdots 17}{21\cdots 38}a^{2}+\frac{21\cdots 33}{63\cdots 14}a-\frac{24\cdots 51}{38\cdots 84}$, $\frac{11\cdots 21}{15\cdots 35}a^{21}-\frac{22\cdots 79}{58\cdots 95}a^{20}-\frac{29\cdots 11}{13\cdots 08}a^{19}+\frac{51\cdots 63}{34\cdots 70}a^{18}+\frac{33\cdots 36}{17\cdots 85}a^{17}-\frac{55\cdots 41}{13\cdots 08}a^{16}+\frac{41\cdots 94}{58\cdots 95}a^{15}+\frac{60\cdots 69}{11\cdots 59}a^{14}-\frac{25\cdots 67}{23\cdots 80}a^{13}-\frac{70\cdots 43}{23\cdots 80}a^{12}+\frac{41\cdots 71}{69\cdots 40}a^{11}+\frac{23\cdots 02}{34\cdots 77}a^{10}-\frac{16\cdots 63}{13\cdots 08}a^{9}+\frac{87\cdots 31}{13\cdots 08}a^{8}-\frac{21\cdots 33}{12\cdots 28}a^{7}-\frac{19\cdots 61}{42\cdots 76}a^{6}+\frac{17\cdots 37}{46\cdots 36}a^{5}+\frac{10\cdots 41}{21\cdots 38}a^{4}-\frac{47\cdots 63}{12\cdots 28}a^{3}-\frac{69\cdots 05}{12\cdots 28}a^{2}+\frac{30\cdots 33}{42\cdots 76}a-\frac{14\cdots 75}{12\cdots 28}$, $\frac{18\cdots 32}{52\cdots 55}a^{21}-\frac{16\cdots 01}{69\cdots 40}a^{20}-\frac{24\cdots 83}{23\cdots 80}a^{19}+\frac{16\cdots 93}{19\cdots 20}a^{18}+\frac{66\cdots 03}{69\cdots 40}a^{17}-\frac{72\cdots 83}{34\cdots 70}a^{16}+\frac{37\cdots 81}{69\cdots 40}a^{15}+\frac{44\cdots 56}{17\cdots 85}a^{14}-\frac{13\cdots 17}{23\cdots 80}a^{13}-\frac{15\cdots 57}{10\cdots 10}a^{12}+\frac{43\cdots 49}{13\cdots 08}a^{11}+\frac{21\cdots 03}{69\cdots 54}a^{10}-\frac{13\cdots 95}{20\cdots 62}a^{9}+\frac{12\cdots 68}{34\cdots 77}a^{8}-\frac{14\cdots 77}{13\cdots 08}a^{7}-\frac{32\cdots 57}{13\cdots 08}a^{6}+\frac{29\cdots 85}{13\cdots 08}a^{5}+\frac{98\cdots 95}{42\cdots 76}a^{4}-\frac{41\cdots 47}{19\cdots 42}a^{3}-\frac{65\cdots 65}{42\cdots 76}a^{2}+\frac{54\cdots 33}{12\cdots 28}a-\frac{29\cdots 33}{38\cdots 84}$, $\frac{45\cdots 13}{69\cdots 40}a^{21}-\frac{16\cdots 83}{58\cdots 95}a^{20}-\frac{12\cdots 53}{63\cdots 40}a^{19}+\frac{82\cdots 47}{69\cdots 40}a^{18}+\frac{12\cdots 53}{69\cdots 40}a^{17}-\frac{73\cdots 59}{21\cdots 80}a^{16}+\frac{14\cdots 83}{34\cdots 70}a^{15}+\frac{16\cdots 99}{34\cdots 70}a^{14}-\frac{65\cdots 47}{69\cdots 40}a^{13}-\frac{29\cdots 83}{10\cdots 90}a^{12}+\frac{81\cdots 66}{15\cdots 35}a^{11}+\frac{91\cdots 79}{13\cdots 08}a^{10}-\frac{13\cdots 17}{13\cdots 08}a^{9}+\frac{17\cdots 78}{34\cdots 77}a^{8}-\frac{31\cdots 49}{23\cdots 18}a^{7}-\frac{14\cdots 59}{34\cdots 77}a^{6}+\frac{41\cdots 11}{13\cdots 08}a^{5}+\frac{58\cdots 07}{12\cdots 28}a^{4}-\frac{18\cdots 49}{63\cdots 14}a^{3}-\frac{37\cdots 05}{63\cdots 14}a^{2}+\frac{12\cdots 53}{21\cdots 38}a-\frac{17\cdots 35}{21\cdots 38}$, $\frac{40\cdots 24}{52\cdots 55}a^{21}-\frac{97\cdots 93}{17\cdots 85}a^{20}-\frac{16\cdots 83}{69\cdots 54}a^{19}+\frac{43\cdots 49}{20\cdots 20}a^{18}+\frac{37\cdots 52}{17\cdots 85}a^{17}-\frac{33\cdots 33}{69\cdots 40}a^{16}+\frac{99\cdots 97}{69\cdots 40}a^{15}+\frac{80\cdots 73}{13\cdots 08}a^{14}-\frac{90\cdots 23}{69\cdots 40}a^{13}-\frac{33\cdots 47}{10\cdots 10}a^{12}+\frac{12\cdots 34}{17\cdots 85}a^{11}+\frac{30\cdots 57}{46\cdots 36}a^{10}-\frac{30\cdots 77}{20\cdots 62}a^{9}+\frac{30\cdots 06}{34\cdots 77}a^{8}-\frac{34\cdots 81}{13\cdots 08}a^{7}-\frac{65\cdots 31}{12\cdots 28}a^{6}+\frac{15\cdots 54}{31\cdots 07}a^{5}+\frac{32\cdots 33}{63\cdots 14}a^{4}-\frac{10\cdots 81}{19\cdots 42}a^{3}-\frac{27\cdots 31}{12\cdots 28}a^{2}+\frac{13\cdots 43}{12\cdots 28}a-\frac{36\cdots 85}{19\cdots 42}$, $\frac{96\cdots 79}{69\cdots 40}a^{21}-\frac{46\cdots 91}{69\cdots 54}a^{20}-\frac{29\cdots 39}{69\cdots 40}a^{19}+\frac{85\cdots 77}{31\cdots 70}a^{18}+\frac{44\cdots 21}{11\cdots 90}a^{17}-\frac{26\cdots 17}{34\cdots 70}a^{16}+\frac{25\cdots 43}{23\cdots 18}a^{15}+\frac{35\cdots 37}{34\cdots 70}a^{14}-\frac{14\cdots 69}{69\cdots 40}a^{13}-\frac{41\cdots 31}{69\cdots 40}a^{12}+\frac{26\cdots 95}{23\cdots 18}a^{11}+\frac{43\cdots 38}{31\cdots 07}a^{10}-\frac{76\cdots 88}{34\cdots 77}a^{9}+\frac{13\cdots 72}{11\cdots 59}a^{8}-\frac{40\cdots 75}{12\cdots 28}a^{7}-\frac{21\cdots 01}{23\cdots 18}a^{6}+\frac{23\cdots 03}{34\cdots 77}a^{5}+\frac{12\cdots 87}{12\cdots 28}a^{4}-\frac{21\cdots 34}{31\cdots 07}a^{3}-\frac{14\cdots 73}{12\cdots 28}a^{2}+\frac{13\cdots 93}{10\cdots 69}a-\frac{25\cdots 93}{12\cdots 28}$, $\frac{42\cdots 59}{69\cdots 40}a^{21}-\frac{11\cdots 59}{34\cdots 70}a^{20}-\frac{19\cdots 27}{10\cdots 90}a^{19}+\frac{42\cdots 21}{31\cdots 70}a^{18}+\frac{38\cdots 89}{23\cdots 80}a^{17}-\frac{23\cdots 47}{69\cdots 40}a^{16}+\frac{16\cdots 71}{23\cdots 80}a^{15}+\frac{30\cdots 47}{69\cdots 40}a^{14}-\frac{65\cdots 27}{69\cdots 40}a^{13}-\frac{17\cdots 57}{69\cdots 40}a^{12}+\frac{11\cdots 61}{23\cdots 80}a^{11}+\frac{65\cdots 91}{11\cdots 59}a^{10}-\frac{71\cdots 11}{69\cdots 54}a^{9}+\frac{26\cdots 73}{46\cdots 36}a^{8}-\frac{51\cdots 98}{31\cdots 07}a^{7}-\frac{92\cdots 31}{23\cdots 18}a^{6}+\frac{11\cdots 91}{34\cdots 77}a^{5}+\frac{52\cdots 85}{12\cdots 28}a^{4}-\frac{42\cdots 67}{12\cdots 28}a^{3}-\frac{12\cdots 10}{31\cdots 07}a^{2}+\frac{67\cdots 60}{10\cdots 69}a-\frac{12\cdots 67}{12\cdots 28}$, $\frac{23\cdots 27}{19\cdots 20}a^{21}-\frac{27\cdots 70}{34\cdots 77}a^{20}-\frac{51\cdots 33}{13\cdots 08}a^{19}+\frac{61\cdots 99}{20\cdots 20}a^{18}+\frac{38\cdots 69}{11\cdots 90}a^{17}-\frac{99\cdots 49}{13\cdots 08}a^{16}+\frac{26\cdots 35}{13\cdots 08}a^{15}+\frac{12\cdots 55}{13\cdots 08}a^{14}-\frac{34\cdots 67}{17\cdots 85}a^{13}-\frac{10\cdots 17}{20\cdots 20}a^{12}+\frac{18\cdots 04}{17\cdots 85}a^{11}+\frac{15\cdots 87}{13\cdots 08}a^{10}-\frac{22\cdots 77}{10\cdots 31}a^{9}+\frac{13\cdots 32}{10\cdots 69}a^{8}-\frac{12\cdots 55}{34\cdots 77}a^{7}-\frac{11\cdots 55}{13\cdots 08}a^{6}+\frac{49\cdots 09}{69\cdots 54}a^{5}+\frac{10\cdots 79}{12\cdots 28}a^{4}-\frac{70\cdots 89}{95\cdots 21}a^{3}-\frac{10\cdots 03}{21\cdots 38}a^{2}+\frac{61\cdots 63}{42\cdots 76}a-\frac{10\cdots 23}{38\cdots 84}$, $\frac{12\cdots 61}{20\cdots 20}a^{21}-\frac{25\cdots 97}{69\cdots 40}a^{20}-\frac{41\cdots 97}{23\cdots 80}a^{19}+\frac{28\cdots 51}{20\cdots 20}a^{18}+\frac{55\cdots 07}{34\cdots 70}a^{17}-\frac{78\cdots 99}{23\cdots 80}a^{16}+\frac{91\cdots 87}{11\cdots 90}a^{15}+\frac{29\cdots 41}{69\cdots 40}a^{14}-\frac{16\cdots 93}{17\cdots 85}a^{13}-\frac{51\cdots 37}{20\cdots 20}a^{12}+\frac{59\cdots 28}{11\cdots 59}a^{11}+\frac{12\cdots 33}{23\cdots 18}a^{10}-\frac{21\cdots 81}{20\cdots 62}a^{9}+\frac{78\cdots 07}{13\cdots 08}a^{8}-\frac{79\cdots 05}{46\cdots 36}a^{7}-\frac{17\cdots 93}{46\cdots 36}a^{6}+\frac{46\cdots 25}{13\cdots 08}a^{5}+\frac{12\cdots 58}{31\cdots 07}a^{4}-\frac{12\cdots 05}{38\cdots 84}a^{3}-\frac{43\cdots 73}{12\cdots 28}a^{2}+\frac{27\cdots 91}{42\cdots 76}a-\frac{10\cdots 29}{95\cdots 21}$, $\frac{10\cdots 93}{69\cdots 40}a^{21}-\frac{36\cdots 21}{69\cdots 40}a^{20}-\frac{14\cdots 92}{31\cdots 07}a^{19}+\frac{53\cdots 77}{23\cdots 80}a^{18}+\frac{14\cdots 61}{34\cdots 70}a^{17}-\frac{26\cdots 69}{34\cdots 70}a^{16}+\frac{96\cdots 83}{34\cdots 70}a^{15}+\frac{50\cdots 75}{46\cdots 36}a^{14}-\frac{48\cdots 57}{23\cdots 80}a^{13}-\frac{23\cdots 91}{34\cdots 70}a^{12}+\frac{51\cdots 53}{46\cdots 36}a^{11}+\frac{37\cdots 15}{23\cdots 18}a^{10}-\frac{29\cdots 05}{13\cdots 08}a^{9}+\frac{68\cdots 91}{69\cdots 54}a^{8}-\frac{84\cdots 74}{34\cdots 77}a^{7}-\frac{34\cdots 31}{34\cdots 77}a^{6}+\frac{68\cdots 65}{11\cdots 59}a^{5}+\frac{11\cdots 54}{10\cdots 69}a^{4}-\frac{36\cdots 53}{63\cdots 14}a^{3}-\frac{54\cdots 63}{31\cdots 07}a^{2}+\frac{34\cdots 20}{31\cdots 07}a-\frac{15\cdots 69}{12\cdots 28}$ 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:  \( 16318467350200 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{5}\cdot 16318467350200 \cdot 1}{2\cdot\sqrt{61117859389674098630012726211547851562500}}\cr\approx \mathstrut & 1.32380675698858 \end{aligned}\] (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 - x^21 - 30*x^20 + 35*x^19 + 265*x^18 - 687*x^17 + 357*x^16 + 7305*x^15 - 18675*x^14 - 35840*x^13 + 103856*x^12 + 58224*x^11 - 211240*x^10 + 163795*x^9 - 64050*x^8 - 651165*x^7 + 823710*x^6 + 466155*x^5 - 852830*x^4 + 161150*x^3 + 139095*x^2 - 61765*x + 6985) 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 - x^21 - 30*x^20 + 35*x^19 + 265*x^18 - 687*x^17 + 357*x^16 + 7305*x^15 - 18675*x^14 - 35840*x^13 + 103856*x^12 + 58224*x^11 - 211240*x^10 + 163795*x^9 - 64050*x^8 - 651165*x^7 + 823710*x^6 + 466155*x^5 - 852830*x^4 + 161150*x^3 + 139095*x^2 - 61765*x + 6985, 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 - x^21 - 30*x^20 + 35*x^19 + 265*x^18 - 687*x^17 + 357*x^16 + 7305*x^15 - 18675*x^14 - 35840*x^13 + 103856*x^12 + 58224*x^11 - 211240*x^10 + 163795*x^9 - 64050*x^8 - 651165*x^7 + 823710*x^6 + 466155*x^5 - 852830*x^4 + 161150*x^3 + 139095*x^2 - 61765*x + 6985); 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 - x^21 - 30*x^20 + 35*x^19 + 265*x^18 - 687*x^17 + 357*x^16 + 7305*x^15 - 18675*x^14 - 35840*x^13 + 103856*x^12 + 58224*x^11 - 211240*x^10 + 163795*x^9 - 64050*x^8 - 651165*x^7 + 823710*x^6 + 466155*x^5 - 852830*x^4 + 161150*x^3 + 139095*x^2 - 61765*x + 6985); 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); G, transitive_group_identification(G)
 
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

11.11.123610132462587890625.1

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: 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 R ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ R ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ $22$ ${\href{/padicField/29.5.0.1}{5} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ $22$ ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.5.0.1}{5} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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$$$[\ ]^{5}$$
2.5.1.0a1.1$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$$$[\ ]^{5}$$
\(3\) Copy content Toggle raw display 3.2.11.20a1.1$x^{22} + 22 x^{21} + 242 x^{20} + 1760 x^{19} + 9460 x^{18} + 39864 x^{17} + 136488 x^{16} + 388608 x^{15} + 934560 x^{14} + 1918400 x^{13} + 3384128 x^{12} + 5149696 x^{11} + 6768256 x^{10} + 7673600 x^{9} + 7476480 x^{8} + 6217728 x^{7} + 4367616 x^{6} + 2551296 x^{5} + 1210880 x^{4} + 450560 x^{3} + 123904 x^{2} + 22528 x + 2051$$11$$2$$20$22T5$$[\ ]_{11}^{10}$$
\(5\) Copy content Toggle raw display 5.1.11.10a1.1$x^{11} + 5$$11$$1$$10$$C_{11}:C_5$$$[\ ]_{11}^{5}$$
5.1.11.10a1.1$x^{11} + 5$$11$$1$$10$$C_{11}:C_5$$$[\ ]_{11}^{5}$$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
11.1.5.4a1.1$x^{5} + 11$$5$$1$$4$$C_5$$$[\ ]_{5}$$
11.1.5.4a1.1$x^{5} + 11$$5$$1$$4$$C_5$$$[\ ]_{5}$$
11.2.5.8a1.2$x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31367 x^{5} + 29970 x^{4} + 14840 x^{3} + 4000 x^{2} + 560 x + 43$$5$$2$$8$$C_{10}$$$[\ ]_{5}^{2}$$

Spectrum of ring of integers

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