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Properties

Label 22.14.319...125.1
Degree $22$
Signature $[14, 4]$
Discriminant $3.192\times 10^{31}$
Root discriminant \(27.04\)
Ramified primes $5,43,547,34374601$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\times S_{11}$ (as 22T47)

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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 - 2*x^21 - 7*x^20 + 14*x^19 + 18*x^18 - 67*x^17 - 101*x^16 + 234*x^15 + 497*x^14 - 273*x^13 - 893*x^12 + 23*x^11 + 753*x^10 + 147*x^9 - 418*x^8 - 88*x^7 + 226*x^6 + 6*x^5 - 90*x^4 + 7*x^3 + 16*x^2 - x - 1)
 
Copy content gp:K = bnfinit(y^22 - 2*y^21 - 7*y^20 + 14*y^19 + 18*y^18 - 67*y^17 - 101*y^16 + 234*y^15 + 497*y^14 - 273*y^13 - 893*y^12 + 23*y^11 + 753*y^10 + 147*y^9 - 418*y^8 - 88*y^7 + 226*y^6 + 6*y^5 - 90*y^4 + 7*y^3 + 16*y^2 - y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^21 - 7*x^20 + 14*x^19 + 18*x^18 - 67*x^17 - 101*x^16 + 234*x^15 + 497*x^14 - 273*x^13 - 893*x^12 + 23*x^11 + 753*x^10 + 147*x^9 - 418*x^8 - 88*x^7 + 226*x^6 + 6*x^5 - 90*x^4 + 7*x^3 + 16*x^2 - x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 2*x^21 - 7*x^20 + 14*x^19 + 18*x^18 - 67*x^17 - 101*x^16 + 234*x^15 + 497*x^14 - 273*x^13 - 893*x^12 + 23*x^11 + 753*x^10 + 147*x^9 - 418*x^8 - 88*x^7 + 226*x^6 + 6*x^5 - 90*x^4 + 7*x^3 + 16*x^2 - x - 1)
 

\( x^{22} - 2 x^{21} - 7 x^{20} + 14 x^{19} + 18 x^{18} - 67 x^{17} - 101 x^{16} + 234 x^{15} + 497 x^{14} + \cdots - 1 \) 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:   \(31919563459480622441144580078125\) \(\medspace = 5^{11}\cdot 43^{2}\cdot 547^{2}\cdot 34374601^{2}\) 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:  \(27.04\)
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:  $5^{1/2}43^{1/2}547^{1/2}34374601^{1/2}\approx 2010627.9990602438$
Ramified primes:   \(5\), \(43\), \(547\), \(34374601\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{13\cdots 21}a^{21}+\frac{44\cdots 35}{13\cdots 21}a^{20}+\frac{60\cdots 87}{13\cdots 21}a^{19}-\frac{46\cdots 86}{13\cdots 21}a^{18}+\frac{19\cdots 09}{13\cdots 21}a^{17}-\frac{24\cdots 86}{13\cdots 21}a^{16}+\frac{37\cdots 58}{13\cdots 21}a^{15}-\frac{62\cdots 98}{13\cdots 21}a^{14}-\frac{51\cdots 72}{13\cdots 21}a^{13}-\frac{35\cdots 82}{13\cdots 21}a^{12}-\frac{25\cdots 57}{13\cdots 21}a^{11}+\frac{62\cdots 28}{13\cdots 21}a^{10}+\frac{53\cdots 13}{13\cdots 21}a^{9}-\frac{54\cdots 27}{13\cdots 21}a^{8}+\frac{12\cdots 85}{13\cdots 21}a^{7}-\frac{19\cdots 19}{13\cdots 21}a^{6}+\frac{35\cdots 02}{13\cdots 21}a^{5}-\frac{42\cdots 40}{13\cdots 21}a^{4}+\frac{13\cdots 62}{13\cdots 21}a^{3}-\frac{49\cdots 13}{13\cdots 21}a^{2}-\frac{38\cdots 93}{13\cdots 21}a-\frac{38\cdots 02}{13\cdots 21}$ 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:  $17$
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:   $\frac{66\cdots 55}{13\cdots 21}a^{21}-\frac{27\cdots 63}{13\cdots 21}a^{20}-\frac{71\cdots 58}{13\cdots 21}a^{19}+\frac{38\cdots 64}{13\cdots 21}a^{18}+\frac{26\cdots 17}{13\cdots 21}a^{17}-\frac{36\cdots 15}{13\cdots 21}a^{16}-\frac{12\cdots 25}{13\cdots 21}a^{15}+\frac{82\cdots 74}{13\cdots 21}a^{14}+\frac{55\cdots 98}{13\cdots 21}a^{13}+\frac{17\cdots 36}{13\cdots 21}a^{12}-\frac{84\cdots 03}{13\cdots 21}a^{11}-\frac{47\cdots 19}{13\cdots 21}a^{10}+\frac{55\cdots 62}{13\cdots 21}a^{9}+\frac{40\cdots 15}{13\cdots 21}a^{8}-\frac{26\cdots 50}{13\cdots 21}a^{7}-\frac{23\cdots 13}{13\cdots 21}a^{6}+\frac{17\cdots 98}{13\cdots 21}a^{5}+\frac{83\cdots 74}{13\cdots 21}a^{4}-\frac{81\cdots 16}{13\cdots 21}a^{3}-\frac{10\cdots 75}{13\cdots 21}a^{2}+\frac{14\cdots 27}{13\cdots 21}a+\frac{25\cdots 59}{13\cdots 21}$, $\frac{10\cdots 47}{13\cdots 21}a^{21}-\frac{25\cdots 73}{13\cdots 21}a^{20}-\frac{53\cdots 06}{13\cdots 21}a^{19}+\frac{14\cdots 27}{13\cdots 21}a^{18}+\frac{76\cdots 70}{13\cdots 21}a^{17}-\frac{61\cdots 70}{13\cdots 21}a^{16}-\frac{71\cdots 96}{13\cdots 21}a^{15}+\frac{22\cdots 63}{13\cdots 21}a^{14}+\frac{35\cdots 51}{13\cdots 21}a^{13}-\frac{25\cdots 88}{13\cdots 21}a^{12}-\frac{49\cdots 84}{13\cdots 21}a^{11}+\frac{58\cdots 47}{13\cdots 21}a^{10}+\frac{30\cdots 30}{13\cdots 21}a^{9}+\frac{11\cdots 40}{13\cdots 21}a^{8}-\frac{17\cdots 73}{13\cdots 21}a^{7}+\frac{28\cdots 68}{13\cdots 21}a^{6}+\frac{79\cdots 44}{13\cdots 21}a^{5}-\frac{22\cdots 66}{13\cdots 21}a^{4}-\frac{14\cdots 60}{13\cdots 21}a^{3}+\frac{37\cdots 47}{13\cdots 21}a^{2}+\frac{91\cdots 14}{13\cdots 21}a+\frac{66\cdots 55}{13\cdots 21}$, $\frac{11\cdots 34}{13\cdots 21}a^{21}-\frac{22\cdots 86}{13\cdots 21}a^{20}-\frac{73\cdots 27}{13\cdots 21}a^{19}+\frac{15\cdots 40}{13\cdots 21}a^{18}+\frac{17\cdots 38}{13\cdots 21}a^{17}-\frac{70\cdots 66}{13\cdots 21}a^{16}-\frac{10\cdots 90}{13\cdots 21}a^{15}+\frac{24\cdots 82}{13\cdots 21}a^{14}+\frac{51\cdots 55}{13\cdots 21}a^{13}-\frac{26\cdots 51}{13\cdots 21}a^{12}-\frac{87\cdots 69}{13\cdots 21}a^{11}+\frac{27\cdots 42}{13\cdots 21}a^{10}+\frac{69\cdots 31}{13\cdots 21}a^{9}+\frac{14\cdots 42}{13\cdots 21}a^{8}-\frac{38\cdots 54}{13\cdots 21}a^{7}-\frac{74\cdots 35}{13\cdots 21}a^{6}+\frac{20\cdots 46}{13\cdots 21}a^{5}+\frac{36\cdots 04}{13\cdots 21}a^{4}-\frac{78\cdots 34}{13\cdots 21}a^{3}+\frac{57\cdots 63}{13\cdots 21}a^{2}+\frac{11\cdots 04}{13\cdots 21}a-\frac{79\cdots 88}{13\cdots 21}$, $\frac{28\cdots 31}{13\cdots 21}a^{21}-\frac{42\cdots 60}{13\cdots 21}a^{20}-\frac{21\cdots 90}{13\cdots 21}a^{19}+\frac{26\cdots 94}{13\cdots 21}a^{18}+\frac{64\cdots 12}{13\cdots 21}a^{17}-\frac{14\cdots 90}{13\cdots 21}a^{16}-\frac{37\cdots 57}{13\cdots 21}a^{15}+\frac{44\cdots 90}{13\cdots 21}a^{14}+\frac{16\cdots 30}{13\cdots 21}a^{13}+\frac{23\cdots 82}{13\cdots 21}a^{12}-\frac{24\cdots 82}{13\cdots 21}a^{11}-\frac{16\cdots 96}{13\cdots 21}a^{10}+\frac{14\cdots 99}{13\cdots 21}a^{9}+\frac{16\cdots 10}{13\cdots 21}a^{8}-\frac{36\cdots 62}{13\cdots 21}a^{7}-\frac{76\cdots 93}{13\cdots 21}a^{6}+\frac{21\cdots 04}{13\cdots 21}a^{5}+\frac{33\cdots 30}{13\cdots 21}a^{4}-\frac{11\cdots 06}{13\cdots 21}a^{3}-\frac{13\cdots 42}{13\cdots 21}a^{2}+\frac{16\cdots 02}{13\cdots 21}a+\frac{16\cdots 96}{13\cdots 21}$, $\frac{19\cdots 12}{13\cdots 21}a^{21}-\frac{32\cdots 13}{13\cdots 21}a^{20}-\frac{14\cdots 51}{13\cdots 21}a^{19}+\frac{22\cdots 60}{13\cdots 21}a^{18}+\frac{40\cdots 83}{13\cdots 21}a^{17}-\frac{11\cdots 46}{13\cdots 21}a^{16}-\frac{22\cdots 26}{13\cdots 21}a^{15}+\frac{37\cdots 28}{13\cdots 21}a^{14}+\frac{10\cdots 72}{13\cdots 21}a^{13}-\frac{17\cdots 64}{13\cdots 21}a^{12}-\frac{16\cdots 44}{13\cdots 21}a^{11}-\frac{47\cdots 01}{13\cdots 21}a^{10}+\frac{11\cdots 08}{13\cdots 21}a^{9}+\frac{59\cdots 44}{13\cdots 21}a^{8}-\frac{51\cdots 38}{13\cdots 21}a^{7}-\frac{27\cdots 44}{13\cdots 21}a^{6}+\frac{29\cdots 07}{13\cdots 21}a^{5}+\frac{77\cdots 50}{13\cdots 21}a^{4}-\frac{11\cdots 87}{13\cdots 21}a^{3}-\frac{17\cdots 71}{13\cdots 21}a^{2}+\frac{12\cdots 35}{13\cdots 21}a+\frac{22\cdots 21}{13\cdots 21}$, $\frac{85\cdots 58}{13\cdots 21}a^{21}-\frac{14\cdots 31}{13\cdots 21}a^{20}-\frac{65\cdots 19}{13\cdots 21}a^{19}+\frac{10\cdots 28}{13\cdots 21}a^{18}+\frac{18\cdots 07}{13\cdots 21}a^{17}-\frac{52\cdots 21}{13\cdots 21}a^{16}-\frac{10\cdots 79}{13\cdots 21}a^{15}+\frac{16\cdots 82}{13\cdots 21}a^{14}+\frac{47\cdots 79}{13\cdots 21}a^{13}-\frac{89\cdots 15}{13\cdots 21}a^{12}-\frac{78\cdots 56}{13\cdots 21}a^{11}-\frac{20\cdots 77}{13\cdots 21}a^{10}+\frac{55\cdots 34}{13\cdots 21}a^{9}+\frac{26\cdots 11}{13\cdots 21}a^{8}-\frac{25\cdots 64}{13\cdots 21}a^{7}-\frac{12\cdots 69}{13\cdots 21}a^{6}+\frac{14\cdots 70}{13\cdots 21}a^{5}+\frac{33\cdots 72}{13\cdots 21}a^{4}-\frac{57\cdots 20}{13\cdots 21}a^{3}-\frac{62\cdots 64}{13\cdots 21}a^{2}+\frac{62\cdots 97}{13\cdots 21}a+\frac{10\cdots 11}{13\cdots 21}$, $\frac{11\cdots 06}{13\cdots 21}a^{21}-\frac{59\cdots 40}{13\cdots 21}a^{20}-\frac{71\cdots 65}{13\cdots 21}a^{19}+\frac{41\cdots 44}{13\cdots 21}a^{18}-\frac{28\cdots 76}{13\cdots 21}a^{17}-\frac{14\cdots 53}{13\cdots 21}a^{16}+\frac{12\cdots 66}{13\cdots 21}a^{15}+\frac{64\cdots 38}{13\cdots 21}a^{14}-\frac{24\cdots 68}{13\cdots 21}a^{13}-\frac{21\cdots 74}{13\cdots 21}a^{12}-\frac{24\cdots 19}{13\cdots 21}a^{11}+\frac{30\cdots 56}{13\cdots 21}a^{10}+\frac{10\cdots 56}{13\cdots 21}a^{9}-\frac{20\cdots 29}{13\cdots 21}a^{8}-\frac{10\cdots 96}{13\cdots 21}a^{7}+\frac{10\cdots 18}{13\cdots 21}a^{6}+\frac{51\cdots 13}{13\cdots 21}a^{5}-\frac{63\cdots 54}{13\cdots 21}a^{4}-\frac{10\cdots 05}{13\cdots 21}a^{3}+\frac{23\cdots 38}{13\cdots 21}a^{2}-\frac{18\cdots 22}{13\cdots 21}a-\frac{22\cdots 82}{13\cdots 21}$, $\frac{33\cdots 49}{13\cdots 21}a^{21}-\frac{41\cdots 03}{13\cdots 21}a^{20}-\frac{35\cdots 00}{13\cdots 21}a^{19}+\frac{39\cdots 61}{13\cdots 21}a^{18}+\frac{14\cdots 85}{13\cdots 21}a^{17}-\frac{11\cdots 29}{13\cdots 21}a^{16}-\frac{73\cdots 99}{13\cdots 21}a^{15}+\frac{14\cdots 06}{13\cdots 21}a^{14}+\frac{30\cdots 22}{13\cdots 21}a^{13}+\frac{21\cdots 94}{13\cdots 21}a^{12}-\frac{42\cdots 77}{13\cdots 21}a^{11}-\frac{51\cdots 27}{13\cdots 21}a^{10}+\frac{19\cdots 26}{13\cdots 21}a^{9}+\frac{44\cdots 77}{13\cdots 21}a^{8}-\frac{78\cdots 02}{13\cdots 21}a^{7}-\frac{21\cdots 90}{13\cdots 21}a^{6}+\frac{15\cdots 36}{13\cdots 21}a^{5}+\frac{10\cdots 06}{13\cdots 21}a^{4}-\frac{20\cdots 58}{13\cdots 21}a^{3}-\frac{37\cdots 07}{13\cdots 21}a^{2}+\frac{41\cdots 39}{13\cdots 21}a+\frac{41\cdots 76}{13\cdots 21}$, $\frac{14\cdots 28}{13\cdots 21}a^{21}-\frac{19\cdots 84}{13\cdots 21}a^{20}-\frac{12\cdots 59}{13\cdots 21}a^{19}+\frac{13\cdots 87}{13\cdots 21}a^{18}+\frac{38\cdots 56}{13\cdots 21}a^{17}-\frac{81\cdots 49}{13\cdots 21}a^{16}-\frac{20\cdots 48}{13\cdots 21}a^{15}+\frac{23\cdots 76}{13\cdots 21}a^{14}+\frac{92\cdots 82}{13\cdots 21}a^{13}+\frac{97\cdots 42}{13\cdots 21}a^{12}-\frac{14\cdots 85}{13\cdots 21}a^{11}-\frac{75\cdots 76}{13\cdots 21}a^{10}+\frac{89\cdots 77}{13\cdots 21}a^{9}+\frac{74\cdots 67}{13\cdots 21}a^{8}-\frac{33\cdots 54}{13\cdots 21}a^{7}-\frac{36\cdots 56}{13\cdots 21}a^{6}+\frac{20\cdots 15}{13\cdots 21}a^{5}+\frac{13\cdots 62}{13\cdots 21}a^{4}-\frac{95\cdots 13}{13\cdots 21}a^{3}-\frac{40\cdots 42}{13\cdots 21}a^{2}+\frac{12\cdots 12}{13\cdots 21}a+\frac{44\cdots 25}{13\cdots 21}$, $\frac{23\cdots 64}{13\cdots 21}a^{21}-\frac{39\cdots 60}{13\cdots 21}a^{20}-\frac{17\cdots 35}{13\cdots 21}a^{19}+\frac{26\cdots 07}{13\cdots 21}a^{18}+\frac{49\cdots 30}{13\cdots 21}a^{17}-\frac{13\cdots 42}{13\cdots 21}a^{16}-\frac{28\cdots 25}{13\cdots 21}a^{15}+\frac{44\cdots 14}{13\cdots 21}a^{14}+\frac{12\cdots 27}{13\cdots 21}a^{13}-\frac{14\cdots 23}{13\cdots 21}a^{12}-\frac{20\cdots 58}{13\cdots 21}a^{11}-\frac{70\cdots 07}{13\cdots 21}a^{10}+\frac{13\cdots 17}{13\cdots 21}a^{9}+\frac{81\cdots 05}{13\cdots 21}a^{8}-\frac{58\cdots 54}{13\cdots 21}a^{7}-\frac{38\cdots 39}{13\cdots 21}a^{6}+\frac{33\cdots 66}{13\cdots 21}a^{5}+\frac{12\cdots 89}{13\cdots 21}a^{4}-\frac{13\cdots 52}{13\cdots 21}a^{3}-\frac{33\cdots 14}{13\cdots 21}a^{2}+\frac{16\cdots 30}{13\cdots 21}a+\frac{37\cdots 48}{13\cdots 21}$, $\frac{12\cdots 19}{13\cdots 21}a^{21}-\frac{19\cdots 24}{13\cdots 21}a^{20}-\frac{96\cdots 99}{13\cdots 21}a^{19}+\frac{12\cdots 99}{13\cdots 21}a^{18}+\frac{28\cdots 60}{13\cdots 21}a^{17}-\frac{70\cdots 94}{13\cdots 21}a^{16}-\frac{15\cdots 03}{13\cdots 21}a^{15}+\frac{21\cdots 41}{13\cdots 21}a^{14}+\frac{72\cdots 49}{13\cdots 21}a^{13}-\frac{10\cdots 21}{13\cdots 21}a^{12}-\frac{11\cdots 93}{13\cdots 21}a^{11}-\frac{48\cdots 26}{13\cdots 21}a^{10}+\frac{73\cdots 63}{13\cdots 21}a^{9}+\frac{52\cdots 89}{13\cdots 21}a^{8}-\frac{29\cdots 59}{13\cdots 21}a^{7}-\frac{25\cdots 51}{13\cdots 21}a^{6}+\frac{17\cdots 90}{13\cdots 21}a^{5}+\frac{89\cdots 44}{13\cdots 21}a^{4}-\frac{75\cdots 38}{13\cdots 21}a^{3}-\frac{25\cdots 81}{13\cdots 21}a^{2}+\frac{93\cdots 97}{13\cdots 21}a+\frac{30\cdots 31}{13\cdots 21}$, $\frac{59\cdots 76}{13\cdots 21}a^{21}-\frac{67\cdots 02}{13\cdots 21}a^{20}-\frac{49\cdots 87}{13\cdots 21}a^{19}+\frac{43\cdots 97}{13\cdots 21}a^{18}+\frac{16\cdots 65}{13\cdots 21}a^{17}-\frac{27\cdots 20}{13\cdots 21}a^{16}-\frac{90\cdots 83}{13\cdots 21}a^{15}+\frac{74\cdots 85}{13\cdots 21}a^{14}+\frac{39\cdots 93}{13\cdots 21}a^{13}+\frac{13\cdots 27}{13\cdots 21}a^{12}-\frac{56\cdots 32}{13\cdots 21}a^{11}-\frac{46\cdots 90}{13\cdots 21}a^{10}+\frac{29\cdots 94}{13\cdots 21}a^{9}+\frac{42\cdots 26}{13\cdots 21}a^{8}-\frac{70\cdots 87}{13\cdots 21}a^{7}-\frac{20\cdots 34}{13\cdots 21}a^{6}+\frac{47\cdots 95}{13\cdots 21}a^{5}+\frac{91\cdots 72}{13\cdots 21}a^{4}-\frac{28\cdots 70}{13\cdots 21}a^{3}-\frac{31\cdots 26}{13\cdots 21}a^{2}+\frac{48\cdots 11}{13\cdots 21}a+\frac{35\cdots 63}{13\cdots 21}$, $\frac{10\cdots 99}{13\cdots 21}a^{21}-\frac{16\cdots 95}{13\cdots 21}a^{20}-\frac{82\cdots 85}{13\cdots 21}a^{19}+\frac{11\cdots 37}{13\cdots 21}a^{18}+\frac{24\cdots 53}{13\cdots 21}a^{17}-\frac{61\cdots 87}{13\cdots 21}a^{16}-\frac{13\cdots 35}{13\cdots 21}a^{15}+\frac{19\cdots 94}{13\cdots 21}a^{14}+\frac{61\cdots 50}{13\cdots 21}a^{13}-\frac{33\cdots 40}{13\cdots 21}a^{12}-\frac{95\cdots 26}{13\cdots 21}a^{11}-\frac{37\cdots 68}{13\cdots 21}a^{10}+\frac{63\cdots 64}{13\cdots 21}a^{9}+\frac{40\cdots 91}{13\cdots 21}a^{8}-\frac{26\cdots 24}{13\cdots 21}a^{7}-\frac{19\cdots 07}{13\cdots 21}a^{6}+\frac{15\cdots 10}{13\cdots 21}a^{5}+\frac{65\cdots 52}{13\cdots 21}a^{4}-\frac{64\cdots 53}{13\cdots 21}a^{3}-\frac{17\cdots 42}{13\cdots 21}a^{2}+\frac{73\cdots 75}{13\cdots 21}a+\frac{18\cdots 52}{13\cdots 21}$, $\frac{12\cdots 85}{13\cdots 21}a^{21}-\frac{24\cdots 21}{13\cdots 21}a^{20}-\frac{91\cdots 66}{13\cdots 21}a^{19}+\frac{16\cdots 64}{13\cdots 21}a^{18}+\frac{24\cdots 36}{13\cdots 21}a^{17}-\frac{80\cdots 08}{13\cdots 21}a^{16}-\frac{13\cdots 36}{13\cdots 21}a^{15}+\frac{27\cdots 92}{13\cdots 21}a^{14}+\frac{65\cdots 58}{13\cdots 21}a^{13}-\frac{22\cdots 84}{13\cdots 21}a^{12}-\frac{10\cdots 90}{13\cdots 21}a^{11}-\frac{15\cdots 83}{13\cdots 21}a^{10}+\frac{80\cdots 50}{13\cdots 21}a^{9}+\frac{28\cdots 48}{13\cdots 21}a^{8}-\frac{38\cdots 37}{13\cdots 21}a^{7}-\frac{13\cdots 85}{13\cdots 21}a^{6}+\frac{21\cdots 86}{13\cdots 21}a^{5}+\frac{23\cdots 19}{13\cdots 21}a^{4}-\frac{82\cdots 24}{13\cdots 21}a^{3}-\frac{10\cdots 58}{13\cdots 21}a^{2}+\frac{93\cdots 29}{13\cdots 21}a+\frac{48\cdots 24}{13\cdots 21}$, $\frac{78\cdots 03}{13\cdots 21}a^{21}-\frac{32\cdots 32}{13\cdots 21}a^{20}-\frac{30\cdots 82}{13\cdots 21}a^{19}+\frac{16\cdots 16}{13\cdots 21}a^{18}-\frac{15\cdots 90}{13\cdots 21}a^{17}-\frac{47\cdots 64}{13\cdots 21}a^{16}+\frac{24\cdots 63}{13\cdots 21}a^{15}+\frac{22\cdots 00}{13\cdots 21}a^{14}-\frac{36\cdots 93}{13\cdots 21}a^{13}-\frac{51\cdots 05}{13\cdots 21}a^{12}+\frac{72\cdots 15}{13\cdots 21}a^{11}+\frac{47\cdots 18}{13\cdots 21}a^{10}-\frac{29\cdots 95}{13\cdots 21}a^{9}-\frac{24\cdots 39}{13\cdots 21}a^{8}+\frac{23\cdots 36}{13\cdots 21}a^{7}+\frac{18\cdots 56}{13\cdots 21}a^{6}-\frac{69\cdots 58}{13\cdots 21}a^{5}-\frac{68\cdots 72}{13\cdots 21}a^{4}+\frac{53\cdots 14}{13\cdots 21}a^{3}-\frac{60\cdots 19}{13\cdots 21}a^{2}-\frac{98\cdots 26}{13\cdots 21}a+\frac{31\cdots 85}{13\cdots 21}$, $\frac{60\cdots 68}{13\cdots 21}a^{21}-\frac{55\cdots 50}{13\cdots 21}a^{20}-\frac{51\cdots 37}{13\cdots 21}a^{19}+\frac{36\cdots 51}{13\cdots 21}a^{18}+\frac{16\cdots 49}{13\cdots 21}a^{17}-\frac{27\cdots 19}{13\cdots 21}a^{16}-\frac{93\cdots 31}{13\cdots 21}a^{15}+\frac{59\cdots 55}{13\cdots 21}a^{14}+\frac{39\cdots 94}{13\cdots 21}a^{13}+\frac{19\cdots 99}{13\cdots 21}a^{12}-\frac{45\cdots 18}{13\cdots 21}a^{11}-\frac{42\cdots 14}{13\cdots 21}a^{10}+\frac{16\cdots 61}{13\cdots 21}a^{9}+\frac{27\cdots 13}{13\cdots 21}a^{8}-\frac{52\cdots 69}{13\cdots 21}a^{7}-\frac{12\cdots 10}{13\cdots 21}a^{6}+\frac{50\cdots 96}{13\cdots 21}a^{5}+\frac{57\cdots 04}{13\cdots 21}a^{4}-\frac{18\cdots 42}{13\cdots 21}a^{3}-\frac{12\cdots 91}{13\cdots 21}a^{2}+\frac{39\cdots 37}{13\cdots 21}a+\frac{30\cdots 23}{13\cdots 21}$, $\frac{18\cdots 65}{13\cdots 21}a^{21}-\frac{30\cdots 40}{13\cdots 21}a^{20}-\frac{13\cdots 45}{13\cdots 21}a^{19}+\frac{21\cdots 33}{13\cdots 21}a^{18}+\frac{39\cdots 13}{13\cdots 21}a^{17}-\frac{10\cdots 76}{13\cdots 21}a^{16}-\frac{21\cdots 30}{13\cdots 21}a^{15}+\frac{35\cdots 65}{13\cdots 21}a^{14}+\frac{10\cdots 21}{13\cdots 21}a^{13}-\frac{15\cdots 76}{13\cdots 21}a^{12}-\frac{16\cdots 60}{13\cdots 21}a^{11}-\frac{48\cdots 48}{13\cdots 21}a^{10}+\frac{11\cdots 78}{13\cdots 21}a^{9}+\frac{59\cdots 04}{13\cdots 21}a^{8}-\frac{49\cdots 65}{13\cdots 21}a^{7}-\frac{27\cdots 12}{13\cdots 21}a^{6}+\frac{28\cdots 63}{13\cdots 21}a^{5}+\frac{80\cdots 16}{13\cdots 21}a^{4}-\frac{11\cdots 27}{13\cdots 21}a^{3}-\frac{17\cdots 18}{13\cdots 21}a^{2}+\frac{12\cdots 21}{13\cdots 21}a+\frac{20\cdots 45}{13\cdots 21}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 73525321.2665 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 73525321.2665 \cdot 1}{2\cdot\sqrt{31919563459480622441144580078125}}\cr\approx \mathstrut & 0.166156706924 \end{aligned}\] (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 - 2*x^21 - 7*x^20 + 14*x^19 + 18*x^18 - 67*x^17 - 101*x^16 + 234*x^15 + 497*x^14 - 273*x^13 - 893*x^12 + 23*x^11 + 753*x^10 + 147*x^9 - 418*x^8 - 88*x^7 + 226*x^6 + 6*x^5 - 90*x^4 + 7*x^3 + 16*x^2 - x - 1) 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 - 2*x^21 - 7*x^20 + 14*x^19 + 18*x^18 - 67*x^17 - 101*x^16 + 234*x^15 + 497*x^14 - 273*x^13 - 893*x^12 + 23*x^11 + 753*x^10 + 147*x^9 - 418*x^8 - 88*x^7 + 226*x^6 + 6*x^5 - 90*x^4 + 7*x^3 + 16*x^2 - x - 1, 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 - 2*x^21 - 7*x^20 + 14*x^19 + 18*x^18 - 67*x^17 - 101*x^16 + 234*x^15 + 497*x^14 - 273*x^13 - 893*x^12 + 23*x^11 + 753*x^10 + 147*x^9 - 418*x^8 - 88*x^7 + 226*x^6 + 6*x^5 - 90*x^4 + 7*x^3 + 16*x^2 - x - 1); 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 - 2*x^21 - 7*x^20 + 14*x^19 + 18*x^18 - 67*x^17 - 101*x^16 + 234*x^15 + 497*x^14 - 273*x^13 - 893*x^12 + 23*x^11 + 753*x^10 + 147*x^9 - 418*x^8 - 88*x^7 + 226*x^6 + 6*x^5 - 90*x^4 + 7*x^3 + 16*x^2 - x - 1); 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

$C_2\times S_{11}$ (as 22T47):

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 non-solvable group of order 79833600
The 112 conjugacy class representatives for $C_2\times S_{11}$
Character table for $C_2\times S_{11}$

Intermediate fields

\(\Q(\sqrt{5}) \), 11.7.808524990121.1

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 sibling: 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 $22$ $22$ R $22$ ${\href{/padicField/11.9.0.1}{9} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.11.0.1}{11} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ $22$ ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ R ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$

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
\(5\) Copy content Toggle raw display 5.11.2.11a1.2$x^{22} + 6 x^{12} + 6 x^{11} + 9 x^{2} + 18 x + 14$$2$$11$$11$22T1$$[\ ]_{2}^{11}$$
\(43\) Copy content Toggle raw display 43.2.1.0a1.1$x^{2} + 42 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
43.2.1.0a1.1$x^{2} + 42 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
43.2.2.2a1.2$x^{4} + 84 x^{3} + 1770 x^{2} + 252 x + 52$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
43.14.1.0a1.1$x^{14} + 38 x^{7} + 22 x^{6} + 24 x^{5} + 37 x^{4} + 18 x^{3} + 4 x^{2} + 19 x + 3$$1$$14$$0$$C_{14}$$$[\ ]^{14}$$
\(547\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $4$$2$$2$$2$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(34374601\) Copy content Toggle raw display $\Q_{34374601}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{34374601}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
Deg $5$$1$$5$$0$$C_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)