Properties

Label 22.22.170...216.1
Degree $22$
Signature $[22, 0]$
Discriminant $1.704\times 10^{47}$
Root discriminant \(140.24\)
Ramified primes $2,43,1277,1297$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.D_{11}$ (as 22T30)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 124*x^20 + 6605*x^18 - 198718*x^16 + 3731509*x^14 - 45556647*x^12 + 364652629*x^10 - 1879790241*x^8 + 5953201774*x^6 - 10615623072*x^4 + 9237457886*x^2 - 3015217921)
 
gp: K = bnfinit(y^22 - 124*y^20 + 6605*y^18 - 198718*y^16 + 3731509*y^14 - 45556647*y^12 + 364652629*y^10 - 1879790241*y^8 + 5953201774*y^6 - 10615623072*y^4 + 9237457886*y^2 - 3015217921, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 124*x^20 + 6605*x^18 - 198718*x^16 + 3731509*x^14 - 45556647*x^12 + 364652629*x^10 - 1879790241*x^8 + 5953201774*x^6 - 10615623072*x^4 + 9237457886*x^2 - 3015217921);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 124*x^20 + 6605*x^18 - 198718*x^16 + 3731509*x^14 - 45556647*x^12 + 364652629*x^10 - 1879790241*x^8 + 5953201774*x^6 - 10615623072*x^4 + 9237457886*x^2 - 3015217921)
 

\( x^{22} - 124 x^{20} + 6605 x^{18} - 198718 x^{16} + 3731509 x^{14} - 45556647 x^{12} + 364652629 x^{10} - 1879790241 x^{8} + 5953201774 x^{6} + \cdots - 3015217921 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[22, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(170364212909807025886845565709641708504367497216\) \(\medspace = 2^{22}\cdot 43^{2}\cdot 1277^{2}\cdot 1297^{10}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(140.24\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(43\), \(1277\), \(1297\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{155}a^{18}+\frac{69}{155}a^{16}-\frac{41}{155}a^{14}+\frac{22}{155}a^{12}+\frac{28}{155}a^{10}-\frac{69}{155}a^{8}+\frac{73}{155}a^{6}-\frac{10}{31}a^{4}+\frac{6}{31}a^{2}+\frac{38}{155}$, $\frac{1}{155}a^{19}+\frac{69}{155}a^{17}-\frac{41}{155}a^{15}+\frac{22}{155}a^{13}+\frac{28}{155}a^{11}-\frac{69}{155}a^{9}+\frac{73}{155}a^{7}-\frac{10}{31}a^{5}+\frac{6}{31}a^{3}+\frac{38}{155}a$, $\frac{1}{50\!\cdots\!15}a^{20}-\frac{72\!\cdots\!23}{50\!\cdots\!15}a^{18}-\frac{65\!\cdots\!94}{50\!\cdots\!15}a^{16}-\frac{20\!\cdots\!51}{50\!\cdots\!15}a^{14}+\frac{28\!\cdots\!69}{50\!\cdots\!15}a^{12}+\frac{20\!\cdots\!53}{10\!\cdots\!83}a^{10}+\frac{17\!\cdots\!11}{50\!\cdots\!15}a^{8}+\frac{55\!\cdots\!59}{50\!\cdots\!15}a^{6}+\frac{49\!\cdots\!69}{10\!\cdots\!83}a^{4}-\frac{83\!\cdots\!24}{29\!\cdots\!55}a^{2}+\frac{43\!\cdots\!89}{92\!\cdots\!65}$, $\frac{1}{50\!\cdots\!15}a^{21}-\frac{72\!\cdots\!23}{50\!\cdots\!15}a^{19}-\frac{65\!\cdots\!94}{50\!\cdots\!15}a^{17}-\frac{20\!\cdots\!51}{50\!\cdots\!15}a^{15}+\frac{28\!\cdots\!69}{50\!\cdots\!15}a^{13}+\frac{20\!\cdots\!53}{10\!\cdots\!83}a^{11}+\frac{17\!\cdots\!11}{50\!\cdots\!15}a^{9}+\frac{55\!\cdots\!59}{50\!\cdots\!15}a^{7}+\frac{49\!\cdots\!69}{10\!\cdots\!83}a^{5}-\frac{83\!\cdots\!24}{29\!\cdots\!55}a^{3}+\frac{43\!\cdots\!89}{92\!\cdots\!65}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{20\!\cdots\!64}{50\!\cdots\!15}a^{20}-\frac{47\!\cdots\!66}{10\!\cdots\!83}a^{18}+\frac{11\!\cdots\!82}{50\!\cdots\!15}a^{16}-\frac{29\!\cdots\!96}{50\!\cdots\!15}a^{14}+\frac{93\!\cdots\!25}{10\!\cdots\!83}a^{12}-\frac{46\!\cdots\!99}{50\!\cdots\!15}a^{10}+\frac{27\!\cdots\!71}{50\!\cdots\!15}a^{8}-\frac{97\!\cdots\!48}{50\!\cdots\!15}a^{6}+\frac{36\!\cdots\!24}{10\!\cdots\!83}a^{4}-\frac{90\!\cdots\!66}{29\!\cdots\!55}a^{2}+\frac{88\!\cdots\!47}{92\!\cdots\!65}$, $\frac{97\!\cdots\!64}{50\!\cdots\!15}a^{20}-\frac{11\!\cdots\!64}{50\!\cdots\!15}a^{18}+\frac{53\!\cdots\!36}{50\!\cdots\!15}a^{16}-\frac{13\!\cdots\!32}{50\!\cdots\!15}a^{14}+\frac{21\!\cdots\!12}{50\!\cdots\!15}a^{12}-\frac{21\!\cdots\!96}{50\!\cdots\!15}a^{10}+\frac{12\!\cdots\!32}{50\!\cdots\!15}a^{8}-\frac{81\!\cdots\!56}{10\!\cdots\!83}a^{6}+\frac{13\!\cdots\!08}{10\!\cdots\!83}a^{4}-\frac{22\!\cdots\!91}{29\!\cdots\!55}a^{2}+\frac{20\!\cdots\!79}{18\!\cdots\!53}$, $\frac{36\!\cdots\!59}{10\!\cdots\!83}a^{20}-\frac{21\!\cdots\!36}{50\!\cdots\!15}a^{18}+\frac{10\!\cdots\!36}{50\!\cdots\!15}a^{16}-\frac{28\!\cdots\!19}{50\!\cdots\!15}a^{14}+\frac{46\!\cdots\!13}{50\!\cdots\!15}a^{12}-\frac{47\!\cdots\!73}{50\!\cdots\!15}a^{10}+\frac{29\!\cdots\!89}{50\!\cdots\!15}a^{8}-\frac{10\!\cdots\!88}{50\!\cdots\!15}a^{6}+\frac{41\!\cdots\!14}{10\!\cdots\!83}a^{4}-\frac{21\!\cdots\!22}{58\!\cdots\!71}a^{2}+\frac{10\!\cdots\!87}{92\!\cdots\!65}$, $\frac{12\!\cdots\!82}{50\!\cdots\!15}a^{20}-\frac{14\!\cdots\!77}{50\!\cdots\!15}a^{18}+\frac{70\!\cdots\!43}{50\!\cdots\!15}a^{16}-\frac{18\!\cdots\!51}{50\!\cdots\!15}a^{14}+\frac{29\!\cdots\!46}{50\!\cdots\!15}a^{12}-\frac{28\!\cdots\!98}{50\!\cdots\!15}a^{10}+\frac{17\!\cdots\!91}{50\!\cdots\!15}a^{8}-\frac{12\!\cdots\!67}{10\!\cdots\!83}a^{6}+\frac{23\!\cdots\!45}{10\!\cdots\!83}a^{4}-\frac{56\!\cdots\!13}{29\!\cdots\!55}a^{2}+\frac{10\!\cdots\!31}{18\!\cdots\!53}$, $\frac{90\!\cdots\!96}{50\!\cdots\!15}a^{20}-\frac{33\!\cdots\!46}{16\!\cdots\!65}a^{18}+\frac{51\!\cdots\!64}{50\!\cdots\!15}a^{16}-\frac{13\!\cdots\!43}{50\!\cdots\!15}a^{14}+\frac{22\!\cdots\!48}{50\!\cdots\!15}a^{12}-\frac{22\!\cdots\!39}{50\!\cdots\!15}a^{10}+\frac{13\!\cdots\!68}{50\!\cdots\!15}a^{8}-\frac{97\!\cdots\!74}{10\!\cdots\!83}a^{6}+\frac{17\!\cdots\!03}{10\!\cdots\!83}a^{4}-\frac{33\!\cdots\!24}{29\!\cdots\!55}a^{2}+\frac{39\!\cdots\!43}{18\!\cdots\!53}$, $\frac{56\!\cdots\!69}{50\!\cdots\!15}a^{20}-\frac{62\!\cdots\!28}{50\!\cdots\!15}a^{18}+\frac{56\!\cdots\!27}{10\!\cdots\!83}a^{16}-\frac{69\!\cdots\!63}{50\!\cdots\!15}a^{14}+\frac{98\!\cdots\!44}{50\!\cdots\!15}a^{12}-\frac{78\!\cdots\!68}{50\!\cdots\!15}a^{10}+\frac{28\!\cdots\!13}{50\!\cdots\!15}a^{8}+\frac{16\!\cdots\!08}{50\!\cdots\!15}a^{6}-\frac{92\!\cdots\!38}{10\!\cdots\!83}a^{4}+\frac{65\!\cdots\!39}{29\!\cdots\!55}a^{2}-\frac{12\!\cdots\!82}{92\!\cdots\!65}$, $\frac{11\!\cdots\!72}{50\!\cdots\!15}a^{20}-\frac{12\!\cdots\!37}{50\!\cdots\!15}a^{18}+\frac{60\!\cdots\!48}{50\!\cdots\!15}a^{16}-\frac{16\!\cdots\!91}{50\!\cdots\!15}a^{14}+\frac{25\!\cdots\!86}{50\!\cdots\!15}a^{12}-\frac{25\!\cdots\!73}{50\!\cdots\!15}a^{10}+\frac{15\!\cdots\!76}{50\!\cdots\!15}a^{8}-\frac{36\!\cdots\!73}{32\!\cdots\!93}a^{6}+\frac{71\!\cdots\!96}{32\!\cdots\!93}a^{4}-\frac{59\!\cdots\!13}{29\!\cdots\!55}a^{2}+\frac{12\!\cdots\!91}{18\!\cdots\!53}$, $\frac{17\!\cdots\!27}{50\!\cdots\!15}a^{20}-\frac{19\!\cdots\!58}{50\!\cdots\!15}a^{18}+\frac{92\!\cdots\!84}{50\!\cdots\!15}a^{16}-\frac{47\!\cdots\!96}{10\!\cdots\!83}a^{14}+\frac{12\!\cdots\!69}{16\!\cdots\!65}a^{12}-\frac{36\!\cdots\!11}{50\!\cdots\!15}a^{10}+\frac{44\!\cdots\!99}{10\!\cdots\!83}a^{8}-\frac{80\!\cdots\!78}{50\!\cdots\!15}a^{6}+\frac{32\!\cdots\!35}{10\!\cdots\!83}a^{4}-\frac{95\!\cdots\!73}{29\!\cdots\!55}a^{2}+\frac{10\!\cdots\!62}{92\!\cdots\!65}$, $\frac{56\!\cdots\!69}{50\!\cdots\!15}a^{20}-\frac{62\!\cdots\!28}{50\!\cdots\!15}a^{18}+\frac{56\!\cdots\!27}{10\!\cdots\!83}a^{16}-\frac{69\!\cdots\!63}{50\!\cdots\!15}a^{14}+\frac{98\!\cdots\!44}{50\!\cdots\!15}a^{12}-\frac{78\!\cdots\!68}{50\!\cdots\!15}a^{10}+\frac{28\!\cdots\!13}{50\!\cdots\!15}a^{8}+\frac{16\!\cdots\!08}{50\!\cdots\!15}a^{6}-\frac{92\!\cdots\!38}{10\!\cdots\!83}a^{4}+\frac{65\!\cdots\!39}{29\!\cdots\!55}a^{2}-\frac{12\!\cdots\!47}{92\!\cdots\!65}$, $\frac{30\!\cdots\!56}{50\!\cdots\!15}a^{20}-\frac{26\!\cdots\!13}{50\!\cdots\!15}a^{18}+\frac{82\!\cdots\!41}{50\!\cdots\!15}a^{16}-\frac{74\!\cdots\!86}{50\!\cdots\!15}a^{14}-\frac{14\!\cdots\!66}{50\!\cdots\!15}a^{12}+\frac{88\!\cdots\!02}{10\!\cdots\!83}a^{10}-\frac{47\!\cdots\!34}{50\!\cdots\!15}a^{8}+\frac{24\!\cdots\!69}{50\!\cdots\!15}a^{6}-\frac{12\!\cdots\!67}{10\!\cdots\!83}a^{4}+\frac{41\!\cdots\!31}{29\!\cdots\!55}a^{2}-\frac{53\!\cdots\!26}{92\!\cdots\!65}$, $\frac{37\!\cdots\!32}{50\!\cdots\!15}a^{20}-\frac{42\!\cdots\!58}{50\!\cdots\!15}a^{18}+\frac{20\!\cdots\!04}{50\!\cdots\!15}a^{16}-\frac{10\!\cdots\!31}{10\!\cdots\!83}a^{14}+\frac{83\!\cdots\!69}{50\!\cdots\!15}a^{12}-\frac{81\!\cdots\!56}{50\!\cdots\!15}a^{10}+\frac{97\!\cdots\!77}{10\!\cdots\!83}a^{8}-\frac{17\!\cdots\!48}{50\!\cdots\!15}a^{6}+\frac{65\!\cdots\!11}{10\!\cdots\!83}a^{4}-\frac{16\!\cdots\!68}{29\!\cdots\!55}a^{2}-a+\frac{17\!\cdots\!37}{92\!\cdots\!65}$, $\frac{34\!\cdots\!56}{50\!\cdots\!15}a^{21}-\frac{18\!\cdots\!14}{10\!\cdots\!83}a^{20}-\frac{43\!\cdots\!89}{50\!\cdots\!15}a^{19}+\frac{10\!\cdots\!72}{50\!\cdots\!15}a^{18}+\frac{23\!\cdots\!37}{50\!\cdots\!15}a^{17}-\frac{48\!\cdots\!02}{50\!\cdots\!15}a^{16}-\frac{13\!\cdots\!12}{10\!\cdots\!83}a^{15}+\frac{12\!\cdots\!93}{50\!\cdots\!15}a^{14}+\frac{11\!\cdots\!87}{50\!\cdots\!15}a^{13}-\frac{18\!\cdots\!96}{50\!\cdots\!15}a^{12}-\frac{12\!\cdots\!83}{50\!\cdots\!15}a^{11}+\frac{17\!\cdots\!36}{50\!\cdots\!15}a^{10}+\frac{14\!\cdots\!75}{10\!\cdots\!83}a^{9}-\frac{10\!\cdots\!58}{50\!\cdots\!15}a^{8}-\frac{22\!\cdots\!44}{50\!\cdots\!15}a^{7}+\frac{36\!\cdots\!81}{50\!\cdots\!15}a^{6}+\frac{36\!\cdots\!08}{10\!\cdots\!83}a^{5}-\frac{13\!\cdots\!62}{10\!\cdots\!83}a^{4}+\frac{57\!\cdots\!66}{94\!\cdots\!05}a^{3}+\frac{72\!\cdots\!64}{58\!\cdots\!71}a^{2}-\frac{58\!\cdots\!74}{92\!\cdots\!65}a-\frac{38\!\cdots\!04}{92\!\cdots\!65}$, $\frac{75\!\cdots\!20}{10\!\cdots\!83}a^{21}-\frac{79\!\cdots\!74}{50\!\cdots\!15}a^{20}-\frac{14\!\cdots\!31}{16\!\cdots\!65}a^{19}+\frac{10\!\cdots\!88}{50\!\cdots\!15}a^{18}+\frac{21\!\cdots\!86}{50\!\cdots\!15}a^{17}-\frac{11\!\cdots\!17}{10\!\cdots\!83}a^{16}-\frac{59\!\cdots\!24}{50\!\cdots\!15}a^{15}+\frac{17\!\cdots\!68}{50\!\cdots\!15}a^{14}+\frac{98\!\cdots\!43}{50\!\cdots\!15}a^{13}-\frac{32\!\cdots\!39}{50\!\cdots\!15}a^{12}-\frac{10\!\cdots\!38}{50\!\cdots\!15}a^{11}+\frac{37\!\cdots\!68}{50\!\cdots\!15}a^{10}+\frac{69\!\cdots\!34}{50\!\cdots\!15}a^{9}-\frac{27\!\cdots\!43}{50\!\cdots\!15}a^{8}-\frac{28\!\cdots\!18}{50\!\cdots\!15}a^{7}+\frac{12\!\cdots\!47}{50\!\cdots\!15}a^{6}+\frac{13\!\cdots\!52}{10\!\cdots\!83}a^{5}-\frac{64\!\cdots\!98}{10\!\cdots\!83}a^{4}-\frac{89\!\cdots\!31}{58\!\cdots\!71}a^{3}+\frac{24\!\cdots\!21}{29\!\cdots\!55}a^{2}+\frac{61\!\cdots\!12}{92\!\cdots\!65}a-\frac{36\!\cdots\!23}{92\!\cdots\!65}$, $\frac{28\!\cdots\!64}{50\!\cdots\!15}a^{21}-\frac{42\!\cdots\!83}{50\!\cdots\!15}a^{20}-\frac{35\!\cdots\!64}{50\!\cdots\!15}a^{19}+\frac{48\!\cdots\!94}{50\!\cdots\!15}a^{18}+\frac{19\!\cdots\!41}{50\!\cdots\!15}a^{17}-\frac{23\!\cdots\!43}{50\!\cdots\!15}a^{16}-\frac{57\!\cdots\!92}{50\!\cdots\!15}a^{15}+\frac{60\!\cdots\!08}{50\!\cdots\!15}a^{14}+\frac{10\!\cdots\!52}{50\!\cdots\!15}a^{13}-\frac{96\!\cdots\!67}{50\!\cdots\!15}a^{12}-\frac{12\!\cdots\!96}{50\!\cdots\!15}a^{11}+\frac{19\!\cdots\!57}{10\!\cdots\!83}a^{10}+\frac{91\!\cdots\!92}{50\!\cdots\!15}a^{9}-\frac{57\!\cdots\!58}{50\!\cdots\!15}a^{8}-\frac{83\!\cdots\!95}{10\!\cdots\!83}a^{7}+\frac{20\!\cdots\!73}{50\!\cdots\!15}a^{6}+\frac{21\!\cdots\!37}{10\!\cdots\!83}a^{5}-\frac{73\!\cdots\!76}{10\!\cdots\!83}a^{4}-\frac{82\!\cdots\!41}{29\!\cdots\!55}a^{3}+\frac{55\!\cdots\!62}{94\!\cdots\!05}a^{2}+\frac{23\!\cdots\!23}{18\!\cdots\!53}a-\frac{15\!\cdots\!77}{92\!\cdots\!65}$, $\frac{13\!\cdots\!38}{12\!\cdots\!77}a^{21}+\frac{45\!\cdots\!58}{10\!\cdots\!83}a^{20}-\frac{16\!\cdots\!49}{12\!\cdots\!77}a^{19}-\frac{52\!\cdots\!45}{10\!\cdots\!83}a^{18}+\frac{79\!\cdots\!34}{12\!\cdots\!77}a^{17}+\frac{25\!\cdots\!83}{10\!\cdots\!83}a^{16}-\frac{21\!\cdots\!37}{12\!\cdots\!77}a^{15}-\frac{68\!\cdots\!44}{10\!\cdots\!83}a^{14}+\frac{36\!\cdots\!88}{12\!\cdots\!77}a^{13}+\frac{11\!\cdots\!49}{10\!\cdots\!83}a^{12}-\frac{37\!\cdots\!79}{12\!\cdots\!77}a^{11}-\frac{11\!\cdots\!08}{10\!\cdots\!83}a^{10}+\frac{24\!\cdots\!35}{12\!\cdots\!77}a^{9}+\frac{71\!\cdots\!57}{10\!\cdots\!83}a^{8}-\frac{91\!\cdots\!02}{12\!\cdots\!77}a^{7}-\frac{26\!\cdots\!64}{10\!\cdots\!83}a^{6}+\frac{19\!\cdots\!36}{12\!\cdots\!77}a^{5}+\frac{53\!\cdots\!68}{10\!\cdots\!83}a^{4}-\frac{10\!\cdots\!15}{74\!\cdots\!49}a^{3}-\frac{29\!\cdots\!59}{58\!\cdots\!71}a^{2}+\frac{12\!\cdots\!72}{23\!\cdots\!07}a+\frac{31\!\cdots\!17}{18\!\cdots\!53}$, $\frac{16\!\cdots\!29}{50\!\cdots\!15}a^{21}-\frac{12\!\cdots\!55}{10\!\cdots\!83}a^{20}-\frac{19\!\cdots\!22}{50\!\cdots\!15}a^{19}+\frac{71\!\cdots\!72}{50\!\cdots\!15}a^{18}+\frac{99\!\cdots\!09}{50\!\cdots\!15}a^{17}-\frac{35\!\cdots\!32}{50\!\cdots\!15}a^{16}-\frac{27\!\cdots\!89}{50\!\cdots\!15}a^{15}+\frac{97\!\cdots\!48}{50\!\cdots\!15}a^{14}+\frac{47\!\cdots\!26}{50\!\cdots\!15}a^{13}-\frac{16\!\cdots\!66}{50\!\cdots\!15}a^{12}-\frac{10\!\cdots\!09}{10\!\cdots\!83}a^{11}+\frac{17\!\cdots\!16}{50\!\cdots\!15}a^{10}+\frac{33\!\cdots\!34}{50\!\cdots\!15}a^{9}-\frac{11\!\cdots\!58}{50\!\cdots\!15}a^{8}-\frac{13\!\cdots\!79}{50\!\cdots\!15}a^{7}+\frac{48\!\cdots\!51}{50\!\cdots\!15}a^{6}+\frac{54\!\cdots\!73}{10\!\cdots\!83}a^{5}-\frac{22\!\cdots\!15}{10\!\cdots\!83}a^{4}-\frac{12\!\cdots\!51}{29\!\cdots\!55}a^{3}+\frac{14\!\cdots\!55}{58\!\cdots\!71}a^{2}+\frac{93\!\cdots\!31}{92\!\cdots\!65}a-\frac{87\!\cdots\!24}{92\!\cdots\!65}$, $\frac{26\!\cdots\!11}{50\!\cdots\!15}a^{21}-\frac{26\!\cdots\!04}{50\!\cdots\!15}a^{20}-\frac{29\!\cdots\!68}{50\!\cdots\!15}a^{19}+\frac{88\!\cdots\!67}{16\!\cdots\!65}a^{18}+\frac{14\!\cdots\!86}{50\!\cdots\!15}a^{17}-\frac{11\!\cdots\!99}{50\!\cdots\!15}a^{16}-\frac{37\!\cdots\!76}{50\!\cdots\!15}a^{15}+\frac{27\!\cdots\!54}{50\!\cdots\!15}a^{14}+\frac{60\!\cdots\!59}{50\!\cdots\!15}a^{13}-\frac{38\!\cdots\!71}{50\!\cdots\!15}a^{12}-\frac{12\!\cdots\!60}{10\!\cdots\!83}a^{11}+\frac{63\!\cdots\!61}{10\!\cdots\!83}a^{10}+\frac{37\!\cdots\!11}{50\!\cdots\!15}a^{9}-\frac{16\!\cdots\!14}{50\!\cdots\!15}a^{8}-\frac{14\!\cdots\!16}{50\!\cdots\!15}a^{7}+\frac{48\!\cdots\!49}{50\!\cdots\!15}a^{6}+\frac{60\!\cdots\!33}{10\!\cdots\!83}a^{5}-\frac{17\!\cdots\!63}{10\!\cdots\!83}a^{4}-\frac{60\!\cdots\!74}{94\!\cdots\!05}a^{3}+\frac{49\!\cdots\!56}{29\!\cdots\!55}a^{2}+\frac{23\!\cdots\!04}{92\!\cdots\!65}a-\frac{59\!\cdots\!16}{92\!\cdots\!65}$, $\frac{35\!\cdots\!53}{50\!\cdots\!15}a^{21}-\frac{92\!\cdots\!68}{50\!\cdots\!15}a^{20}-\frac{41\!\cdots\!77}{50\!\cdots\!15}a^{19}+\frac{10\!\cdots\!01}{50\!\cdots\!15}a^{18}+\frac{20\!\cdots\!41}{50\!\cdots\!15}a^{17}-\frac{10\!\cdots\!68}{10\!\cdots\!83}a^{16}-\frac{11\!\cdots\!56}{10\!\cdots\!83}a^{15}+\frac{14\!\cdots\!11}{50\!\cdots\!15}a^{14}+\frac{93\!\cdots\!91}{50\!\cdots\!15}a^{13}-\frac{22\!\cdots\!08}{50\!\cdots\!15}a^{12}-\frac{96\!\cdots\!44}{50\!\cdots\!15}a^{11}+\frac{23\!\cdots\!11}{50\!\cdots\!15}a^{10}+\frac{12\!\cdots\!13}{10\!\cdots\!83}a^{9}-\frac{14\!\cdots\!16}{50\!\cdots\!15}a^{8}-\frac{24\!\cdots\!12}{50\!\cdots\!15}a^{7}+\frac{52\!\cdots\!89}{50\!\cdots\!15}a^{6}+\frac{10\!\cdots\!07}{10\!\cdots\!83}a^{5}-\frac{20\!\cdots\!67}{10\!\cdots\!83}a^{4}-\frac{33\!\cdots\!47}{29\!\cdots\!55}a^{3}+\frac{49\!\cdots\!47}{29\!\cdots\!55}a^{2}+\frac{42\!\cdots\!68}{92\!\cdots\!65}a-\frac{46\!\cdots\!76}{92\!\cdots\!65}$, $\frac{27\!\cdots\!88}{50\!\cdots\!15}a^{21}-\frac{43\!\cdots\!51}{10\!\cdots\!83}a^{20}-\frac{30\!\cdots\!41}{50\!\cdots\!15}a^{19}+\frac{25\!\cdots\!56}{50\!\cdots\!15}a^{18}+\frac{28\!\cdots\!58}{10\!\cdots\!83}a^{17}-\frac{11\!\cdots\!06}{50\!\cdots\!15}a^{16}-\frac{36\!\cdots\!11}{50\!\cdots\!15}a^{15}+\frac{31\!\cdots\!89}{50\!\cdots\!15}a^{14}+\frac{55\!\cdots\!88}{50\!\cdots\!15}a^{13}-\frac{50\!\cdots\!03}{50\!\cdots\!15}a^{12}-\frac{51\!\cdots\!86}{50\!\cdots\!15}a^{11}+\frac{49\!\cdots\!78}{50\!\cdots\!15}a^{10}+\frac{30\!\cdots\!91}{50\!\cdots\!15}a^{9}-\frac{30\!\cdots\!94}{50\!\cdots\!15}a^{8}-\frac{10\!\cdots\!09}{50\!\cdots\!15}a^{7}+\frac{10\!\cdots\!88}{50\!\cdots\!15}a^{6}+\frac{37\!\cdots\!70}{10\!\cdots\!83}a^{5}-\frac{42\!\cdots\!51}{10\!\cdots\!83}a^{4}-\frac{94\!\cdots\!42}{29\!\cdots\!55}a^{3}+\frac{22\!\cdots\!36}{58\!\cdots\!71}a^{2}+\frac{93\!\cdots\!81}{92\!\cdots\!65}a-\frac{11\!\cdots\!72}{92\!\cdots\!65}$, $\frac{11\!\cdots\!98}{50\!\cdots\!15}a^{21}-\frac{18\!\cdots\!67}{50\!\cdots\!15}a^{20}-\frac{13\!\cdots\!22}{50\!\cdots\!15}a^{19}+\frac{21\!\cdots\!01}{50\!\cdots\!15}a^{18}+\frac{64\!\cdots\!06}{50\!\cdots\!15}a^{17}-\frac{10\!\cdots\!87}{50\!\cdots\!15}a^{16}-\frac{35\!\cdots\!36}{10\!\cdots\!83}a^{15}+\frac{28\!\cdots\!72}{50\!\cdots\!15}a^{14}+\frac{28\!\cdots\!51}{50\!\cdots\!15}a^{13}-\frac{46\!\cdots\!73}{50\!\cdots\!15}a^{12}-\frac{29\!\cdots\!69}{50\!\cdots\!15}a^{11}+\frac{96\!\cdots\!60}{10\!\cdots\!83}a^{10}+\frac{37\!\cdots\!30}{10\!\cdots\!83}a^{9}-\frac{30\!\cdots\!87}{50\!\cdots\!15}a^{8}-\frac{68\!\cdots\!47}{50\!\cdots\!15}a^{7}+\frac{11\!\cdots\!62}{50\!\cdots\!15}a^{6}+\frac{27\!\cdots\!80}{10\!\cdots\!83}a^{5}-\frac{46\!\cdots\!01}{10\!\cdots\!83}a^{4}-\frac{72\!\cdots\!37}{29\!\cdots\!55}a^{3}+\frac{12\!\cdots\!78}{29\!\cdots\!55}a^{2}+\frac{75\!\cdots\!58}{92\!\cdots\!65}a-\frac{13\!\cdots\!33}{92\!\cdots\!65}$, $\frac{25\!\cdots\!17}{29\!\cdots\!55}a^{21}-\frac{12\!\cdots\!86}{10\!\cdots\!83}a^{20}-\frac{96\!\cdots\!86}{94\!\cdots\!05}a^{19}+\frac{70\!\cdots\!74}{50\!\cdots\!15}a^{18}+\frac{14\!\cdots\!42}{29\!\cdots\!55}a^{17}-\frac{33\!\cdots\!59}{50\!\cdots\!15}a^{16}-\frac{38\!\cdots\!77}{29\!\cdots\!55}a^{15}+\frac{88\!\cdots\!71}{50\!\cdots\!15}a^{14}+\frac{62\!\cdots\!18}{29\!\cdots\!55}a^{13}-\frac{13\!\cdots\!62}{50\!\cdots\!15}a^{12}-\frac{12\!\cdots\!89}{58\!\cdots\!71}a^{11}+\frac{13\!\cdots\!47}{50\!\cdots\!15}a^{10}+\frac{38\!\cdots\!27}{29\!\cdots\!55}a^{9}-\frac{83\!\cdots\!06}{50\!\cdots\!15}a^{8}-\frac{14\!\cdots\!27}{29\!\cdots\!55}a^{7}+\frac{30\!\cdots\!32}{50\!\cdots\!15}a^{6}+\frac{56\!\cdots\!99}{58\!\cdots\!71}a^{5}-\frac{11\!\cdots\!53}{10\!\cdots\!83}a^{4}-\frac{26\!\cdots\!59}{29\!\cdots\!55}a^{3}+\frac{65\!\cdots\!65}{58\!\cdots\!71}a^{2}+\frac{16\!\cdots\!48}{53\!\cdots\!05}a-\frac{35\!\cdots\!08}{92\!\cdots\!65}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 41214642130400000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{22}\cdot(2\pi)^{0}\cdot 41214642130400000 \cdot 1}{2\cdot\sqrt{170364212909807025886845565709641708504367497216}}\cr\approx \mathstrut & 0.209407511832245 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - 124*x^20 + 6605*x^18 - 198718*x^16 + 3731509*x^14 - 45556647*x^12 + 364652629*x^10 - 1879790241*x^8 + 5953201774*x^6 - 10615623072*x^4 + 9237457886*x^2 - 3015217921)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^22 - 124*x^20 + 6605*x^18 - 198718*x^16 + 3731509*x^14 - 45556647*x^12 + 364652629*x^10 - 1879790241*x^8 + 5953201774*x^6 - 10615623072*x^4 + 9237457886*x^2 - 3015217921, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^22 - 124*x^20 + 6605*x^18 - 198718*x^16 + 3731509*x^14 - 45556647*x^12 + 364652629*x^10 - 1879790241*x^8 + 5953201774*x^6 - 10615623072*x^4 + 9237457886*x^2 - 3015217921);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 124*x^20 + 6605*x^18 - 198718*x^16 + 3731509*x^14 - 45556647*x^12 + 364652629*x^10 - 1879790241*x^8 + 5953201774*x^6 - 10615623072*x^4 + 9237457886*x^2 - 3015217921);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 22528
The 100 conjugacy class representatives for $C_2^{10}.D_{11}$ are not computed
Character table for $C_2^{10}.D_{11}$ is not computed

Intermediate fields

11.11.3670285774226257.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 22 siblings: data not computed
Degree 44 siblings: data not computed
Minimal sibling: 22.22.220962384144019712575238698725405295930164643889152.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.11.0.1}{11} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{5}{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.11.0.1}{11} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.11.0.1}{11} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.11.0.1}{11} }^{2}$ ${\href{/padicField/23.11.0.1}{11} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{9}$ ${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ R ${\href{/padicField/47.11.0.1}{11} }^{2}$ ${\href{/padicField/53.11.0.1}{11} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 Deg $22$$2$$11$$22$
\(43\) Copy content Toggle raw display 43.2.0.1$x^{2} + 42 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.2.2$x^{4} - 1806 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.0.1$x^{4} + 5 x^{2} + 42 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.0.1$x^{4} + 5 x^{2} + 42 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.0.1$x^{4} + 5 x^{2} + 42 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.0.1$x^{4} + 5 x^{2} + 42 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
\(1277\) 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$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$2$$2$$2$
\(1297\) Copy content Toggle raw display $\Q_{1297}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1297}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$