Properties

Label 22.10.412...904.6
Degree $22$
Signature $[10, 6]$
Discriminant $4.129\times 10^{45}$
Root discriminant \(118.43\)
Ramified primes $2,74843$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.\PSL(2,11)$ (as 22T39)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^20 - 439*x^18 - 1549*x^16 + 5292*x^14 + 11519*x^12 - 25067*x^10 - 2315*x^8 + 10967*x^6 + 2679*x^4 - 156*x^2 - 49)
 
gp: K = bnfinit(y^22 - 4*y^20 - 439*y^18 - 1549*y^16 + 5292*y^14 + 11519*y^12 - 25067*y^10 - 2315*y^8 + 10967*y^6 + 2679*y^4 - 156*y^2 - 49, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 4*x^20 - 439*x^18 - 1549*x^16 + 5292*x^14 + 11519*x^12 - 25067*x^10 - 2315*x^8 + 10967*x^6 + 2679*x^4 - 156*x^2 - 49);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 4*x^20 - 439*x^18 - 1549*x^16 + 5292*x^14 + 11519*x^12 - 25067*x^10 - 2315*x^8 + 10967*x^6 + 2679*x^4 - 156*x^2 - 49)
 

\( x^{22} - 4 x^{20} - 439 x^{18} - 1549 x^{16} + 5292 x^{14} + 11519 x^{12} - 25067 x^{10} - 2315 x^{8} + 10967 x^{6} + 2679 x^{4} - 156 x^{2} - 49 \) 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:  $[10, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(4129233136056857981979443884256982828952059904\) \(\medspace = 2^{22}\cdot 74843^{8}\) 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:  \(118.43\)
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:   \(2\), \(74843\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{9}a^{16}+\frac{2}{9}a^{14}+\frac{2}{9}a^{12}+\frac{4}{9}a^{10}-\frac{2}{9}a^{8}-\frac{4}{9}$, $\frac{1}{9}a^{17}+\frac{2}{9}a^{15}+\frac{2}{9}a^{13}+\frac{4}{9}a^{11}-\frac{2}{9}a^{9}-\frac{4}{9}a$, $\frac{1}{81}a^{18}+\frac{1}{27}a^{16}-\frac{5}{81}a^{14}-\frac{7}{27}a^{12}+\frac{2}{81}a^{10}+\frac{25}{81}a^{8}-\frac{1}{9}a^{6}+\frac{1}{9}a^{4}+\frac{23}{81}a^{2}-\frac{4}{81}$, $\frac{1}{81}a^{19}+\frac{1}{27}a^{17}-\frac{5}{81}a^{15}-\frac{7}{27}a^{13}+\frac{2}{81}a^{11}+\frac{25}{81}a^{9}-\frac{1}{9}a^{7}+\frac{1}{9}a^{5}+\frac{23}{81}a^{3}-\frac{4}{81}a$, $\frac{1}{84\!\cdots\!99}a^{20}+\frac{16\!\cdots\!60}{84\!\cdots\!99}a^{18}-\frac{15\!\cdots\!20}{84\!\cdots\!99}a^{16}+\frac{23\!\cdots\!84}{84\!\cdots\!99}a^{14}+\frac{33\!\cdots\!24}{84\!\cdots\!99}a^{12}+\frac{21\!\cdots\!76}{94\!\cdots\!11}a^{10}+\frac{12\!\cdots\!71}{84\!\cdots\!99}a^{8}+\frac{15\!\cdots\!46}{34\!\cdots\!93}a^{6}+\frac{27\!\cdots\!23}{84\!\cdots\!99}a^{4}+\frac{14\!\cdots\!71}{84\!\cdots\!99}a^{2}-\frac{18\!\cdots\!58}{84\!\cdots\!99}$, $\frac{1}{59\!\cdots\!93}a^{21}+\frac{33\!\cdots\!97}{59\!\cdots\!93}a^{19}-\frac{15\!\cdots\!20}{59\!\cdots\!93}a^{17}+\frac{73\!\cdots\!76}{59\!\cdots\!93}a^{15}+\frac{48\!\cdots\!32}{84\!\cdots\!99}a^{13}+\frac{17\!\cdots\!68}{65\!\cdots\!77}a^{11}+\frac{35\!\cdots\!17}{84\!\cdots\!99}a^{9}+\frac{24\!\cdots\!67}{81\!\cdots\!17}a^{7}+\frac{31\!\cdots\!56}{59\!\cdots\!93}a^{5}+\frac{23\!\cdots\!23}{59\!\cdots\!93}a^{3}+\frac{27\!\cdots\!35}{59\!\cdots\!93}a$ 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:  $15$
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:   $\frac{25\!\cdots\!67}{84\!\cdots\!99}a^{20}-\frac{10\!\cdots\!59}{84\!\cdots\!99}a^{18}-\frac{11\!\cdots\!01}{84\!\cdots\!99}a^{16}-\frac{38\!\cdots\!07}{84\!\cdots\!99}a^{14}+\frac{14\!\cdots\!35}{84\!\cdots\!99}a^{12}+\frac{30\!\cdots\!54}{94\!\cdots\!11}a^{10}-\frac{68\!\cdots\!47}{84\!\cdots\!99}a^{8}+\frac{77\!\cdots\!11}{10\!\cdots\!79}a^{6}+\frac{24\!\cdots\!36}{84\!\cdots\!99}a^{4}+\frac{25\!\cdots\!38}{84\!\cdots\!99}a^{2}-\frac{14\!\cdots\!13}{84\!\cdots\!99}$, $\frac{93\!\cdots\!91}{28\!\cdots\!33}a^{20}-\frac{43\!\cdots\!28}{28\!\cdots\!33}a^{18}-\frac{40\!\cdots\!82}{28\!\cdots\!33}a^{16}-\frac{11\!\cdots\!60}{28\!\cdots\!33}a^{14}+\frac{59\!\cdots\!77}{28\!\cdots\!33}a^{12}+\frac{29\!\cdots\!72}{10\!\cdots\!79}a^{10}-\frac{30\!\cdots\!65}{28\!\cdots\!33}a^{8}+\frac{13\!\cdots\!34}{34\!\cdots\!93}a^{6}+\frac{93\!\cdots\!01}{28\!\cdots\!33}a^{4}-\frac{25\!\cdots\!39}{28\!\cdots\!33}a^{2}-\frac{36\!\cdots\!24}{28\!\cdots\!33}$, $\frac{38\!\cdots\!80}{28\!\cdots\!33}a^{20}-\frac{15\!\cdots\!97}{28\!\cdots\!33}a^{18}-\frac{16\!\cdots\!49}{28\!\cdots\!33}a^{16}-\frac{57\!\cdots\!18}{28\!\cdots\!33}a^{14}+\frac{20\!\cdots\!35}{28\!\cdots\!33}a^{12}+\frac{13\!\cdots\!09}{94\!\cdots\!11}a^{10}-\frac{10\!\cdots\!15}{28\!\cdots\!33}a^{8}+\frac{23\!\cdots\!20}{10\!\cdots\!79}a^{6}+\frac{39\!\cdots\!80}{28\!\cdots\!33}a^{4}+\frac{38\!\cdots\!69}{28\!\cdots\!33}a^{2}-\frac{10\!\cdots\!73}{28\!\cdots\!33}$, $\frac{12\!\cdots\!62}{59\!\cdots\!93}a^{21}-\frac{52\!\cdots\!50}{59\!\cdots\!93}a^{19}-\frac{56\!\cdots\!75}{59\!\cdots\!93}a^{17}-\frac{19\!\cdots\!30}{59\!\cdots\!93}a^{15}+\frac{10\!\cdots\!61}{84\!\cdots\!99}a^{13}+\frac{51\!\cdots\!33}{21\!\cdots\!59}a^{11}-\frac{48\!\cdots\!45}{84\!\cdots\!99}a^{9}+\frac{12\!\cdots\!32}{73\!\cdots\!53}a^{7}+\frac{14\!\cdots\!57}{59\!\cdots\!93}a^{5}+\frac{14\!\cdots\!91}{59\!\cdots\!93}a^{3}-\frac{44\!\cdots\!02}{59\!\cdots\!93}a$, $\frac{64\!\cdots\!27}{59\!\cdots\!93}a^{21}-\frac{27\!\cdots\!07}{59\!\cdots\!93}a^{19}-\frac{28\!\cdots\!89}{59\!\cdots\!93}a^{17}-\frac{91\!\cdots\!69}{59\!\cdots\!93}a^{15}+\frac{52\!\cdots\!15}{84\!\cdots\!99}a^{13}+\frac{70\!\cdots\!85}{65\!\cdots\!77}a^{11}-\frac{25\!\cdots\!37}{84\!\cdots\!99}a^{9}+\frac{48\!\cdots\!04}{81\!\cdots\!17}a^{7}+\frac{59\!\cdots\!37}{59\!\cdots\!93}a^{5}+\frac{51\!\cdots\!93}{59\!\cdots\!93}a^{3}-\frac{11\!\cdots\!00}{59\!\cdots\!93}a$, $\frac{25\!\cdots\!32}{59\!\cdots\!93}a^{21}-\frac{23\!\cdots\!03}{59\!\cdots\!93}a^{19}-\frac{10\!\cdots\!54}{59\!\cdots\!93}a^{17}+\frac{19\!\cdots\!68}{59\!\cdots\!93}a^{15}+\frac{52\!\cdots\!55}{84\!\cdots\!99}a^{13}-\frac{34\!\cdots\!90}{65\!\cdots\!77}a^{11}-\frac{31\!\cdots\!49}{84\!\cdots\!99}a^{9}+\frac{34\!\cdots\!07}{81\!\cdots\!17}a^{7}-\frac{80\!\cdots\!90}{59\!\cdots\!93}a^{5}-\frac{21\!\cdots\!81}{59\!\cdots\!93}a^{3}+\frac{16\!\cdots\!46}{59\!\cdots\!93}a$, $\frac{34\!\cdots\!79}{65\!\cdots\!77}a^{21}-\frac{56\!\cdots\!87}{24\!\cdots\!51}a^{19}-\frac{15\!\cdots\!31}{65\!\cdots\!77}a^{17}-\frac{16\!\cdots\!49}{21\!\cdots\!59}a^{15}+\frac{28\!\cdots\!89}{94\!\cdots\!11}a^{13}+\frac{33\!\cdots\!08}{65\!\cdots\!77}a^{11}-\frac{47\!\cdots\!92}{31\!\cdots\!37}a^{9}+\frac{28\!\cdots\!67}{73\!\cdots\!53}a^{7}+\frac{31\!\cdots\!73}{65\!\cdots\!77}a^{5}-\frac{17\!\cdots\!39}{65\!\cdots\!77}a^{3}-\frac{13\!\cdots\!97}{21\!\cdots\!59}a$, $\frac{73\!\cdots\!63}{84\!\cdots\!99}a^{20}-\frac{30\!\cdots\!82}{84\!\cdots\!99}a^{18}-\frac{32\!\cdots\!34}{84\!\cdots\!99}a^{16}-\frac{10\!\cdots\!76}{84\!\cdots\!99}a^{14}+\frac{40\!\cdots\!95}{84\!\cdots\!99}a^{12}+\frac{32\!\cdots\!24}{34\!\cdots\!93}a^{10}-\frac{19\!\cdots\!09}{84\!\cdots\!99}a^{8}+\frac{16\!\cdots\!98}{10\!\cdots\!79}a^{6}+\frac{78\!\cdots\!52}{84\!\cdots\!99}a^{4}+\frac{76\!\cdots\!03}{84\!\cdots\!99}a^{2}-\frac{23\!\cdots\!02}{84\!\cdots\!99}$, $\frac{34\!\cdots\!82}{84\!\cdots\!99}a^{20}-\frac{15\!\cdots\!77}{84\!\cdots\!99}a^{18}-\frac{15\!\cdots\!93}{84\!\cdots\!99}a^{16}-\frac{47\!\cdots\!53}{84\!\cdots\!99}a^{14}+\frac{20\!\cdots\!36}{84\!\cdots\!99}a^{12}+\frac{11\!\cdots\!25}{31\!\cdots\!37}a^{10}-\frac{10\!\cdots\!21}{84\!\cdots\!99}a^{8}+\frac{36\!\cdots\!65}{10\!\cdots\!79}a^{6}+\frac{30\!\cdots\!06}{84\!\cdots\!99}a^{4}+\frac{41\!\cdots\!65}{84\!\cdots\!99}a^{2}-\frac{52\!\cdots\!00}{84\!\cdots\!99}$, $\frac{45\!\cdots\!24}{28\!\cdots\!33}a^{20}-\frac{20\!\cdots\!27}{28\!\cdots\!33}a^{18}-\frac{19\!\cdots\!71}{28\!\cdots\!33}a^{16}-\frac{61\!\cdots\!09}{28\!\cdots\!33}a^{14}+\frac{26\!\cdots\!20}{28\!\cdots\!33}a^{12}+\frac{44\!\cdots\!51}{31\!\cdots\!37}a^{10}-\frac{13\!\cdots\!97}{28\!\cdots\!33}a^{8}+\frac{20\!\cdots\!59}{11\!\cdots\!31}a^{6}+\frac{34\!\cdots\!61}{28\!\cdots\!33}a^{4}-\frac{70\!\cdots\!47}{28\!\cdots\!33}a^{2}-\frac{77\!\cdots\!20}{28\!\cdots\!33}$, $\frac{17\!\cdots\!56}{19\!\cdots\!31}a^{21}-\frac{16\!\cdots\!06}{28\!\cdots\!33}a^{20}-\frac{68\!\cdots\!05}{19\!\cdots\!31}a^{19}+\frac{64\!\cdots\!90}{28\!\cdots\!33}a^{18}-\frac{79\!\cdots\!60}{19\!\cdots\!31}a^{17}+\frac{71\!\cdots\!04}{28\!\cdots\!33}a^{16}-\frac{29\!\cdots\!88}{19\!\cdots\!31}a^{15}+\frac{25\!\cdots\!58}{28\!\cdots\!33}a^{14}+\frac{12\!\cdots\!36}{28\!\cdots\!33}a^{13}-\frac{85\!\cdots\!11}{28\!\cdots\!33}a^{12}+\frac{25\!\cdots\!43}{21\!\cdots\!59}a^{11}-\frac{21\!\cdots\!23}{31\!\cdots\!37}a^{10}-\frac{58\!\cdots\!88}{28\!\cdots\!33}a^{9}+\frac{39\!\cdots\!91}{28\!\cdots\!33}a^{8}-\frac{17\!\cdots\!60}{24\!\cdots\!51}a^{7}+\frac{74\!\cdots\!56}{34\!\cdots\!93}a^{6}+\frac{19\!\cdots\!57}{19\!\cdots\!31}a^{5}-\frac{16\!\cdots\!27}{28\!\cdots\!33}a^{4}+\frac{94\!\cdots\!06}{19\!\cdots\!31}a^{3}-\frac{55\!\cdots\!46}{28\!\cdots\!33}a^{2}+\frac{11\!\cdots\!60}{19\!\cdots\!31}a-\frac{38\!\cdots\!92}{28\!\cdots\!33}$, $\frac{12\!\cdots\!51}{59\!\cdots\!93}a^{21}-\frac{18\!\cdots\!48}{84\!\cdots\!99}a^{20}-\frac{33\!\cdots\!37}{59\!\cdots\!93}a^{19}+\frac{53\!\cdots\!51}{84\!\cdots\!99}a^{18}-\frac{53\!\cdots\!42}{59\!\cdots\!93}a^{17}+\frac{83\!\cdots\!58}{84\!\cdots\!99}a^{16}-\frac{24\!\cdots\!37}{59\!\cdots\!93}a^{15}+\frac{38\!\cdots\!95}{84\!\cdots\!99}a^{14}+\frac{49\!\cdots\!88}{84\!\cdots\!99}a^{13}-\frac{54\!\cdots\!92}{84\!\cdots\!99}a^{12}+\frac{20\!\cdots\!04}{65\!\cdots\!77}a^{11}-\frac{31\!\cdots\!36}{94\!\cdots\!11}a^{10}-\frac{12\!\cdots\!72}{84\!\cdots\!99}a^{9}+\frac{14\!\cdots\!48}{84\!\cdots\!99}a^{8}-\frac{17\!\cdots\!28}{73\!\cdots\!53}a^{7}+\frac{26\!\cdots\!35}{10\!\cdots\!79}a^{6}-\frac{32\!\cdots\!71}{59\!\cdots\!93}a^{5}+\frac{44\!\cdots\!67}{84\!\cdots\!99}a^{4}+\frac{68\!\cdots\!41}{59\!\cdots\!93}a^{3}-\frac{22\!\cdots\!29}{84\!\cdots\!99}a^{2}+\frac{32\!\cdots\!28}{59\!\cdots\!93}a-\frac{71\!\cdots\!27}{84\!\cdots\!99}$, $\frac{78\!\cdots\!69}{59\!\cdots\!93}a^{21}-\frac{11\!\cdots\!57}{31\!\cdots\!37}a^{20}-\frac{29\!\cdots\!07}{59\!\cdots\!93}a^{19}+\frac{15\!\cdots\!99}{94\!\cdots\!11}a^{18}-\frac{34\!\cdots\!55}{59\!\cdots\!93}a^{17}+\frac{16\!\cdots\!49}{10\!\cdots\!79}a^{16}-\frac{12\!\cdots\!61}{59\!\cdots\!93}a^{15}+\frac{44\!\cdots\!00}{94\!\cdots\!11}a^{14}+\frac{53\!\cdots\!43}{84\!\cdots\!99}a^{13}-\frac{24\!\cdots\!02}{10\!\cdots\!79}a^{12}+\frac{10\!\cdots\!86}{65\!\cdots\!77}a^{11}-\frac{42\!\cdots\!04}{94\!\cdots\!11}a^{10}-\frac{24\!\cdots\!70}{84\!\cdots\!99}a^{9}+\frac{99\!\cdots\!78}{94\!\cdots\!11}a^{8}-\frac{52\!\cdots\!22}{73\!\cdots\!53}a^{7}+\frac{19\!\cdots\!30}{10\!\cdots\!79}a^{6}+\frac{75\!\cdots\!59}{59\!\cdots\!93}a^{5}-\frac{15\!\cdots\!83}{31\!\cdots\!37}a^{4}+\frac{29\!\cdots\!38}{59\!\cdots\!93}a^{3}-\frac{17\!\cdots\!08}{94\!\cdots\!11}a^{2}+\frac{27\!\cdots\!18}{59\!\cdots\!93}a-\frac{15\!\cdots\!38}{94\!\cdots\!11}$, $\frac{72\!\cdots\!65}{59\!\cdots\!93}a^{21}+\frac{14\!\cdots\!27}{84\!\cdots\!99}a^{20}-\frac{12\!\cdots\!22}{59\!\cdots\!93}a^{19}-\frac{24\!\cdots\!18}{84\!\cdots\!99}a^{18}-\frac{32\!\cdots\!97}{59\!\cdots\!93}a^{17}-\frac{65\!\cdots\!58}{84\!\cdots\!99}a^{16}-\frac{18\!\cdots\!19}{59\!\cdots\!93}a^{15}-\frac{38\!\cdots\!26}{84\!\cdots\!99}a^{14}-\frac{36\!\cdots\!42}{84\!\cdots\!99}a^{13}-\frac{97\!\cdots\!87}{84\!\cdots\!99}a^{12}+\frac{84\!\cdots\!79}{65\!\cdots\!77}a^{11}+\frac{17\!\cdots\!27}{94\!\cdots\!11}a^{10}-\frac{18\!\cdots\!70}{84\!\cdots\!99}a^{9}-\frac{63\!\cdots\!47}{84\!\cdots\!99}a^{8}-\frac{15\!\cdots\!42}{24\!\cdots\!51}a^{7}-\frac{35\!\cdots\!21}{34\!\cdots\!93}a^{6}-\frac{71\!\cdots\!64}{59\!\cdots\!93}a^{5}-\frac{22\!\cdots\!36}{84\!\cdots\!99}a^{4}+\frac{54\!\cdots\!24}{59\!\cdots\!93}a^{3}+\frac{11\!\cdots\!41}{84\!\cdots\!99}a^{2}+\frac{10\!\cdots\!05}{59\!\cdots\!93}a+\frac{41\!\cdots\!86}{84\!\cdots\!99}$, $\frac{65\!\cdots\!31}{19\!\cdots\!31}a^{21}-\frac{31\!\cdots\!61}{84\!\cdots\!99}a^{20}-\frac{17\!\cdots\!14}{19\!\cdots\!31}a^{19}+\frac{86\!\cdots\!12}{84\!\cdots\!99}a^{18}-\frac{29\!\cdots\!63}{19\!\cdots\!31}a^{17}+\frac{14\!\cdots\!13}{84\!\cdots\!99}a^{16}-\frac{13\!\cdots\!23}{19\!\cdots\!31}a^{15}+\frac{67\!\cdots\!03}{84\!\cdots\!99}a^{14}+\frac{24\!\cdots\!45}{28\!\cdots\!33}a^{13}-\frac{83\!\cdots\!64}{84\!\cdots\!99}a^{12}+\frac{32\!\cdots\!90}{65\!\cdots\!77}a^{11}-\frac{52\!\cdots\!50}{94\!\cdots\!11}a^{10}-\frac{58\!\cdots\!49}{28\!\cdots\!33}a^{9}+\frac{19\!\cdots\!03}{84\!\cdots\!99}a^{8}-\frac{24\!\cdots\!47}{73\!\cdots\!53}a^{7}+\frac{40\!\cdots\!13}{10\!\cdots\!79}a^{6}-\frac{13\!\cdots\!08}{19\!\cdots\!31}a^{5}+\frac{63\!\cdots\!33}{84\!\cdots\!99}a^{4}+\frac{10\!\cdots\!62}{19\!\cdots\!31}a^{3}-\frac{48\!\cdots\!77}{84\!\cdots\!99}a^{2}+\frac{25\!\cdots\!40}{19\!\cdots\!31}a-\frac{12\!\cdots\!90}{84\!\cdots\!99}$ 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:  \( 944497361130000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 944497361130000 \cdot 1}{2\cdot\sqrt{4129233136056857981979443884256982828952059904}}\cr\approx \mathstrut & 0.463036490821387 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^20 - 439*x^18 - 1549*x^16 + 5292*x^14 + 11519*x^12 - 25067*x^10 - 2315*x^8 + 10967*x^6 + 2679*x^4 - 156*x^2 - 49)
 
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 - 4*x^20 - 439*x^18 - 1549*x^16 + 5292*x^14 + 11519*x^12 - 25067*x^10 - 2315*x^8 + 10967*x^6 + 2679*x^4 - 156*x^2 - 49, 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 - 4*x^20 - 439*x^18 - 1549*x^16 + 5292*x^14 + 11519*x^12 - 25067*x^10 - 2315*x^8 + 10967*x^6 + 2679*x^4 - 156*x^2 - 49);
 
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 - 4*x^20 - 439*x^18 - 1549*x^16 + 5292*x^14 + 11519*x^12 - 25067*x^10 - 2315*x^8 + 10967*x^6 + 2679*x^4 - 156*x^2 - 49);
 
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

$C_2^{10}.\PSL(2,11)$ (as 22T39):

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 non-solvable group of order 675840
The 56 conjugacy class representatives for $C_2^{10}.\PSL(2,11)$ are not computed
Character table for $C_2^{10}.\PSL(2,11)$ is not computed

Intermediate fields

11.11.31376518243389673201.1

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 sibling: data not computed
Minimal sibling: 22.18.4129233136056857981979443884256982828952059904.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.11.0.1}{11} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.11.0.1}{11} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.11.0.1}{11} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.11.0.1}{11} }^{2}$ ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.11.0.1}{11} }^{2}$ ${\href{/padicField/43.5.0.1}{5} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.11.0.1}{11} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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$
\(74843\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $8$$2$$4$$4$