Properties

Label 22.2.275...000.1
Degree $22$
Signature $[2, 10]$
Discriminant $2.755\times 10^{45}$
Root discriminant \(116.27\)
Ramified primes $2,3,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.M_{11}$ (as 22T43)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 78*x^20 + 2355*x^18 + 30360*x^16 + 156915*x^14 + 293802*x^12 + 1279641*x^10 + 795000*x^8 - 582480*x^6 + 170360*x^4 + 14292*x^2 - 5184)
 
gp: K = bnfinit(y^22 + 78*y^20 + 2355*y^18 + 30360*y^16 + 156915*y^14 + 293802*y^12 + 1279641*y^10 + 795000*y^8 - 582480*y^6 + 170360*y^4 + 14292*y^2 - 5184, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 78*x^20 + 2355*x^18 + 30360*x^16 + 156915*x^14 + 293802*x^12 + 1279641*x^10 + 795000*x^8 - 582480*x^6 + 170360*x^4 + 14292*x^2 - 5184);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 78*x^20 + 2355*x^18 + 30360*x^16 + 156915*x^14 + 293802*x^12 + 1279641*x^10 + 795000*x^8 - 582480*x^6 + 170360*x^4 + 14292*x^2 - 5184)
 

\( x^{22} + 78 x^{20} + 2355 x^{18} + 30360 x^{16} + 156915 x^{14} + 293802 x^{12} + 1279641 x^{10} + 795000 x^{8} - 582480 x^{6} + 170360 x^{4} + 14292 x^{2} + \cdots - 5184 \) 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:  $[2, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(2754990144000000000000000000000000000000000000\) \(\medspace = 2^{42}\cdot 3^{16}\cdot 5^{36}\) 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:  \(116.27\)
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:  $2^{139/48}3^{7/8}5^{39/20}\approx 448.95757350728394$
Ramified primes:   \(2\), \(3\), \(5\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{3}{8}a^{8}-\frac{1}{2}a^{7}+\frac{1}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{3}{8}a^{9}-\frac{1}{2}a^{8}+\frac{1}{8}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}$, $\frac{1}{8}a^{14}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{15}+\frac{1}{4}a^{9}-\frac{1}{8}a^{7}+\frac{1}{4}a^{3}$, $\frac{1}{8}a^{16}-\frac{1}{4}a^{10}+\frac{3}{8}a^{8}-\frac{1}{2}a^{6}-\frac{1}{4}a^{4}$, $\frac{1}{48}a^{17}-\frac{1}{24}a^{15}-\frac{1}{24}a^{13}-\frac{1}{24}a^{11}+\frac{7}{48}a^{9}+\frac{5}{24}a^{7}+\frac{1}{3}a^{5}-\frac{1}{2}a^{4}+\frac{1}{12}a^{3}+\frac{1}{12}a$, $\frac{1}{48}a^{18}-\frac{1}{24}a^{16}-\frac{1}{24}a^{14}-\frac{1}{24}a^{12}+\frac{7}{48}a^{10}+\frac{5}{24}a^{8}+\frac{1}{3}a^{6}-\frac{1}{2}a^{5}+\frac{1}{12}a^{4}+\frac{1}{12}a^{2}$, $\frac{1}{48}a^{19}-\frac{1}{16}a^{11}+\frac{3}{8}a^{9}-\frac{1}{2}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}+\frac{1}{6}a$, $\frac{1}{22\!\cdots\!40}a^{20}-\frac{95\!\cdots\!35}{14\!\cdots\!56}a^{18}+\frac{11\!\cdots\!31}{36\!\cdots\!64}a^{16}+\frac{47\!\cdots\!55}{18\!\cdots\!82}a^{14}-\frac{19\!\cdots\!45}{49\!\cdots\!52}a^{12}-\frac{14\!\cdots\!11}{73\!\cdots\!80}a^{10}-\frac{96\!\cdots\!13}{18\!\cdots\!82}a^{8}-\frac{28\!\cdots\!39}{73\!\cdots\!28}a^{6}-\frac{1}{2}a^{5}-\frac{13\!\cdots\!25}{12\!\cdots\!88}a^{4}-\frac{1}{2}a^{3}-\frac{23\!\cdots\!13}{55\!\cdots\!46}a^{2}+\frac{12\!\cdots\!58}{15\!\cdots\!35}$, $\frac{1}{13\!\cdots\!40}a^{21}+\frac{41\!\cdots\!03}{44\!\cdots\!68}a^{19}-\frac{45\!\cdots\!17}{88\!\cdots\!36}a^{17}+\frac{47\!\cdots\!55}{11\!\cdots\!92}a^{15}-\frac{65\!\cdots\!15}{98\!\cdots\!04}a^{13}-\frac{49\!\cdots\!03}{22\!\cdots\!40}a^{11}-\frac{1}{4}a^{10}+\frac{26\!\cdots\!85}{88\!\cdots\!36}a^{9}-\frac{1}{4}a^{8}-\frac{28\!\cdots\!31}{22\!\cdots\!84}a^{7}-\frac{1}{4}a^{6}-\frac{24\!\cdots\!29}{61\!\cdots\!94}a^{5}-\frac{1}{4}a^{4}-\frac{40\!\cdots\!07}{83\!\cdots\!69}a^{3}-\frac{1}{2}a^{2}-\frac{41\!\cdots\!73}{36\!\cdots\!40}a-\frac{1}{2}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$

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:  $11$
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:   $\frac{28\!\cdots\!47}{13\!\cdots\!40}a^{21}+\frac{57\!\cdots\!21}{24\!\cdots\!60}a^{20}+\frac{74\!\cdots\!13}{44\!\cdots\!68}a^{19}+\frac{89\!\cdots\!97}{49\!\cdots\!52}a^{18}+\frac{44\!\cdots\!07}{88\!\cdots\!36}a^{17}+\frac{33\!\cdots\!09}{61\!\cdots\!94}a^{16}+\frac{72\!\cdots\!63}{11\!\cdots\!92}a^{15}+\frac{17\!\cdots\!27}{24\!\cdots\!76}a^{14}+\frac{10\!\cdots\!53}{29\!\cdots\!12}a^{13}+\frac{18\!\cdots\!07}{49\!\cdots\!52}a^{12}+\frac{14\!\cdots\!59}{22\!\cdots\!40}a^{11}+\frac{18\!\cdots\!57}{24\!\cdots\!60}a^{10}+\frac{25\!\cdots\!57}{88\!\cdots\!36}a^{9}+\frac{37\!\cdots\!23}{12\!\cdots\!88}a^{8}+\frac{22\!\cdots\!37}{11\!\cdots\!92}a^{7}+\frac{70\!\cdots\!75}{30\!\cdots\!47}a^{6}-\frac{33\!\cdots\!19}{36\!\cdots\!64}a^{5}-\frac{66\!\cdots\!35}{61\!\cdots\!94}a^{4}+\frac{13\!\cdots\!79}{33\!\cdots\!76}a^{3}+\frac{16\!\cdots\!67}{12\!\cdots\!88}a^{2}+\frac{12\!\cdots\!73}{12\!\cdots\!80}a+\frac{17\!\cdots\!79}{30\!\cdots\!70}$, $\frac{10\!\cdots\!67}{16\!\cdots\!80}a^{21}+\frac{52\!\cdots\!37}{14\!\cdots\!56}a^{20}+\frac{11\!\cdots\!43}{22\!\cdots\!84}a^{19}+\frac{40\!\cdots\!53}{14\!\cdots\!56}a^{18}+\frac{34\!\cdots\!43}{22\!\cdots\!84}a^{17}+\frac{62\!\cdots\!27}{73\!\cdots\!28}a^{16}+\frac{45\!\cdots\!03}{22\!\cdots\!84}a^{15}+\frac{40\!\cdots\!25}{36\!\cdots\!64}a^{14}+\frac{81\!\cdots\!03}{73\!\cdots\!28}a^{13}+\frac{86\!\cdots\!49}{14\!\cdots\!56}a^{12}+\frac{12\!\cdots\!43}{55\!\cdots\!60}a^{11}+\frac{17\!\cdots\!11}{14\!\cdots\!56}a^{10}+\frac{25\!\cdots\!62}{27\!\cdots\!23}a^{9}+\frac{17\!\cdots\!59}{36\!\cdots\!64}a^{8}+\frac{23\!\cdots\!01}{27\!\cdots\!23}a^{7}+\frac{15\!\cdots\!99}{36\!\cdots\!64}a^{6}-\frac{11\!\cdots\!75}{92\!\cdots\!41}a^{5}-\frac{45\!\cdots\!79}{36\!\cdots\!64}a^{4}-\frac{11\!\cdots\!48}{83\!\cdots\!69}a^{3}-\frac{54\!\cdots\!77}{36\!\cdots\!64}a^{2}+\frac{85\!\cdots\!43}{15\!\cdots\!35}a+\frac{45\!\cdots\!89}{30\!\cdots\!47}$, $\frac{15\!\cdots\!83}{24\!\cdots\!60}a^{21}+\frac{99\!\cdots\!79}{44\!\cdots\!68}a^{20}+\frac{45\!\cdots\!91}{92\!\cdots\!41}a^{19}+\frac{42\!\cdots\!77}{24\!\cdots\!76}a^{18}+\frac{27\!\cdots\!21}{18\!\cdots\!82}a^{17}+\frac{64\!\cdots\!21}{12\!\cdots\!88}a^{16}+\frac{71\!\cdots\!27}{36\!\cdots\!64}a^{15}+\frac{49\!\cdots\!69}{73\!\cdots\!28}a^{14}+\frac{14\!\cdots\!61}{14\!\cdots\!56}a^{13}+\frac{48\!\cdots\!35}{14\!\cdots\!56}a^{12}+\frac{68\!\cdots\!39}{36\!\cdots\!40}a^{11}+\frac{12\!\cdots\!51}{24\!\cdots\!76}a^{10}+\frac{14\!\cdots\!71}{18\!\cdots\!82}a^{9}+\frac{18\!\cdots\!91}{73\!\cdots\!28}a^{8}+\frac{88\!\cdots\!13}{18\!\cdots\!82}a^{7}+\frac{23\!\cdots\!93}{36\!\cdots\!64}a^{6}-\frac{88\!\cdots\!39}{18\!\cdots\!82}a^{5}-\frac{51\!\cdots\!33}{18\!\cdots\!82}a^{4}-\frac{21\!\cdots\!61}{18\!\cdots\!82}a^{3}-\frac{31\!\cdots\!80}{27\!\cdots\!23}a^{2}+\frac{11\!\cdots\!61}{92\!\cdots\!10}a+\frac{22\!\cdots\!37}{30\!\cdots\!47}$, $\frac{83\!\cdots\!99}{66\!\cdots\!20}a^{21}-\frac{13\!\cdots\!59}{46\!\cdots\!05}a^{20}+\frac{21\!\cdots\!37}{22\!\cdots\!84}a^{19}-\frac{33\!\cdots\!01}{14\!\cdots\!56}a^{18}+\frac{32\!\cdots\!55}{11\!\cdots\!92}a^{17}-\frac{62\!\cdots\!10}{92\!\cdots\!41}a^{16}+\frac{20\!\cdots\!13}{55\!\cdots\!46}a^{15}-\frac{61\!\cdots\!11}{73\!\cdots\!28}a^{14}+\frac{28\!\cdots\!79}{14\!\cdots\!56}a^{13}-\frac{27\!\cdots\!61}{73\!\cdots\!28}a^{12}+\frac{18\!\cdots\!39}{55\!\cdots\!60}a^{11}-\frac{27\!\cdots\!83}{73\!\cdots\!80}a^{10}+\frac{16\!\cdots\!95}{11\!\cdots\!92}a^{9}-\frac{98\!\cdots\!11}{36\!\cdots\!64}a^{8}+\frac{39\!\cdots\!69}{55\!\cdots\!46}a^{7}+\frac{67\!\cdots\!61}{36\!\cdots\!64}a^{6}-\frac{11\!\cdots\!98}{92\!\cdots\!41}a^{5}+\frac{57\!\cdots\!19}{92\!\cdots\!41}a^{4}-\frac{13\!\cdots\!67}{83\!\cdots\!69}a^{3}-\frac{24\!\cdots\!43}{36\!\cdots\!64}a^{2}+\frac{96\!\cdots\!93}{30\!\cdots\!70}a-\frac{25\!\cdots\!11}{15\!\cdots\!35}$, $\frac{72\!\cdots\!51}{66\!\cdots\!20}a^{21}+\frac{94\!\cdots\!43}{11\!\cdots\!92}a^{19}+\frac{11\!\cdots\!03}{44\!\cdots\!68}a^{17}+\frac{74\!\cdots\!57}{22\!\cdots\!84}a^{15}+\frac{87\!\cdots\!87}{49\!\cdots\!52}a^{13}+\frac{97\!\cdots\!63}{27\!\cdots\!30}a^{11}+\frac{63\!\cdots\!49}{44\!\cdots\!68}a^{9}+\frac{24\!\cdots\!71}{22\!\cdots\!84}a^{7}-\frac{38\!\cdots\!15}{61\!\cdots\!94}a^{5}-\frac{42\!\cdots\!01}{33\!\cdots\!76}a^{3}+\frac{15\!\cdots\!89}{61\!\cdots\!40}a$, $\frac{13\!\cdots\!96}{41\!\cdots\!45}a^{21}-\frac{19\!\cdots\!87}{73\!\cdots\!80}a^{20}+\frac{28\!\cdots\!07}{11\!\cdots\!92}a^{19}-\frac{38\!\cdots\!19}{18\!\cdots\!82}a^{18}+\frac{33\!\cdots\!59}{44\!\cdots\!68}a^{17}-\frac{22\!\cdots\!55}{36\!\cdots\!64}a^{16}+\frac{20\!\cdots\!85}{22\!\cdots\!84}a^{15}-\frac{69\!\cdots\!66}{92\!\cdots\!41}a^{14}+\frac{30\!\cdots\!71}{73\!\cdots\!28}a^{13}-\frac{49\!\cdots\!07}{14\!\cdots\!56}a^{12}+\frac{55\!\cdots\!87}{11\!\cdots\!20}a^{11}-\frac{14\!\cdots\!97}{36\!\cdots\!40}a^{10}+\frac{16\!\cdots\!69}{44\!\cdots\!68}a^{9}-\frac{11\!\cdots\!95}{36\!\cdots\!64}a^{8}-\frac{36\!\cdots\!17}{22\!\cdots\!84}a^{7}+\frac{54\!\cdots\!65}{36\!\cdots\!64}a^{6}+\frac{29\!\cdots\!39}{92\!\cdots\!41}a^{5}-\frac{11\!\cdots\!87}{36\!\cdots\!64}a^{4}+\frac{21\!\cdots\!97}{33\!\cdots\!76}a^{3}-\frac{13\!\cdots\!07}{30\!\cdots\!47}a^{2}-\frac{48\!\cdots\!81}{61\!\cdots\!40}a+\frac{15\!\cdots\!22}{15\!\cdots\!35}$, $\frac{82\!\cdots\!41}{22\!\cdots\!40}a^{21}+\frac{13\!\cdots\!19}{46\!\cdots\!05}a^{20}+\frac{43\!\cdots\!99}{14\!\cdots\!56}a^{19}+\frac{58\!\cdots\!15}{24\!\cdots\!76}a^{18}+\frac{66\!\cdots\!07}{73\!\cdots\!28}a^{17}+\frac{88\!\cdots\!81}{12\!\cdots\!88}a^{16}+\frac{72\!\cdots\!69}{61\!\cdots\!94}a^{15}+\frac{23\!\cdots\!27}{24\!\cdots\!76}a^{14}+\frac{95\!\cdots\!07}{14\!\cdots\!56}a^{13}+\frac{15\!\cdots\!35}{30\!\cdots\!47}a^{12}+\frac{10\!\cdots\!89}{73\!\cdots\!80}a^{11}+\frac{13\!\cdots\!23}{12\!\cdots\!80}a^{10}+\frac{66\!\cdots\!75}{12\!\cdots\!88}a^{9}+\frac{51\!\cdots\!31}{12\!\cdots\!88}a^{8}+\frac{69\!\cdots\!31}{12\!\cdots\!88}a^{7}+\frac{93\!\cdots\!09}{24\!\cdots\!76}a^{6}+\frac{28\!\cdots\!97}{92\!\cdots\!41}a^{5}-\frac{45\!\cdots\!33}{61\!\cdots\!94}a^{4}-\frac{18\!\cdots\!15}{11\!\cdots\!92}a^{3}-\frac{37\!\cdots\!17}{36\!\cdots\!64}a^{2}-\frac{42\!\cdots\!09}{30\!\cdots\!70}a+\frac{26\!\cdots\!81}{15\!\cdots\!35}$, $\frac{82\!\cdots\!41}{22\!\cdots\!40}a^{21}-\frac{13\!\cdots\!19}{46\!\cdots\!05}a^{20}+\frac{43\!\cdots\!99}{14\!\cdots\!56}a^{19}-\frac{58\!\cdots\!15}{24\!\cdots\!76}a^{18}+\frac{66\!\cdots\!07}{73\!\cdots\!28}a^{17}-\frac{88\!\cdots\!81}{12\!\cdots\!88}a^{16}+\frac{72\!\cdots\!69}{61\!\cdots\!94}a^{15}-\frac{23\!\cdots\!27}{24\!\cdots\!76}a^{14}+\frac{95\!\cdots\!07}{14\!\cdots\!56}a^{13}-\frac{15\!\cdots\!35}{30\!\cdots\!47}a^{12}+\frac{10\!\cdots\!89}{73\!\cdots\!80}a^{11}-\frac{13\!\cdots\!23}{12\!\cdots\!80}a^{10}+\frac{66\!\cdots\!75}{12\!\cdots\!88}a^{9}-\frac{51\!\cdots\!31}{12\!\cdots\!88}a^{8}+\frac{69\!\cdots\!31}{12\!\cdots\!88}a^{7}-\frac{93\!\cdots\!09}{24\!\cdots\!76}a^{6}+\frac{28\!\cdots\!97}{92\!\cdots\!41}a^{5}+\frac{45\!\cdots\!33}{61\!\cdots\!94}a^{4}-\frac{18\!\cdots\!15}{11\!\cdots\!92}a^{3}+\frac{37\!\cdots\!17}{36\!\cdots\!64}a^{2}-\frac{42\!\cdots\!09}{30\!\cdots\!70}a-\frac{26\!\cdots\!81}{15\!\cdots\!35}$, $\frac{10\!\cdots\!29}{26\!\cdots\!08}a^{21}-\frac{79\!\cdots\!49}{22\!\cdots\!40}a^{20}+\frac{13\!\cdots\!03}{44\!\cdots\!68}a^{19}-\frac{13\!\cdots\!37}{49\!\cdots\!52}a^{18}+\frac{83\!\cdots\!09}{88\!\cdots\!36}a^{17}-\frac{10\!\cdots\!99}{12\!\cdots\!88}a^{16}+\frac{27\!\cdots\!29}{22\!\cdots\!84}a^{15}-\frac{81\!\cdots\!93}{73\!\cdots\!28}a^{14}+\frac{19\!\cdots\!51}{29\!\cdots\!12}a^{13}-\frac{85\!\cdots\!95}{14\!\cdots\!56}a^{12}+\frac{56\!\cdots\!23}{44\!\cdots\!68}a^{11}-\frac{28\!\cdots\!87}{24\!\cdots\!60}a^{10}+\frac{47\!\cdots\!83}{88\!\cdots\!36}a^{9}-\frac{35\!\cdots\!09}{73\!\cdots\!28}a^{8}+\frac{11\!\cdots\!72}{27\!\cdots\!23}a^{7}-\frac{27\!\cdots\!73}{73\!\cdots\!28}a^{6}-\frac{30\!\cdots\!69}{18\!\cdots\!82}a^{5}+\frac{53\!\cdots\!99}{36\!\cdots\!64}a^{4}+\frac{12\!\cdots\!55}{33\!\cdots\!76}a^{3}-\frac{21\!\cdots\!07}{55\!\cdots\!46}a^{2}+\frac{87\!\cdots\!85}{73\!\cdots\!28}a-\frac{35\!\cdots\!39}{30\!\cdots\!70}$, $\frac{18\!\cdots\!69}{66\!\cdots\!20}a^{21}-\frac{29\!\cdots\!31}{11\!\cdots\!20}a^{20}+\frac{24\!\cdots\!67}{11\!\cdots\!92}a^{19}-\frac{31\!\cdots\!45}{14\!\cdots\!56}a^{18}+\frac{15\!\cdots\!35}{22\!\cdots\!84}a^{17}-\frac{47\!\cdots\!21}{73\!\cdots\!28}a^{16}+\frac{19\!\cdots\!93}{22\!\cdots\!84}a^{15}-\frac{20\!\cdots\!27}{24\!\cdots\!76}a^{14}+\frac{24\!\cdots\!05}{49\!\cdots\!52}a^{13}-\frac{16\!\cdots\!65}{36\!\cdots\!64}a^{12}+\frac{14\!\cdots\!41}{13\!\cdots\!15}a^{11}-\frac{67\!\cdots\!23}{73\!\cdots\!80}a^{10}+\frac{22\!\cdots\!85}{55\!\cdots\!46}a^{9}-\frac{91\!\cdots\!31}{24\!\cdots\!76}a^{8}+\frac{45\!\cdots\!25}{11\!\cdots\!92}a^{7}-\frac{95\!\cdots\!31}{30\!\cdots\!47}a^{6}+\frac{26\!\cdots\!60}{30\!\cdots\!47}a^{5}+\frac{21\!\cdots\!73}{36\!\cdots\!64}a^{4}-\frac{22\!\cdots\!73}{16\!\cdots\!38}a^{3}+\frac{74\!\cdots\!81}{11\!\cdots\!92}a^{2}-\frac{88\!\cdots\!71}{92\!\cdots\!10}a-\frac{24\!\cdots\!41}{15\!\cdots\!35}$, $\frac{10\!\cdots\!71}{13\!\cdots\!40}a^{21}+\frac{32\!\cdots\!89}{36\!\cdots\!40}a^{20}+\frac{27\!\cdots\!13}{44\!\cdots\!68}a^{19}+\frac{10\!\cdots\!43}{14\!\cdots\!56}a^{18}+\frac{16\!\cdots\!97}{88\!\cdots\!36}a^{17}+\frac{15\!\cdots\!33}{73\!\cdots\!28}a^{16}+\frac{49\!\cdots\!37}{22\!\cdots\!84}a^{15}+\frac{20\!\cdots\!05}{73\!\cdots\!28}a^{14}+\frac{95\!\cdots\!83}{98\!\cdots\!04}a^{13}+\frac{52\!\cdots\!11}{36\!\cdots\!64}a^{12}+\frac{17\!\cdots\!07}{22\!\cdots\!40}a^{11}+\frac{21\!\cdots\!61}{73\!\cdots\!80}a^{10}+\frac{64\!\cdots\!15}{88\!\cdots\!36}a^{9}+\frac{95\!\cdots\!29}{73\!\cdots\!28}a^{8}-\frac{13\!\cdots\!82}{27\!\cdots\!23}a^{7}+\frac{45\!\cdots\!01}{36\!\cdots\!64}a^{6}-\frac{17\!\cdots\!65}{12\!\cdots\!88}a^{5}-\frac{37\!\cdots\!37}{92\!\cdots\!41}a^{4}+\frac{10\!\cdots\!29}{33\!\cdots\!76}a^{3}+\frac{56\!\cdots\!77}{36\!\cdots\!64}a^{2}-\frac{38\!\cdots\!03}{36\!\cdots\!40}a+\frac{21\!\cdots\!99}{30\!\cdots\!70}$ 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:  \( 4373557224090000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 4373557224090000 \cdot 1}{2\cdot\sqrt{2754990144000000000000000000000000000000000000}}\cr\approx \mathstrut & 15.9809821989944 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 + 78*x^20 + 2355*x^18 + 30360*x^16 + 156915*x^14 + 293802*x^12 + 1279641*x^10 + 795000*x^8 - 582480*x^6 + 170360*x^4 + 14292*x^2 - 5184)
 
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 + 78*x^20 + 2355*x^18 + 30360*x^16 + 156915*x^14 + 293802*x^12 + 1279641*x^10 + 795000*x^8 - 582480*x^6 + 170360*x^4 + 14292*x^2 - 5184, 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 + 78*x^20 + 2355*x^18 + 30360*x^16 + 156915*x^14 + 293802*x^12 + 1279641*x^10 + 795000*x^8 - 582480*x^6 + 170360*x^4 + 14292*x^2 - 5184);
 
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 + 78*x^20 + 2355*x^18 + 30360*x^16 + 156915*x^14 + 293802*x^12 + 1279641*x^10 + 795000*x^8 - 582480*x^6 + 170360*x^4 + 14292*x^2 - 5184);
 
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

$C_2^{10}.M_{11}$ (as 22T43):

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 8110080
The 52 conjugacy class representatives for $C_2^{10}.M_{11}$ are not computed
Character table for $C_2^{10}.M_{11}$ is not computed

Intermediate fields

11.3.6561000000000000000000.1

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
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 R R R ${\href{/padicField/7.11.0.1}{11} }^{2}$ ${\href{/padicField/11.11.0.1}{11} }^{2}$ ${\href{/padicField/13.5.0.1}{5} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.4.0.1}{4} }$ ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.5.0.1}{5} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.11.0.1}{11} }^{2}$ ${\href{/padicField/37.11.0.1}{11} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.11.0.1}{11} }^{2}$ $16{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$

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 2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.6.10.4$x^{6} + 2 x^{5} + 4 x^{3} + 4 x + 2$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
2.6.10.4$x^{6} + 2 x^{5} + 4 x^{3} + 4 x + 2$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
2.8.22.144$x^{8} + 4 x^{7} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 2$$8$$1$$22$$Q_8:S_4$$[4/3, 4/3, 8/3, 8/3, 7/2]_{3}^{2}$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
\(5\) Copy content Toggle raw display 5.2.0.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.5.9.4$x^{5} + 25 x^{2} + 25 x + 5$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.4$x^{5} + 25 x^{2} + 25 x + 5$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.18.2$x^{10} - 50 x^{7} - 50 x^{6} - 140 x^{5} - 4375 x^{4} - 8750 x^{3} - 10875 x^{2} - 6500 x - 100$$5$$2$$18$$F_{5}\times C_2$$[9/4]_{4}^{2}$