Properties

Label 22.22.170...216.2
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)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 140*x^20 + 8475*x^18 - 291035*x^16 + 6250874*x^14 - 87255611*x^12 + 795482411*x^10 - 4618766140*x^8 + 15964116759*x^6 - 28133091175*x^4 + 16687617633*x^2 - 3015217921)
 
gp: K = bnfinit(y^22 - 140*y^20 + 8475*y^18 - 291035*y^16 + 6250874*y^14 - 87255611*y^12 + 795482411*y^10 - 4618766140*y^8 + 15964116759*y^6 - 28133091175*y^4 + 16687617633*y^2 - 3015217921, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 140*x^20 + 8475*x^18 - 291035*x^16 + 6250874*x^14 - 87255611*x^12 + 795482411*x^10 - 4618766140*x^8 + 15964116759*x^6 - 28133091175*x^4 + 16687617633*x^2 - 3015217921);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 140*x^20 + 8475*x^18 - 291035*x^16 + 6250874*x^14 - 87255611*x^12 + 795482411*x^10 - 4618766140*x^8 + 15964116759*x^6 - 28133091175*x^4 + 16687617633*x^2 - 3015217921)
 

\( x^{22} - 140 x^{20} + 8475 x^{18} - 291035 x^{16} + 6250874 x^{14} - 87255611 x^{12} + 795482411 x^{10} - 4618766140 x^{8} + 15964116759 x^{6} + \cdots - 3015217921 \) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5}a^{14}+\frac{2}{5}a^{10}-\frac{2}{5}a^{8}+\frac{1}{5}a^{4}-\frac{1}{5}a^{2}-\frac{2}{5}$, $\frac{1}{5}a^{15}+\frac{2}{5}a^{11}-\frac{2}{5}a^{9}+\frac{1}{5}a^{5}-\frac{1}{5}a^{3}-\frac{2}{5}a$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{12}-\frac{2}{5}a^{10}+\frac{1}{5}a^{6}-\frac{1}{5}a^{4}-\frac{2}{5}a^{2}$, $\frac{1}{5}a^{17}+\frac{2}{5}a^{13}-\frac{2}{5}a^{11}+\frac{1}{5}a^{7}-\frac{1}{5}a^{5}-\frac{2}{5}a^{3}$, $\frac{1}{145}a^{18}-\frac{3}{145}a^{16}-\frac{14}{145}a^{14}-\frac{63}{145}a^{12}+\frac{14}{145}a^{10}-\frac{37}{145}a^{8}+\frac{71}{145}a^{6}+\frac{11}{29}a^{4}+\frac{42}{145}a^{2}-\frac{63}{145}$, $\frac{1}{145}a^{19}-\frac{3}{145}a^{17}-\frac{14}{145}a^{15}-\frac{63}{145}a^{13}+\frac{14}{145}a^{11}-\frac{37}{145}a^{9}+\frac{71}{145}a^{7}+\frac{11}{29}a^{5}+\frac{42}{145}a^{3}-\frac{63}{145}a$, $\frac{1}{20\!\cdots\!05}a^{20}-\frac{60\!\cdots\!22}{20\!\cdots\!05}a^{18}+\frac{59\!\cdots\!86}{40\!\cdots\!61}a^{16}+\frac{10\!\cdots\!38}{20\!\cdots\!05}a^{14}-\frac{11\!\cdots\!68}{40\!\cdots\!61}a^{12}-\frac{23\!\cdots\!42}{20\!\cdots\!05}a^{10}+\frac{64\!\cdots\!19}{20\!\cdots\!05}a^{8}-\frac{99\!\cdots\!97}{20\!\cdots\!05}a^{6}+\frac{57\!\cdots\!40}{40\!\cdots\!61}a^{4}-\frac{40\!\cdots\!34}{40\!\cdots\!61}a^{2}+\frac{19\!\cdots\!72}{36\!\cdots\!55}$, $\frac{1}{20\!\cdots\!05}a^{21}-\frac{60\!\cdots\!22}{20\!\cdots\!05}a^{19}+\frac{59\!\cdots\!86}{40\!\cdots\!61}a^{17}+\frac{10\!\cdots\!38}{20\!\cdots\!05}a^{15}-\frac{11\!\cdots\!68}{40\!\cdots\!61}a^{13}-\frac{23\!\cdots\!42}{20\!\cdots\!05}a^{11}+\frac{64\!\cdots\!19}{20\!\cdots\!05}a^{9}-\frac{99\!\cdots\!97}{20\!\cdots\!05}a^{7}+\frac{57\!\cdots\!40}{40\!\cdots\!61}a^{5}-\frac{40\!\cdots\!34}{40\!\cdots\!61}a^{3}+\frac{19\!\cdots\!72}{36\!\cdots\!55}a$ 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:   $\frac{74\!\cdots\!52}{20\!\cdots\!05}a^{20}-\frac{19\!\cdots\!39}{40\!\cdots\!61}a^{18}+\frac{55\!\cdots\!01}{20\!\cdots\!05}a^{16}-\frac{17\!\cdots\!31}{20\!\cdots\!05}a^{14}+\frac{33\!\cdots\!69}{20\!\cdots\!05}a^{12}-\frac{40\!\cdots\!06}{20\!\cdots\!05}a^{10}+\frac{29\!\cdots\!32}{20\!\cdots\!05}a^{8}-\frac{12\!\cdots\!82}{20\!\cdots\!05}a^{6}+\frac{25\!\cdots\!16}{20\!\cdots\!05}a^{4}-\frac{16\!\cdots\!57}{20\!\cdots\!05}a^{2}+\frac{54\!\cdots\!44}{36\!\cdots\!55}$, $\frac{50\!\cdots\!82}{20\!\cdots\!05}a^{20}-\frac{67\!\cdots\!24}{20\!\cdots\!05}a^{18}+\frac{37\!\cdots\!28}{20\!\cdots\!05}a^{16}-\frac{11\!\cdots\!91}{20\!\cdots\!05}a^{14}+\frac{22\!\cdots\!36}{20\!\cdots\!05}a^{12}-\frac{26\!\cdots\!54}{20\!\cdots\!05}a^{10}+\frac{19\!\cdots\!42}{20\!\cdots\!05}a^{8}-\frac{82\!\cdots\!86}{20\!\cdots\!05}a^{6}+\frac{34\!\cdots\!82}{40\!\cdots\!61}a^{4}-\frac{10\!\cdots\!19}{20\!\cdots\!05}a^{2}+\frac{35\!\cdots\!88}{36\!\cdots\!55}$, $\frac{10\!\cdots\!28}{20\!\cdots\!05}a^{20}-\frac{13\!\cdots\!49}{20\!\cdots\!05}a^{18}+\frac{75\!\cdots\!42}{20\!\cdots\!05}a^{16}-\frac{23\!\cdots\!32}{20\!\cdots\!05}a^{14}+\frac{89\!\cdots\!74}{40\!\cdots\!61}a^{12}-\frac{10\!\cdots\!45}{40\!\cdots\!61}a^{10}+\frac{38\!\cdots\!34}{20\!\cdots\!05}a^{8}-\frac{16\!\cdots\!16}{20\!\cdots\!05}a^{6}+\frac{33\!\cdots\!64}{20\!\cdots\!05}a^{4}-\frac{21\!\cdots\!44}{20\!\cdots\!05}a^{2}+\frac{70\!\cdots\!41}{36\!\cdots\!55}$, $\frac{39\!\cdots\!96}{20\!\cdots\!05}a^{20}-\frac{52\!\cdots\!13}{20\!\cdots\!05}a^{18}+\frac{29\!\cdots\!59}{20\!\cdots\!05}a^{16}-\frac{18\!\cdots\!37}{40\!\cdots\!61}a^{14}+\frac{34\!\cdots\!54}{40\!\cdots\!61}a^{12}-\frac{20\!\cdots\!42}{20\!\cdots\!05}a^{10}+\frac{29\!\cdots\!16}{40\!\cdots\!61}a^{8}-\frac{61\!\cdots\!77}{20\!\cdots\!05}a^{6}+\frac{12\!\cdots\!12}{20\!\cdots\!05}a^{4}-\frac{78\!\cdots\!92}{20\!\cdots\!05}a^{2}+\frac{26\!\cdots\!09}{36\!\cdots\!55}$, $\frac{57\!\cdots\!44}{40\!\cdots\!61}a^{20}-\frac{38\!\cdots\!28}{20\!\cdots\!05}a^{18}+\frac{21\!\cdots\!71}{20\!\cdots\!05}a^{16}-\frac{67\!\cdots\!51}{20\!\cdots\!05}a^{14}+\frac{12\!\cdots\!28}{20\!\cdots\!05}a^{12}-\frac{15\!\cdots\!42}{20\!\cdots\!05}a^{10}+\frac{11\!\cdots\!17}{20\!\cdots\!05}a^{8}-\frac{46\!\cdots\!16}{20\!\cdots\!05}a^{6}+\frac{19\!\cdots\!00}{40\!\cdots\!61}a^{4}-\frac{59\!\cdots\!77}{20\!\cdots\!05}a^{2}+\frac{40\!\cdots\!90}{72\!\cdots\!51}$, $\frac{43\!\cdots\!64}{20\!\cdots\!05}a^{20}-\frac{57\!\cdots\!24}{20\!\cdots\!05}a^{18}+\frac{32\!\cdots\!58}{20\!\cdots\!05}a^{16}-\frac{20\!\cdots\!05}{40\!\cdots\!61}a^{14}+\frac{19\!\cdots\!73}{20\!\cdots\!05}a^{12}-\frac{22\!\cdots\!09}{20\!\cdots\!05}a^{10}+\frac{33\!\cdots\!75}{40\!\cdots\!61}a^{8}-\frac{69\!\cdots\!74}{20\!\cdots\!05}a^{6}+\frac{14\!\cdots\!54}{20\!\cdots\!05}a^{4}-\frac{90\!\cdots\!16}{20\!\cdots\!05}a^{2}+\frac{30\!\cdots\!28}{36\!\cdots\!55}$, $\frac{11\!\cdots\!89}{20\!\cdots\!05}a^{20}-\frac{15\!\cdots\!38}{20\!\cdots\!05}a^{18}+\frac{85\!\cdots\!32}{20\!\cdots\!05}a^{16}-\frac{26\!\cdots\!03}{20\!\cdots\!05}a^{14}+\frac{50\!\cdots\!64}{20\!\cdots\!05}a^{12}-\frac{60\!\cdots\!97}{20\!\cdots\!05}a^{10}+\frac{43\!\cdots\!46}{20\!\cdots\!05}a^{8}-\frac{18\!\cdots\!96}{20\!\cdots\!05}a^{6}+\frac{37\!\cdots\!48}{20\!\cdots\!05}a^{4}-\frac{23\!\cdots\!34}{20\!\cdots\!05}a^{2}+\frac{79\!\cdots\!88}{36\!\cdots\!55}$, $\frac{16\!\cdots\!96}{20\!\cdots\!05}a^{20}-\frac{21\!\cdots\!92}{20\!\cdots\!05}a^{18}+\frac{12\!\cdots\!88}{20\!\cdots\!05}a^{16}-\frac{75\!\cdots\!27}{40\!\cdots\!61}a^{14}+\frac{72\!\cdots\!76}{20\!\cdots\!05}a^{12}-\frac{85\!\cdots\!89}{20\!\cdots\!05}a^{10}+\frac{12\!\cdots\!60}{40\!\cdots\!61}a^{8}-\frac{26\!\cdots\!59}{20\!\cdots\!05}a^{6}+\frac{53\!\cdots\!34}{20\!\cdots\!05}a^{4}-\frac{33\!\cdots\!13}{20\!\cdots\!05}a^{2}+\frac{11\!\cdots\!38}{36\!\cdots\!55}$, $\frac{43\!\cdots\!64}{20\!\cdots\!05}a^{20}-\frac{57\!\cdots\!24}{20\!\cdots\!05}a^{18}+\frac{32\!\cdots\!58}{20\!\cdots\!05}a^{16}-\frac{20\!\cdots\!05}{40\!\cdots\!61}a^{14}+\frac{19\!\cdots\!73}{20\!\cdots\!05}a^{12}-\frac{22\!\cdots\!09}{20\!\cdots\!05}a^{10}+\frac{33\!\cdots\!75}{40\!\cdots\!61}a^{8}-\frac{69\!\cdots\!74}{20\!\cdots\!05}a^{6}+\frac{14\!\cdots\!54}{20\!\cdots\!05}a^{4}-\frac{90\!\cdots\!16}{20\!\cdots\!05}a^{2}+\frac{30\!\cdots\!73}{36\!\cdots\!55}$, $\frac{46\!\cdots\!32}{46\!\cdots\!35}a^{20}-\frac{61\!\cdots\!89}{46\!\cdots\!35}a^{18}+\frac{34\!\cdots\!71}{46\!\cdots\!35}a^{16}-\frac{10\!\cdots\!41}{46\!\cdots\!35}a^{14}+\frac{20\!\cdots\!62}{46\!\cdots\!35}a^{12}-\frac{49\!\cdots\!57}{93\!\cdots\!27}a^{10}+\frac{17\!\cdots\!57}{46\!\cdots\!35}a^{8}-\frac{75\!\cdots\!73}{46\!\cdots\!35}a^{6}+\frac{15\!\cdots\!02}{46\!\cdots\!35}a^{4}-\frac{19\!\cdots\!49}{93\!\cdots\!27}a^{2}+\frac{14\!\cdots\!69}{36\!\cdots\!55}$, $\frac{63\!\cdots\!22}{20\!\cdots\!05}a^{21}-\frac{20\!\cdots\!32}{20\!\cdots\!05}a^{20}-\frac{82\!\cdots\!36}{20\!\cdots\!05}a^{19}+\frac{54\!\cdots\!96}{40\!\cdots\!61}a^{18}+\frac{45\!\cdots\!34}{20\!\cdots\!05}a^{17}-\frac{15\!\cdots\!43}{20\!\cdots\!05}a^{16}-\frac{13\!\cdots\!36}{20\!\cdots\!05}a^{15}+\frac{48\!\cdots\!48}{20\!\cdots\!05}a^{14}+\frac{25\!\cdots\!07}{20\!\cdots\!05}a^{13}-\frac{91\!\cdots\!28}{20\!\cdots\!05}a^{12}-\frac{29\!\cdots\!83}{20\!\cdots\!05}a^{11}+\frac{10\!\cdots\!54}{20\!\cdots\!05}a^{10}+\frac{20\!\cdots\!82}{20\!\cdots\!05}a^{9}-\frac{79\!\cdots\!26}{20\!\cdots\!05}a^{8}-\frac{78\!\cdots\!38}{20\!\cdots\!05}a^{7}+\frac{66\!\cdots\!34}{40\!\cdots\!61}a^{6}+\frac{14\!\cdots\!97}{20\!\cdots\!05}a^{5}-\frac{68\!\cdots\!82}{20\!\cdots\!05}a^{4}-\frac{13\!\cdots\!45}{40\!\cdots\!61}a^{3}+\frac{42\!\cdots\!24}{20\!\cdots\!05}a^{2}+\frac{35\!\cdots\!42}{72\!\cdots\!51}a-\frac{14\!\cdots\!93}{36\!\cdots\!55}$, $\frac{98\!\cdots\!52}{20\!\cdots\!05}a^{21}+\frac{29\!\cdots\!95}{40\!\cdots\!61}a^{20}-\frac{13\!\cdots\!31}{20\!\cdots\!05}a^{19}-\frac{19\!\cdots\!69}{20\!\cdots\!05}a^{18}+\frac{72\!\cdots\!41}{20\!\cdots\!05}a^{17}+\frac{21\!\cdots\!05}{40\!\cdots\!61}a^{16}-\frac{22\!\cdots\!52}{20\!\cdots\!05}a^{15}-\frac{33\!\cdots\!91}{20\!\cdots\!05}a^{14}+\frac{42\!\cdots\!91}{20\!\cdots\!05}a^{13}+\frac{63\!\cdots\!43}{20\!\cdots\!05}a^{12}-\frac{50\!\cdots\!44}{20\!\cdots\!05}a^{11}-\frac{73\!\cdots\!71}{20\!\cdots\!05}a^{10}+\frac{36\!\cdots\!24}{20\!\cdots\!05}a^{9}+\frac{52\!\cdots\!87}{20\!\cdots\!05}a^{8}-\frac{15\!\cdots\!46}{20\!\cdots\!05}a^{7}-\frac{21\!\cdots\!26}{20\!\cdots\!05}a^{6}+\frac{30\!\cdots\!14}{20\!\cdots\!05}a^{5}+\frac{88\!\cdots\!39}{40\!\cdots\!61}a^{4}-\frac{19\!\cdots\!93}{20\!\cdots\!05}a^{3}-\frac{27\!\cdots\!02}{20\!\cdots\!05}a^{2}+\frac{65\!\cdots\!82}{36\!\cdots\!55}a+\frac{92\!\cdots\!91}{36\!\cdots\!55}$, $\frac{30\!\cdots\!93}{20\!\cdots\!05}a^{21}+\frac{87\!\cdots\!79}{20\!\cdots\!05}a^{20}-\frac{40\!\cdots\!67}{20\!\cdots\!05}a^{19}-\frac{11\!\cdots\!98}{20\!\cdots\!05}a^{18}+\frac{23\!\cdots\!23}{20\!\cdots\!05}a^{17}+\frac{13\!\cdots\!21}{40\!\cdots\!61}a^{16}-\frac{72\!\cdots\!52}{20\!\cdots\!05}a^{15}-\frac{20\!\cdots\!73}{20\!\cdots\!05}a^{14}+\frac{13\!\cdots\!13}{20\!\cdots\!05}a^{13}+\frac{78\!\cdots\!47}{40\!\cdots\!61}a^{12}-\frac{32\!\cdots\!90}{40\!\cdots\!61}a^{11}-\frac{46\!\cdots\!03}{20\!\cdots\!05}a^{10}+\frac{12\!\cdots\!89}{20\!\cdots\!05}a^{9}+\frac{34\!\cdots\!56}{20\!\cdots\!05}a^{8}-\frac{50\!\cdots\!42}{20\!\cdots\!05}a^{7}-\frac{14\!\cdots\!18}{20\!\cdots\!05}a^{6}+\frac{21\!\cdots\!20}{40\!\cdots\!61}a^{5}+\frac{59\!\cdots\!25}{40\!\cdots\!61}a^{4}-\frac{66\!\cdots\!22}{20\!\cdots\!05}a^{3}-\frac{37\!\cdots\!28}{40\!\cdots\!61}a^{2}+\frac{22\!\cdots\!84}{36\!\cdots\!55}a+\frac{63\!\cdots\!98}{36\!\cdots\!55}$, $\frac{58\!\cdots\!64}{20\!\cdots\!05}a^{21}+\frac{16\!\cdots\!02}{20\!\cdots\!05}a^{20}-\frac{77\!\cdots\!21}{20\!\cdots\!05}a^{19}-\frac{21\!\cdots\!59}{20\!\cdots\!05}a^{18}+\frac{43\!\cdots\!86}{20\!\cdots\!05}a^{17}+\frac{12\!\cdots\!42}{20\!\cdots\!05}a^{16}-\frac{13\!\cdots\!38}{20\!\cdots\!05}a^{15}-\frac{77\!\cdots\!53}{40\!\cdots\!61}a^{14}+\frac{26\!\cdots\!18}{20\!\cdots\!05}a^{13}+\frac{73\!\cdots\!84}{20\!\cdots\!05}a^{12}-\frac{31\!\cdots\!98}{20\!\cdots\!05}a^{11}-\frac{17\!\cdots\!09}{40\!\cdots\!61}a^{10}+\frac{22\!\cdots\!36}{20\!\cdots\!05}a^{9}+\frac{12\!\cdots\!38}{40\!\cdots\!61}a^{8}-\frac{96\!\cdots\!44}{20\!\cdots\!05}a^{7}-\frac{27\!\cdots\!82}{20\!\cdots\!05}a^{6}+\frac{19\!\cdots\!46}{20\!\cdots\!05}a^{5}+\frac{56\!\cdots\!57}{20\!\cdots\!05}a^{4}-\frac{12\!\cdots\!33}{20\!\cdots\!05}a^{3}-\frac{35\!\cdots\!78}{20\!\cdots\!05}a^{2}+\frac{42\!\cdots\!46}{36\!\cdots\!55}a+\frac{11\!\cdots\!96}{36\!\cdots\!55}$, $\frac{17\!\cdots\!09}{40\!\cdots\!61}a^{21}+\frac{26\!\cdots\!76}{20\!\cdots\!05}a^{20}-\frac{22\!\cdots\!64}{40\!\cdots\!61}a^{19}-\frac{34\!\cdots\!72}{20\!\cdots\!05}a^{18}+\frac{63\!\cdots\!64}{20\!\cdots\!05}a^{17}+\frac{19\!\cdots\!53}{20\!\cdots\!05}a^{16}-\frac{19\!\cdots\!04}{20\!\cdots\!05}a^{15}-\frac{61\!\cdots\!51}{20\!\cdots\!05}a^{14}+\frac{37\!\cdots\!03}{20\!\cdots\!05}a^{13}+\frac{11\!\cdots\!96}{20\!\cdots\!05}a^{12}-\frac{45\!\cdots\!06}{20\!\cdots\!05}a^{11}-\frac{13\!\cdots\!41}{20\!\cdots\!05}a^{10}+\frac{33\!\cdots\!88}{20\!\cdots\!05}a^{9}+\frac{10\!\cdots\!32}{20\!\cdots\!05}a^{8}-\frac{13\!\cdots\!71}{20\!\cdots\!05}a^{7}-\frac{42\!\cdots\!64}{20\!\cdots\!05}a^{6}+\frac{28\!\cdots\!37}{20\!\cdots\!05}a^{5}+\frac{88\!\cdots\!38}{20\!\cdots\!05}a^{4}-\frac{17\!\cdots\!99}{20\!\cdots\!05}a^{3}-\frac{55\!\cdots\!07}{20\!\cdots\!05}a^{2}+\frac{58\!\cdots\!88}{36\!\cdots\!55}a+\frac{37\!\cdots\!85}{72\!\cdots\!51}$, $\frac{11\!\cdots\!86}{20\!\cdots\!05}a^{21}+\frac{51\!\cdots\!96}{20\!\cdots\!05}a^{20}-\frac{14\!\cdots\!28}{20\!\cdots\!05}a^{19}-\frac{68\!\cdots\!02}{20\!\cdots\!05}a^{18}+\frac{16\!\cdots\!12}{40\!\cdots\!61}a^{17}+\frac{38\!\cdots\!07}{20\!\cdots\!05}a^{16}-\frac{52\!\cdots\!38}{40\!\cdots\!61}a^{15}-\frac{12\!\cdots\!06}{20\!\cdots\!05}a^{14}+\frac{49\!\cdots\!92}{20\!\cdots\!05}a^{13}+\frac{23\!\cdots\!69}{20\!\cdots\!05}a^{12}-\frac{59\!\cdots\!29}{20\!\cdots\!05}a^{11}-\frac{27\!\cdots\!54}{20\!\cdots\!05}a^{10}+\frac{88\!\cdots\!33}{40\!\cdots\!61}a^{9}+\frac{20\!\cdots\!02}{20\!\cdots\!05}a^{8}-\frac{18\!\cdots\!06}{20\!\cdots\!05}a^{7}-\frac{17\!\cdots\!30}{40\!\cdots\!61}a^{6}+\frac{39\!\cdots\!41}{20\!\cdots\!05}a^{5}+\frac{17\!\cdots\!04}{20\!\cdots\!05}a^{4}-\frac{24\!\cdots\!59}{20\!\cdots\!05}a^{3}-\frac{22\!\cdots\!43}{40\!\cdots\!61}a^{2}+\frac{83\!\cdots\!84}{36\!\cdots\!55}a+\frac{76\!\cdots\!10}{72\!\cdots\!51}$, $\frac{41\!\cdots\!51}{31\!\cdots\!93}a^{21}+\frac{71\!\cdots\!82}{20\!\cdots\!05}a^{20}-\frac{27\!\cdots\!32}{15\!\cdots\!65}a^{19}-\frac{94\!\cdots\!72}{20\!\cdots\!05}a^{18}+\frac{30\!\cdots\!21}{31\!\cdots\!93}a^{17}+\frac{10\!\cdots\!73}{40\!\cdots\!61}a^{16}-\frac{48\!\cdots\!97}{15\!\cdots\!65}a^{15}-\frac{33\!\cdots\!87}{40\!\cdots\!61}a^{14}+\frac{92\!\cdots\!49}{15\!\cdots\!65}a^{13}+\frac{32\!\cdots\!41}{20\!\cdots\!05}a^{12}-\frac{11\!\cdots\!21}{15\!\cdots\!65}a^{11}-\frac{38\!\cdots\!79}{20\!\cdots\!05}a^{10}+\frac{81\!\cdots\!09}{15\!\cdots\!65}a^{9}+\frac{56\!\cdots\!12}{40\!\cdots\!61}a^{8}-\frac{34\!\cdots\!23}{15\!\cdots\!65}a^{7}-\frac{11\!\cdots\!61}{20\!\cdots\!05}a^{6}+\frac{71\!\cdots\!21}{15\!\cdots\!65}a^{5}+\frac{24\!\cdots\!11}{20\!\cdots\!05}a^{4}-\frac{44\!\cdots\!67}{15\!\cdots\!65}a^{3}-\frac{30\!\cdots\!34}{40\!\cdots\!61}a^{2}+\frac{19\!\cdots\!67}{36\!\cdots\!55}a+\frac{51\!\cdots\!24}{36\!\cdots\!55}$, $\frac{29\!\cdots\!79}{20\!\cdots\!05}a^{21}-\frac{11\!\cdots\!09}{20\!\cdots\!05}a^{20}-\frac{77\!\cdots\!81}{40\!\cdots\!61}a^{19}+\frac{15\!\cdots\!24}{20\!\cdots\!05}a^{18}+\frac{21\!\cdots\!59}{20\!\cdots\!05}a^{17}-\frac{85\!\cdots\!07}{20\!\cdots\!05}a^{16}-\frac{13\!\cdots\!05}{40\!\cdots\!61}a^{15}+\frac{27\!\cdots\!74}{20\!\cdots\!05}a^{14}+\frac{12\!\cdots\!52}{20\!\cdots\!05}a^{13}-\frac{52\!\cdots\!86}{20\!\cdots\!05}a^{12}-\frac{14\!\cdots\!52}{20\!\cdots\!05}a^{11}+\frac{12\!\cdots\!81}{40\!\cdots\!61}a^{10}+\frac{21\!\cdots\!03}{40\!\cdots\!61}a^{9}-\frac{46\!\cdots\!18}{20\!\cdots\!05}a^{8}-\frac{45\!\cdots\!52}{20\!\cdots\!05}a^{7}+\frac{39\!\cdots\!54}{40\!\cdots\!61}a^{6}+\frac{92\!\cdots\!67}{20\!\cdots\!05}a^{5}-\frac{41\!\cdots\!16}{20\!\cdots\!05}a^{4}-\frac{11\!\cdots\!09}{40\!\cdots\!61}a^{3}+\frac{51\!\cdots\!98}{40\!\cdots\!61}a^{2}+\frac{19\!\cdots\!94}{36\!\cdots\!55}a-\frac{87\!\cdots\!31}{36\!\cdots\!55}$, $\frac{76\!\cdots\!57}{20\!\cdots\!05}a^{21}+\frac{41\!\cdots\!04}{40\!\cdots\!61}a^{20}-\frac{10\!\cdots\!42}{20\!\cdots\!05}a^{19}-\frac{55\!\cdots\!22}{40\!\cdots\!61}a^{18}+\frac{57\!\cdots\!37}{20\!\cdots\!05}a^{17}+\frac{15\!\cdots\!42}{20\!\cdots\!05}a^{16}-\frac{17\!\cdots\!62}{20\!\cdots\!05}a^{15}-\frac{97\!\cdots\!30}{40\!\cdots\!61}a^{14}+\frac{68\!\cdots\!68}{40\!\cdots\!61}a^{13}+\frac{93\!\cdots\!29}{20\!\cdots\!05}a^{12}-\frac{40\!\cdots\!87}{20\!\cdots\!05}a^{11}-\frac{11\!\cdots\!74}{20\!\cdots\!05}a^{10}+\frac{29\!\cdots\!24}{20\!\cdots\!05}a^{9}+\frac{16\!\cdots\!84}{40\!\cdots\!61}a^{8}-\frac{12\!\cdots\!09}{20\!\cdots\!05}a^{7}-\frac{34\!\cdots\!88}{20\!\cdots\!05}a^{6}+\frac{25\!\cdots\!57}{20\!\cdots\!05}a^{5}+\frac{70\!\cdots\!68}{20\!\cdots\!05}a^{4}-\frac{16\!\cdots\!97}{20\!\cdots\!05}a^{3}-\frac{44\!\cdots\!39}{20\!\cdots\!05}a^{2}+\frac{54\!\cdots\!53}{36\!\cdots\!55}a+\frac{29\!\cdots\!56}{72\!\cdots\!51}$, $\frac{35\!\cdots\!68}{20\!\cdots\!05}a^{21}+\frac{20\!\cdots\!86}{40\!\cdots\!61}a^{20}-\frac{47\!\cdots\!31}{20\!\cdots\!05}a^{19}-\frac{13\!\cdots\!77}{20\!\cdots\!05}a^{18}+\frac{26\!\cdots\!21}{20\!\cdots\!05}a^{17}+\frac{77\!\cdots\!09}{20\!\cdots\!05}a^{16}-\frac{82\!\cdots\!92}{20\!\cdots\!05}a^{15}-\frac{48\!\cdots\!65}{40\!\cdots\!61}a^{14}+\frac{15\!\cdots\!97}{20\!\cdots\!05}a^{13}+\frac{46\!\cdots\!27}{20\!\cdots\!05}a^{12}-\frac{37\!\cdots\!10}{40\!\cdots\!61}a^{11}-\frac{11\!\cdots\!06}{40\!\cdots\!61}a^{10}+\frac{13\!\cdots\!04}{20\!\cdots\!05}a^{9}+\frac{81\!\cdots\!13}{40\!\cdots\!61}a^{8}-\frac{11\!\cdots\!37}{40\!\cdots\!61}a^{7}-\frac{17\!\cdots\!84}{20\!\cdots\!05}a^{6}+\frac{11\!\cdots\!23}{20\!\cdots\!05}a^{5}+\frac{35\!\cdots\!04}{20\!\cdots\!05}a^{4}-\frac{69\!\cdots\!61}{20\!\cdots\!05}a^{3}-\frac{22\!\cdots\!27}{20\!\cdots\!05}a^{2}+\frac{21\!\cdots\!98}{36\!\cdots\!55}a+\frac{76\!\cdots\!67}{36\!\cdots\!55}$, $\frac{17\!\cdots\!02}{20\!\cdots\!05}a^{21}-\frac{10\!\cdots\!77}{40\!\cdots\!61}a^{20}-\frac{23\!\cdots\!48}{20\!\cdots\!05}a^{19}+\frac{69\!\cdots\!43}{20\!\cdots\!05}a^{18}+\frac{13\!\cdots\!91}{20\!\cdots\!05}a^{17}-\frac{38\!\cdots\!42}{20\!\cdots\!05}a^{16}-\frac{81\!\cdots\!20}{40\!\cdots\!61}a^{15}+\frac{12\!\cdots\!72}{20\!\cdots\!05}a^{14}+\frac{15\!\cdots\!67}{40\!\cdots\!61}a^{13}-\frac{45\!\cdots\!98}{40\!\cdots\!61}a^{12}-\frac{90\!\cdots\!32}{20\!\cdots\!05}a^{11}+\frac{26\!\cdots\!06}{20\!\cdots\!05}a^{10}+\frac{13\!\cdots\!99}{40\!\cdots\!61}a^{9}-\frac{19\!\cdots\!29}{20\!\cdots\!05}a^{8}-\frac{27\!\cdots\!89}{20\!\cdots\!05}a^{7}+\frac{16\!\cdots\!15}{40\!\cdots\!61}a^{6}+\frac{55\!\cdots\!24}{20\!\cdots\!05}a^{5}-\frac{16\!\cdots\!83}{20\!\cdots\!05}a^{4}-\frac{34\!\cdots\!19}{20\!\cdots\!05}a^{3}+\frac{10\!\cdots\!28}{20\!\cdots\!05}a^{2}+\frac{23\!\cdots\!50}{72\!\cdots\!51}a-\frac{34\!\cdots\!12}{36\!\cdots\!55}$ 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:  \( 49537806486500000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{22}\cdot(2\pi)^{0}\cdot 49537806486500000 \cdot 1}{2\cdot\sqrt{170364212909807025886845565709641708504367497216}}\cr\approx \mathstrut & 0.251696685006847 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - 140*x^20 + 8475*x^18 - 291035*x^16 + 6250874*x^14 - 87255611*x^12 + 795482411*x^10 - 4618766140*x^8 + 15964116759*x^6 - 28133091175*x^4 + 16687617633*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 - 140*x^20 + 8475*x^18 - 291035*x^16 + 6250874*x^14 - 87255611*x^12 + 795482411*x^10 - 4618766140*x^8 + 15964116759*x^6 - 28133091175*x^4 + 16687617633*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 - 140*x^20 + 8475*x^18 - 291035*x^16 + 6250874*x^14 - 87255611*x^12 + 795482411*x^10 - 4618766140*x^8 + 15964116759*x^6 - 28133091175*x^4 + 16687617633*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 - 140*x^20 + 8475*x^18 - 291035*x^16 + 6250874*x^14 - 87255611*x^12 + 795482411*x^10 - 4618766140*x^8 + 15964116759*x^6 - 28133091175*x^4 + 16687617633*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 ${\href{/padicField/3.11.0.1}{11} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{9}$ ${\href{/padicField/7.11.0.1}{11} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{5}$ ${\href{/padicField/13.11.0.1}{11} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.11.0.1}{11} }^{2}$ ${\href{/padicField/23.11.0.1}{11} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ R ${\href{/padicField/47.11.0.1}{11} }^{2}$ ${\href{/padicField/53.11.0.1}{11} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}$

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$$[\ ]^{2}$
43.2.0.1$x^{2} + 42 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} + 42 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} + 42 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} + 42 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} + 42 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} + 42 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.2.1$x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
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 $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$