Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 55*x^20 + 1221*x^18 - 14333*x^16 + 98318*x^14 - 409706*x^12 + 1039852*x^10 - 1564134*x^8 + 1316293*x^6 - 561011*x^4 + 96041*x^2 - 1369)
gp: K = bnfinit(y^22 - 55*y^20 + 1221*y^18 - 14333*y^16 + 98318*y^14 - 409706*y^12 + 1039852*y^10 - 1564134*y^8 + 1316293*y^6 - 561011*y^4 + 96041*y^2 - 1369, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 55*x^20 + 1221*x^18 - 14333*x^16 + 98318*x^14 - 409706*x^12 + 1039852*x^10 - 1564134*x^8 + 1316293*x^6 - 561011*x^4 + 96041*x^2 - 1369);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 55*x^20 + 1221*x^18 - 14333*x^16 + 98318*x^14 - 409706*x^12 + 1039852*x^10 - 1564134*x^8 + 1316293*x^6 - 561011*x^4 + 96041*x^2 - 1369)
\( x^{22} - 55 x^{20} + 1221 x^{18} - 14333 x^{16} + 98318 x^{14} - 409706 x^{12} + 1039852 x^{10} + \cdots - 1369 \)
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: |
\(101129833334963577202973160253082380445679616\)
\(\medspace = 2^{42}\cdot 7^{10}\cdot 11^{22}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(100.05\) | 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: |
\(2\), \(7\), \(11\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{62\!\cdots\!69}a^{20}+\frac{18\!\cdots\!63}{62\!\cdots\!69}a^{18}-\frac{14\!\cdots\!03}{62\!\cdots\!69}a^{16}+\frac{27\!\cdots\!82}{62\!\cdots\!69}a^{14}-\frac{20\!\cdots\!13}{62\!\cdots\!69}a^{12}+\frac{14\!\cdots\!15}{62\!\cdots\!69}a^{10}-\frac{88\!\cdots\!44}{62\!\cdots\!69}a^{8}-\frac{23\!\cdots\!49}{62\!\cdots\!69}a^{6}-\frac{10\!\cdots\!04}{62\!\cdots\!69}a^{4}-\frac{13\!\cdots\!00}{62\!\cdots\!69}a^{2}-\frac{21\!\cdots\!89}{62\!\cdots\!69}$, $\frac{1}{23\!\cdots\!53}a^{21}+\frac{18\!\cdots\!63}{23\!\cdots\!53}a^{19}+\frac{47\!\cdots\!92}{62\!\cdots\!69}a^{17}+\frac{46\!\cdots\!65}{23\!\cdots\!53}a^{15}+\frac{79\!\cdots\!84}{23\!\cdots\!53}a^{13}+\frac{10\!\cdots\!19}{23\!\cdots\!53}a^{11}+\frac{87\!\cdots\!22}{23\!\cdots\!53}a^{9}-\frac{77\!\cdots\!77}{23\!\cdots\!53}a^{7}-\frac{51\!\cdots\!56}{23\!\cdots\!53}a^{5}-\frac{10\!\cdots\!73}{23\!\cdots\!53}a^{3}-\frac{21\!\cdots\!96}{23\!\cdots\!53}a$
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 $2$)
| 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: |
$\frac{10\!\cdots\!05}{23\!\cdots\!53}a^{21}-\frac{57\!\cdots\!52}{23\!\cdots\!53}a^{19}+\frac{35\!\cdots\!16}{62\!\cdots\!69}a^{17}-\frac{15\!\cdots\!84}{23\!\cdots\!53}a^{15}+\frac{11\!\cdots\!26}{23\!\cdots\!53}a^{13}-\frac{47\!\cdots\!41}{23\!\cdots\!53}a^{11}+\frac{12\!\cdots\!99}{23\!\cdots\!53}a^{9}-\frac{17\!\cdots\!77}{23\!\cdots\!53}a^{7}+\frac{12\!\cdots\!44}{23\!\cdots\!53}a^{5}-\frac{33\!\cdots\!31}{23\!\cdots\!53}a^{3}-\frac{18\!\cdots\!86}{23\!\cdots\!53}a$, $\frac{83\!\cdots\!82}{23\!\cdots\!53}a^{21}-\frac{45\!\cdots\!18}{23\!\cdots\!53}a^{19}+\frac{27\!\cdots\!24}{62\!\cdots\!69}a^{17}-\frac{11\!\cdots\!45}{23\!\cdots\!53}a^{15}+\frac{77\!\cdots\!34}{23\!\cdots\!53}a^{13}-\frac{30\!\cdots\!12}{23\!\cdots\!53}a^{11}+\frac{73\!\cdots\!35}{23\!\cdots\!53}a^{9}-\frac{97\!\cdots\!49}{23\!\cdots\!53}a^{7}+\frac{64\!\cdots\!01}{23\!\cdots\!53}a^{5}-\frac{15\!\cdots\!67}{23\!\cdots\!53}a^{3}+\frac{10\!\cdots\!86}{23\!\cdots\!53}a$, $\frac{45\!\cdots\!37}{23\!\cdots\!53}a^{21}-\frac{24\!\cdots\!41}{23\!\cdots\!53}a^{19}+\frac{14\!\cdots\!07}{62\!\cdots\!69}a^{17}-\frac{62\!\cdots\!22}{23\!\cdots\!53}a^{15}+\frac{41\!\cdots\!92}{23\!\cdots\!53}a^{13}-\frac{16\!\cdots\!47}{23\!\cdots\!53}a^{11}+\frac{38\!\cdots\!67}{23\!\cdots\!53}a^{9}-\frac{50\!\cdots\!26}{23\!\cdots\!53}a^{7}+\frac{33\!\cdots\!62}{23\!\cdots\!53}a^{5}-\frac{86\!\cdots\!97}{23\!\cdots\!53}a^{3}+\frac{23\!\cdots\!31}{23\!\cdots\!53}a$, $\frac{99\!\cdots\!95}{23\!\cdots\!53}a^{21}-\frac{54\!\cdots\!71}{23\!\cdots\!53}a^{19}+\frac{32\!\cdots\!43}{62\!\cdots\!69}a^{17}-\frac{13\!\cdots\!73}{23\!\cdots\!53}a^{15}+\frac{90\!\cdots\!48}{23\!\cdots\!53}a^{13}-\frac{35\!\cdots\!67}{23\!\cdots\!53}a^{11}+\frac{84\!\cdots\!76}{23\!\cdots\!53}a^{9}-\frac{11\!\cdots\!37}{23\!\cdots\!53}a^{7}+\frac{72\!\cdots\!21}{23\!\cdots\!53}a^{5}-\frac{18\!\cdots\!41}{23\!\cdots\!53}a^{3}+\frac{54\!\cdots\!57}{23\!\cdots\!53}a$, $\frac{65\!\cdots\!89}{62\!\cdots\!69}a^{20}-\frac{35\!\cdots\!24}{62\!\cdots\!69}a^{18}+\frac{76\!\cdots\!71}{62\!\cdots\!69}a^{16}-\frac{86\!\cdots\!97}{62\!\cdots\!69}a^{14}+\frac{56\!\cdots\!35}{62\!\cdots\!69}a^{12}-\frac{22\!\cdots\!99}{62\!\cdots\!69}a^{10}+\frac{51\!\cdots\!24}{62\!\cdots\!69}a^{8}-\frac{67\!\cdots\!55}{62\!\cdots\!69}a^{6}+\frac{44\!\cdots\!19}{62\!\cdots\!69}a^{4}-\frac{11\!\cdots\!12}{62\!\cdots\!69}a^{2}+\frac{13\!\cdots\!30}{62\!\cdots\!69}$, $\frac{45\!\cdots\!98}{23\!\cdots\!53}a^{21}-\frac{24\!\cdots\!43}{23\!\cdots\!53}a^{19}+\frac{14\!\cdots\!65}{62\!\cdots\!69}a^{17}-\frac{60\!\cdots\!09}{23\!\cdots\!53}a^{15}+\frac{40\!\cdots\!05}{23\!\cdots\!53}a^{13}-\frac{15\!\cdots\!14}{23\!\cdots\!53}a^{11}+\frac{36\!\cdots\!22}{23\!\cdots\!53}a^{9}-\frac{48\!\cdots\!33}{23\!\cdots\!53}a^{7}+\frac{31\!\cdots\!60}{23\!\cdots\!53}a^{5}-\frac{78\!\cdots\!51}{23\!\cdots\!53}a^{3}+\frac{14\!\cdots\!17}{23\!\cdots\!53}a$, $\frac{40\!\cdots\!12}{62\!\cdots\!69}a^{20}-\frac{21\!\cdots\!34}{62\!\cdots\!69}a^{18}+\frac{47\!\cdots\!46}{62\!\cdots\!69}a^{16}-\frac{55\!\cdots\!82}{62\!\cdots\!69}a^{14}+\frac{36\!\cdots\!95}{62\!\cdots\!69}a^{12}-\frac{14\!\cdots\!04}{62\!\cdots\!69}a^{10}+\frac{34\!\cdots\!46}{62\!\cdots\!69}a^{8}-\frac{45\!\cdots\!75}{62\!\cdots\!69}a^{6}+\frac{29\!\cdots\!87}{62\!\cdots\!69}a^{4}-\frac{72\!\cdots\!70}{62\!\cdots\!69}a^{2}+\frac{10\!\cdots\!52}{62\!\cdots\!69}$, $\frac{13\!\cdots\!23}{23\!\cdots\!53}a^{21}-\frac{73\!\cdots\!04}{23\!\cdots\!53}a^{19}+\frac{44\!\cdots\!65}{62\!\cdots\!69}a^{17}-\frac{19\!\cdots\!95}{23\!\cdots\!53}a^{15}+\frac{13\!\cdots\!14}{23\!\cdots\!53}a^{13}-\frac{57\!\cdots\!77}{23\!\cdots\!53}a^{11}+\frac{14\!\cdots\!66}{23\!\cdots\!53}a^{9}-\frac{21\!\cdots\!89}{23\!\cdots\!53}a^{7}+\frac{16\!\cdots\!91}{23\!\cdots\!53}a^{5}-\frac{47\!\cdots\!87}{23\!\cdots\!53}a^{3}+\frac{18\!\cdots\!24}{23\!\cdots\!53}a$, $\frac{67\!\cdots\!40}{62\!\cdots\!69}a^{20}-\frac{36\!\cdots\!94}{62\!\cdots\!69}a^{18}+\frac{80\!\cdots\!01}{62\!\cdots\!69}a^{16}-\frac{93\!\cdots\!91}{62\!\cdots\!69}a^{14}+\frac{62\!\cdots\!38}{62\!\cdots\!69}a^{12}-\frac{25\!\cdots\!63}{62\!\cdots\!69}a^{10}+\frac{60\!\cdots\!92}{62\!\cdots\!69}a^{8}-\frac{82\!\cdots\!19}{62\!\cdots\!69}a^{6}+\frac{58\!\cdots\!60}{62\!\cdots\!69}a^{4}-\frac{18\!\cdots\!27}{62\!\cdots\!69}a^{2}+\frac{15\!\cdots\!46}{62\!\cdots\!69}$, $\frac{32\!\cdots\!66}{23\!\cdots\!53}a^{21}-\frac{20\!\cdots\!47}{62\!\cdots\!69}a^{20}-\frac{17\!\cdots\!06}{23\!\cdots\!53}a^{19}+\frac{11\!\cdots\!26}{62\!\cdots\!69}a^{18}+\frac{99\!\cdots\!41}{62\!\cdots\!69}a^{17}-\frac{24\!\cdots\!97}{62\!\cdots\!69}a^{16}-\frac{41\!\cdots\!12}{23\!\cdots\!53}a^{15}+\frac{27\!\cdots\!16}{62\!\cdots\!69}a^{14}+\frac{26\!\cdots\!04}{23\!\cdots\!53}a^{13}-\frac{17\!\cdots\!14}{62\!\cdots\!69}a^{12}-\frac{10\!\cdots\!72}{23\!\cdots\!53}a^{11}+\frac{70\!\cdots\!02}{62\!\cdots\!69}a^{10}+\frac{23\!\cdots\!60}{23\!\cdots\!53}a^{9}-\frac{16\!\cdots\!53}{62\!\cdots\!69}a^{8}-\frac{30\!\cdots\!16}{23\!\cdots\!53}a^{7}+\frac{21\!\cdots\!38}{62\!\cdots\!69}a^{6}+\frac{19\!\cdots\!01}{23\!\cdots\!53}a^{5}-\frac{13\!\cdots\!81}{62\!\cdots\!69}a^{4}-\frac{49\!\cdots\!94}{23\!\cdots\!53}a^{3}+\frac{34\!\cdots\!95}{62\!\cdots\!69}a^{2}+\frac{45\!\cdots\!97}{23\!\cdots\!53}a-\frac{51\!\cdots\!29}{62\!\cdots\!69}$, $\frac{62\!\cdots\!00}{23\!\cdots\!53}a^{21}+\frac{89\!\cdots\!18}{62\!\cdots\!69}a^{20}-\frac{33\!\cdots\!79}{23\!\cdots\!53}a^{19}-\frac{49\!\cdots\!84}{62\!\cdots\!69}a^{18}+\frac{20\!\cdots\!32}{62\!\cdots\!69}a^{17}+\frac{10\!\cdots\!88}{62\!\cdots\!69}a^{16}-\frac{85\!\cdots\!63}{23\!\cdots\!53}a^{15}-\frac{12\!\cdots\!38}{62\!\cdots\!69}a^{14}+\frac{57\!\cdots\!28}{23\!\cdots\!53}a^{13}+\frac{81\!\cdots\!48}{62\!\cdots\!69}a^{12}-\frac{22\!\cdots\!29}{23\!\cdots\!53}a^{11}-\frac{32\!\cdots\!45}{62\!\cdots\!69}a^{10}+\frac{53\!\cdots\!00}{23\!\cdots\!53}a^{9}+\frac{76\!\cdots\!07}{62\!\cdots\!69}a^{8}-\frac{71\!\cdots\!72}{23\!\cdots\!53}a^{7}-\frac{10\!\cdots\!03}{62\!\cdots\!69}a^{6}+\frac{48\!\cdots\!25}{23\!\cdots\!53}a^{5}+\frac{67\!\cdots\!07}{62\!\cdots\!69}a^{4}-\frac{12\!\cdots\!60}{23\!\cdots\!53}a^{3}-\frac{17\!\cdots\!47}{62\!\cdots\!69}a^{2}+\frac{37\!\cdots\!35}{23\!\cdots\!53}a+\frac{36\!\cdots\!75}{62\!\cdots\!69}$, $\frac{62\!\cdots\!00}{23\!\cdots\!53}a^{21}-\frac{89\!\cdots\!18}{62\!\cdots\!69}a^{20}-\frac{33\!\cdots\!79}{23\!\cdots\!53}a^{19}+\frac{49\!\cdots\!84}{62\!\cdots\!69}a^{18}+\frac{20\!\cdots\!32}{62\!\cdots\!69}a^{17}-\frac{10\!\cdots\!88}{62\!\cdots\!69}a^{16}-\frac{85\!\cdots\!63}{23\!\cdots\!53}a^{15}+\frac{12\!\cdots\!38}{62\!\cdots\!69}a^{14}+\frac{57\!\cdots\!28}{23\!\cdots\!53}a^{13}-\frac{81\!\cdots\!48}{62\!\cdots\!69}a^{12}-\frac{22\!\cdots\!29}{23\!\cdots\!53}a^{11}+\frac{32\!\cdots\!45}{62\!\cdots\!69}a^{10}+\frac{53\!\cdots\!00}{23\!\cdots\!53}a^{9}-\frac{76\!\cdots\!07}{62\!\cdots\!69}a^{8}-\frac{71\!\cdots\!72}{23\!\cdots\!53}a^{7}+\frac{10\!\cdots\!03}{62\!\cdots\!69}a^{6}+\frac{48\!\cdots\!25}{23\!\cdots\!53}a^{5}-\frac{67\!\cdots\!07}{62\!\cdots\!69}a^{4}-\frac{12\!\cdots\!60}{23\!\cdots\!53}a^{3}+\frac{17\!\cdots\!47}{62\!\cdots\!69}a^{2}+\frac{37\!\cdots\!35}{23\!\cdots\!53}a-\frac{36\!\cdots\!75}{62\!\cdots\!69}$, $\frac{35\!\cdots\!49}{23\!\cdots\!53}a^{21}-\frac{31\!\cdots\!23}{62\!\cdots\!69}a^{20}-\frac{19\!\cdots\!27}{23\!\cdots\!53}a^{19}+\frac{17\!\cdots\!18}{62\!\cdots\!69}a^{18}+\frac{11\!\cdots\!81}{62\!\cdots\!69}a^{17}-\frac{39\!\cdots\!25}{62\!\cdots\!69}a^{16}-\frac{48\!\cdots\!73}{23\!\cdots\!53}a^{15}+\frac{46\!\cdots\!97}{62\!\cdots\!69}a^{14}+\frac{32\!\cdots\!06}{23\!\cdots\!53}a^{13}-\frac{32\!\cdots\!90}{62\!\cdots\!69}a^{12}-\frac{12\!\cdots\!34}{23\!\cdots\!53}a^{11}+\frac{13\!\cdots\!20}{62\!\cdots\!69}a^{10}+\frac{31\!\cdots\!52}{23\!\cdots\!53}a^{9}-\frac{33\!\cdots\!56}{62\!\cdots\!69}a^{8}-\frac{43\!\cdots\!68}{23\!\cdots\!53}a^{7}+\frac{46\!\cdots\!05}{62\!\cdots\!69}a^{6}+\frac{32\!\cdots\!13}{23\!\cdots\!53}a^{5}-\frac{33\!\cdots\!90}{62\!\cdots\!69}a^{4}-\frac{10\!\cdots\!30}{23\!\cdots\!53}a^{3}+\frac{95\!\cdots\!12}{62\!\cdots\!69}a^{2}+\frac{13\!\cdots\!33}{23\!\cdots\!53}a-\frac{57\!\cdots\!63}{62\!\cdots\!69}$, $\frac{45\!\cdots\!01}{23\!\cdots\!53}a^{21}-\frac{44\!\cdots\!70}{62\!\cdots\!69}a^{20}-\frac{24\!\cdots\!13}{23\!\cdots\!53}a^{19}+\frac{24\!\cdots\!71}{62\!\cdots\!69}a^{18}+\frac{14\!\cdots\!20}{62\!\cdots\!69}a^{17}-\frac{52\!\cdots\!03}{62\!\cdots\!69}a^{16}-\frac{64\!\cdots\!05}{23\!\cdots\!53}a^{15}+\frac{59\!\cdots\!13}{62\!\cdots\!69}a^{14}+\frac{43\!\cdots\!80}{23\!\cdots\!53}a^{13}-\frac{39\!\cdots\!96}{62\!\cdots\!69}a^{12}-\frac{18\!\cdots\!15}{23\!\cdots\!53}a^{11}+\frac{15\!\cdots\!76}{62\!\cdots\!69}a^{10}+\frac{45\!\cdots\!41}{23\!\cdots\!53}a^{9}-\frac{34\!\cdots\!45}{62\!\cdots\!69}a^{8}-\frac{68\!\cdots\!07}{23\!\cdots\!53}a^{7}+\frac{42\!\cdots\!47}{62\!\cdots\!69}a^{6}+\frac{56\!\cdots\!58}{23\!\cdots\!53}a^{5}-\frac{23\!\cdots\!30}{62\!\cdots\!69}a^{4}-\frac{23\!\cdots\!82}{23\!\cdots\!53}a^{3}+\frac{20\!\cdots\!69}{62\!\cdots\!69}a^{2}+\frac{37\!\cdots\!67}{23\!\cdots\!53}a+\frac{13\!\cdots\!76}{62\!\cdots\!69}$, $\frac{50\!\cdots\!52}{23\!\cdots\!53}a^{21}+\frac{55\!\cdots\!36}{62\!\cdots\!69}a^{20}-\frac{29\!\cdots\!38}{23\!\cdots\!53}a^{19}-\frac{28\!\cdots\!49}{62\!\cdots\!69}a^{18}+\frac{18\!\cdots\!69}{62\!\cdots\!69}a^{17}+\frac{58\!\cdots\!19}{62\!\cdots\!69}a^{16}-\frac{87\!\cdots\!96}{23\!\cdots\!53}a^{15}-\frac{61\!\cdots\!03}{62\!\cdots\!69}a^{14}+\frac{64\!\cdots\!59}{23\!\cdots\!53}a^{13}+\frac{35\!\cdots\!70}{62\!\cdots\!69}a^{12}-\frac{28\!\cdots\!03}{23\!\cdots\!53}a^{11}-\frac{12\!\cdots\!51}{62\!\cdots\!69}a^{10}+\frac{73\!\cdots\!74}{23\!\cdots\!53}a^{9}+\frac{23\!\cdots\!82}{62\!\cdots\!69}a^{8}-\frac{10\!\cdots\!32}{23\!\cdots\!53}a^{7}-\frac{23\!\cdots\!06}{62\!\cdots\!69}a^{6}+\frac{71\!\cdots\!80}{23\!\cdots\!53}a^{5}+\frac{10\!\cdots\!50}{62\!\cdots\!69}a^{4}-\frac{14\!\cdots\!23}{23\!\cdots\!53}a^{3}-\frac{18\!\cdots\!27}{62\!\cdots\!69}a^{2}-\frac{14\!\cdots\!35}{23\!\cdots\!53}a+\frac{80\!\cdots\!97}{62\!\cdots\!69}$, $\frac{39\!\cdots\!60}{23\!\cdots\!53}a^{21}+\frac{38\!\cdots\!83}{62\!\cdots\!69}a^{20}-\frac{21\!\cdots\!93}{23\!\cdots\!53}a^{19}-\frac{21\!\cdots\!35}{62\!\cdots\!69}a^{18}+\frac{12\!\cdots\!04}{62\!\cdots\!69}a^{17}+\frac{48\!\cdots\!18}{62\!\cdots\!69}a^{16}-\frac{54\!\cdots\!53}{23\!\cdots\!53}a^{15}-\frac{58\!\cdots\!76}{62\!\cdots\!69}a^{14}+\frac{36\!\cdots\!06}{23\!\cdots\!53}a^{13}+\frac{41\!\cdots\!13}{62\!\cdots\!69}a^{12}-\frac{14\!\cdots\!67}{23\!\cdots\!53}a^{11}-\frac{17\!\cdots\!46}{62\!\cdots\!69}a^{10}+\frac{34\!\cdots\!11}{23\!\cdots\!53}a^{9}+\frac{44\!\cdots\!45}{62\!\cdots\!69}a^{8}-\frac{46\!\cdots\!89}{23\!\cdots\!53}a^{7}-\frac{63\!\cdots\!82}{62\!\cdots\!69}a^{6}+\frac{30\!\cdots\!49}{23\!\cdots\!53}a^{5}+\frac{45\!\cdots\!28}{62\!\cdots\!69}a^{4}-\frac{72\!\cdots\!32}{23\!\cdots\!53}a^{3}-\frac{11\!\cdots\!83}{62\!\cdots\!69}a^{2}+\frac{39\!\cdots\!56}{23\!\cdots\!53}a+\frac{17\!\cdots\!76}{62\!\cdots\!69}$, $\frac{26\!\cdots\!60}{23\!\cdots\!53}a^{21}+\frac{26\!\cdots\!94}{62\!\cdots\!69}a^{20}-\frac{14\!\cdots\!00}{23\!\cdots\!53}a^{19}-\frac{14\!\cdots\!56}{62\!\cdots\!69}a^{18}+\frac{85\!\cdots\!86}{62\!\cdots\!69}a^{17}+\frac{31\!\cdots\!47}{62\!\cdots\!69}a^{16}-\frac{36\!\cdots\!73}{23\!\cdots\!53}a^{15}-\frac{36\!\cdots\!30}{62\!\cdots\!69}a^{14}+\frac{23\!\cdots\!62}{23\!\cdots\!53}a^{13}+\frac{23\!\cdots\!36}{62\!\cdots\!69}a^{12}-\frac{92\!\cdots\!53}{23\!\cdots\!53}a^{11}-\frac{94\!\cdots\!73}{62\!\cdots\!69}a^{10}+\frac{20\!\cdots\!38}{23\!\cdots\!53}a^{9}+\frac{22\!\cdots\!27}{62\!\cdots\!69}a^{8}-\frac{25\!\cdots\!96}{23\!\cdots\!53}a^{7}-\frac{29\!\cdots\!48}{62\!\cdots\!69}a^{6}+\frac{13\!\cdots\!92}{23\!\cdots\!53}a^{5}+\frac{19\!\cdots\!97}{62\!\cdots\!69}a^{4}-\frac{13\!\cdots\!89}{23\!\cdots\!53}a^{3}-\frac{47\!\cdots\!34}{62\!\cdots\!69}a^{2}-\frac{45\!\cdots\!33}{23\!\cdots\!53}a+\frac{81\!\cdots\!33}{62\!\cdots\!69}$, $\frac{33\!\cdots\!48}{23\!\cdots\!53}a^{21}+\frac{70\!\cdots\!22}{62\!\cdots\!69}a^{20}-\frac{18\!\cdots\!33}{23\!\cdots\!53}a^{19}-\frac{38\!\cdots\!64}{62\!\cdots\!69}a^{18}+\frac{10\!\cdots\!92}{62\!\cdots\!69}a^{17}+\frac{83\!\cdots\!06}{62\!\cdots\!69}a^{16}-\frac{45\!\cdots\!58}{23\!\cdots\!53}a^{15}-\frac{96\!\cdots\!78}{62\!\cdots\!69}a^{14}+\frac{30\!\cdots\!52}{23\!\cdots\!53}a^{13}+\frac{64\!\cdots\!69}{62\!\cdots\!69}a^{12}-\frac{12\!\cdots\!43}{23\!\cdots\!53}a^{11}-\frac{25\!\cdots\!24}{62\!\cdots\!69}a^{10}+\frac{28\!\cdots\!94}{23\!\cdots\!53}a^{9}+\frac{60\!\cdots\!21}{62\!\cdots\!69}a^{8}-\frac{37\!\cdots\!41}{23\!\cdots\!53}a^{7}-\frac{80\!\cdots\!73}{62\!\cdots\!69}a^{6}+\frac{24\!\cdots\!55}{23\!\cdots\!53}a^{5}+\frac{52\!\cdots\!46}{62\!\cdots\!69}a^{4}-\frac{59\!\cdots\!59}{23\!\cdots\!53}a^{3}-\frac{13\!\cdots\!31}{62\!\cdots\!69}a^{2}-\frac{84\!\cdots\!70}{23\!\cdots\!53}a+\frac{24\!\cdots\!56}{62\!\cdots\!69}$, $\frac{22\!\cdots\!61}{23\!\cdots\!53}a^{21}+\frac{42\!\cdots\!05}{62\!\cdots\!69}a^{20}-\frac{12\!\cdots\!02}{23\!\cdots\!53}a^{19}-\frac{23\!\cdots\!63}{62\!\cdots\!69}a^{18}+\frac{72\!\cdots\!23}{62\!\cdots\!69}a^{17}+\frac{50\!\cdots\!64}{62\!\cdots\!69}a^{16}-\frac{30\!\cdots\!20}{23\!\cdots\!53}a^{15}-\frac{58\!\cdots\!42}{62\!\cdots\!69}a^{14}+\frac{20\!\cdots\!03}{23\!\cdots\!53}a^{13}+\frac{39\!\cdots\!67}{62\!\cdots\!69}a^{12}-\frac{83\!\cdots\!32}{23\!\cdots\!53}a^{11}-\frac{15\!\cdots\!06}{62\!\cdots\!69}a^{10}+\frac{19\!\cdots\!98}{23\!\cdots\!53}a^{9}+\frac{37\!\cdots\!66}{62\!\cdots\!69}a^{8}-\frac{26\!\cdots\!31}{23\!\cdots\!53}a^{7}-\frac{49\!\cdots\!79}{62\!\cdots\!69}a^{6}+\frac{18\!\cdots\!51}{23\!\cdots\!53}a^{5}+\frac{32\!\cdots\!73}{62\!\cdots\!69}a^{4}-\frac{50\!\cdots\!57}{23\!\cdots\!53}a^{3}-\frac{79\!\cdots\!33}{62\!\cdots\!69}a^{2}+\frac{23\!\cdots\!87}{23\!\cdots\!53}a+\frac{63\!\cdots\!84}{62\!\cdots\!69}$, $\frac{21\!\cdots\!86}{23\!\cdots\!53}a^{21}-\frac{37\!\cdots\!76}{62\!\cdots\!69}a^{20}-\frac{11\!\cdots\!45}{23\!\cdots\!53}a^{19}+\frac{20\!\cdots\!35}{62\!\cdots\!69}a^{18}+\frac{68\!\cdots\!17}{62\!\cdots\!69}a^{17}-\frac{45\!\cdots\!31}{62\!\cdots\!69}a^{16}-\frac{28\!\cdots\!06}{23\!\cdots\!53}a^{15}+\frac{53\!\cdots\!80}{62\!\cdots\!69}a^{14}+\frac{19\!\cdots\!11}{23\!\cdots\!53}a^{13}-\frac{35\!\cdots\!16}{62\!\cdots\!69}a^{12}-\frac{75\!\cdots\!82}{23\!\cdots\!53}a^{11}+\frac{14\!\cdots\!58}{62\!\cdots\!69}a^{10}+\frac{17\!\cdots\!59}{23\!\cdots\!53}a^{9}-\frac{34\!\cdots\!76}{62\!\cdots\!69}a^{8}-\frac{23\!\cdots\!97}{23\!\cdots\!53}a^{7}+\frac{46\!\cdots\!26}{62\!\cdots\!69}a^{6}+\frac{15\!\cdots\!74}{23\!\cdots\!53}a^{5}-\frac{30\!\cdots\!92}{62\!\cdots\!69}a^{4}-\frac{36\!\cdots\!00}{23\!\cdots\!53}a^{3}+\frac{75\!\cdots\!69}{62\!\cdots\!69}a^{2}+\frac{41\!\cdots\!69}{23\!\cdots\!53}a-\frac{11\!\cdots\!10}{62\!\cdots\!69}$, $\frac{19\!\cdots\!62}{23\!\cdots\!53}a^{21}+\frac{56\!\cdots\!57}{62\!\cdots\!69}a^{20}-\frac{10\!\cdots\!83}{23\!\cdots\!53}a^{19}-\frac{31\!\cdots\!48}{62\!\cdots\!69}a^{18}+\frac{62\!\cdots\!42}{62\!\cdots\!69}a^{17}+\frac{67\!\cdots\!15}{62\!\cdots\!69}a^{16}-\frac{26\!\cdots\!73}{23\!\cdots\!53}a^{15}-\frac{77\!\cdots\!98}{62\!\cdots\!69}a^{14}+\frac{17\!\cdots\!62}{23\!\cdots\!53}a^{13}+\frac{51\!\cdots\!17}{62\!\cdots\!69}a^{12}-\frac{68\!\cdots\!13}{23\!\cdots\!53}a^{11}-\frac{20\!\cdots\!01}{62\!\cdots\!69}a^{10}+\frac{15\!\cdots\!83}{23\!\cdots\!53}a^{9}+\frac{47\!\cdots\!63}{62\!\cdots\!69}a^{8}-\frac{20\!\cdots\!12}{23\!\cdots\!53}a^{7}-\frac{62\!\cdots\!72}{62\!\cdots\!69}a^{6}+\frac{12\!\cdots\!23}{23\!\cdots\!53}a^{5}+\frac{39\!\cdots\!92}{62\!\cdots\!69}a^{4}-\frac{27\!\cdots\!83}{23\!\cdots\!53}a^{3}-\frac{94\!\cdots\!03}{62\!\cdots\!69}a^{2}-\frac{25\!\cdots\!59}{23\!\cdots\!53}a+\frac{11\!\cdots\!42}{62\!\cdots\!69}$
| 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: | \( 3806572397390000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{22}\cdot(2\pi)^{0}\cdot 3806572397390000 \cdot 1}{2\cdot\sqrt{101129833334963577202973160253082380445679616}}\cr\approx \mathstrut & 0.793824241815585 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 55*x^20 + 1221*x^18 - 14333*x^16 + 98318*x^14 - 409706*x^12 + 1039852*x^10 - 1564134*x^8 + 1316293*x^6 - 561011*x^4 + 96041*x^2 - 1369)
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 - 55*x^20 + 1221*x^18 - 14333*x^16 + 98318*x^14 - 409706*x^12 + 1039852*x^10 - 1564134*x^8 + 1316293*x^6 - 561011*x^4 + 96041*x^2 - 1369, 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 - 55*x^20 + 1221*x^18 - 14333*x^16 + 98318*x^14 - 409706*x^12 + 1039852*x^10 - 1564134*x^8 + 1316293*x^6 - 561011*x^4 + 96041*x^2 - 1369);
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 - 55*x^20 + 1221*x^18 - 14333*x^16 + 98318*x^14 - 409706*x^12 + 1039852*x^10 - 1564134*x^8 + 1316293*x^6 - 561011*x^4 + 96041*x^2 - 1369);
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
$C_2^{10}.F_{11}$ (as 22T34):
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 112640 |
| The 44 conjugacy class representatives for $C_2^{10}.F_{11}$ |
| Character table for $C_2^{10}.F_{11}$ |
Intermediate fields
| 11.11.4910318845910094848.1 |
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 siblings: | 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 | $20{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | R | ${\href{/padicField/13.5.0.1}{5} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $22$ | $1$ | $42$ | |||
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.20.10.1 | $x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ | |
|
\(11\)
| 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |
| 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |