Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^20 + 33*x^18 - 66*x^17 - 44*x^16 + 308*x^15 + 253*x^14 - 44*x^13 + 913*x^12 + 1840*x^11 + 1309*x^10 - 814*x^9 + 2904*x^8 + 4752*x^7 + 7117*x^6 - 4004*x^5 + 825*x^4 - 44*x^3 + 6721*x^2 - 5434*x + 1644)
gp: K = bnfinit(y^22 - 11*y^20 + 33*y^18 - 66*y^17 - 44*y^16 + 308*y^15 + 253*y^14 - 44*y^13 + 913*y^12 + 1840*y^11 + 1309*y^10 - 814*y^9 + 2904*y^8 + 4752*y^7 + 7117*y^6 - 4004*y^5 + 825*y^4 - 44*y^3 + 6721*y^2 - 5434*y + 1644, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^20 + 33*x^18 - 66*x^17 - 44*x^16 + 308*x^15 + 253*x^14 - 44*x^13 + 913*x^12 + 1840*x^11 + 1309*x^10 - 814*x^9 + 2904*x^8 + 4752*x^7 + 7117*x^6 - 4004*x^5 + 825*x^4 - 44*x^3 + 6721*x^2 - 5434*x + 1644);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^20 + 33*x^18 - 66*x^17 - 44*x^16 + 308*x^15 + 253*x^14 - 44*x^13 + 913*x^12 + 1840*x^11 + 1309*x^10 - 814*x^9 + 2904*x^8 + 4752*x^7 + 7117*x^6 - 4004*x^5 + 825*x^4 - 44*x^3 + 6721*x^2 - 5434*x + 1644)
\( x^{22} - 11 x^{20} + 33 x^{18} - 66 x^{17} - 44 x^{16} + 308 x^{15} + 253 x^{14} - 44 x^{13} + \cdots + 1644 \)
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: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-61533497770089105512253259111202816\)
\(\medspace = -\,2^{36}\cdot 11^{23}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(38.13\) | 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: |
\(2\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{12}+\frac{1}{4}a^{8}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{9}+\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}+\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{18}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}+\frac{1}{8}a^{10}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{3}-\frac{3}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{19}+\frac{1}{8}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{3}{8}a^{3}$, $\frac{1}{16}a^{20}-\frac{1}{16}a^{19}-\frac{1}{16}a^{18}-\frac{1}{8}a^{17}-\frac{3}{16}a^{12}+\frac{3}{16}a^{11}-\frac{1}{16}a^{10}+\frac{1}{8}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{16}a^{4}-\frac{1}{16}a^{3}-\frac{5}{16}a^{2}-\frac{3}{8}a-\frac{1}{4}$, $\frac{1}{15\!\cdots\!84}a^{21}+\frac{35\!\cdots\!21}{15\!\cdots\!84}a^{20}+\frac{13\!\cdots\!83}{42\!\cdots\!32}a^{19}+\frac{12\!\cdots\!39}{26\!\cdots\!77}a^{18}+\frac{60\!\cdots\!24}{97\!\cdots\!49}a^{17}-\frac{31\!\cdots\!97}{39\!\cdots\!96}a^{16}-\frac{92\!\cdots\!87}{39\!\cdots\!96}a^{15}+\frac{20\!\cdots\!57}{19\!\cdots\!98}a^{14}-\frac{36\!\cdots\!91}{15\!\cdots\!84}a^{13}+\frac{16\!\cdots\!57}{42\!\cdots\!32}a^{12}-\frac{38\!\cdots\!21}{15\!\cdots\!84}a^{11}+\frac{40\!\cdots\!29}{19\!\cdots\!98}a^{10}-\frac{48\!\cdots\!43}{19\!\cdots\!98}a^{9}+\frac{21\!\cdots\!11}{39\!\cdots\!96}a^{8}+\frac{27\!\cdots\!43}{39\!\cdots\!96}a^{7}+\frac{19\!\cdots\!85}{39\!\cdots\!96}a^{6}-\frac{15\!\cdots\!11}{15\!\cdots\!84}a^{5}+\frac{66\!\cdots\!45}{15\!\cdots\!84}a^{4}+\frac{12\!\cdots\!15}{15\!\cdots\!84}a^{3}+\frac{27\!\cdots\!98}{97\!\cdots\!49}a^{2}+\frac{48\!\cdots\!76}{97\!\cdots\!49}a+\frac{78\!\cdots\!55}{19\!\cdots\!98}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $10$ | 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: |
$\frac{38\!\cdots\!87}{78\!\cdots\!92}a^{21}+\frac{11\!\cdots\!03}{15\!\cdots\!84}a^{20}-\frac{21\!\cdots\!49}{42\!\cdots\!32}a^{19}-\frac{32\!\cdots\!57}{42\!\cdots\!32}a^{18}+\frac{10\!\cdots\!03}{78\!\cdots\!92}a^{17}-\frac{39\!\cdots\!85}{39\!\cdots\!96}a^{16}-\frac{11\!\cdots\!53}{19\!\cdots\!98}a^{15}+\frac{40\!\cdots\!51}{39\!\cdots\!96}a^{14}+\frac{25\!\cdots\!15}{78\!\cdots\!92}a^{13}+\frac{10\!\cdots\!47}{42\!\cdots\!32}a^{12}+\frac{79\!\cdots\!47}{15\!\cdots\!84}a^{11}+\frac{24\!\cdots\!39}{15\!\cdots\!84}a^{10}+\frac{17\!\cdots\!21}{78\!\cdots\!92}a^{9}+\frac{41\!\cdots\!23}{39\!\cdots\!96}a^{8}+\frac{12\!\cdots\!61}{97\!\cdots\!49}a^{7}+\frac{17\!\cdots\!07}{39\!\cdots\!96}a^{6}+\frac{59\!\cdots\!83}{78\!\cdots\!92}a^{5}+\frac{62\!\cdots\!51}{15\!\cdots\!84}a^{4}+\frac{53\!\cdots\!07}{15\!\cdots\!84}a^{3}+\frac{12\!\cdots\!79}{15\!\cdots\!84}a^{2}+\frac{23\!\cdots\!81}{78\!\cdots\!92}a+\frac{36\!\cdots\!23}{39\!\cdots\!96}$, $\frac{43\!\cdots\!11}{15\!\cdots\!84}a^{21}+\frac{17\!\cdots\!81}{15\!\cdots\!84}a^{20}-\frac{13\!\cdots\!17}{42\!\cdots\!32}a^{19}-\frac{13\!\cdots\!79}{10\!\cdots\!08}a^{18}+\frac{19\!\cdots\!89}{19\!\cdots\!98}a^{17}-\frac{55\!\cdots\!59}{39\!\cdots\!96}a^{16}-\frac{40\!\cdots\!97}{19\!\cdots\!98}a^{15}+\frac{16\!\cdots\!99}{19\!\cdots\!98}a^{14}+\frac{17\!\cdots\!35}{15\!\cdots\!84}a^{13}+\frac{16\!\cdots\!53}{42\!\cdots\!32}a^{12}+\frac{34\!\cdots\!27}{15\!\cdots\!84}a^{11}+\frac{23\!\cdots\!93}{39\!\cdots\!96}a^{10}+\frac{19\!\cdots\!21}{39\!\cdots\!96}a^{9}-\frac{96\!\cdots\!09}{39\!\cdots\!96}a^{8}+\frac{19\!\cdots\!55}{39\!\cdots\!96}a^{7}+\frac{56\!\cdots\!31}{39\!\cdots\!96}a^{6}+\frac{36\!\cdots\!79}{15\!\cdots\!84}a^{5}-\frac{11\!\cdots\!03}{15\!\cdots\!84}a^{4}-\frac{13\!\cdots\!85}{15\!\cdots\!84}a^{3}-\frac{21\!\cdots\!17}{39\!\cdots\!96}a^{2}+\frac{34\!\cdots\!11}{19\!\cdots\!98}a-\frac{17\!\cdots\!91}{19\!\cdots\!98}$, $\frac{95\!\cdots\!38}{97\!\cdots\!49}a^{21}-\frac{14\!\cdots\!25}{15\!\cdots\!84}a^{20}-\frac{48\!\cdots\!35}{42\!\cdots\!32}a^{19}+\frac{25\!\cdots\!29}{42\!\cdots\!32}a^{18}+\frac{31\!\cdots\!75}{78\!\cdots\!92}a^{17}-\frac{12\!\cdots\!75}{19\!\cdots\!98}a^{16}-\frac{22\!\cdots\!95}{39\!\cdots\!96}a^{15}+\frac{13\!\cdots\!11}{39\!\cdots\!96}a^{14}+\frac{50\!\cdots\!53}{19\!\cdots\!98}a^{13}-\frac{10\!\cdots\!21}{42\!\cdots\!32}a^{12}+\frac{88\!\cdots\!13}{15\!\cdots\!84}a^{11}+\frac{21\!\cdots\!81}{15\!\cdots\!84}a^{10}+\frac{32\!\cdots\!71}{78\!\cdots\!92}a^{9}-\frac{51\!\cdots\!37}{19\!\cdots\!98}a^{8}+\frac{16\!\cdots\!85}{39\!\cdots\!96}a^{7}+\frac{79\!\cdots\!01}{19\!\cdots\!98}a^{6}+\frac{18\!\cdots\!29}{39\!\cdots\!96}a^{5}-\frac{13\!\cdots\!45}{15\!\cdots\!84}a^{4}-\frac{12\!\cdots\!95}{15\!\cdots\!84}a^{3}-\frac{23\!\cdots\!31}{15\!\cdots\!84}a^{2}+\frac{56\!\cdots\!15}{78\!\cdots\!92}a-\frac{12\!\cdots\!19}{39\!\cdots\!96}$, $\frac{22\!\cdots\!31}{39\!\cdots\!96}a^{21}+\frac{68\!\cdots\!13}{78\!\cdots\!92}a^{20}-\frac{13\!\cdots\!15}{21\!\cdots\!16}a^{19}-\frac{33\!\cdots\!85}{21\!\cdots\!16}a^{18}+\frac{21\!\cdots\!15}{97\!\cdots\!49}a^{17}-\frac{54\!\cdots\!37}{19\!\cdots\!98}a^{16}-\frac{40\!\cdots\!27}{97\!\cdots\!49}a^{15}+\frac{66\!\cdots\!41}{39\!\cdots\!96}a^{14}+\frac{22\!\cdots\!19}{97\!\cdots\!49}a^{13}-\frac{17\!\cdots\!73}{21\!\cdots\!16}a^{12}+\frac{20\!\cdots\!75}{78\!\cdots\!92}a^{11}+\frac{83\!\cdots\!03}{78\!\cdots\!92}a^{10}+\frac{11\!\cdots\!55}{19\!\cdots\!98}a^{9}-\frac{55\!\cdots\!59}{39\!\cdots\!96}a^{8}+\frac{23\!\cdots\!61}{39\!\cdots\!96}a^{7}+\frac{25\!\cdots\!86}{97\!\cdots\!49}a^{6}+\frac{16\!\cdots\!35}{39\!\cdots\!96}a^{5}-\frac{31\!\cdots\!65}{78\!\cdots\!92}a^{4}-\frac{25\!\cdots\!65}{78\!\cdots\!92}a^{3}-\frac{23\!\cdots\!91}{78\!\cdots\!92}a^{2}+\frac{19\!\cdots\!43}{39\!\cdots\!96}a-\frac{37\!\cdots\!73}{19\!\cdots\!98}$, $\frac{10\!\cdots\!87}{78\!\cdots\!92}a^{21}-\frac{58\!\cdots\!21}{39\!\cdots\!96}a^{20}-\frac{32\!\cdots\!87}{21\!\cdots\!16}a^{19}+\frac{74\!\cdots\!41}{21\!\cdots\!16}a^{18}+\frac{89\!\cdots\!29}{19\!\cdots\!98}a^{17}-\frac{11\!\cdots\!75}{97\!\cdots\!49}a^{16}-\frac{66\!\cdots\!93}{19\!\cdots\!98}a^{15}+\frac{19\!\cdots\!45}{39\!\cdots\!96}a^{14}+\frac{82\!\cdots\!69}{78\!\cdots\!92}a^{13}-\frac{13\!\cdots\!91}{26\!\cdots\!77}a^{12}+\frac{14\!\cdots\!71}{78\!\cdots\!92}a^{11}+\frac{15\!\cdots\!93}{78\!\cdots\!92}a^{10}+\frac{25\!\cdots\!19}{19\!\cdots\!98}a^{9}+\frac{20\!\cdots\!53}{39\!\cdots\!96}a^{8}+\frac{16\!\cdots\!83}{39\!\cdots\!96}a^{7}+\frac{64\!\cdots\!15}{97\!\cdots\!49}a^{6}+\frac{55\!\cdots\!87}{78\!\cdots\!92}a^{5}-\frac{11\!\cdots\!89}{97\!\cdots\!49}a^{4}+\frac{19\!\cdots\!47}{78\!\cdots\!92}a^{3}+\frac{45\!\cdots\!55}{78\!\cdots\!92}a^{2}-\frac{11\!\cdots\!59}{39\!\cdots\!96}a+\frac{58\!\cdots\!61}{19\!\cdots\!98}$, $\frac{79\!\cdots\!55}{15\!\cdots\!84}a^{21}+\frac{25\!\cdots\!31}{15\!\cdots\!84}a^{20}-\frac{23\!\cdots\!73}{42\!\cdots\!32}a^{19}-\frac{19\!\cdots\!43}{10\!\cdots\!08}a^{18}+\frac{16\!\cdots\!70}{97\!\cdots\!49}a^{17}-\frac{10\!\cdots\!11}{39\!\cdots\!96}a^{16}-\frac{31\!\cdots\!66}{97\!\cdots\!49}a^{15}+\frac{28\!\cdots\!27}{19\!\cdots\!98}a^{14}+\frac{28\!\cdots\!19}{15\!\cdots\!84}a^{13}+\frac{11\!\cdots\!27}{42\!\cdots\!32}a^{12}+\frac{70\!\cdots\!67}{15\!\cdots\!84}a^{11}+\frac{41\!\cdots\!31}{39\!\cdots\!96}a^{10}+\frac{38\!\cdots\!91}{39\!\cdots\!96}a^{9}-\frac{74\!\cdots\!49}{39\!\cdots\!96}a^{8}+\frac{51\!\cdots\!79}{39\!\cdots\!96}a^{7}+\frac{10\!\cdots\!75}{39\!\cdots\!96}a^{6}+\frac{68\!\cdots\!43}{15\!\cdots\!84}a^{5}-\frac{13\!\cdots\!53}{15\!\cdots\!84}a^{4}-\frac{37\!\cdots\!57}{15\!\cdots\!84}a^{3}-\frac{91\!\cdots\!79}{39\!\cdots\!96}a^{2}+\frac{63\!\cdots\!65}{19\!\cdots\!98}a-\frac{32\!\cdots\!59}{19\!\cdots\!98}$, $\frac{91\!\cdots\!93}{78\!\cdots\!92}a^{21}+\frac{11\!\cdots\!33}{15\!\cdots\!84}a^{20}-\frac{56\!\cdots\!67}{42\!\cdots\!32}a^{19}-\frac{29\!\cdots\!47}{42\!\cdots\!32}a^{18}+\frac{34\!\cdots\!29}{78\!\cdots\!92}a^{17}-\frac{12\!\cdots\!37}{19\!\cdots\!98}a^{16}-\frac{21\!\cdots\!41}{19\!\cdots\!98}a^{15}+\frac{15\!\cdots\!55}{39\!\cdots\!96}a^{14}+\frac{36\!\cdots\!49}{78\!\cdots\!92}a^{13}-\frac{24\!\cdots\!63}{42\!\cdots\!32}a^{12}+\frac{18\!\cdots\!01}{15\!\cdots\!84}a^{11}+\frac{43\!\cdots\!65}{15\!\cdots\!84}a^{10}+\frac{16\!\cdots\!31}{78\!\cdots\!92}a^{9}-\frac{11\!\cdots\!23}{19\!\cdots\!98}a^{8}+\frac{59\!\cdots\!99}{19\!\cdots\!98}a^{7}+\frac{79\!\cdots\!09}{97\!\cdots\!49}a^{6}+\frac{76\!\cdots\!21}{78\!\cdots\!92}a^{5}+\frac{54\!\cdots\!65}{15\!\cdots\!84}a^{4}-\frac{40\!\cdots\!99}{15\!\cdots\!84}a^{3}+\frac{58\!\cdots\!73}{15\!\cdots\!84}a^{2}+\frac{24\!\cdots\!83}{78\!\cdots\!92}a-\frac{75\!\cdots\!35}{39\!\cdots\!96}$, $\frac{20\!\cdots\!67}{15\!\cdots\!84}a^{21}+\frac{95\!\cdots\!93}{15\!\cdots\!84}a^{20}-\frac{59\!\cdots\!33}{42\!\cdots\!32}a^{19}-\frac{18\!\cdots\!19}{26\!\cdots\!77}a^{18}+\frac{16\!\cdots\!93}{39\!\cdots\!96}a^{17}-\frac{11\!\cdots\!19}{19\!\cdots\!98}a^{16}-\frac{18\!\cdots\!01}{19\!\cdots\!98}a^{15}+\frac{35\!\cdots\!65}{97\!\cdots\!49}a^{14}+\frac{82\!\cdots\!07}{15\!\cdots\!84}a^{13}+\frac{53\!\cdots\!25}{42\!\cdots\!32}a^{12}+\frac{16\!\cdots\!31}{15\!\cdots\!84}a^{11}+\frac{10\!\cdots\!79}{39\!\cdots\!96}a^{10}+\frac{51\!\cdots\!81}{19\!\cdots\!98}a^{9}-\frac{65\!\cdots\!18}{97\!\cdots\!49}a^{8}+\frac{23\!\cdots\!39}{97\!\cdots\!49}a^{7}+\frac{66\!\cdots\!20}{97\!\cdots\!49}a^{6}+\frac{19\!\cdots\!47}{15\!\cdots\!84}a^{5}-\frac{66\!\cdots\!59}{15\!\cdots\!84}a^{4}-\frac{19\!\cdots\!65}{15\!\cdots\!84}a^{3}-\frac{55\!\cdots\!61}{39\!\cdots\!96}a^{2}+\frac{16\!\cdots\!69}{19\!\cdots\!98}a-\frac{54\!\cdots\!83}{19\!\cdots\!98}$, $\frac{19\!\cdots\!05}{78\!\cdots\!92}a^{21}+\frac{72\!\cdots\!63}{78\!\cdots\!92}a^{20}-\frac{17\!\cdots\!21}{10\!\cdots\!08}a^{19}-\frac{52\!\cdots\!95}{52\!\cdots\!54}a^{18}-\frac{97\!\cdots\!31}{39\!\cdots\!96}a^{17}+\frac{87\!\cdots\!97}{97\!\cdots\!49}a^{16}-\frac{20\!\cdots\!57}{39\!\cdots\!96}a^{15}-\frac{17\!\cdots\!41}{19\!\cdots\!98}a^{14}+\frac{26\!\cdots\!69}{78\!\cdots\!92}a^{13}+\frac{91\!\cdots\!23}{21\!\cdots\!16}a^{12}+\frac{58\!\cdots\!91}{97\!\cdots\!49}a^{11}+\frac{17\!\cdots\!61}{97\!\cdots\!49}a^{10}+\frac{28\!\cdots\!63}{97\!\cdots\!49}a^{9}+\frac{49\!\cdots\!11}{19\!\cdots\!98}a^{8}+\frac{75\!\cdots\!45}{39\!\cdots\!96}a^{7}+\frac{24\!\cdots\!39}{97\!\cdots\!49}a^{6}+\frac{39\!\cdots\!55}{78\!\cdots\!92}a^{5}+\frac{26\!\cdots\!47}{78\!\cdots\!92}a^{4}+\frac{90\!\cdots\!15}{39\!\cdots\!96}a^{3}-\frac{93\!\cdots\!73}{19\!\cdots\!98}a^{2}+\frac{10\!\cdots\!03}{19\!\cdots\!98}a-\frac{85\!\cdots\!18}{97\!\cdots\!49}$, $\frac{39\!\cdots\!35}{78\!\cdots\!92}a^{21}+\frac{19\!\cdots\!79}{15\!\cdots\!84}a^{20}-\frac{23\!\cdots\!75}{42\!\cdots\!32}a^{19}-\frac{58\!\cdots\!27}{42\!\cdots\!32}a^{18}+\frac{13\!\cdots\!95}{78\!\cdots\!92}a^{17}-\frac{11\!\cdots\!61}{39\!\cdots\!96}a^{16}-\frac{62\!\cdots\!49}{19\!\cdots\!98}a^{15}+\frac{15\!\cdots\!10}{97\!\cdots\!49}a^{14}+\frac{12\!\cdots\!41}{78\!\cdots\!92}a^{13}-\frac{42\!\cdots\!57}{42\!\cdots\!32}a^{12}+\frac{68\!\cdots\!09}{15\!\cdots\!84}a^{11}+\frac{15\!\cdots\!13}{15\!\cdots\!84}a^{10}+\frac{62\!\cdots\!93}{78\!\cdots\!92}a^{9}-\frac{73\!\cdots\!67}{19\!\cdots\!98}a^{8}+\frac{45\!\cdots\!75}{39\!\cdots\!96}a^{7}+\frac{10\!\cdots\!55}{39\!\cdots\!96}a^{6}+\frac{30\!\cdots\!15}{78\!\cdots\!92}a^{5}-\frac{21\!\cdots\!33}{15\!\cdots\!84}a^{4}-\frac{12\!\cdots\!03}{15\!\cdots\!84}a^{3}-\frac{24\!\cdots\!79}{15\!\cdots\!84}a^{2}+\frac{23\!\cdots\!71}{78\!\cdots\!92}a-\frac{75\!\cdots\!51}{39\!\cdots\!96}$
| 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: | \( 984813482.893 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 984813482.893 \cdot 1}{2\cdot\sqrt{61533497770089105512253259111202816}}\cr\approx \mathstrut & 1.19604285451 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^20 + 33*x^18 - 66*x^17 - 44*x^16 + 308*x^15 + 253*x^14 - 44*x^13 + 913*x^12 + 1840*x^11 + 1309*x^10 - 814*x^9 + 2904*x^8 + 4752*x^7 + 7117*x^6 - 4004*x^5 + 825*x^4 - 44*x^3 + 6721*x^2 - 5434*x + 1644)
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 - 11*x^20 + 33*x^18 - 66*x^17 - 44*x^16 + 308*x^15 + 253*x^14 - 44*x^13 + 913*x^12 + 1840*x^11 + 1309*x^10 - 814*x^9 + 2904*x^8 + 4752*x^7 + 7117*x^6 - 4004*x^5 + 825*x^4 - 44*x^3 + 6721*x^2 - 5434*x + 1644, 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 - 11*x^20 + 33*x^18 - 66*x^17 - 44*x^16 + 308*x^15 + 253*x^14 - 44*x^13 + 913*x^12 + 1840*x^11 + 1309*x^10 - 814*x^9 + 2904*x^8 + 4752*x^7 + 7117*x^6 - 4004*x^5 + 825*x^4 - 44*x^3 + 6721*x^2 - 5434*x + 1644);
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 - 11*x^20 + 33*x^18 - 66*x^17 - 44*x^16 + 308*x^15 + 253*x^14 - 44*x^13 + 913*x^12 + 1840*x^11 + 1309*x^10 - 814*x^9 + 2904*x^8 + 4752*x^7 + 7117*x^6 - 4004*x^5 + 825*x^4 - 44*x^3 + 6721*x^2 - 5434*x + 1644);
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
$\PGL(2,11)$ (as 22T14):
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 1320 |
| The 13 conjugacy class representatives for $\PGL(2,11)$ |
| Character table for $\PGL(2,11)$ |
Intermediate fields
| \(\Q(\sqrt{-11}) \) |
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 12 sibling: | data not computed |
| Degree 24 sibling: | data not computed |
| Minimal sibling: | 12.2.19146942100646395904.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{4}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | R | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.5.0.1}{5} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.8.16.8 | $x^{8} + 4 x^{7} + 10 x^{6} + 24 x^{5} + 51 x^{4} + 48 x^{3} - 18 x^{2} + 63$ | $4$ | $2$ | $16$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
| 2.12.20.37 | $x^{12} + 10 x^{11} + 51 x^{10} + 176 x^{9} + 450 x^{8} + 870 x^{7} + 1299 x^{6} + 1516 x^{5} + 1250 x^{4} + 542 x^{3} + 67 x^{2} - 56 x + 7$ | $6$ | $2$ | $20$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
|
\(11\)
| Deg $22$ | $22$ | $1$ | $23$ |