Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 55*x^20 - 154*x^19 + 187*x^18 + 363*x^17 - 2255*x^16 + 5016*x^15 - 5082*x^14 - 2046*x^13 + 13442*x^12 - 15368*x^11 + 6358*x^10 - 33242*x^9 + 159434*x^8 - 361944*x^7 + 546073*x^6 - 684519*x^5 + 779031*x^4 - 754798*x^3 + 577951*x^2 - 314545*x + 86485)
gp: K = bnfinit(y^22 - 11*y^21 + 55*y^20 - 154*y^19 + 187*y^18 + 363*y^17 - 2255*y^16 + 5016*y^15 - 5082*y^14 - 2046*y^13 + 13442*y^12 - 15368*y^11 + 6358*y^10 - 33242*y^9 + 159434*y^8 - 361944*y^7 + 546073*y^6 - 684519*y^5 + 779031*y^4 - 754798*y^3 + 577951*y^2 - 314545*y + 86485, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 + 55*x^20 - 154*x^19 + 187*x^18 + 363*x^17 - 2255*x^16 + 5016*x^15 - 5082*x^14 - 2046*x^13 + 13442*x^12 - 15368*x^11 + 6358*x^10 - 33242*x^9 + 159434*x^8 - 361944*x^7 + 546073*x^6 - 684519*x^5 + 779031*x^4 - 754798*x^3 + 577951*x^2 - 314545*x + 86485);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^21 + 55*x^20 - 154*x^19 + 187*x^18 + 363*x^17 - 2255*x^16 + 5016*x^15 - 5082*x^14 - 2046*x^13 + 13442*x^12 - 15368*x^11 + 6358*x^10 - 33242*x^9 + 159434*x^8 - 361944*x^7 + 546073*x^6 - 684519*x^5 + 779031*x^4 - 754798*x^3 + 577951*x^2 - 314545*x + 86485)
\( x^{22} - 11 x^{21} + 55 x^{20} - 154 x^{19} + 187 x^{18} + 363 x^{17} - 2255 x^{16} + 5016 x^{15} + \cdots + 86485 \)
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: |
\(-366768227637345227672179097600000000\)
\(\medspace = -\,2^{20}\cdot 5^{8}\cdot 11^{23}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(41.36\) | 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: |
\(2\), \(5\), \(11\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{22}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{3}{11}$, $\frac{1}{44}a^{12}-\frac{1}{44}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{5}{44}a+\frac{17}{44}$, $\frac{1}{44}a^{13}-\frac{1}{44}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{17}{44}a^{2}+\frac{17}{44}$, $\frac{1}{44}a^{14}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{3}{22}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{44}a^{15}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}-\frac{3}{22}a^{4}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{88}a^{16}-\frac{1}{88}a^{15}-\frac{1}{88}a^{13}+\frac{1}{88}a^{11}-\frac{1}{4}a^{10}-\frac{1}{8}a^{9}-\frac{1}{4}a^{8}+\frac{3}{8}a^{7}-\frac{1}{2}a^{6}+\frac{5}{88}a^{5}-\frac{2}{11}a^{4}-\frac{3}{8}a^{3}-\frac{19}{44}a^{2}-\frac{3}{8}a+\frac{5}{88}$, $\frac{1}{88}a^{17}-\frac{1}{88}a^{15}-\frac{1}{88}a^{14}-\frac{1}{88}a^{13}-\frac{1}{88}a^{12}+\frac{1}{88}a^{11}+\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{3}{8}a^{7}+\frac{5}{88}a^{6}+\frac{3}{8}a^{5}+\frac{17}{88}a^{4}+\frac{39}{88}a^{3}+\frac{17}{88}a^{2}-\frac{19}{44}a+\frac{27}{88}$, $\frac{1}{88}a^{18}-\frac{1}{88}a^{14}-\frac{1}{88}a^{12}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{1}{8}a^{8}+\frac{2}{11}a^{7}+\frac{3}{8}a^{6}-\frac{1}{2}a^{5}+\frac{3}{8}a^{4}-\frac{2}{11}a^{3}-\frac{1}{4}a^{2}-\frac{19}{44}a-\frac{1}{8}$, $\frac{1}{88}a^{19}-\frac{1}{88}a^{15}-\frac{1}{88}a^{13}+\frac{1}{88}a^{11}-\frac{1}{4}a^{10}+\frac{1}{8}a^{9}+\frac{2}{11}a^{8}-\frac{1}{8}a^{7}-\frac{1}{2}a^{6}-\frac{1}{8}a^{5}-\frac{2}{11}a^{4}-\frac{1}{4}a^{3}-\frac{19}{44}a^{2}+\frac{3}{8}a+\frac{2}{11}$, $\frac{1}{1496}a^{20}-\frac{1}{748}a^{19}+\frac{5}{1496}a^{17}-\frac{3}{748}a^{16}-\frac{3}{374}a^{15}-\frac{2}{187}a^{14}+\frac{3}{374}a^{13}+\frac{1}{374}a^{12}-\frac{3}{748}a^{11}+\frac{13}{68}a^{10}+\frac{35}{187}a^{9}-\frac{26}{187}a^{8}-\frac{5}{34}a^{7}-\frac{167}{374}a^{6}+\frac{163}{374}a^{5}+\frac{23}{88}a^{4}-\frac{21}{187}a^{3}+\frac{63}{748}a^{2}+\frac{603}{1496}a+\frac{205}{748}$, $\frac{1}{58\!\cdots\!16}a^{21}-\frac{49\!\cdots\!75}{29\!\cdots\!08}a^{20}+\frac{12\!\cdots\!07}{58\!\cdots\!16}a^{19}+\frac{39\!\cdots\!29}{83\!\cdots\!88}a^{18}-\frac{85\!\cdots\!19}{29\!\cdots\!08}a^{17}-\frac{14\!\cdots\!35}{58\!\cdots\!16}a^{16}+\frac{95\!\cdots\!27}{29\!\cdots\!08}a^{15}+\frac{23\!\cdots\!89}{29\!\cdots\!08}a^{14}+\frac{35\!\cdots\!43}{41\!\cdots\!44}a^{13}-\frac{33\!\cdots\!96}{72\!\cdots\!27}a^{12}-\frac{16\!\cdots\!17}{72\!\cdots\!27}a^{11}+\frac{99\!\cdots\!72}{72\!\cdots\!27}a^{10}-\frac{28\!\cdots\!97}{14\!\cdots\!54}a^{9}-\frac{99\!\cdots\!65}{17\!\cdots\!24}a^{8}-\frac{90\!\cdots\!23}{29\!\cdots\!08}a^{7}+\frac{14\!\cdots\!37}{29\!\cdots\!08}a^{6}-\frac{19\!\cdots\!71}{58\!\cdots\!16}a^{5}+\frac{20\!\cdots\!13}{29\!\cdots\!08}a^{4}+\frac{18\!\cdots\!85}{58\!\cdots\!16}a^{3}+\frac{17\!\cdots\!93}{58\!\cdots\!16}a^{2}+\frac{50\!\cdots\!93}{29\!\cdots\!08}a-\frac{23\!\cdots\!29}{23\!\cdots\!96}$
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: |
$\frac{31\!\cdots\!81}{58\!\cdots\!16}a^{21}-\frac{31\!\cdots\!65}{58\!\cdots\!16}a^{20}+\frac{40\!\cdots\!35}{17\!\cdots\!24}a^{19}-\frac{11\!\cdots\!99}{20\!\cdots\!22}a^{18}+\frac{51\!\cdots\!15}{14\!\cdots\!54}a^{17}+\frac{13\!\cdots\!79}{58\!\cdots\!16}a^{16}-\frac{27\!\cdots\!51}{29\!\cdots\!08}a^{15}+\frac{11\!\cdots\!27}{72\!\cdots\!27}a^{14}-\frac{17\!\cdots\!41}{20\!\cdots\!22}a^{13}-\frac{62\!\cdots\!31}{29\!\cdots\!08}a^{12}+\frac{34\!\cdots\!33}{72\!\cdots\!27}a^{11}-\frac{19\!\cdots\!88}{72\!\cdots\!27}a^{10}+\frac{38\!\cdots\!59}{14\!\cdots\!54}a^{9}-\frac{51\!\cdots\!81}{29\!\cdots\!08}a^{8}+\frac{19\!\cdots\!03}{29\!\cdots\!08}a^{7}-\frac{86\!\cdots\!59}{72\!\cdots\!27}a^{6}+\frac{53\!\cdots\!39}{34\!\cdots\!48}a^{5}-\frac{10\!\cdots\!93}{58\!\cdots\!16}a^{4}+\frac{58\!\cdots\!01}{29\!\cdots\!08}a^{3}-\frac{24\!\cdots\!67}{14\!\cdots\!54}a^{2}+\frac{15\!\cdots\!57}{14\!\cdots\!54}a-\frac{54\!\cdots\!79}{13\!\cdots\!88}$, $\frac{67\!\cdots\!45}{72\!\cdots\!27}a^{21}-\frac{26\!\cdots\!57}{29\!\cdots\!08}a^{20}+\frac{23\!\cdots\!27}{58\!\cdots\!16}a^{19}-\frac{39\!\cdots\!71}{41\!\cdots\!44}a^{18}+\frac{35\!\cdots\!23}{58\!\cdots\!16}a^{17}+\frac{11\!\cdots\!69}{29\!\cdots\!08}a^{16}-\frac{46\!\cdots\!75}{29\!\cdots\!08}a^{15}+\frac{15\!\cdots\!49}{58\!\cdots\!16}a^{14}-\frac{35\!\cdots\!85}{24\!\cdots\!32}a^{13}-\frac{20\!\cdots\!69}{58\!\cdots\!16}a^{12}+\frac{11\!\cdots\!07}{14\!\cdots\!54}a^{11}-\frac{26\!\cdots\!93}{58\!\cdots\!16}a^{10}+\frac{13\!\cdots\!21}{29\!\cdots\!08}a^{9}-\frac{17\!\cdots\!03}{58\!\cdots\!16}a^{8}+\frac{32\!\cdots\!97}{29\!\cdots\!08}a^{7}-\frac{11\!\cdots\!45}{58\!\cdots\!16}a^{6}+\frac{76\!\cdots\!37}{29\!\cdots\!08}a^{5}-\frac{18\!\cdots\!61}{58\!\cdots\!16}a^{4}+\frac{19\!\cdots\!09}{58\!\cdots\!16}a^{3}-\frac{99\!\cdots\!77}{34\!\cdots\!48}a^{2}+\frac{10\!\cdots\!43}{58\!\cdots\!16}a-\frac{15\!\cdots\!89}{23\!\cdots\!96}$, $\frac{75\!\cdots\!30}{72\!\cdots\!27}a^{21}-\frac{14\!\cdots\!77}{14\!\cdots\!54}a^{20}+\frac{65\!\cdots\!11}{14\!\cdots\!54}a^{19}-\frac{11\!\cdots\!53}{10\!\cdots\!61}a^{18}+\frac{38\!\cdots\!15}{58\!\cdots\!16}a^{17}+\frac{26\!\cdots\!73}{58\!\cdots\!16}a^{16}-\frac{13\!\cdots\!65}{72\!\cdots\!27}a^{15}+\frac{17\!\cdots\!39}{58\!\cdots\!16}a^{14}-\frac{39\!\cdots\!61}{24\!\cdots\!32}a^{13}-\frac{23\!\cdots\!75}{58\!\cdots\!16}a^{12}+\frac{26\!\cdots\!85}{29\!\cdots\!08}a^{11}-\frac{29\!\cdots\!11}{58\!\cdots\!16}a^{10}+\frac{81\!\cdots\!33}{14\!\cdots\!54}a^{9}-\frac{19\!\cdots\!23}{58\!\cdots\!16}a^{8}+\frac{91\!\cdots\!65}{72\!\cdots\!27}a^{7}-\frac{13\!\cdots\!95}{58\!\cdots\!16}a^{6}+\frac{86\!\cdots\!37}{29\!\cdots\!08}a^{5}-\frac{20\!\cdots\!09}{58\!\cdots\!16}a^{4}+\frac{11\!\cdots\!41}{29\!\cdots\!08}a^{3}-\frac{11\!\cdots\!69}{34\!\cdots\!48}a^{2}+\frac{12\!\cdots\!37}{58\!\cdots\!16}a-\frac{88\!\cdots\!39}{11\!\cdots\!48}$, $\frac{79\!\cdots\!33}{58\!\cdots\!16}a^{21}-\frac{39\!\cdots\!27}{29\!\cdots\!08}a^{20}+\frac{34\!\cdots\!39}{58\!\cdots\!16}a^{19}-\frac{58\!\cdots\!19}{41\!\cdots\!44}a^{18}+\frac{66\!\cdots\!98}{72\!\cdots\!27}a^{17}+\frac{34\!\cdots\!17}{58\!\cdots\!16}a^{16}-\frac{17\!\cdots\!57}{72\!\cdots\!27}a^{15}+\frac{23\!\cdots\!09}{58\!\cdots\!16}a^{14}-\frac{91\!\cdots\!97}{41\!\cdots\!44}a^{13}-\frac{31\!\cdots\!31}{58\!\cdots\!16}a^{12}+\frac{35\!\cdots\!41}{29\!\cdots\!08}a^{11}-\frac{40\!\cdots\!25}{58\!\cdots\!16}a^{10}+\frac{10\!\cdots\!43}{17\!\cdots\!24}a^{9}-\frac{25\!\cdots\!47}{58\!\cdots\!16}a^{8}+\frac{12\!\cdots\!57}{72\!\cdots\!27}a^{7}-\frac{17\!\cdots\!13}{58\!\cdots\!16}a^{6}+\frac{23\!\cdots\!29}{58\!\cdots\!16}a^{5}-\frac{27\!\cdots\!11}{58\!\cdots\!16}a^{4}+\frac{29\!\cdots\!75}{58\!\cdots\!16}a^{3}-\frac{25\!\cdots\!45}{58\!\cdots\!16}a^{2}+\frac{82\!\cdots\!73}{29\!\cdots\!08}a-\frac{11\!\cdots\!01}{11\!\cdots\!48}$, $\frac{71\!\cdots\!29}{58\!\cdots\!16}a^{21}-\frac{69\!\cdots\!51}{58\!\cdots\!16}a^{20}+\frac{17\!\cdots\!87}{34\!\cdots\!48}a^{19}-\frac{10\!\cdots\!87}{83\!\cdots\!88}a^{18}+\frac{54\!\cdots\!51}{72\!\cdots\!27}a^{17}+\frac{15\!\cdots\!51}{29\!\cdots\!08}a^{16}-\frac{60\!\cdots\!15}{29\!\cdots\!08}a^{15}+\frac{20\!\cdots\!03}{58\!\cdots\!16}a^{14}-\frac{37\!\cdots\!65}{20\!\cdots\!22}a^{13}-\frac{27\!\cdots\!39}{58\!\cdots\!16}a^{12}+\frac{15\!\cdots\!51}{14\!\cdots\!54}a^{11}-\frac{32\!\cdots\!93}{58\!\cdots\!16}a^{10}+\frac{99\!\cdots\!11}{14\!\cdots\!54}a^{9}-\frac{23\!\cdots\!19}{58\!\cdots\!16}a^{8}+\frac{42\!\cdots\!11}{29\!\cdots\!08}a^{7}-\frac{15\!\cdots\!63}{58\!\cdots\!16}a^{6}+\frac{11\!\cdots\!87}{34\!\cdots\!48}a^{5}-\frac{11\!\cdots\!87}{29\!\cdots\!08}a^{4}+\frac{25\!\cdots\!69}{58\!\cdots\!16}a^{3}-\frac{10\!\cdots\!25}{29\!\cdots\!08}a^{2}+\frac{34\!\cdots\!31}{14\!\cdots\!54}a-\frac{11\!\cdots\!09}{13\!\cdots\!88}$, $\frac{40\!\cdots\!01}{58\!\cdots\!16}a^{21}-\frac{10\!\cdots\!67}{14\!\cdots\!54}a^{20}+\frac{17\!\cdots\!05}{58\!\cdots\!16}a^{19}-\frac{29\!\cdots\!77}{41\!\cdots\!44}a^{18}+\frac{13\!\cdots\!61}{29\!\cdots\!08}a^{17}+\frac{17\!\cdots\!49}{58\!\cdots\!16}a^{16}-\frac{17\!\cdots\!63}{14\!\cdots\!54}a^{15}+\frac{12\!\cdots\!21}{58\!\cdots\!16}a^{14}-\frac{23\!\cdots\!59}{20\!\cdots\!22}a^{13}-\frac{15\!\cdots\!55}{58\!\cdots\!16}a^{12}+\frac{90\!\cdots\!89}{14\!\cdots\!54}a^{11}-\frac{20\!\cdots\!41}{58\!\cdots\!16}a^{10}+\frac{10\!\cdots\!83}{29\!\cdots\!08}a^{9}-\frac{13\!\cdots\!99}{58\!\cdots\!16}a^{8}+\frac{12\!\cdots\!01}{14\!\cdots\!54}a^{7}-\frac{89\!\cdots\!49}{58\!\cdots\!16}a^{6}+\frac{11\!\cdots\!15}{58\!\cdots\!16}a^{5}-\frac{14\!\cdots\!37}{58\!\cdots\!16}a^{4}+\frac{15\!\cdots\!91}{58\!\cdots\!16}a^{3}-\frac{12\!\cdots\!29}{58\!\cdots\!16}a^{2}+\frac{61\!\cdots\!08}{42\!\cdots\!31}a-\frac{60\!\cdots\!71}{11\!\cdots\!48}$, $\frac{30\!\cdots\!83}{29\!\cdots\!08}a^{21}-\frac{30\!\cdots\!57}{29\!\cdots\!08}a^{20}+\frac{27\!\cdots\!49}{58\!\cdots\!16}a^{19}-\frac{92\!\cdots\!29}{83\!\cdots\!88}a^{18}+\frac{52\!\cdots\!03}{72\!\cdots\!27}a^{17}+\frac{27\!\cdots\!15}{58\!\cdots\!16}a^{16}-\frac{13\!\cdots\!64}{72\!\cdots\!27}a^{15}+\frac{18\!\cdots\!53}{58\!\cdots\!16}a^{14}-\frac{70\!\cdots\!33}{41\!\cdots\!44}a^{13}-\frac{24\!\cdots\!83}{58\!\cdots\!16}a^{12}+\frac{69\!\cdots\!38}{72\!\cdots\!27}a^{11}-\frac{17\!\cdots\!71}{34\!\cdots\!48}a^{10}+\frac{27\!\cdots\!63}{29\!\cdots\!08}a^{9}-\frac{20\!\cdots\!19}{58\!\cdots\!16}a^{8}+\frac{94\!\cdots\!98}{72\!\cdots\!27}a^{7}-\frac{13\!\cdots\!21}{58\!\cdots\!16}a^{6}+\frac{44\!\cdots\!11}{14\!\cdots\!54}a^{5}-\frac{21\!\cdots\!15}{58\!\cdots\!16}a^{4}+\frac{22\!\cdots\!35}{58\!\cdots\!16}a^{3}-\frac{97\!\cdots\!89}{29\!\cdots\!08}a^{2}+\frac{63\!\cdots\!19}{29\!\cdots\!08}a-\frac{22\!\cdots\!02}{29\!\cdots\!37}$, $\frac{15\!\cdots\!51}{52\!\cdots\!56}a^{21}-\frac{14\!\cdots\!85}{52\!\cdots\!56}a^{20}+\frac{15\!\cdots\!27}{13\!\cdots\!14}a^{19}-\frac{24\!\cdots\!62}{94\!\cdots\!51}a^{18}+\frac{24\!\cdots\!29}{26\!\cdots\!28}a^{17}+\frac{16\!\cdots\!15}{12\!\cdots\!74}a^{16}-\frac{24\!\cdots\!61}{52\!\cdots\!56}a^{15}+\frac{91\!\cdots\!13}{13\!\cdots\!14}a^{14}-\frac{13\!\cdots\!63}{75\!\cdots\!08}a^{13}-\frac{31\!\cdots\!13}{26\!\cdots\!28}a^{12}+\frac{10\!\cdots\!89}{52\!\cdots\!56}a^{11}-\frac{11\!\cdots\!43}{26\!\cdots\!28}a^{10}+\frac{14\!\cdots\!87}{31\!\cdots\!68}a^{9}-\frac{13\!\cdots\!87}{13\!\cdots\!14}a^{8}+\frac{17\!\cdots\!97}{52\!\cdots\!56}a^{7}-\frac{33\!\cdots\!99}{66\!\cdots\!57}a^{6}+\frac{15\!\cdots\!37}{24\!\cdots\!48}a^{5}-\frac{42\!\cdots\!65}{52\!\cdots\!56}a^{4}+\frac{44\!\cdots\!15}{52\!\cdots\!56}a^{3}-\frac{18\!\cdots\!39}{26\!\cdots\!28}a^{2}+\frac{23\!\cdots\!95}{52\!\cdots\!56}a-\frac{38\!\cdots\!38}{26\!\cdots\!67}$, $\frac{71\!\cdots\!25}{58\!\cdots\!16}a^{21}-\frac{70\!\cdots\!11}{58\!\cdots\!16}a^{20}+\frac{31\!\cdots\!97}{58\!\cdots\!16}a^{19}-\frac{10\!\cdots\!63}{83\!\cdots\!88}a^{18}+\frac{46\!\cdots\!83}{58\!\cdots\!16}a^{17}+\frac{15\!\cdots\!19}{29\!\cdots\!08}a^{16}-\frac{12\!\cdots\!09}{58\!\cdots\!16}a^{15}+\frac{52\!\cdots\!91}{14\!\cdots\!54}a^{14}-\frac{15\!\cdots\!23}{83\!\cdots\!88}a^{13}-\frac{35\!\cdots\!84}{72\!\cdots\!27}a^{12}+\frac{62\!\cdots\!75}{58\!\cdots\!16}a^{11}-\frac{86\!\cdots\!41}{14\!\cdots\!54}a^{10}+\frac{33\!\cdots\!21}{58\!\cdots\!16}a^{9}-\frac{68\!\cdots\!69}{17\!\cdots\!24}a^{8}+\frac{86\!\cdots\!85}{58\!\cdots\!16}a^{7}-\frac{39\!\cdots\!27}{14\!\cdots\!54}a^{6}+\frac{51\!\cdots\!67}{14\!\cdots\!54}a^{5}-\frac{24\!\cdots\!61}{58\!\cdots\!16}a^{4}+\frac{33\!\cdots\!63}{72\!\cdots\!27}a^{3}-\frac{22\!\cdots\!91}{58\!\cdots\!16}a^{2}+\frac{18\!\cdots\!11}{72\!\cdots\!27}a-\frac{51\!\cdots\!11}{58\!\cdots\!74}$, $\frac{10\!\cdots\!49}{14\!\cdots\!54}a^{21}-\frac{41\!\cdots\!73}{58\!\cdots\!16}a^{20}+\frac{23\!\cdots\!03}{72\!\cdots\!27}a^{19}-\frac{15\!\cdots\!31}{20\!\cdots\!22}a^{18}+\frac{16\!\cdots\!85}{34\!\cdots\!48}a^{17}+\frac{23\!\cdots\!59}{72\!\cdots\!27}a^{16}-\frac{36\!\cdots\!99}{29\!\cdots\!08}a^{15}+\frac{63\!\cdots\!05}{29\!\cdots\!08}a^{14}-\frac{24\!\cdots\!55}{20\!\cdots\!22}a^{13}-\frac{82\!\cdots\!63}{29\!\cdots\!08}a^{12}+\frac{11\!\cdots\!71}{17\!\cdots\!24}a^{11}-\frac{10\!\cdots\!97}{29\!\cdots\!08}a^{10}+\frac{11\!\cdots\!99}{29\!\cdots\!08}a^{9}-\frac{34\!\cdots\!35}{14\!\cdots\!54}a^{8}+\frac{25\!\cdots\!81}{29\!\cdots\!08}a^{7}-\frac{27\!\cdots\!03}{17\!\cdots\!24}a^{6}+\frac{15\!\cdots\!34}{72\!\cdots\!27}a^{5}-\frac{14\!\cdots\!77}{58\!\cdots\!16}a^{4}+\frac{79\!\cdots\!79}{29\!\cdots\!08}a^{3}-\frac{67\!\cdots\!55}{29\!\cdots\!08}a^{2}+\frac{87\!\cdots\!89}{58\!\cdots\!16}a-\frac{32\!\cdots\!09}{58\!\cdots\!74}$
| 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: | \( 3435709897.46 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 3435709897.46 \cdot 1}{2\cdot\sqrt{366768227637345227672179097600000000}}\cr\approx \mathstrut & 1.70910674831 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 55*x^20 - 154*x^19 + 187*x^18 + 363*x^17 - 2255*x^16 + 5016*x^15 - 5082*x^14 - 2046*x^13 + 13442*x^12 - 15368*x^11 + 6358*x^10 - 33242*x^9 + 159434*x^8 - 361944*x^7 + 546073*x^6 - 684519*x^5 + 779031*x^4 - 754798*x^3 + 577951*x^2 - 314545*x + 86485)
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^21 + 55*x^20 - 154*x^19 + 187*x^18 + 363*x^17 - 2255*x^16 + 5016*x^15 - 5082*x^14 - 2046*x^13 + 13442*x^12 - 15368*x^11 + 6358*x^10 - 33242*x^9 + 159434*x^8 - 361944*x^7 + 546073*x^6 - 684519*x^5 + 779031*x^4 - 754798*x^3 + 577951*x^2 - 314545*x + 86485, 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^21 + 55*x^20 - 154*x^19 + 187*x^18 + 363*x^17 - 2255*x^16 + 5016*x^15 - 5082*x^14 - 2046*x^13 + 13442*x^12 - 15368*x^11 + 6358*x^10 - 33242*x^9 + 159434*x^8 - 361944*x^7 + 546073*x^6 - 684519*x^5 + 779031*x^4 - 754798*x^3 + 577951*x^2 - 314545*x + 86485);
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^21 + 55*x^20 - 154*x^19 + 187*x^18 + 363*x^17 - 2255*x^16 + 5016*x^15 - 5082*x^14 - 2046*x^13 + 13442*x^12 - 15368*x^11 + 6358*x^10 - 33242*x^9 + 159434*x^8 - 361944*x^7 + 546073*x^6 - 684519*x^5 + 779031*x^4 - 754798*x^3 + 577951*x^2 - 314545*x + 86485);
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: | 12.2.73039787676416000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{4}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/7.10.0.1}{10} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.2.0.1}{2} }^{11}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.11.0.1}{11} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.3.0.1}{3} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.8.8.11 | $x^{8} + 4 x^{7} + 14 x^{6} + 32 x^{5} + 55 x^{4} + 60 x^{3} + 36 x^{2} + 18 x + 9$ | $4$ | $2$ | $8$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
| 2.12.12.28 | $x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 130 x^{7} + 159 x^{6} + 132 x^{5} + 10 x^{4} - 100 x^{3} - 53 x^{2} + 22 x + 19$ | $6$ | $2$ | $12$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
|
\(5\)
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(11\)
| Deg $22$ | $22$ | $1$ | $23$ |