Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + x^20 - x^19 - 7*x^17 + 5*x^16 + 45*x^15 + 31*x^14 - 38*x^13 - 205*x^12 - 278*x^11 - 123*x^10 + 249*x^9 + 532*x^8 + 371*x^7 + 205*x^6 + 153*x^5 + 236*x^4 + 94*x^3 + 34*x^2 + 5*x + 25)
gp: K = bnfinit(y^22 - 2*y^21 + y^20 - y^19 - 7*y^17 + 5*y^16 + 45*y^15 + 31*y^14 - 38*y^13 - 205*y^12 - 278*y^11 - 123*y^10 + 249*y^9 + 532*y^8 + 371*y^7 + 205*y^6 + 153*y^5 + 236*y^4 + 94*y^3 + 34*y^2 + 5*y + 25, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^21 + x^20 - x^19 - 7*x^17 + 5*x^16 + 45*x^15 + 31*x^14 - 38*x^13 - 205*x^12 - 278*x^11 - 123*x^10 + 249*x^9 + 532*x^8 + 371*x^7 + 205*x^6 + 153*x^5 + 236*x^4 + 94*x^3 + 34*x^2 + 5*x + 25);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 2*x^21 + x^20 - x^19 - 7*x^17 + 5*x^16 + 45*x^15 + 31*x^14 - 38*x^13 - 205*x^12 - 278*x^11 - 123*x^10 + 249*x^9 + 532*x^8 + 371*x^7 + 205*x^6 + 153*x^5 + 236*x^4 + 94*x^3 + 34*x^2 + 5*x + 25)
\( x^{22} - 2 x^{21} + x^{20} - x^{19} - 7 x^{17} + 5 x^{16} + 45 x^{15} + 31 x^{14} - 38 x^{13} + \cdots + 25 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(31524414620673611592542191616\)
\(\medspace = 2^{24}\cdot 11^{5}\cdot 19^{4}\cdot 547^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.74\) | 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: |
\(2\), \(11\), \(19\), \(547\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{15\!\cdots\!75}a^{21}+\frac{65\!\cdots\!88}{15\!\cdots\!75}a^{20}+\frac{53\!\cdots\!71}{15\!\cdots\!75}a^{19}-\frac{39\!\cdots\!86}{15\!\cdots\!75}a^{18}-\frac{11\!\cdots\!43}{31\!\cdots\!75}a^{17}-\frac{13\!\cdots\!32}{15\!\cdots\!75}a^{16}+\frac{18\!\cdots\!82}{62\!\cdots\!95}a^{15}-\frac{45\!\cdots\!66}{31\!\cdots\!75}a^{14}-\frac{45\!\cdots\!19}{15\!\cdots\!75}a^{13}-\frac{51\!\cdots\!98}{15\!\cdots\!75}a^{12}-\frac{17\!\cdots\!73}{62\!\cdots\!95}a^{11}+\frac{43\!\cdots\!97}{15\!\cdots\!75}a^{10}+\frac{78\!\cdots\!57}{15\!\cdots\!75}a^{9}-\frac{72\!\cdots\!71}{15\!\cdots\!75}a^{8}-\frac{24\!\cdots\!58}{15\!\cdots\!75}a^{7}-\frac{72\!\cdots\!49}{15\!\cdots\!75}a^{6}-\frac{11\!\cdots\!71}{31\!\cdots\!75}a^{5}-\frac{45\!\cdots\!47}{15\!\cdots\!75}a^{4}-\frac{53\!\cdots\!69}{15\!\cdots\!75}a^{3}-\frac{14\!\cdots\!41}{15\!\cdots\!75}a^{2}-\frac{77\!\cdots\!31}{15\!\cdots\!75}a-\frac{87\!\cdots\!27}{31\!\cdots\!75}$
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: | $11$ | 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: |
$\frac{13\!\cdots\!37}{15\!\cdots\!75}a^{21}-\frac{11\!\cdots\!94}{15\!\cdots\!75}a^{20}+\frac{21\!\cdots\!77}{15\!\cdots\!75}a^{19}-\frac{18\!\cdots\!57}{15\!\cdots\!75}a^{18}+\frac{38\!\cdots\!84}{31\!\cdots\!75}a^{17}-\frac{16\!\cdots\!34}{15\!\cdots\!75}a^{16}+\frac{27\!\cdots\!74}{62\!\cdots\!95}a^{15}+\frac{12\!\cdots\!58}{31\!\cdots\!75}a^{14}-\frac{34\!\cdots\!53}{15\!\cdots\!75}a^{13}-\frac{22\!\cdots\!01}{15\!\cdots\!75}a^{12}-\frac{14\!\cdots\!01}{62\!\cdots\!95}a^{11}+\frac{13\!\cdots\!64}{15\!\cdots\!75}a^{10}+\frac{20\!\cdots\!59}{15\!\cdots\!75}a^{9}+\frac{12\!\cdots\!48}{15\!\cdots\!75}a^{8}-\frac{13\!\cdots\!71}{15\!\cdots\!75}a^{7}-\frac{39\!\cdots\!38}{15\!\cdots\!75}a^{6}-\frac{55\!\cdots\!27}{31\!\cdots\!75}a^{5}-\frac{18\!\cdots\!39}{15\!\cdots\!75}a^{4}-\frac{54\!\cdots\!03}{15\!\cdots\!75}a^{3}-\frac{10\!\cdots\!42}{15\!\cdots\!75}a^{2}-\frac{23\!\cdots\!22}{15\!\cdots\!75}a-\frac{48\!\cdots\!74}{31\!\cdots\!75}$, $\frac{11\!\cdots\!91}{15\!\cdots\!75}a^{21}-\frac{26\!\cdots\!42}{15\!\cdots\!75}a^{20}+\frac{13\!\cdots\!36}{15\!\cdots\!75}a^{19}+\frac{11\!\cdots\!49}{15\!\cdots\!75}a^{18}-\frac{45\!\cdots\!63}{31\!\cdots\!75}a^{17}-\frac{56\!\cdots\!12}{15\!\cdots\!75}a^{16}+\frac{25\!\cdots\!27}{62\!\cdots\!95}a^{15}+\frac{10\!\cdots\!44}{31\!\cdots\!75}a^{14}+\frac{18\!\cdots\!71}{15\!\cdots\!75}a^{13}-\frac{65\!\cdots\!93}{15\!\cdots\!75}a^{12}-\frac{85\!\cdots\!93}{62\!\cdots\!95}a^{11}-\frac{26\!\cdots\!98}{15\!\cdots\!75}a^{10}-\frac{21\!\cdots\!88}{15\!\cdots\!75}a^{9}+\frac{33\!\cdots\!64}{15\!\cdots\!75}a^{8}+\frac{52\!\cdots\!72}{15\!\cdots\!75}a^{7}+\frac{24\!\cdots\!66}{15\!\cdots\!75}a^{6}+\frac{21\!\cdots\!64}{31\!\cdots\!75}a^{5}+\frac{16\!\cdots\!48}{15\!\cdots\!75}a^{4}+\frac{12\!\cdots\!71}{15\!\cdots\!75}a^{3}+\frac{81\!\cdots\!44}{15\!\cdots\!75}a^{2}-\frac{17\!\cdots\!71}{15\!\cdots\!75}a+\frac{25\!\cdots\!93}{31\!\cdots\!75}$, $\frac{55\!\cdots\!71}{15\!\cdots\!75}a^{21}-\frac{13\!\cdots\!52}{15\!\cdots\!75}a^{20}+\frac{60\!\cdots\!41}{15\!\cdots\!75}a^{19}-\frac{65\!\cdots\!81}{15\!\cdots\!75}a^{18}-\frac{34\!\cdots\!78}{31\!\cdots\!75}a^{17}-\frac{29\!\cdots\!47}{15\!\cdots\!75}a^{16}+\frac{14\!\cdots\!02}{62\!\cdots\!95}a^{15}+\frac{53\!\cdots\!14}{31\!\cdots\!75}a^{14}+\frac{62\!\cdots\!51}{15\!\cdots\!75}a^{13}-\frac{46\!\cdots\!33}{15\!\cdots\!75}a^{12}-\frac{48\!\cdots\!93}{62\!\cdots\!95}a^{11}-\frac{99\!\cdots\!88}{15\!\cdots\!75}a^{10}+\frac{84\!\cdots\!97}{15\!\cdots\!75}a^{9}+\frac{30\!\cdots\!59}{15\!\cdots\!75}a^{8}+\frac{31\!\cdots\!57}{15\!\cdots\!75}a^{7}-\frac{10\!\cdots\!04}{15\!\cdots\!75}a^{6}-\frac{49\!\cdots\!91}{31\!\cdots\!75}a^{5}-\frac{19\!\cdots\!37}{15\!\cdots\!75}a^{4}-\frac{49\!\cdots\!99}{15\!\cdots\!75}a^{3}-\frac{23\!\cdots\!36}{15\!\cdots\!75}a^{2}-\frac{66\!\cdots\!26}{15\!\cdots\!75}a-\frac{71\!\cdots\!17}{31\!\cdots\!75}$, $\frac{38\!\cdots\!77}{15\!\cdots\!75}a^{21}+\frac{83\!\cdots\!51}{15\!\cdots\!75}a^{20}-\frac{20\!\cdots\!33}{15\!\cdots\!75}a^{19}+\frac{16\!\cdots\!03}{15\!\cdots\!75}a^{18}-\frac{35\!\cdots\!86}{31\!\cdots\!75}a^{17}-\frac{12\!\cdots\!89}{15\!\cdots\!75}a^{16}-\frac{20\!\cdots\!96}{62\!\cdots\!95}a^{15}+\frac{12\!\cdots\!18}{31\!\cdots\!75}a^{14}+\frac{39\!\cdots\!37}{15\!\cdots\!75}a^{13}+\frac{16\!\cdots\!54}{15\!\cdots\!75}a^{12}-\frac{14\!\cdots\!91}{62\!\cdots\!95}a^{11}-\frac{18\!\cdots\!81}{15\!\cdots\!75}a^{10}-\frac{23\!\cdots\!86}{15\!\cdots\!75}a^{9}-\frac{81\!\cdots\!67}{15\!\cdots\!75}a^{8}+\frac{21\!\cdots\!59}{15\!\cdots\!75}a^{7}+\frac{45\!\cdots\!02}{15\!\cdots\!75}a^{6}+\frac{64\!\cdots\!83}{31\!\cdots\!75}a^{5}+\frac{21\!\cdots\!06}{15\!\cdots\!75}a^{4}+\frac{10\!\cdots\!12}{15\!\cdots\!75}a^{3}+\frac{13\!\cdots\!68}{15\!\cdots\!75}a^{2}+\frac{50\!\cdots\!38}{15\!\cdots\!75}a+\frac{57\!\cdots\!46}{31\!\cdots\!75}$, $\frac{93\!\cdots\!51}{15\!\cdots\!75}a^{21}-\frac{18\!\cdots\!37}{15\!\cdots\!75}a^{20}+\frac{86\!\cdots\!71}{15\!\cdots\!75}a^{19}-\frac{13\!\cdots\!86}{15\!\cdots\!75}a^{18}+\frac{16\!\cdots\!82}{31\!\cdots\!75}a^{17}-\frac{67\!\cdots\!57}{15\!\cdots\!75}a^{16}+\frac{18\!\cdots\!72}{62\!\cdots\!95}a^{15}+\frac{85\!\cdots\!84}{31\!\cdots\!75}a^{14}+\frac{33\!\cdots\!81}{15\!\cdots\!75}a^{13}-\frac{38\!\cdots\!98}{15\!\cdots\!75}a^{12}-\frac{86\!\cdots\!63}{62\!\cdots\!95}a^{11}-\frac{27\!\cdots\!78}{15\!\cdots\!75}a^{10}-\frac{96\!\cdots\!18}{15\!\cdots\!75}a^{9}+\frac{33\!\cdots\!79}{15\!\cdots\!75}a^{8}+\frac{64\!\cdots\!17}{15\!\cdots\!75}a^{7}+\frac{39\!\cdots\!01}{15\!\cdots\!75}a^{6}+\frac{90\!\cdots\!79}{31\!\cdots\!75}a^{5}-\frac{14\!\cdots\!22}{15\!\cdots\!75}a^{4}+\frac{61\!\cdots\!56}{15\!\cdots\!75}a^{3}+\frac{40\!\cdots\!59}{15\!\cdots\!75}a^{2}+\frac{79\!\cdots\!94}{15\!\cdots\!75}a-\frac{11\!\cdots\!52}{31\!\cdots\!75}$, $\frac{21\!\cdots\!07}{15\!\cdots\!75}a^{21}-\frac{56\!\cdots\!84}{15\!\cdots\!75}a^{20}+\frac{58\!\cdots\!72}{15\!\cdots\!75}a^{19}-\frac{55\!\cdots\!77}{15\!\cdots\!75}a^{18}+\frac{38\!\cdots\!24}{31\!\cdots\!75}a^{17}-\frac{14\!\cdots\!74}{15\!\cdots\!75}a^{16}+\frac{71\!\cdots\!59}{62\!\cdots\!95}a^{15}+\frac{17\!\cdots\!38}{31\!\cdots\!75}a^{14}+\frac{10\!\cdots\!42}{15\!\cdots\!75}a^{13}-\frac{78\!\cdots\!11}{15\!\cdots\!75}a^{12}-\frac{15\!\cdots\!06}{62\!\cdots\!95}a^{11}-\frac{38\!\cdots\!71}{15\!\cdots\!75}a^{10}-\frac{56\!\cdots\!01}{15\!\cdots\!75}a^{9}+\frac{50\!\cdots\!78}{15\!\cdots\!75}a^{8}+\frac{84\!\cdots\!44}{15\!\cdots\!75}a^{7}+\frac{38\!\cdots\!57}{15\!\cdots\!75}a^{6}+\frac{69\!\cdots\!53}{31\!\cdots\!75}a^{5}+\frac{14\!\cdots\!71}{15\!\cdots\!75}a^{4}+\frac{24\!\cdots\!17}{15\!\cdots\!75}a^{3}-\frac{18\!\cdots\!62}{15\!\cdots\!75}a^{2}+\frac{58\!\cdots\!33}{15\!\cdots\!75}a-\frac{15\!\cdots\!89}{31\!\cdots\!75}$, $\frac{27\!\cdots\!04}{15\!\cdots\!75}a^{21}-\frac{41\!\cdots\!48}{15\!\cdots\!75}a^{20}+\frac{82\!\cdots\!09}{15\!\cdots\!75}a^{19}-\frac{10\!\cdots\!44}{15\!\cdots\!75}a^{18}-\frac{58\!\cdots\!97}{31\!\cdots\!75}a^{17}-\frac{17\!\cdots\!78}{15\!\cdots\!75}a^{16}+\frac{12\!\cdots\!33}{62\!\cdots\!95}a^{15}+\frac{26\!\cdots\!11}{31\!\cdots\!75}a^{14}+\frac{14\!\cdots\!74}{15\!\cdots\!75}a^{13}-\frac{56\!\cdots\!17}{15\!\cdots\!75}a^{12}-\frac{24\!\cdots\!37}{62\!\cdots\!95}a^{11}-\frac{10\!\cdots\!37}{15\!\cdots\!75}a^{10}-\frac{76\!\cdots\!72}{15\!\cdots\!75}a^{9}+\frac{47\!\cdots\!16}{15\!\cdots\!75}a^{8}+\frac{18\!\cdots\!43}{15\!\cdots\!75}a^{7}+\frac{19\!\cdots\!54}{15\!\cdots\!75}a^{6}+\frac{24\!\cdots\!41}{31\!\cdots\!75}a^{5}+\frac{70\!\cdots\!87}{15\!\cdots\!75}a^{4}+\frac{59\!\cdots\!74}{15\!\cdots\!75}a^{3}+\frac{40\!\cdots\!86}{15\!\cdots\!75}a^{2}+\frac{15\!\cdots\!01}{15\!\cdots\!75}a+\frac{70\!\cdots\!17}{31\!\cdots\!75}$, $\frac{32\!\cdots\!42}{15\!\cdots\!75}a^{21}-\frac{10\!\cdots\!29}{15\!\cdots\!75}a^{20}+\frac{13\!\cdots\!07}{15\!\cdots\!75}a^{19}-\frac{64\!\cdots\!62}{15\!\cdots\!75}a^{18}-\frac{14\!\cdots\!81}{31\!\cdots\!75}a^{17}-\frac{87\!\cdots\!19}{15\!\cdots\!75}a^{16}+\frac{13\!\cdots\!79}{62\!\cdots\!95}a^{15}+\frac{23\!\cdots\!53}{31\!\cdots\!75}a^{14}-\frac{11\!\cdots\!48}{15\!\cdots\!75}a^{13}-\frac{13\!\cdots\!16}{15\!\cdots\!75}a^{12}-\frac{11\!\cdots\!96}{62\!\cdots\!95}a^{11}-\frac{12\!\cdots\!76}{15\!\cdots\!75}a^{10}+\frac{28\!\cdots\!19}{15\!\cdots\!75}a^{9}+\frac{20\!\cdots\!68}{15\!\cdots\!75}a^{8}-\frac{59\!\cdots\!86}{15\!\cdots\!75}a^{7}-\frac{42\!\cdots\!58}{15\!\cdots\!75}a^{6}+\frac{20\!\cdots\!93}{31\!\cdots\!75}a^{5}+\frac{15\!\cdots\!01}{15\!\cdots\!75}a^{4}+\frac{47\!\cdots\!77}{15\!\cdots\!75}a^{3}-\frac{55\!\cdots\!47}{15\!\cdots\!75}a^{2}+\frac{20\!\cdots\!98}{15\!\cdots\!75}a+\frac{75\!\cdots\!16}{31\!\cdots\!75}$, $\frac{14\!\cdots\!84}{15\!\cdots\!75}a^{21}-\frac{35\!\cdots\!83}{15\!\cdots\!75}a^{20}+\frac{27\!\cdots\!89}{15\!\cdots\!75}a^{19}-\frac{26\!\cdots\!49}{15\!\cdots\!75}a^{18}+\frac{37\!\cdots\!88}{31\!\cdots\!75}a^{17}-\frac{10\!\cdots\!63}{15\!\cdots\!75}a^{16}+\frac{45\!\cdots\!83}{62\!\cdots\!95}a^{15}+\frac{12\!\cdots\!31}{31\!\cdots\!75}a^{14}+\frac{17\!\cdots\!04}{15\!\cdots\!75}a^{13}-\frac{80\!\cdots\!57}{15\!\cdots\!75}a^{12}-\frac{12\!\cdots\!57}{62\!\cdots\!95}a^{11}-\frac{28\!\cdots\!52}{15\!\cdots\!75}a^{10}+\frac{58\!\cdots\!63}{15\!\cdots\!75}a^{9}+\frac{59\!\cdots\!36}{15\!\cdots\!75}a^{8}+\frac{79\!\cdots\!78}{15\!\cdots\!75}a^{7}+\frac{20\!\cdots\!34}{15\!\cdots\!75}a^{6}-\frac{43\!\cdots\!64}{31\!\cdots\!75}a^{5}-\frac{34\!\cdots\!23}{15\!\cdots\!75}a^{4}+\frac{75\!\cdots\!54}{15\!\cdots\!75}a^{3}-\frac{48\!\cdots\!94}{15\!\cdots\!75}a^{2}-\frac{53\!\cdots\!79}{15\!\cdots\!75}a-\frac{18\!\cdots\!93}{31\!\cdots\!75}$, $\frac{10\!\cdots\!48}{15\!\cdots\!75}a^{21}-\frac{22\!\cdots\!01}{15\!\cdots\!75}a^{20}+\frac{16\!\cdots\!83}{15\!\cdots\!75}a^{19}+\frac{49\!\cdots\!97}{15\!\cdots\!75}a^{18}-\frac{13\!\cdots\!64}{31\!\cdots\!75}a^{17}-\frac{73\!\cdots\!61}{15\!\cdots\!75}a^{16}+\frac{25\!\cdots\!86}{62\!\cdots\!95}a^{15}+\frac{11\!\cdots\!57}{31\!\cdots\!75}a^{14}+\frac{29\!\cdots\!63}{15\!\cdots\!75}a^{13}-\frac{91\!\cdots\!29}{15\!\cdots\!75}a^{12}-\frac{10\!\cdots\!69}{62\!\cdots\!95}a^{11}-\frac{26\!\cdots\!69}{15\!\cdots\!75}a^{10}+\frac{10\!\cdots\!61}{15\!\cdots\!75}a^{9}+\frac{64\!\cdots\!42}{15\!\cdots\!75}a^{8}+\frac{78\!\cdots\!41}{15\!\cdots\!75}a^{7}+\frac{13\!\cdots\!98}{15\!\cdots\!75}a^{6}-\frac{94\!\cdots\!08}{31\!\cdots\!75}a^{5}-\frac{43\!\cdots\!06}{15\!\cdots\!75}a^{4}-\frac{49\!\cdots\!37}{15\!\cdots\!75}a^{3}-\frac{33\!\cdots\!68}{15\!\cdots\!75}a^{2}-\frac{11\!\cdots\!13}{15\!\cdots\!75}a-\frac{17\!\cdots\!46}{31\!\cdots\!75}$, $\frac{16\!\cdots\!69}{15\!\cdots\!75}a^{21}-\frac{29\!\cdots\!28}{15\!\cdots\!75}a^{20}-\frac{91\!\cdots\!76}{15\!\cdots\!75}a^{19}+\frac{36\!\cdots\!41}{15\!\cdots\!75}a^{18}-\frac{10\!\cdots\!67}{31\!\cdots\!75}a^{17}-\frac{74\!\cdots\!33}{15\!\cdots\!75}a^{16}+\frac{22\!\cdots\!13}{62\!\cdots\!95}a^{15}+\frac{17\!\cdots\!71}{31\!\cdots\!75}a^{14}+\frac{51\!\cdots\!39}{15\!\cdots\!75}a^{13}-\frac{13\!\cdots\!87}{15\!\cdots\!75}a^{12}-\frac{14\!\cdots\!12}{62\!\cdots\!95}a^{11}-\frac{44\!\cdots\!32}{15\!\cdots\!75}a^{10}+\frac{37\!\cdots\!33}{15\!\cdots\!75}a^{9}+\frac{71\!\cdots\!01}{15\!\cdots\!75}a^{8}+\frac{97\!\cdots\!48}{15\!\cdots\!75}a^{7}+\frac{32\!\cdots\!19}{15\!\cdots\!75}a^{6}-\frac{55\!\cdots\!99}{31\!\cdots\!75}a^{5}-\frac{11\!\cdots\!43}{15\!\cdots\!75}a^{4}+\frac{15\!\cdots\!89}{15\!\cdots\!75}a^{3}+\frac{47\!\cdots\!46}{15\!\cdots\!75}a^{2}-\frac{14\!\cdots\!89}{15\!\cdots\!75}a+\frac{16\!\cdots\!12}{31\!\cdots\!75}$
| 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: | \( 221981.001358 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 221981.001358 \cdot 1}{2\cdot\sqrt{31524414620673611592542191616}}\cr\approx \mathstrut & 0.239784424746 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + x^20 - x^19 - 7*x^17 + 5*x^16 + 45*x^15 + 31*x^14 - 38*x^13 - 205*x^12 - 278*x^11 - 123*x^10 + 249*x^9 + 532*x^8 + 371*x^7 + 205*x^6 + 153*x^5 + 236*x^4 + 94*x^3 + 34*x^2 + 5*x + 25)
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 - 2*x^21 + x^20 - x^19 - 7*x^17 + 5*x^16 + 45*x^15 + 31*x^14 - 38*x^13 - 205*x^12 - 278*x^11 - 123*x^10 + 249*x^9 + 532*x^8 + 371*x^7 + 205*x^6 + 153*x^5 + 236*x^4 + 94*x^3 + 34*x^2 + 5*x + 25, 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 - 2*x^21 + x^20 - x^19 - 7*x^17 + 5*x^16 + 45*x^15 + 31*x^14 - 38*x^13 - 205*x^12 - 278*x^11 - 123*x^10 + 249*x^9 + 532*x^8 + 371*x^7 + 205*x^6 + 153*x^5 + 236*x^4 + 94*x^3 + 34*x^2 + 5*x + 25);
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 - 2*x^21 + x^20 - x^19 - 7*x^17 + 5*x^16 + 45*x^15 + 31*x^14 - 38*x^13 - 205*x^12 - 278*x^11 - 123*x^10 + 249*x^9 + 532*x^8 + 371*x^7 + 205*x^6 + 153*x^5 + 236*x^4 + 94*x^3 + 34*x^2 + 5*x + 25);
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
$C_2^{11}.A_{11}$ (as 22T52):
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 40874803200 |
| The 400 conjugacy class representatives for $C_2^{11}.A_{11}$ are not computed |
| Character table for $C_2^{11}.A_{11}$ is not computed |
Intermediate fields
| 11.3.836463893056.1 |
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 44 sibling: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $22$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | R | $22$ | ${\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.18.24.39 | $x^{18} + 6 x^{17} + 16 x^{16} + 14 x^{15} + 30 x^{14} + 56 x^{13} + 42 x^{12} + 80 x^{11} + 92 x^{10} + 24 x^{9} + 80 x^{8} + 88 x^{7} + 60 x^{6} + 56 x^{5} - 8 x^{4} + 120 x^{3} + 120 x^{2} + 72$ | $6$ | $3$ | $24$ | 18T175 | $[4/3, 4/3, 4/3, 4/3, 2]_{3}^{6}$ | |
|
\(11\)
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.6.5.2 | $x^{6} + 11$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ | |
|
\(19\)
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.7.0.1 | $x^{7} + 6 x + 17$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 19.7.0.1 | $x^{7} + 6 x + 17$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
|
\(547\)
| $\Q_{547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |