Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 + 12*x^19 - 22*x^18 + 41*x^17 - 54*x^16 + 79*x^15 - 60*x^14 + 57*x^13 + 46*x^12 - 70*x^11 + 222*x^10 - 182*x^9 + 306*x^8 - 168*x^7 + 207*x^6 - 48*x^5 + 42*x^4 + 5*x^3 - 17*x^2 - 15*x - 1)
gp: K = bnfinit(y^21 - 4*y^20 + 12*y^19 - 22*y^18 + 41*y^17 - 54*y^16 + 79*y^15 - 60*y^14 + 57*y^13 + 46*y^12 - 70*y^11 + 222*y^10 - 182*y^9 + 306*y^8 - 168*y^7 + 207*y^6 - 48*y^5 + 42*y^4 + 5*y^3 - 17*y^2 - 15*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 4*x^20 + 12*x^19 - 22*x^18 + 41*x^17 - 54*x^16 + 79*x^15 - 60*x^14 + 57*x^13 + 46*x^12 - 70*x^11 + 222*x^10 - 182*x^9 + 306*x^8 - 168*x^7 + 207*x^6 - 48*x^5 + 42*x^4 + 5*x^3 - 17*x^2 - 15*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 4*x^20 + 12*x^19 - 22*x^18 + 41*x^17 - 54*x^16 + 79*x^15 - 60*x^14 + 57*x^13 + 46*x^12 - 70*x^11 + 222*x^10 - 182*x^9 + 306*x^8 - 168*x^7 + 207*x^6 - 48*x^5 + 42*x^4 + 5*x^3 - 17*x^2 - 15*x - 1)
\( x^{21} - 4 x^{20} + 12 x^{19} - 22 x^{18} + 41 x^{17} - 54 x^{16} + 79 x^{15} - 60 x^{14} + 57 x^{13} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[3, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-65701728236743660173798567\)
\(\medspace = -\,3^{24}\cdot 7^{17}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.96\) | 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: | $3^{4/3}7^{5/6}\approx 21.898281770364438$ | ||
| Ramified primes: |
\(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{16\!\cdots\!39}a^{20}-\frac{73\!\cdots\!73}{16\!\cdots\!39}a^{19}+\frac{49\!\cdots\!40}{16\!\cdots\!39}a^{18}-\frac{25\!\cdots\!28}{16\!\cdots\!39}a^{17}-\frac{50\!\cdots\!47}{16\!\cdots\!39}a^{16}+\frac{18\!\cdots\!12}{16\!\cdots\!39}a^{15}-\frac{19\!\cdots\!57}{16\!\cdots\!39}a^{14}-\frac{52\!\cdots\!02}{16\!\cdots\!39}a^{13}+\frac{51\!\cdots\!11}{16\!\cdots\!39}a^{12}-\frac{73\!\cdots\!09}{16\!\cdots\!39}a^{11}+\frac{84\!\cdots\!80}{16\!\cdots\!39}a^{10}+\frac{75\!\cdots\!81}{16\!\cdots\!39}a^{9}+\frac{37\!\cdots\!40}{16\!\cdots\!39}a^{8}-\frac{48\!\cdots\!29}{16\!\cdots\!39}a^{7}+\frac{82\!\cdots\!20}{16\!\cdots\!39}a^{6}+\frac{66\!\cdots\!79}{16\!\cdots\!39}a^{5}+\frac{31\!\cdots\!94}{16\!\cdots\!39}a^{4}+\frac{46\!\cdots\!27}{16\!\cdots\!39}a^{3}-\frac{12\!\cdots\!35}{16\!\cdots\!39}a^{2}-\frac{76\!\cdots\!09}{16\!\cdots\!39}a-\frac{64\!\cdots\!25}{16\!\cdots\!39}$
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: |
$\frac{62\!\cdots\!78}{16\!\cdots\!39}a^{20}-\frac{20\!\cdots\!13}{16\!\cdots\!39}a^{19}+\frac{56\!\cdots\!94}{16\!\cdots\!39}a^{18}-\frac{82\!\cdots\!81}{16\!\cdots\!39}a^{17}+\frac{15\!\cdots\!23}{16\!\cdots\!39}a^{16}-\frac{16\!\cdots\!29}{16\!\cdots\!39}a^{15}+\frac{26\!\cdots\!13}{16\!\cdots\!39}a^{14}-\frac{84\!\cdots\!82}{16\!\cdots\!39}a^{13}+\frac{14\!\cdots\!06}{16\!\cdots\!39}a^{12}+\frac{36\!\cdots\!18}{16\!\cdots\!39}a^{11}-\frac{10\!\cdots\!20}{16\!\cdots\!39}a^{10}+\frac{81\!\cdots\!10}{16\!\cdots\!39}a^{9}-\frac{16\!\cdots\!03}{16\!\cdots\!39}a^{8}+\frac{11\!\cdots\!60}{16\!\cdots\!39}a^{7}-\frac{68\!\cdots\!86}{16\!\cdots\!39}a^{6}+\frac{83\!\cdots\!46}{16\!\cdots\!39}a^{5}+\frac{10\!\cdots\!76}{16\!\cdots\!39}a^{4}+\frac{57\!\cdots\!21}{16\!\cdots\!39}a^{3}-\frac{89\!\cdots\!00}{16\!\cdots\!39}a^{2}+\frac{27\!\cdots\!58}{16\!\cdots\!39}a-\frac{81\!\cdots\!89}{16\!\cdots\!39}$, $\frac{33\!\cdots\!27}{16\!\cdots\!39}a^{20}-\frac{13\!\cdots\!49}{16\!\cdots\!39}a^{19}+\frac{42\!\cdots\!29}{16\!\cdots\!39}a^{18}-\frac{78\!\cdots\!60}{16\!\cdots\!39}a^{17}+\frac{14\!\cdots\!96}{16\!\cdots\!39}a^{16}-\frac{20\!\cdots\!50}{16\!\cdots\!39}a^{15}+\frac{29\!\cdots\!58}{16\!\cdots\!39}a^{14}-\frac{24\!\cdots\!59}{16\!\cdots\!39}a^{13}+\frac{23\!\cdots\!32}{16\!\cdots\!39}a^{12}+\frac{10\!\cdots\!43}{16\!\cdots\!39}a^{11}-\frac{22\!\cdots\!89}{16\!\cdots\!39}a^{10}+\frac{75\!\cdots\!37}{16\!\cdots\!39}a^{9}-\frac{70\!\cdots\!66}{16\!\cdots\!39}a^{8}+\frac{11\!\cdots\!48}{16\!\cdots\!39}a^{7}-\frac{75\!\cdots\!21}{16\!\cdots\!39}a^{6}+\frac{86\!\cdots\!42}{16\!\cdots\!39}a^{5}-\frac{32\!\cdots\!87}{16\!\cdots\!39}a^{4}+\frac{25\!\cdots\!79}{16\!\cdots\!39}a^{3}-\frac{44\!\cdots\!18}{16\!\cdots\!39}a^{2}-\frac{36\!\cdots\!12}{16\!\cdots\!39}a-\frac{24\!\cdots\!07}{16\!\cdots\!39}$, $\frac{30\!\cdots\!87}{16\!\cdots\!39}a^{20}-\frac{12\!\cdots\!60}{16\!\cdots\!39}a^{19}+\frac{39\!\cdots\!06}{16\!\cdots\!39}a^{18}-\frac{73\!\cdots\!78}{16\!\cdots\!39}a^{17}+\frac{13\!\cdots\!12}{16\!\cdots\!39}a^{16}-\frac{18\!\cdots\!17}{16\!\cdots\!39}a^{15}+\frac{26\!\cdots\!04}{16\!\cdots\!39}a^{14}-\frac{22\!\cdots\!99}{16\!\cdots\!39}a^{13}+\frac{20\!\cdots\!46}{16\!\cdots\!39}a^{12}+\frac{11\!\cdots\!71}{16\!\cdots\!39}a^{11}-\frac{23\!\cdots\!94}{16\!\cdots\!39}a^{10}+\frac{71\!\cdots\!73}{16\!\cdots\!39}a^{9}-\frac{65\!\cdots\!23}{16\!\cdots\!39}a^{8}+\frac{10\!\cdots\!31}{16\!\cdots\!39}a^{7}-\frac{65\!\cdots\!15}{16\!\cdots\!39}a^{6}+\frac{75\!\cdots\!84}{16\!\cdots\!39}a^{5}-\frac{22\!\cdots\!23}{16\!\cdots\!39}a^{4}+\frac{18\!\cdots\!24}{16\!\cdots\!39}a^{3}+\frac{74\!\cdots\!09}{16\!\cdots\!39}a^{2}-\frac{42\!\cdots\!31}{16\!\cdots\!39}a-\frac{33\!\cdots\!22}{16\!\cdots\!39}$, $\frac{19\!\cdots\!74}{16\!\cdots\!39}a^{20}-\frac{11\!\cdots\!53}{16\!\cdots\!39}a^{19}+\frac{37\!\cdots\!43}{16\!\cdots\!39}a^{18}-\frac{81\!\cdots\!81}{16\!\cdots\!39}a^{17}+\frac{14\!\cdots\!90}{16\!\cdots\!39}a^{16}-\frac{22\!\cdots\!16}{16\!\cdots\!39}a^{15}+\frac{28\!\cdots\!59}{16\!\cdots\!39}a^{14}-\frac{32\!\cdots\!83}{16\!\cdots\!39}a^{13}+\frac{18\!\cdots\!75}{16\!\cdots\!39}a^{12}-\frac{28\!\cdots\!69}{16\!\cdots\!39}a^{11}-\frac{41\!\cdots\!18}{16\!\cdots\!39}a^{10}+\frac{56\!\cdots\!99}{16\!\cdots\!39}a^{9}-\frac{10\!\cdots\!92}{16\!\cdots\!39}a^{8}+\frac{83\!\cdots\!48}{16\!\cdots\!39}a^{7}-\frac{11\!\cdots\!40}{16\!\cdots\!39}a^{6}+\frac{41\!\cdots\!00}{16\!\cdots\!39}a^{5}-\frac{55\!\cdots\!99}{16\!\cdots\!39}a^{4}+\frac{45\!\cdots\!37}{16\!\cdots\!39}a^{3}-\frac{22\!\cdots\!65}{16\!\cdots\!39}a^{2}-\frac{59\!\cdots\!14}{16\!\cdots\!39}a-\frac{79\!\cdots\!92}{16\!\cdots\!39}$, $\frac{97721335297616}{582192140674087}a^{20}-\frac{399443896722011}{582192140674087}a^{19}+\frac{11\!\cdots\!70}{582192140674087}a^{18}-\frac{21\!\cdots\!09}{582192140674087}a^{17}+\frac{40\!\cdots\!02}{582192140674087}a^{16}-\frac{53\!\cdots\!80}{582192140674087}a^{15}+\frac{76\!\cdots\!81}{582192140674087}a^{14}-\frac{58\!\cdots\!89}{582192140674087}a^{13}+\frac{52\!\cdots\!02}{582192140674087}a^{12}+\frac{45\!\cdots\!37}{582192140674087}a^{11}-\frac{74\!\cdots\!74}{582192140674087}a^{10}+\frac{21\!\cdots\!81}{582192140674087}a^{9}-\frac{18\!\cdots\!73}{582192140674087}a^{8}+\frac{28\!\cdots\!54}{582192140674087}a^{7}-\frac{16\!\cdots\!64}{582192140674087}a^{6}+\frac{19\!\cdots\!79}{582192140674087}a^{5}-\frac{44\!\cdots\!09}{582192140674087}a^{4}+\frac{42\!\cdots\!34}{582192140674087}a^{3}+\frac{516970772568214}{582192140674087}a^{2}-\frac{11\!\cdots\!05}{582192140674087}a-\frac{808905272281207}{582192140674087}$, $\frac{13\!\cdots\!22}{16\!\cdots\!39}a^{20}-\frac{62\!\cdots\!13}{16\!\cdots\!39}a^{19}+\frac{20\!\cdots\!88}{16\!\cdots\!39}a^{18}-\frac{41\!\cdots\!77}{16\!\cdots\!39}a^{17}+\frac{78\!\cdots\!70}{16\!\cdots\!39}a^{16}-\frac{11\!\cdots\!14}{16\!\cdots\!39}a^{15}+\frac{16\!\cdots\!07}{16\!\cdots\!39}a^{14}-\frac{16\!\cdots\!26}{16\!\cdots\!39}a^{13}+\frac{14\!\cdots\!50}{16\!\cdots\!39}a^{12}+\frac{61\!\cdots\!73}{16\!\cdots\!39}a^{11}-\frac{13\!\cdots\!57}{16\!\cdots\!39}a^{10}+\frac{38\!\cdots\!53}{16\!\cdots\!39}a^{9}-\frac{46\!\cdots\!26}{16\!\cdots\!39}a^{8}+\frac{61\!\cdots\!40}{16\!\cdots\!39}a^{7}-\frac{52\!\cdots\!81}{16\!\cdots\!39}a^{6}+\frac{47\!\cdots\!82}{16\!\cdots\!39}a^{5}-\frac{26\!\cdots\!99}{16\!\cdots\!39}a^{4}+\frac{11\!\cdots\!26}{16\!\cdots\!39}a^{3}-\frac{35\!\cdots\!72}{16\!\cdots\!39}a^{2}-\frac{38\!\cdots\!23}{16\!\cdots\!39}a-\frac{15\!\cdots\!04}{16\!\cdots\!39}$, $\frac{11\!\cdots\!68}{16\!\cdots\!39}a^{20}-\frac{50\!\cdots\!66}{16\!\cdots\!39}a^{19}+\frac{15\!\cdots\!39}{16\!\cdots\!39}a^{18}-\frac{32\!\cdots\!34}{16\!\cdots\!39}a^{17}+\frac{62\!\cdots\!57}{16\!\cdots\!39}a^{16}-\frac{92\!\cdots\!90}{16\!\cdots\!39}a^{15}+\frac{13\!\cdots\!82}{16\!\cdots\!39}a^{14}-\frac{13\!\cdots\!93}{16\!\cdots\!39}a^{13}+\frac{12\!\cdots\!66}{16\!\cdots\!39}a^{12}-\frac{95\!\cdots\!79}{16\!\cdots\!39}a^{11}-\frac{83\!\cdots\!97}{16\!\cdots\!39}a^{10}+\frac{30\!\cdots\!71}{16\!\cdots\!39}a^{9}-\frac{37\!\cdots\!53}{16\!\cdots\!39}a^{8}+\frac{53\!\cdots\!92}{16\!\cdots\!39}a^{7}-\frac{45\!\cdots\!38}{16\!\cdots\!39}a^{6}+\frac{43\!\cdots\!08}{16\!\cdots\!39}a^{5}-\frac{23\!\cdots\!35}{16\!\cdots\!39}a^{4}+\frac{13\!\cdots\!52}{16\!\cdots\!39}a^{3}-\frac{23\!\cdots\!04}{16\!\cdots\!39}a^{2}-\frac{43\!\cdots\!20}{16\!\cdots\!39}a-\frac{88\!\cdots\!82}{16\!\cdots\!39}$, $\frac{59\!\cdots\!86}{16\!\cdots\!39}a^{20}-\frac{24\!\cdots\!00}{16\!\cdots\!39}a^{19}+\frac{73\!\cdots\!35}{16\!\cdots\!39}a^{18}-\frac{13\!\cdots\!64}{16\!\cdots\!39}a^{17}+\frac{25\!\cdots\!84}{16\!\cdots\!39}a^{16}-\frac{33\!\cdots\!04}{16\!\cdots\!39}a^{15}+\frac{48\!\cdots\!37}{16\!\cdots\!39}a^{14}-\frac{37\!\cdots\!99}{16\!\cdots\!39}a^{13}+\frac{34\!\cdots\!59}{16\!\cdots\!39}a^{12}+\frac{28\!\cdots\!46}{16\!\cdots\!39}a^{11}-\frac{44\!\cdots\!60}{16\!\cdots\!39}a^{10}+\frac{13\!\cdots\!67}{16\!\cdots\!39}a^{9}-\frac{11\!\cdots\!20}{16\!\cdots\!39}a^{8}+\frac{18\!\cdots\!03}{16\!\cdots\!39}a^{7}-\frac{10\!\cdots\!08}{16\!\cdots\!39}a^{6}+\frac{12\!\cdots\!25}{16\!\cdots\!39}a^{5}-\frac{24\!\cdots\!55}{16\!\cdots\!39}a^{4}+\frac{22\!\cdots\!19}{16\!\cdots\!39}a^{3}+\frac{92\!\cdots\!12}{16\!\cdots\!39}a^{2}-\frac{11\!\cdots\!74}{16\!\cdots\!39}a-\frac{59\!\cdots\!29}{16\!\cdots\!39}$, $\frac{19\!\cdots\!76}{16\!\cdots\!39}a^{20}-\frac{80\!\cdots\!53}{16\!\cdots\!39}a^{19}+\frac{24\!\cdots\!29}{16\!\cdots\!39}a^{18}-\frac{45\!\cdots\!16}{16\!\cdots\!39}a^{17}+\frac{83\!\cdots\!26}{16\!\cdots\!39}a^{16}-\frac{11\!\cdots\!72}{16\!\cdots\!39}a^{15}+\frac{16\!\cdots\!26}{16\!\cdots\!39}a^{14}-\frac{12\!\cdots\!30}{16\!\cdots\!39}a^{13}+\frac{11\!\cdots\!32}{16\!\cdots\!39}a^{12}+\frac{87\!\cdots\!17}{16\!\cdots\!39}a^{11}-\frac{15\!\cdots\!61}{16\!\cdots\!39}a^{10}+\frac{45\!\cdots\!84}{16\!\cdots\!39}a^{9}-\frac{39\!\cdots\!12}{16\!\cdots\!39}a^{8}+\frac{62\!\cdots\!18}{16\!\cdots\!39}a^{7}-\frac{35\!\cdots\!12}{16\!\cdots\!39}a^{6}+\frac{41\!\cdots\!54}{16\!\cdots\!39}a^{5}-\frac{89\!\cdots\!12}{16\!\cdots\!39}a^{4}+\frac{78\!\cdots\!07}{16\!\cdots\!39}a^{3}+\frac{19\!\cdots\!43}{16\!\cdots\!39}a^{2}-\frac{28\!\cdots\!44}{16\!\cdots\!39}a-\frac{27\!\cdots\!19}{16\!\cdots\!39}$, $\frac{16\!\cdots\!60}{16\!\cdots\!39}a^{20}-\frac{76\!\cdots\!23}{16\!\cdots\!39}a^{19}+\frac{24\!\cdots\!34}{16\!\cdots\!39}a^{18}-\frac{48\!\cdots\!34}{16\!\cdots\!39}a^{17}+\frac{89\!\cdots\!42}{16\!\cdots\!39}a^{16}-\frac{12\!\cdots\!25}{16\!\cdots\!39}a^{15}+\frac{17\!\cdots\!11}{16\!\cdots\!39}a^{14}-\frac{16\!\cdots\!34}{16\!\cdots\!39}a^{13}+\frac{14\!\cdots\!59}{16\!\cdots\!39}a^{12}+\frac{43\!\cdots\!39}{16\!\cdots\!39}a^{11}-\frac{17\!\cdots\!64}{16\!\cdots\!39}a^{10}+\frac{45\!\cdots\!39}{16\!\cdots\!39}a^{9}-\frac{49\!\cdots\!56}{16\!\cdots\!39}a^{8}+\frac{66\!\cdots\!82}{16\!\cdots\!39}a^{7}-\frac{51\!\cdots\!86}{16\!\cdots\!39}a^{6}+\frac{48\!\cdots\!76}{16\!\cdots\!39}a^{5}-\frac{20\!\cdots\!61}{16\!\cdots\!39}a^{4}+\frac{10\!\cdots\!83}{16\!\cdots\!39}a^{3}-\frac{57\!\cdots\!71}{16\!\cdots\!39}a^{2}-\frac{13\!\cdots\!44}{16\!\cdots\!39}a-\frac{18\!\cdots\!51}{16\!\cdots\!39}$, $\frac{24\!\cdots\!25}{16\!\cdots\!39}a^{20}-\frac{10\!\cdots\!16}{16\!\cdots\!39}a^{19}+\frac{32\!\cdots\!52}{16\!\cdots\!39}a^{18}-\frac{63\!\cdots\!58}{16\!\cdots\!39}a^{17}+\frac{11\!\cdots\!27}{16\!\cdots\!39}a^{16}-\frac{16\!\cdots\!41}{16\!\cdots\!39}a^{15}+\frac{24\!\cdots\!99}{16\!\cdots\!39}a^{14}-\frac{22\!\cdots\!51}{16\!\cdots\!39}a^{13}+\frac{20\!\cdots\!21}{16\!\cdots\!39}a^{12}+\frac{49\!\cdots\!12}{16\!\cdots\!39}a^{11}-\frac{19\!\cdots\!40}{16\!\cdots\!39}a^{10}+\frac{60\!\cdots\!84}{16\!\cdots\!39}a^{9}-\frac{63\!\cdots\!95}{16\!\cdots\!39}a^{8}+\frac{94\!\cdots\!89}{16\!\cdots\!39}a^{7}-\frac{70\!\cdots\!65}{16\!\cdots\!39}a^{6}+\frac{71\!\cdots\!58}{16\!\cdots\!39}a^{5}-\frac{32\!\cdots\!56}{16\!\cdots\!39}a^{4}+\frac{18\!\cdots\!44}{16\!\cdots\!39}a^{3}-\frac{39\!\cdots\!51}{16\!\cdots\!39}a^{2}-\frac{60\!\cdots\!12}{16\!\cdots\!39}a-\frac{20\!\cdots\!44}{16\!\cdots\!39}$
| 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: | \( 31647.6272279 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 31647.6272279 \cdot 1}{2\cdot\sqrt{65701728236743660173798567}}\cr\approx \mathstrut & 0.238359005225 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 + 12*x^19 - 22*x^18 + 41*x^17 - 54*x^16 + 79*x^15 - 60*x^14 + 57*x^13 + 46*x^12 - 70*x^11 + 222*x^10 - 182*x^9 + 306*x^8 - 168*x^7 + 207*x^6 - 48*x^5 + 42*x^4 + 5*x^3 - 17*x^2 - 15*x - 1)
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^21 - 4*x^20 + 12*x^19 - 22*x^18 + 41*x^17 - 54*x^16 + 79*x^15 - 60*x^14 + 57*x^13 + 46*x^12 - 70*x^11 + 222*x^10 - 182*x^9 + 306*x^8 - 168*x^7 + 207*x^6 - 48*x^5 + 42*x^4 + 5*x^3 - 17*x^2 - 15*x - 1, 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^21 - 4*x^20 + 12*x^19 - 22*x^18 + 41*x^17 - 54*x^16 + 79*x^15 - 60*x^14 + 57*x^13 + 46*x^12 - 70*x^11 + 222*x^10 - 182*x^9 + 306*x^8 - 168*x^7 + 207*x^6 - 48*x^5 + 42*x^4 + 5*x^3 - 17*x^2 - 15*x - 1);
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^21 - 4*x^20 + 12*x^19 - 22*x^18 + 41*x^17 - 54*x^16 + 79*x^15 - 60*x^14 + 57*x^13 + 46*x^12 - 70*x^11 + 222*x^10 - 182*x^9 + 306*x^8 - 168*x^7 + 207*x^6 - 48*x^5 + 42*x^4 + 5*x^3 - 17*x^2 - 15*x - 1);
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
$C_3\times F_7$ (as 21T9):
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 126 |
| The 21 conjugacy class representatives for $C_3\times F_7$ |
| Character table for $C_3\times F_7$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 7.1.110270727.1 |
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 21 siblings: | data not computed |
| Degree 42 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 | $21$ | R | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{7}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }^{7}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{7}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.3.0.1}{3} }^{7}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| Deg $18$ | $3$ | $6$ | $24$ | ||||
|
\(7\)
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.18.15.5 | $x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$ | $6$ | $3$ | $15$ | $C_6 \times C_3$ | $[\ ]_{6}^{3}$ |