Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 3*x^19 + 15*x^18 - 14*x^17 - 42*x^16 + 112*x^15 + 113*x^14 + 10*x^13 - 937*x^12 - 860*x^11 + 1848*x^10 + 3892*x^9 - 2944*x^8 - 2961*x^7 + 1339*x^6 - 176*x^5 - 182*x^4 + 74*x^3 - 4*x^2 - 6*x + 1)
gp: K = bnfinit(y^21 - y^20 - 3*y^19 + 15*y^18 - 14*y^17 - 42*y^16 + 112*y^15 + 113*y^14 + 10*y^13 - 937*y^12 - 860*y^11 + 1848*y^10 + 3892*y^9 - 2944*y^8 - 2961*y^7 + 1339*y^6 - 176*y^5 - 182*y^4 + 74*y^3 - 4*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - x^20 - 3*x^19 + 15*x^18 - 14*x^17 - 42*x^16 + 112*x^15 + 113*x^14 + 10*x^13 - 937*x^12 - 860*x^11 + 1848*x^10 + 3892*x^9 - 2944*x^8 - 2961*x^7 + 1339*x^6 - 176*x^5 - 182*x^4 + 74*x^3 - 4*x^2 - 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - x^20 - 3*x^19 + 15*x^18 - 14*x^17 - 42*x^16 + 112*x^15 + 113*x^14 + 10*x^13 - 937*x^12 - 860*x^11 + 1848*x^10 + 3892*x^9 - 2944*x^8 - 2961*x^7 + 1339*x^6 - 176*x^5 - 182*x^4 + 74*x^3 - 4*x^2 - 6*x + 1)
\( x^{21} - x^{20} - 3 x^{19} + 15 x^{18} - 14 x^{17} - 42 x^{16} + 112 x^{15} + 113 x^{14} + 10 x^{13} + \cdots + 1 \)
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: |
\(-11238530872385902708009257189376\)
\(\medspace = -\,2^{14}\cdot 37^{7}\cdot 193327^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.10\) | 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: | $2^{2/3}37^{1/2}193327^{1/2}\approx 4245.548257194108$ | ||
| Ramified primes: |
\(2\), \(37\), \(193327\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-7153099}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{66\!\cdots\!75}a^{20}-\frac{58\!\cdots\!22}{13\!\cdots\!55}a^{19}-\frac{15\!\cdots\!63}{66\!\cdots\!75}a^{18}-\frac{27\!\cdots\!43}{66\!\cdots\!75}a^{17}+\frac{51\!\cdots\!97}{11\!\cdots\!25}a^{16}-\frac{14\!\cdots\!24}{66\!\cdots\!75}a^{15}+\frac{23\!\cdots\!03}{66\!\cdots\!75}a^{14}+\frac{85\!\cdots\!36}{66\!\cdots\!75}a^{13}+\frac{17\!\cdots\!36}{66\!\cdots\!75}a^{12}+\frac{78\!\cdots\!39}{66\!\cdots\!75}a^{11}+\frac{12\!\cdots\!89}{66\!\cdots\!75}a^{10}-\frac{15\!\cdots\!03}{66\!\cdots\!75}a^{9}-\frac{12\!\cdots\!31}{66\!\cdots\!75}a^{8}+\frac{31\!\cdots\!97}{13\!\cdots\!55}a^{7}-\frac{23\!\cdots\!01}{66\!\cdots\!75}a^{6}-\frac{17\!\cdots\!52}{66\!\cdots\!75}a^{5}+\frac{16\!\cdots\!92}{66\!\cdots\!75}a^{4}+\frac{63\!\cdots\!58}{13\!\cdots\!55}a^{3}-\frac{23\!\cdots\!36}{66\!\cdots\!75}a^{2}-\frac{52\!\cdots\!46}{13\!\cdots\!55}a+\frac{30\!\cdots\!39}{66\!\cdots\!75}$
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: |
$\frac{24\!\cdots\!16}{66\!\cdots\!75}a^{20}-\frac{22\!\cdots\!32}{13\!\cdots\!55}a^{19}+\frac{18\!\cdots\!92}{66\!\cdots\!75}a^{18}+\frac{55\!\cdots\!37}{66\!\cdots\!75}a^{17}-\frac{27\!\cdots\!98}{11\!\cdots\!25}a^{16}+\frac{46\!\cdots\!91}{66\!\cdots\!75}a^{15}+\frac{53\!\cdots\!23}{66\!\cdots\!75}a^{14}-\frac{64\!\cdots\!74}{66\!\cdots\!75}a^{13}-\frac{59\!\cdots\!49}{66\!\cdots\!75}a^{12}-\frac{29\!\cdots\!26}{66\!\cdots\!75}a^{11}+\frac{50\!\cdots\!99}{66\!\cdots\!75}a^{10}+\frac{11\!\cdots\!02}{66\!\cdots\!75}a^{9}-\frac{95\!\cdots\!71}{66\!\cdots\!75}a^{8}-\frac{65\!\cdots\!98}{13\!\cdots\!55}a^{7}+\frac{10\!\cdots\!84}{66\!\cdots\!75}a^{6}-\frac{52\!\cdots\!82}{66\!\cdots\!75}a^{5}-\frac{19\!\cdots\!28}{66\!\cdots\!75}a^{4}+\frac{45\!\cdots\!73}{13\!\cdots\!55}a^{3}+\frac{20\!\cdots\!49}{66\!\cdots\!75}a^{2}-\frac{11\!\cdots\!16}{13\!\cdots\!55}a+\frac{67\!\cdots\!24}{66\!\cdots\!75}$, $\frac{26\!\cdots\!43}{66\!\cdots\!75}a^{20}-\frac{55\!\cdots\!36}{13\!\cdots\!55}a^{19}-\frac{59\!\cdots\!84}{66\!\cdots\!75}a^{18}+\frac{39\!\cdots\!76}{66\!\cdots\!75}a^{17}-\frac{76\!\cdots\!29}{11\!\cdots\!25}a^{16}-\frac{83\!\cdots\!82}{66\!\cdots\!75}a^{15}+\frac{28\!\cdots\!54}{66\!\cdots\!75}a^{14}+\frac{20\!\cdots\!98}{66\!\cdots\!75}a^{13}+\frac{21\!\cdots\!48}{66\!\cdots\!75}a^{12}-\frac{22\!\cdots\!48}{66\!\cdots\!75}a^{11}-\frac{21\!\cdots\!98}{66\!\cdots\!75}a^{10}+\frac{32\!\cdots\!96}{66\!\cdots\!75}a^{9}+\frac{80\!\cdots\!17}{66\!\cdots\!75}a^{8}-\frac{10\!\cdots\!44}{13\!\cdots\!55}a^{7}+\frac{92\!\cdots\!32}{66\!\cdots\!75}a^{6}+\frac{48\!\cdots\!14}{66\!\cdots\!75}a^{5}-\frac{73\!\cdots\!94}{66\!\cdots\!75}a^{4}+\frac{98\!\cdots\!34}{13\!\cdots\!55}a^{3}+\frac{15\!\cdots\!02}{66\!\cdots\!75}a^{2}-\frac{43\!\cdots\!48}{13\!\cdots\!55}a+\frac{66\!\cdots\!77}{66\!\cdots\!75}$, $\frac{20\!\cdots\!99}{66\!\cdots\!75}a^{20}-\frac{30\!\cdots\!08}{13\!\cdots\!55}a^{19}-\frac{65\!\cdots\!62}{66\!\cdots\!75}a^{18}+\frac{29\!\cdots\!43}{66\!\cdots\!75}a^{17}-\frac{36\!\cdots\!97}{11\!\cdots\!25}a^{16}-\frac{91\!\cdots\!51}{66\!\cdots\!75}a^{15}+\frac{20\!\cdots\!47}{66\!\cdots\!75}a^{14}+\frac{28\!\cdots\!14}{66\!\cdots\!75}a^{13}+\frac{10\!\cdots\!39}{66\!\cdots\!75}a^{12}-\frac{19\!\cdots\!64}{66\!\cdots\!75}a^{11}-\frac{23\!\cdots\!89}{66\!\cdots\!75}a^{10}+\frac{31\!\cdots\!28}{66\!\cdots\!75}a^{9}+\frac{89\!\cdots\!31}{66\!\cdots\!75}a^{8}-\frac{71\!\cdots\!82}{13\!\cdots\!55}a^{7}-\frac{69\!\cdots\!24}{66\!\cdots\!75}a^{6}+\frac{54\!\cdots\!52}{66\!\cdots\!75}a^{5}-\frac{21\!\cdots\!17}{66\!\cdots\!75}a^{4}-\frac{41\!\cdots\!53}{13\!\cdots\!55}a^{3}+\frac{17\!\cdots\!61}{66\!\cdots\!75}a^{2}-\frac{69\!\cdots\!94}{13\!\cdots\!55}a-\frac{65\!\cdots\!64}{66\!\cdots\!75}$, $\frac{31\!\cdots\!27}{66\!\cdots\!75}a^{20}-\frac{16\!\cdots\!59}{13\!\cdots\!55}a^{19}-\frac{29\!\cdots\!26}{66\!\cdots\!75}a^{18}+\frac{56\!\cdots\!14}{66\!\cdots\!75}a^{17}-\frac{20\!\cdots\!81}{11\!\cdots\!25}a^{16}-\frac{29\!\cdots\!73}{66\!\cdots\!75}a^{15}+\frac{47\!\cdots\!81}{66\!\cdots\!75}a^{14}-\frac{19\!\cdots\!53}{66\!\cdots\!75}a^{13}-\frac{24\!\cdots\!03}{66\!\cdots\!75}a^{12}-\frac{34\!\cdots\!97}{66\!\cdots\!75}a^{11}+\frac{16\!\cdots\!78}{66\!\cdots\!75}a^{10}+\frac{82\!\cdots\!44}{66\!\cdots\!75}a^{9}+\frac{58\!\cdots\!38}{66\!\cdots\!75}a^{8}-\frac{44\!\cdots\!61}{13\!\cdots\!55}a^{7}+\frac{66\!\cdots\!23}{66\!\cdots\!75}a^{6}+\frac{35\!\cdots\!46}{66\!\cdots\!75}a^{5}-\frac{13\!\cdots\!41}{66\!\cdots\!75}a^{4}+\frac{77\!\cdots\!11}{13\!\cdots\!55}a^{3}+\frac{13\!\cdots\!28}{66\!\cdots\!75}a^{2}-\frac{76\!\cdots\!82}{13\!\cdots\!55}a-\frac{20\!\cdots\!47}{66\!\cdots\!75}$, $\frac{75\!\cdots\!64}{66\!\cdots\!75}a^{20}-\frac{17\!\cdots\!33}{13\!\cdots\!55}a^{19}-\frac{13\!\cdots\!07}{66\!\cdots\!75}a^{18}+\frac{11\!\cdots\!73}{66\!\cdots\!75}a^{17}-\frac{23\!\cdots\!17}{11\!\cdots\!25}a^{16}-\frac{19\!\cdots\!86}{66\!\cdots\!75}a^{15}+\frac{78\!\cdots\!67}{66\!\cdots\!75}a^{14}+\frac{49\!\cdots\!54}{66\!\cdots\!75}a^{13}+\frac{69\!\cdots\!29}{66\!\cdots\!75}a^{12}-\frac{63\!\cdots\!54}{66\!\cdots\!75}a^{11}-\frac{44\!\cdots\!04}{66\!\cdots\!75}a^{10}+\frac{85\!\cdots\!58}{66\!\cdots\!75}a^{9}+\frac{20\!\cdots\!16}{66\!\cdots\!75}a^{8}-\frac{40\!\cdots\!82}{13\!\cdots\!55}a^{7}+\frac{44\!\cdots\!86}{66\!\cdots\!75}a^{6}+\frac{12\!\cdots\!47}{66\!\cdots\!75}a^{5}-\frac{11\!\cdots\!37}{66\!\cdots\!75}a^{4}+\frac{49\!\cdots\!12}{13\!\cdots\!55}a^{3}+\frac{39\!\cdots\!46}{66\!\cdots\!75}a^{2}-\frac{58\!\cdots\!64}{13\!\cdots\!55}a-\frac{12\!\cdots\!54}{66\!\cdots\!75}$, $\frac{54\!\cdots\!86}{66\!\cdots\!75}a^{20}+\frac{32\!\cdots\!43}{13\!\cdots\!55}a^{19}-\frac{17\!\cdots\!93}{66\!\cdots\!75}a^{18}+\frac{56\!\cdots\!77}{66\!\cdots\!75}a^{17}+\frac{16\!\cdots\!42}{11\!\cdots\!25}a^{16}-\frac{24\!\cdots\!64}{66\!\cdots\!75}a^{15}+\frac{25\!\cdots\!33}{66\!\cdots\!75}a^{14}+\frac{11\!\cdots\!96}{66\!\cdots\!75}a^{13}+\frac{14\!\cdots\!21}{66\!\cdots\!75}a^{12}-\frac{40\!\cdots\!21}{66\!\cdots\!75}a^{11}-\frac{10\!\cdots\!46}{66\!\cdots\!75}a^{10}-\frac{16\!\cdots\!33}{66\!\cdots\!75}a^{9}+\frac{26\!\cdots\!34}{66\!\cdots\!75}a^{8}+\frac{40\!\cdots\!92}{13\!\cdots\!55}a^{7}-\frac{90\!\cdots\!61}{66\!\cdots\!75}a^{6}-\frac{21\!\cdots\!47}{66\!\cdots\!75}a^{5}-\frac{14\!\cdots\!13}{66\!\cdots\!75}a^{4}-\frac{54\!\cdots\!97}{13\!\cdots\!55}a^{3}-\frac{63\!\cdots\!71}{66\!\cdots\!75}a^{2}-\frac{85\!\cdots\!71}{13\!\cdots\!55}a-\frac{74\!\cdots\!96}{66\!\cdots\!75}$, $\frac{86\!\cdots\!62}{66\!\cdots\!75}a^{20}-\frac{13\!\cdots\!54}{13\!\cdots\!55}a^{19}-\frac{27\!\cdots\!81}{66\!\cdots\!75}a^{18}+\frac{12\!\cdots\!34}{66\!\cdots\!75}a^{17}-\frac{16\!\cdots\!36}{11\!\cdots\!25}a^{16}-\frac{38\!\cdots\!13}{66\!\cdots\!75}a^{15}+\frac{88\!\cdots\!36}{66\!\cdots\!75}a^{14}+\frac{11\!\cdots\!32}{66\!\cdots\!75}a^{13}+\frac{35\!\cdots\!32}{66\!\cdots\!75}a^{12}-\frac{80\!\cdots\!07}{66\!\cdots\!75}a^{11}-\frac{91\!\cdots\!57}{66\!\cdots\!75}a^{10}+\frac{13\!\cdots\!89}{66\!\cdots\!75}a^{9}+\frac{36\!\cdots\!28}{66\!\cdots\!75}a^{8}-\frac{34\!\cdots\!71}{13\!\cdots\!55}a^{7}-\frac{28\!\cdots\!87}{66\!\cdots\!75}a^{6}+\frac{44\!\cdots\!01}{66\!\cdots\!75}a^{5}-\frac{10\!\cdots\!96}{66\!\cdots\!75}a^{4}-\frac{20\!\cdots\!59}{13\!\cdots\!55}a^{3}+\frac{35\!\cdots\!18}{66\!\cdots\!75}a^{2}+\frac{98\!\cdots\!93}{13\!\cdots\!55}a-\frac{12\!\cdots\!07}{66\!\cdots\!75}$, $\frac{18\!\cdots\!89}{66\!\cdots\!75}a^{20}+\frac{31\!\cdots\!32}{13\!\cdots\!55}a^{19}-\frac{15\!\cdots\!07}{66\!\cdots\!75}a^{18}-\frac{32\!\cdots\!27}{66\!\cdots\!75}a^{17}+\frac{36\!\cdots\!33}{11\!\cdots\!25}a^{16}-\frac{21\!\cdots\!61}{66\!\cdots\!75}a^{15}-\frac{63\!\cdots\!83}{66\!\cdots\!75}a^{14}+\frac{18\!\cdots\!04}{66\!\cdots\!75}a^{13}+\frac{28\!\cdots\!04}{66\!\cdots\!75}a^{12}-\frac{63\!\cdots\!04}{66\!\cdots\!75}a^{11}-\frac{18\!\cdots\!79}{66\!\cdots\!75}a^{10}-\frac{18\!\cdots\!92}{66\!\cdots\!75}a^{9}+\frac{33\!\cdots\!91}{66\!\cdots\!75}a^{8}+\frac{15\!\cdots\!18}{13\!\cdots\!55}a^{7}-\frac{27\!\cdots\!89}{66\!\cdots\!75}a^{6}-\frac{71\!\cdots\!53}{66\!\cdots\!75}a^{5}-\frac{24\!\cdots\!37}{66\!\cdots\!75}a^{4}+\frac{13\!\cdots\!97}{13\!\cdots\!55}a^{3}-\frac{63\!\cdots\!29}{66\!\cdots\!75}a^{2}-\frac{10\!\cdots\!69}{13\!\cdots\!55}a-\frac{24\!\cdots\!04}{66\!\cdots\!75}$, $\frac{39\!\cdots\!26}{66\!\cdots\!75}a^{20}-\frac{57\!\cdots\!32}{13\!\cdots\!55}a^{19}-\frac{12\!\cdots\!13}{66\!\cdots\!75}a^{18}+\frac{55\!\cdots\!32}{66\!\cdots\!75}a^{17}-\frac{66\!\cdots\!28}{11\!\cdots\!25}a^{16}-\frac{17\!\cdots\!74}{66\!\cdots\!75}a^{15}+\frac{39\!\cdots\!78}{66\!\cdots\!75}a^{14}+\frac{55\!\cdots\!86}{66\!\cdots\!75}a^{13}+\frac{17\!\cdots\!86}{66\!\cdots\!75}a^{12}-\frac{36\!\cdots\!61}{66\!\cdots\!75}a^{11}-\frac{43\!\cdots\!86}{66\!\cdots\!75}a^{10}+\frac{62\!\cdots\!47}{66\!\cdots\!75}a^{9}+\frac{17\!\cdots\!44}{66\!\cdots\!75}a^{8}-\frac{14\!\cdots\!88}{13\!\cdots\!55}a^{7}-\frac{14\!\cdots\!76}{66\!\cdots\!75}a^{6}+\frac{15\!\cdots\!23}{66\!\cdots\!75}a^{5}+\frac{43\!\cdots\!92}{66\!\cdots\!75}a^{4}-\frac{11\!\cdots\!22}{13\!\cdots\!55}a^{3}+\frac{17\!\cdots\!39}{66\!\cdots\!75}a^{2}+\frac{13\!\cdots\!14}{13\!\cdots\!55}a-\frac{10\!\cdots\!11}{66\!\cdots\!75}$, $\frac{18\!\cdots\!23}{66\!\cdots\!75}a^{20}-\frac{28\!\cdots\!06}{13\!\cdots\!55}a^{19}-\frac{57\!\cdots\!49}{66\!\cdots\!75}a^{18}+\frac{25\!\cdots\!61}{66\!\cdots\!75}a^{17}-\frac{33\!\cdots\!19}{11\!\cdots\!25}a^{16}-\frac{79\!\cdots\!02}{66\!\cdots\!75}a^{15}+\frac{18\!\cdots\!69}{66\!\cdots\!75}a^{14}+\frac{24\!\cdots\!53}{66\!\cdots\!75}a^{13}+\frac{67\!\cdots\!03}{66\!\cdots\!75}a^{12}-\frac{16\!\cdots\!78}{66\!\cdots\!75}a^{11}-\frac{18\!\cdots\!28}{66\!\cdots\!75}a^{10}+\frac{29\!\cdots\!06}{66\!\cdots\!75}a^{9}+\frac{76\!\cdots\!12}{66\!\cdots\!75}a^{8}-\frac{75\!\cdots\!39}{13\!\cdots\!55}a^{7}-\frac{60\!\cdots\!48}{66\!\cdots\!75}a^{6}+\frac{11\!\cdots\!29}{66\!\cdots\!75}a^{5}-\frac{88\!\cdots\!34}{66\!\cdots\!75}a^{4}-\frac{68\!\cdots\!56}{13\!\cdots\!55}a^{3}+\frac{64\!\cdots\!22}{66\!\cdots\!75}a^{2}+\frac{12\!\cdots\!17}{13\!\cdots\!55}a-\frac{92\!\cdots\!78}{66\!\cdots\!75}$, $\frac{70\!\cdots\!73}{66\!\cdots\!75}a^{20}-\frac{18\!\cdots\!36}{13\!\cdots\!55}a^{19}-\frac{18\!\cdots\!24}{66\!\cdots\!75}a^{18}+\frac{11\!\cdots\!61}{66\!\cdots\!75}a^{17}-\frac{23\!\cdots\!19}{11\!\cdots\!25}a^{16}-\frac{25\!\cdots\!52}{66\!\cdots\!75}a^{15}+\frac{88\!\cdots\!19}{66\!\cdots\!75}a^{14}+\frac{51\!\cdots\!03}{66\!\cdots\!75}a^{13}-\frac{14\!\cdots\!47}{66\!\cdots\!75}a^{12}-\frac{66\!\cdots\!03}{66\!\cdots\!75}a^{11}-\frac{38\!\cdots\!28}{66\!\cdots\!75}a^{10}+\frac{14\!\cdots\!31}{66\!\cdots\!75}a^{9}+\frac{22\!\cdots\!87}{66\!\cdots\!75}a^{8}-\frac{58\!\cdots\!54}{13\!\cdots\!55}a^{7}-\frac{12\!\cdots\!73}{66\!\cdots\!75}a^{6}+\frac{13\!\cdots\!04}{66\!\cdots\!75}a^{5}-\frac{46\!\cdots\!09}{66\!\cdots\!75}a^{4}+\frac{12\!\cdots\!74}{13\!\cdots\!55}a^{3}+\frac{36\!\cdots\!72}{66\!\cdots\!75}a^{2}-\frac{20\!\cdots\!83}{13\!\cdots\!55}a+\frac{63\!\cdots\!72}{66\!\cdots\!75}$
| 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: | \( 13187708.2201 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 13187708.2201 \cdot 1}{2\cdot\sqrt{11238530872385902708009257189376}}\cr\approx \mathstrut & 0.240155957651 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 3*x^19 + 15*x^18 - 14*x^17 - 42*x^16 + 112*x^15 + 113*x^14 + 10*x^13 - 937*x^12 - 860*x^11 + 1848*x^10 + 3892*x^9 - 2944*x^8 - 2961*x^7 + 1339*x^6 - 176*x^5 - 182*x^4 + 74*x^3 - 4*x^2 - 6*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 - x^20 - 3*x^19 + 15*x^18 - 14*x^17 - 42*x^16 + 112*x^15 + 113*x^14 + 10*x^13 - 937*x^12 - 860*x^11 + 1848*x^10 + 3892*x^9 - 2944*x^8 - 2961*x^7 + 1339*x^6 - 176*x^5 - 182*x^4 + 74*x^3 - 4*x^2 - 6*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 - x^20 - 3*x^19 + 15*x^18 - 14*x^17 - 42*x^16 + 112*x^15 + 113*x^14 + 10*x^13 - 937*x^12 - 860*x^11 + 1848*x^10 + 3892*x^9 - 2944*x^8 - 2961*x^7 + 1339*x^6 - 176*x^5 - 182*x^4 + 74*x^3 - 4*x^2 - 6*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 - x^20 - 3*x^19 + 15*x^18 - 14*x^17 - 42*x^16 + 112*x^15 + 113*x^14 + 10*x^13 - 937*x^12 - 860*x^11 + 1848*x^10 + 3892*x^9 - 2944*x^8 - 2961*x^7 + 1339*x^6 - 176*x^5 - 182*x^4 + 74*x^3 - 4*x^2 - 6*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
$S_3\times S_7$ (as 21T74):
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 30240 |
| The 45 conjugacy class representatives for $S_3\times S_7$ |
| Character table for $S_3\times S_7$ |
Intermediate fields
| 3.3.148.1, 7.1.193327.1 |
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 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 | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{3}$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | $15{,}\,{\href{/padicField/7.6.0.1}{6} }$ | $21$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | R | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $21$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.21.14.1 | $x^{21} + 14 x^{18} + 87 x^{15} + 3 x^{14} - 14 x^{12} - 462 x^{11} + 1655 x^{9} + 4290 x^{8} + 3 x^{7} + 2982 x^{6} - 6090 x^{5} + 210 x^{4} - 1651 x^{3} + 1263 x^{2} + 87 x + 251$ | $3$ | $7$ | $14$ | 21T6 | $[\ ]_{3}^{14}$ |
|
\(37\)
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.5.0.1 | $x^{5} + 10 x + 35$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 37.10.5.1 | $x^{10} + 4625 x^{9} + 8556435 x^{8} + 7915215750 x^{7} + 3661420460585 x^{6} + 677772793435945 x^{5} + 135472642761580 x^{4} + 10915368272100 x^{3} + 37311922841505 x^{2} + 6899973656637600 x + 23967251414048817$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(193327\)
| $\Q_{193327}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |