Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 23*x^19 - 46*x^18 + 59*x^17 - 38*x^16 - 51*x^15 + 276*x^14 - 725*x^13 + 1392*x^12 - 2056*x^11 + 2397*x^10 - 2311*x^9 + 1987*x^8 - 1615*x^7 + 1208*x^6 - 783*x^5 + 436*x^4 - 201*x^3 + 66*x^2 - 12*x + 1)
gp: K = bnfinit(y^21 - 7*y^20 + 23*y^19 - 46*y^18 + 59*y^17 - 38*y^16 - 51*y^15 + 276*y^14 - 725*y^13 + 1392*y^12 - 2056*y^11 + 2397*y^10 - 2311*y^9 + 1987*y^8 - 1615*y^7 + 1208*y^6 - 783*y^5 + 436*y^4 - 201*y^3 + 66*y^2 - 12*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 7*x^20 + 23*x^19 - 46*x^18 + 59*x^17 - 38*x^16 - 51*x^15 + 276*x^14 - 725*x^13 + 1392*x^12 - 2056*x^11 + 2397*x^10 - 2311*x^9 + 1987*x^8 - 1615*x^7 + 1208*x^6 - 783*x^5 + 436*x^4 - 201*x^3 + 66*x^2 - 12*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 7*x^20 + 23*x^19 - 46*x^18 + 59*x^17 - 38*x^16 - 51*x^15 + 276*x^14 - 725*x^13 + 1392*x^12 - 2056*x^11 + 2397*x^10 - 2311*x^9 + 1987*x^8 - 1615*x^7 + 1208*x^6 - 783*x^5 + 436*x^4 - 201*x^3 + 66*x^2 - 12*x + 1)
\( x^{21} - 7 x^{20} + 23 x^{19} - 46 x^{18} + 59 x^{17} - 38 x^{16} - 51 x^{15} + 276 x^{14} - 725 x^{13} + \cdots + 1 \)
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: |
\(-167297208611536987435565056\)
\(\medspace = -\,2^{18}\cdot 17^{6}\cdot 31^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.73\) | 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: | $2^{4/3}17^{2/3}31^{1/2}\approx 92.75844411611145$ | ||
| Ramified primes: |
\(2\), \(17\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{10\!\cdots\!93}a^{20}-\frac{44\!\cdots\!00}{10\!\cdots\!93}a^{19}-\frac{986992814227005}{10\!\cdots\!93}a^{18}+\frac{30\!\cdots\!62}{10\!\cdots\!93}a^{17}-\frac{31\!\cdots\!10}{10\!\cdots\!93}a^{16}-\frac{30\!\cdots\!09}{10\!\cdots\!93}a^{15}+\frac{21\!\cdots\!58}{10\!\cdots\!93}a^{14}+\frac{18\!\cdots\!86}{10\!\cdots\!93}a^{13}-\frac{10\!\cdots\!69}{10\!\cdots\!93}a^{12}+\frac{28\!\cdots\!73}{10\!\cdots\!93}a^{11}+\frac{34\!\cdots\!07}{10\!\cdots\!93}a^{10}+\frac{41\!\cdots\!94}{10\!\cdots\!93}a^{9}+\frac{38\!\cdots\!35}{10\!\cdots\!93}a^{8}-\frac{200976020094346}{10\!\cdots\!93}a^{7}-\frac{14\!\cdots\!08}{10\!\cdots\!93}a^{6}-\frac{556945127368886}{10\!\cdots\!93}a^{5}-\frac{51\!\cdots\!05}{10\!\cdots\!93}a^{4}+\frac{26\!\cdots\!91}{10\!\cdots\!93}a^{3}+\frac{12\!\cdots\!80}{10\!\cdots\!93}a^{2}-\frac{22\!\cdots\!04}{10\!\cdots\!93}a+\frac{44\!\cdots\!19}{10\!\cdots\!93}$
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: |
$\frac{14214110687511}{197530954277581}a^{20}-\frac{174019812778914}{197530954277581}a^{19}+\frac{773600421560737}{197530954277581}a^{18}-\frac{19\!\cdots\!00}{197530954277581}a^{17}+\frac{29\!\cdots\!21}{197530954277581}a^{16}-\frac{26\!\cdots\!24}{197530954277581}a^{15}-\frac{239559994210908}{197530954277581}a^{14}+\frac{83\!\cdots\!02}{197530954277581}a^{13}-\frac{26\!\cdots\!29}{197530954277581}a^{12}+\frac{57\!\cdots\!74}{197530954277581}a^{11}-\frac{93\!\cdots\!95}{197530954277581}a^{10}+\frac{11\!\cdots\!18}{197530954277581}a^{9}-\frac{12\!\cdots\!63}{197530954277581}a^{8}+\frac{10\!\cdots\!70}{197530954277581}a^{7}-\frac{86\!\cdots\!93}{197530954277581}a^{6}+\frac{67\!\cdots\!28}{197530954277581}a^{5}-\frac{44\!\cdots\!67}{197530954277581}a^{4}+\frac{25\!\cdots\!26}{197530954277581}a^{3}-\frac{12\!\cdots\!03}{197530954277581}a^{2}+\frac{41\!\cdots\!56}{197530954277581}a-\frac{574269762958658}{197530954277581}$, $\frac{15\!\cdots\!59}{10\!\cdots\!93}a^{20}-\frac{10\!\cdots\!30}{10\!\cdots\!93}a^{19}+\frac{31\!\cdots\!12}{10\!\cdots\!93}a^{18}-\frac{57\!\cdots\!82}{10\!\cdots\!93}a^{17}+\frac{66\!\cdots\!15}{10\!\cdots\!93}a^{16}-\frac{28\!\cdots\!17}{10\!\cdots\!93}a^{15}-\frac{94\!\cdots\!11}{10\!\cdots\!93}a^{14}+\frac{38\!\cdots\!59}{10\!\cdots\!93}a^{13}-\frac{95\!\cdots\!97}{10\!\cdots\!93}a^{12}+\frac{17\!\cdots\!25}{10\!\cdots\!93}a^{11}-\frac{24\!\cdots\!57}{10\!\cdots\!93}a^{10}+\frac{26\!\cdots\!24}{10\!\cdots\!93}a^{9}-\frac{23\!\cdots\!32}{10\!\cdots\!93}a^{8}+\frac{19\!\cdots\!73}{10\!\cdots\!93}a^{7}-\frac{15\!\cdots\!96}{10\!\cdots\!93}a^{6}+\frac{11\!\cdots\!59}{10\!\cdots\!93}a^{5}-\frac{66\!\cdots\!09}{10\!\cdots\!93}a^{4}+\frac{33\!\cdots\!28}{10\!\cdots\!93}a^{3}-\frac{13\!\cdots\!04}{10\!\cdots\!93}a^{2}+\frac{28\!\cdots\!55}{10\!\cdots\!93}a-\frac{18\!\cdots\!70}{10\!\cdots\!93}$, $\frac{83669028763494}{197530954277581}a^{20}-\frac{509604161449513}{197530954277581}a^{19}+\frac{14\!\cdots\!69}{197530954277581}a^{18}-\frac{23\!\cdots\!13}{197530954277581}a^{17}+\frac{23\!\cdots\!23}{197530954277581}a^{16}-\frac{313545452772371}{197530954277581}a^{15}-\frac{53\!\cdots\!09}{197530954277581}a^{14}+\frac{18\!\cdots\!24}{197530954277581}a^{13}-\frac{42\!\cdots\!19}{197530954277581}a^{12}+\frac{72\!\cdots\!06}{197530954277581}a^{11}-\frac{93\!\cdots\!41}{197530954277581}a^{10}+\frac{93\!\cdots\!34}{197530954277581}a^{9}-\frac{78\!\cdots\!62}{197530954277581}a^{8}+\frac{64\!\cdots\!84}{197530954277581}a^{7}-\frac{50\!\cdots\!95}{197530954277581}a^{6}+\frac{33\!\cdots\!84}{197530954277581}a^{5}-\frac{17\!\cdots\!95}{197530954277581}a^{4}+\frac{84\!\cdots\!78}{197530954277581}a^{3}-\frac{25\!\cdots\!99}{197530954277581}a^{2}+\frac{14280416387257}{197530954277581}a+\frac{59844318513709}{197530954277581}$, $\frac{57\!\cdots\!80}{10\!\cdots\!93}a^{20}-\frac{38\!\cdots\!02}{10\!\cdots\!93}a^{19}+\frac{12\!\cdots\!30}{10\!\cdots\!93}a^{18}-\frac{23\!\cdots\!74}{10\!\cdots\!93}a^{17}+\frac{29\!\cdots\!10}{10\!\cdots\!93}a^{16}-\frac{16\!\cdots\!81}{10\!\cdots\!93}a^{15}-\frac{31\!\cdots\!24}{10\!\cdots\!93}a^{14}+\frac{14\!\cdots\!45}{10\!\cdots\!93}a^{13}-\frac{38\!\cdots\!72}{10\!\cdots\!93}a^{12}+\frac{71\!\cdots\!62}{10\!\cdots\!93}a^{11}-\frac{10\!\cdots\!43}{10\!\cdots\!93}a^{10}+\frac{11\!\cdots\!34}{10\!\cdots\!93}a^{9}-\frac{11\!\cdots\!15}{10\!\cdots\!93}a^{8}+\frac{94\!\cdots\!43}{10\!\cdots\!93}a^{7}-\frac{76\!\cdots\!15}{10\!\cdots\!93}a^{6}+\frac{56\!\cdots\!68}{10\!\cdots\!93}a^{5}-\frac{35\!\cdots\!75}{10\!\cdots\!93}a^{4}+\frac{19\!\cdots\!73}{10\!\cdots\!93}a^{3}-\frac{83\!\cdots\!94}{10\!\cdots\!93}a^{2}+\frac{25\!\cdots\!18}{10\!\cdots\!93}a-\frac{33\!\cdots\!89}{10\!\cdots\!93}$, $a-1$, $\frac{61\!\cdots\!58}{10\!\cdots\!93}a^{20}-\frac{43\!\cdots\!20}{10\!\cdots\!93}a^{19}+\frac{14\!\cdots\!21}{10\!\cdots\!93}a^{18}-\frac{28\!\cdots\!65}{10\!\cdots\!93}a^{17}+\frac{36\!\cdots\!17}{10\!\cdots\!93}a^{16}-\frac{23\!\cdots\!99}{10\!\cdots\!93}a^{15}-\frac{32\!\cdots\!25}{10\!\cdots\!93}a^{14}+\frac{17\!\cdots\!51}{10\!\cdots\!93}a^{13}-\frac{45\!\cdots\!69}{10\!\cdots\!93}a^{12}+\frac{87\!\cdots\!52}{10\!\cdots\!93}a^{11}-\frac{12\!\cdots\!02}{10\!\cdots\!93}a^{10}+\frac{14\!\cdots\!57}{10\!\cdots\!93}a^{9}-\frac{14\!\cdots\!82}{10\!\cdots\!93}a^{8}+\frac{12\!\cdots\!64}{10\!\cdots\!93}a^{7}-\frac{97\!\cdots\!39}{10\!\cdots\!93}a^{6}+\frac{72\!\cdots\!78}{10\!\cdots\!93}a^{5}-\frac{46\!\cdots\!78}{10\!\cdots\!93}a^{4}+\frac{25\!\cdots\!31}{10\!\cdots\!93}a^{3}-\frac{11\!\cdots\!10}{10\!\cdots\!93}a^{2}+\frac{33\!\cdots\!55}{10\!\cdots\!93}a-\frac{42\!\cdots\!65}{10\!\cdots\!93}$, $\frac{30\!\cdots\!55}{10\!\cdots\!93}a^{20}-\frac{22\!\cdots\!34}{10\!\cdots\!93}a^{19}+\frac{76\!\cdots\!12}{10\!\cdots\!93}a^{18}-\frac{15\!\cdots\!54}{10\!\cdots\!93}a^{17}+\frac{20\!\cdots\!19}{10\!\cdots\!93}a^{16}-\frac{12\!\cdots\!59}{10\!\cdots\!93}a^{15}-\frac{17\!\cdots\!18}{10\!\cdots\!93}a^{14}+\frac{92\!\cdots\!42}{10\!\cdots\!93}a^{13}-\frac{24\!\cdots\!20}{10\!\cdots\!93}a^{12}+\frac{46\!\cdots\!15}{10\!\cdots\!93}a^{11}-\frac{69\!\cdots\!08}{10\!\cdots\!93}a^{10}+\frac{79\!\cdots\!03}{10\!\cdots\!93}a^{9}-\frac{74\!\cdots\!19}{10\!\cdots\!93}a^{8}+\frac{61\!\cdots\!62}{10\!\cdots\!93}a^{7}-\frac{49\!\cdots\!53}{10\!\cdots\!93}a^{6}+\frac{37\!\cdots\!49}{10\!\cdots\!93}a^{5}-\frac{23\!\cdots\!28}{10\!\cdots\!93}a^{4}+\frac{12\!\cdots\!52}{10\!\cdots\!93}a^{3}-\frac{52\!\cdots\!99}{10\!\cdots\!93}a^{2}+\frac{12\!\cdots\!39}{10\!\cdots\!93}a+\frac{63\!\cdots\!02}{10\!\cdots\!93}$, $\frac{28\!\cdots\!60}{10\!\cdots\!93}a^{20}-\frac{20\!\cdots\!35}{10\!\cdots\!93}a^{19}+\frac{68\!\cdots\!03}{10\!\cdots\!93}a^{18}-\frac{14\!\cdots\!22}{10\!\cdots\!93}a^{17}+\frac{18\!\cdots\!84}{10\!\cdots\!93}a^{16}-\frac{12\!\cdots\!54}{10\!\cdots\!93}a^{15}-\frac{13\!\cdots\!80}{10\!\cdots\!93}a^{14}+\frac{81\!\cdots\!80}{10\!\cdots\!93}a^{13}-\frac{21\!\cdots\!09}{10\!\cdots\!93}a^{12}+\frac{42\!\cdots\!79}{10\!\cdots\!93}a^{11}-\frac{63\!\cdots\!42}{10\!\cdots\!93}a^{10}+\frac{75\!\cdots\!57}{10\!\cdots\!93}a^{9}-\frac{74\!\cdots\!78}{10\!\cdots\!93}a^{8}+\frac{64\!\cdots\!05}{10\!\cdots\!93}a^{7}-\frac{52\!\cdots\!18}{10\!\cdots\!93}a^{6}+\frac{39\!\cdots\!77}{10\!\cdots\!93}a^{5}-\frac{26\!\cdots\!62}{10\!\cdots\!93}a^{4}+\frac{14\!\cdots\!04}{10\!\cdots\!93}a^{3}-\frac{70\!\cdots\!31}{10\!\cdots\!93}a^{2}+\frac{23\!\cdots\!24}{10\!\cdots\!93}a-\frac{44\!\cdots\!53}{10\!\cdots\!93}$, $\frac{32\!\cdots\!88}{10\!\cdots\!93}a^{20}-\frac{23\!\cdots\!15}{10\!\cdots\!93}a^{19}+\frac{79\!\cdots\!29}{10\!\cdots\!93}a^{18}-\frac{15\!\cdots\!92}{10\!\cdots\!93}a^{17}+\frac{18\!\cdots\!35}{10\!\cdots\!93}a^{16}-\frac{98\!\cdots\!49}{10\!\cdots\!93}a^{15}-\frac{21\!\cdots\!95}{10\!\cdots\!93}a^{14}+\frac{97\!\cdots\!32}{10\!\cdots\!93}a^{13}-\frac{24\!\cdots\!93}{10\!\cdots\!93}a^{12}+\frac{46\!\cdots\!97}{10\!\cdots\!93}a^{11}-\frac{66\!\cdots\!66}{10\!\cdots\!93}a^{10}+\frac{73\!\cdots\!03}{10\!\cdots\!93}a^{9}-\frac{65\!\cdots\!65}{10\!\cdots\!93}a^{8}+\frac{53\!\cdots\!86}{10\!\cdots\!93}a^{7}-\frac{42\!\cdots\!51}{10\!\cdots\!93}a^{6}+\frac{30\!\cdots\!51}{10\!\cdots\!93}a^{5}-\frac{18\!\cdots\!26}{10\!\cdots\!93}a^{4}+\frac{86\!\cdots\!89}{10\!\cdots\!93}a^{3}-\frac{33\!\cdots\!38}{10\!\cdots\!93}a^{2}+\frac{46\!\cdots\!65}{10\!\cdots\!93}a+\frac{17\!\cdots\!63}{10\!\cdots\!93}$, $\frac{944663849116539}{10\!\cdots\!93}a^{20}-\frac{51\!\cdots\!23}{10\!\cdots\!93}a^{19}+\frac{12\!\cdots\!82}{10\!\cdots\!93}a^{18}-\frac{16\!\cdots\!25}{10\!\cdots\!93}a^{17}+\frac{80\!\cdots\!04}{10\!\cdots\!93}a^{16}+\frac{13\!\cdots\!75}{10\!\cdots\!93}a^{15}-\frac{61\!\cdots\!43}{10\!\cdots\!93}a^{14}+\frac{16\!\cdots\!02}{10\!\cdots\!93}a^{13}-\frac{33\!\cdots\!40}{10\!\cdots\!93}a^{12}+\frac{49\!\cdots\!53}{10\!\cdots\!93}a^{11}-\frac{50\!\cdots\!10}{10\!\cdots\!93}a^{10}+\frac{35\!\cdots\!36}{10\!\cdots\!93}a^{9}-\frac{20\!\cdots\!42}{10\!\cdots\!93}a^{8}+\frac{15\!\cdots\!50}{10\!\cdots\!93}a^{7}-\frac{10\!\cdots\!07}{10\!\cdots\!93}a^{6}-\frac{82\!\cdots\!86}{10\!\cdots\!93}a^{5}+\frac{57\!\cdots\!88}{10\!\cdots\!93}a^{4}-\frac{30\!\cdots\!85}{10\!\cdots\!93}a^{3}+\frac{10\!\cdots\!60}{10\!\cdots\!93}a^{2}-\frac{20\!\cdots\!91}{10\!\cdots\!93}a+\frac{51\!\cdots\!74}{10\!\cdots\!93}$, $\frac{15\!\cdots\!39}{10\!\cdots\!93}a^{20}-\frac{10\!\cdots\!05}{10\!\cdots\!93}a^{19}+\frac{36\!\cdots\!43}{10\!\cdots\!93}a^{18}-\frac{72\!\cdots\!00}{10\!\cdots\!93}a^{17}+\frac{91\!\cdots\!48}{10\!\cdots\!93}a^{16}-\frac{57\!\cdots\!99}{10\!\cdots\!93}a^{15}-\frac{83\!\cdots\!30}{10\!\cdots\!93}a^{14}+\frac{43\!\cdots\!99}{10\!\cdots\!93}a^{13}-\frac{11\!\cdots\!94}{10\!\cdots\!93}a^{12}+\frac{21\!\cdots\!92}{10\!\cdots\!93}a^{11}-\frac{31\!\cdots\!77}{10\!\cdots\!93}a^{10}+\frac{36\!\cdots\!62}{10\!\cdots\!93}a^{9}-\frac{35\!\cdots\!38}{10\!\cdots\!93}a^{8}+\frac{29\!\cdots\!71}{10\!\cdots\!93}a^{7}-\frac{24\!\cdots\!73}{10\!\cdots\!93}a^{6}+\frac{17\!\cdots\!19}{10\!\cdots\!93}a^{5}-\frac{11\!\cdots\!82}{10\!\cdots\!93}a^{4}+\frac{61\!\cdots\!47}{10\!\cdots\!93}a^{3}-\frac{27\!\cdots\!62}{10\!\cdots\!93}a^{2}+\frac{80\!\cdots\!81}{10\!\cdots\!93}a-\frac{10\!\cdots\!21}{10\!\cdots\!93}$
| 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: | \( 48967.484032 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 48967.484032 \cdot 1}{2\cdot\sqrt{167297208611536987435565056}}\cr\approx \mathstrut & 0.23112251226 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 23*x^19 - 46*x^18 + 59*x^17 - 38*x^16 - 51*x^15 + 276*x^14 - 725*x^13 + 1392*x^12 - 2056*x^11 + 2397*x^10 - 2311*x^9 + 1987*x^8 - 1615*x^7 + 1208*x^6 - 783*x^5 + 436*x^4 - 201*x^3 + 66*x^2 - 12*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 - 7*x^20 + 23*x^19 - 46*x^18 + 59*x^17 - 38*x^16 - 51*x^15 + 276*x^14 - 725*x^13 + 1392*x^12 - 2056*x^11 + 2397*x^10 - 2311*x^9 + 1987*x^8 - 1615*x^7 + 1208*x^6 - 783*x^5 + 436*x^4 - 201*x^3 + 66*x^2 - 12*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 - 7*x^20 + 23*x^19 - 46*x^18 + 59*x^17 - 38*x^16 - 51*x^15 + 276*x^14 - 725*x^13 + 1392*x^12 - 2056*x^11 + 2397*x^10 - 2311*x^9 + 1987*x^8 - 1615*x^7 + 1208*x^6 - 783*x^5 + 436*x^4 - 201*x^3 + 66*x^2 - 12*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 - 7*x^20 + 23*x^19 - 46*x^18 + 59*x^17 - 38*x^16 - 51*x^15 + 276*x^14 - 725*x^13 + 1392*x^12 - 2056*x^11 + 2397*x^10 - 2311*x^9 + 1987*x^8 - 1615*x^7 + 1208*x^6 - 783*x^5 + 436*x^4 - 201*x^3 + 66*x^2 - 12*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
$S_3\times A_7$ (as 21T57):
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 15120 |
| The 27 conjugacy class representatives for $S_3\times A_7$ |
| Character table for $S_3\times A_7$ |
Intermediate fields
| 3.1.31.1, 7.3.17774656.1 |
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 sibling: | data not computed |
| Degree 45 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 | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.7.0.1}{7} }$ | $21$ | $21$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.7.0.1}{7} }$ | R | $21$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.7.0.1}{7} }$ | R | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.7.0.1}{7} }$ | $15{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $21$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
| 2.12.12.25 | $x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$ | $2$ | $6$ | $12$ | $C_{12}$ | $[2]^{6}$ | |
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.3.2.1 | $x^{3} + 17$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 17.6.4.1 | $x^{6} + 48 x^{5} + 879 x^{4} + 7682 x^{3} + 44916 x^{2} + 265428 x + 127425$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(31\)
| 31.3.0.1 | $x^{3} + x + 28$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.6.3.2 | $x^{6} + 95 x^{4} + 56 x^{3} + 2884 x^{2} - 5152 x + 28684$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |