Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 + 16*x^19 - 6*x^18 + 60*x^17 + 24*x^16 - 262*x^15 + 630*x^14 - 2088*x^13 + 2404*x^12 - 2916*x^11 - 68*x^10 + 6420*x^9 - 11568*x^8 + 16832*x^7 - 14508*x^6 + 9552*x^5 - 4776*x^4 + 204*x^3 + 360*x^2 - 600*x + 500)
gp: K = bnfinit(y^21 + 16*y^19 - 6*y^18 + 60*y^17 + 24*y^16 - 262*y^15 + 630*y^14 - 2088*y^13 + 2404*y^12 - 2916*y^11 - 68*y^10 + 6420*y^9 - 11568*y^8 + 16832*y^7 - 14508*y^6 + 9552*y^5 - 4776*y^4 + 204*y^3 + 360*y^2 - 600*y + 500, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 + 16*x^19 - 6*x^18 + 60*x^17 + 24*x^16 - 262*x^15 + 630*x^14 - 2088*x^13 + 2404*x^12 - 2916*x^11 - 68*x^10 + 6420*x^9 - 11568*x^8 + 16832*x^7 - 14508*x^6 + 9552*x^5 - 4776*x^4 + 204*x^3 + 360*x^2 - 600*x + 500);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 + 16*x^19 - 6*x^18 + 60*x^17 + 24*x^16 - 262*x^15 + 630*x^14 - 2088*x^13 + 2404*x^12 - 2916*x^11 - 68*x^10 + 6420*x^9 - 11568*x^8 + 16832*x^7 - 14508*x^6 + 9552*x^5 - 4776*x^4 + 204*x^3 + 360*x^2 - 600*x + 500)
\( x^{21} + 16 x^{19} - 6 x^{18} + 60 x^{17} + 24 x^{16} - 262 x^{15} + 630 x^{14} - 2088 x^{13} + \cdots + 500 \)
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: | $[7, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-467975407174598390024135662204092416\)
\(\medspace = -\,2^{20}\cdot 11^{7}\cdot 73^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(49.96\) | 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: | $2^{20/21}11^{1/2}73^{2/3}\approx 112.10031078443632$ | ||
| Ramified primes: |
\(2\), \(11\), \(73\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{10}a^{19}+\frac{1}{10}a^{17}-\frac{1}{10}a^{16}-\frac{1}{10}a^{14}-\frac{1}{5}a^{13}+\frac{1}{5}a^{11}+\frac{2}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{32\!\cdots\!50}a^{20}-\frac{44\!\cdots\!81}{65\!\cdots\!90}a^{19}+\frac{27\!\cdots\!33}{16\!\cdots\!25}a^{18}+\frac{11\!\cdots\!57}{16\!\cdots\!25}a^{17}+\frac{76\!\cdots\!63}{65\!\cdots\!90}a^{16}+\frac{88\!\cdots\!49}{32\!\cdots\!50}a^{15}-\frac{24\!\cdots\!91}{16\!\cdots\!25}a^{14}+\frac{45\!\cdots\!44}{32\!\cdots\!45}a^{13}-\frac{88\!\cdots\!63}{32\!\cdots\!50}a^{12}+\frac{29\!\cdots\!22}{16\!\cdots\!25}a^{11}+\frac{36\!\cdots\!32}{16\!\cdots\!25}a^{10}-\frac{57\!\cdots\!44}{16\!\cdots\!25}a^{9}-\frac{87\!\cdots\!84}{32\!\cdots\!45}a^{8}-\frac{64\!\cdots\!34}{16\!\cdots\!25}a^{7}-\frac{34\!\cdots\!89}{16\!\cdots\!25}a^{6}+\frac{58\!\cdots\!16}{16\!\cdots\!25}a^{5}+\frac{69\!\cdots\!96}{16\!\cdots\!25}a^{4}+\frac{79\!\cdots\!82}{16\!\cdots\!25}a^{3}+\frac{36\!\cdots\!92}{16\!\cdots\!25}a^{2}+\frac{88\!\cdots\!19}{32\!\cdots\!45}a+\frac{31\!\cdots\!88}{65\!\cdots\!69}$
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: | $13$ | 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: |
$\frac{35\!\cdots\!79}{32\!\cdots\!50}a^{20}+\frac{77\!\cdots\!27}{65\!\cdots\!69}a^{19}+\frac{27\!\cdots\!07}{16\!\cdots\!25}a^{18}+\frac{19\!\cdots\!88}{16\!\cdots\!25}a^{17}+\frac{16\!\cdots\!04}{32\!\cdots\!45}a^{16}+\frac{15\!\cdots\!48}{16\!\cdots\!25}a^{15}-\frac{88\!\cdots\!23}{32\!\cdots\!50}a^{14}+\frac{10\!\cdots\!17}{32\!\cdots\!45}a^{13}-\frac{23\!\cdots\!76}{16\!\cdots\!25}a^{12}+\frac{20\!\cdots\!58}{16\!\cdots\!25}a^{11}+\frac{22\!\cdots\!93}{16\!\cdots\!25}a^{10}-\frac{55\!\cdots\!36}{16\!\cdots\!25}a^{9}+\frac{21\!\cdots\!98}{32\!\cdots\!45}a^{8}-\frac{57\!\cdots\!61}{16\!\cdots\!25}a^{7}+\frac{37\!\cdots\!64}{16\!\cdots\!25}a^{6}+\frac{83\!\cdots\!34}{16\!\cdots\!25}a^{5}-\frac{11\!\cdots\!96}{16\!\cdots\!25}a^{4}+\frac{53\!\cdots\!98}{16\!\cdots\!25}a^{3}-\frac{45\!\cdots\!42}{16\!\cdots\!25}a^{2}-\frac{16\!\cdots\!06}{32\!\cdots\!45}a+\frac{45\!\cdots\!71}{65\!\cdots\!69}$, $\frac{59\!\cdots\!05}{65\!\cdots\!69}a^{20}+\frac{10\!\cdots\!95}{65\!\cdots\!69}a^{19}+\frac{20\!\cdots\!45}{13\!\cdots\!38}a^{18}+\frac{30\!\cdots\!97}{13\!\cdots\!38}a^{17}+\frac{44\!\cdots\!80}{65\!\cdots\!69}a^{16}+\frac{19\!\cdots\!69}{13\!\cdots\!38}a^{15}-\frac{57\!\cdots\!06}{65\!\cdots\!69}a^{14}+\frac{18\!\cdots\!27}{65\!\cdots\!69}a^{13}-\frac{12\!\cdots\!69}{13\!\cdots\!38}a^{12}-\frac{52\!\cdots\!49}{65\!\cdots\!69}a^{11}-\frac{55\!\cdots\!52}{65\!\cdots\!69}a^{10}-\frac{28\!\cdots\!29}{65\!\cdots\!69}a^{9}+\frac{11\!\cdots\!03}{65\!\cdots\!69}a^{8}-\frac{12\!\cdots\!18}{65\!\cdots\!69}a^{7}+\frac{36\!\cdots\!91}{65\!\cdots\!69}a^{6}+\frac{40\!\cdots\!16}{65\!\cdots\!69}a^{5}+\frac{69\!\cdots\!64}{65\!\cdots\!69}a^{4}+\frac{16\!\cdots\!97}{65\!\cdots\!69}a^{3}+\frac{25\!\cdots\!30}{65\!\cdots\!69}a^{2}+\frac{41\!\cdots\!50}{65\!\cdots\!69}a-\frac{20\!\cdots\!73}{65\!\cdots\!69}$, $\frac{77\!\cdots\!27}{65\!\cdots\!69}a^{20}-\frac{22\!\cdots\!25}{65\!\cdots\!69}a^{19}+\frac{12\!\cdots\!65}{65\!\cdots\!69}a^{18}-\frac{82\!\cdots\!74}{65\!\cdots\!69}a^{17}+\frac{43\!\cdots\!88}{65\!\cdots\!69}a^{16}+\frac{14\!\cdots\!55}{13\!\cdots\!38}a^{15}-\frac{22\!\cdots\!32}{65\!\cdots\!69}a^{14}+\frac{54\!\cdots\!68}{65\!\cdots\!69}a^{13}-\frac{16\!\cdots\!12}{65\!\cdots\!69}a^{12}+\frac{21\!\cdots\!11}{65\!\cdots\!69}a^{11}-\frac{21\!\cdots\!98}{65\!\cdots\!69}a^{10}-\frac{14\!\cdots\!44}{65\!\cdots\!69}a^{9}+\frac{58\!\cdots\!71}{65\!\cdots\!69}a^{8}-\frac{10\!\cdots\!52}{65\!\cdots\!69}a^{7}+\frac{13\!\cdots\!60}{65\!\cdots\!69}a^{6}-\frac{11\!\cdots\!84}{65\!\cdots\!69}a^{5}+\frac{54\!\cdots\!42}{65\!\cdots\!69}a^{4}-\frac{19\!\cdots\!24}{65\!\cdots\!69}a^{3}-\frac{57\!\cdots\!10}{65\!\cdots\!69}a^{2}+\frac{21\!\cdots\!50}{65\!\cdots\!69}a+\frac{30\!\cdots\!79}{65\!\cdots\!69}$, $\frac{53\!\cdots\!51}{65\!\cdots\!69}a^{20}-\frac{65\!\cdots\!49}{65\!\cdots\!69}a^{19}+\frac{84\!\cdots\!98}{65\!\cdots\!69}a^{18}-\frac{41\!\cdots\!94}{65\!\cdots\!69}a^{17}+\frac{30\!\cdots\!15}{65\!\cdots\!69}a^{16}+\frac{22\!\cdots\!21}{13\!\cdots\!38}a^{15}-\frac{14\!\cdots\!06}{65\!\cdots\!69}a^{14}+\frac{35\!\cdots\!21}{65\!\cdots\!69}a^{13}-\frac{22\!\cdots\!85}{13\!\cdots\!38}a^{12}+\frac{13\!\cdots\!75}{65\!\cdots\!69}a^{11}-\frac{14\!\cdots\!84}{65\!\cdots\!69}a^{10}-\frac{28\!\cdots\!01}{65\!\cdots\!69}a^{9}+\frac{38\!\cdots\!71}{65\!\cdots\!69}a^{8}-\frac{65\!\cdots\!50}{65\!\cdots\!69}a^{7}+\frac{86\!\cdots\!73}{65\!\cdots\!69}a^{6}-\frac{66\!\cdots\!64}{65\!\cdots\!69}a^{5}+\frac{35\!\cdots\!22}{65\!\cdots\!69}a^{4}-\frac{10\!\cdots\!30}{65\!\cdots\!69}a^{3}-\frac{33\!\cdots\!00}{65\!\cdots\!69}a^{2}+\frac{13\!\cdots\!50}{65\!\cdots\!69}a-\frac{32\!\cdots\!51}{65\!\cdots\!69}$, $\frac{65\!\cdots\!61}{65\!\cdots\!69}a^{20}+\frac{15\!\cdots\!41}{65\!\cdots\!69}a^{19}+\frac{10\!\cdots\!43}{65\!\cdots\!69}a^{18}-\frac{11\!\cdots\!97}{65\!\cdots\!69}a^{17}+\frac{39\!\cdots\!75}{65\!\cdots\!69}a^{16}+\frac{60\!\cdots\!59}{13\!\cdots\!38}a^{15}-\frac{16\!\cdots\!18}{65\!\cdots\!69}a^{14}+\frac{39\!\cdots\!75}{65\!\cdots\!69}a^{13}-\frac{24\!\cdots\!23}{13\!\cdots\!38}a^{12}+\frac{12\!\cdots\!77}{65\!\cdots\!69}a^{11}-\frac{15\!\cdots\!88}{65\!\cdots\!69}a^{10}-\frac{85\!\cdots\!59}{65\!\cdots\!69}a^{9}+\frac{40\!\cdots\!77}{65\!\cdots\!69}a^{8}-\frac{68\!\cdots\!86}{65\!\cdots\!69}a^{7}+\frac{86\!\cdots\!55}{65\!\cdots\!69}a^{6}-\frac{58\!\cdots\!32}{65\!\cdots\!69}a^{5}+\frac{36\!\cdots\!50}{65\!\cdots\!69}a^{4}-\frac{74\!\cdots\!36}{65\!\cdots\!69}a^{3}-\frac{28\!\cdots\!40}{65\!\cdots\!69}a^{2}+\frac{14\!\cdots\!50}{65\!\cdots\!69}a-\frac{31\!\cdots\!59}{65\!\cdots\!69}$, $\frac{50\!\cdots\!42}{65\!\cdots\!69}a^{20}+\frac{21\!\cdots\!28}{65\!\cdots\!69}a^{19}+\frac{81\!\cdots\!67}{65\!\cdots\!69}a^{18}+\frac{71\!\cdots\!49}{65\!\cdots\!69}a^{17}+\frac{30\!\cdots\!73}{65\!\cdots\!69}a^{16}+\frac{59\!\cdots\!07}{13\!\cdots\!38}a^{15}-\frac{11\!\cdots\!24}{65\!\cdots\!69}a^{14}+\frac{28\!\cdots\!89}{65\!\cdots\!69}a^{13}-\frac{18\!\cdots\!09}{13\!\cdots\!38}a^{12}+\frac{76\!\cdots\!75}{65\!\cdots\!69}a^{11}-\frac{10\!\cdots\!86}{65\!\cdots\!69}a^{10}-\frac{86\!\cdots\!64}{65\!\cdots\!69}a^{9}+\frac{29\!\cdots\!81}{65\!\cdots\!69}a^{8}-\frac{47\!\cdots\!42}{65\!\cdots\!69}a^{7}+\frac{59\!\cdots\!87}{65\!\cdots\!69}a^{6}-\frac{34\!\cdots\!38}{65\!\cdots\!69}a^{5}+\frac{25\!\cdots\!66}{65\!\cdots\!69}a^{4}-\frac{31\!\cdots\!58}{65\!\cdots\!69}a^{3}-\frac{16\!\cdots\!70}{65\!\cdots\!69}a^{2}+\frac{99\!\cdots\!50}{65\!\cdots\!69}a-\frac{59\!\cdots\!93}{65\!\cdots\!69}$, $\frac{86\!\cdots\!18}{65\!\cdots\!69}a^{20}+\frac{17\!\cdots\!09}{65\!\cdots\!69}a^{19}+\frac{13\!\cdots\!80}{65\!\cdots\!69}a^{18}-\frac{38\!\cdots\!99}{13\!\cdots\!38}a^{17}+\frac{51\!\cdots\!66}{65\!\cdots\!69}a^{16}+\frac{76\!\cdots\!45}{13\!\cdots\!38}a^{15}-\frac{21\!\cdots\!62}{65\!\cdots\!69}a^{14}+\frac{52\!\cdots\!01}{65\!\cdots\!69}a^{13}-\frac{16\!\cdots\!08}{65\!\cdots\!69}a^{12}+\frac{16\!\cdots\!37}{65\!\cdots\!69}a^{11}-\frac{20\!\cdots\!66}{65\!\cdots\!69}a^{10}-\frac{10\!\cdots\!21}{65\!\cdots\!69}a^{9}+\frac{54\!\cdots\!21}{65\!\cdots\!69}a^{8}-\frac{91\!\cdots\!14}{65\!\cdots\!69}a^{7}+\frac{11\!\cdots\!29}{65\!\cdots\!69}a^{6}-\frac{79\!\cdots\!54}{65\!\cdots\!69}a^{5}+\frac{49\!\cdots\!86}{65\!\cdots\!69}a^{4}-\frac{10\!\cdots\!19}{65\!\cdots\!69}a^{3}-\frac{39\!\cdots\!10}{65\!\cdots\!69}a^{2}+\frac{19\!\cdots\!50}{65\!\cdots\!69}a-\frac{60\!\cdots\!19}{65\!\cdots\!69}$, $\frac{29\!\cdots\!09}{32\!\cdots\!50}a^{20}+\frac{41\!\cdots\!69}{32\!\cdots\!45}a^{19}+\frac{49\!\cdots\!69}{32\!\cdots\!50}a^{18}+\frac{23\!\cdots\!93}{16\!\cdots\!25}a^{17}+\frac{35\!\cdots\!87}{65\!\cdots\!69}a^{16}+\frac{28\!\cdots\!91}{32\!\cdots\!50}a^{15}-\frac{29\!\cdots\!49}{16\!\cdots\!25}a^{14}+\frac{77\!\cdots\!44}{32\!\cdots\!45}a^{13}-\frac{40\!\cdots\!17}{32\!\cdots\!50}a^{12}+\frac{83\!\cdots\!83}{16\!\cdots\!25}a^{11}-\frac{11\!\cdots\!17}{16\!\cdots\!25}a^{10}-\frac{32\!\cdots\!76}{16\!\cdots\!25}a^{9}+\frac{12\!\cdots\!01}{32\!\cdots\!45}a^{8}-\frac{44\!\cdots\!31}{16\!\cdots\!25}a^{7}+\frac{71\!\cdots\!84}{16\!\cdots\!25}a^{6}-\frac{96\!\cdots\!21}{16\!\cdots\!25}a^{5}+\frac{78\!\cdots\!24}{16\!\cdots\!25}a^{4}-\frac{55\!\cdots\!52}{16\!\cdots\!25}a^{3}-\frac{15\!\cdots\!02}{16\!\cdots\!25}a^{2}-\frac{49\!\cdots\!64}{65\!\cdots\!69}a-\frac{17\!\cdots\!93}{65\!\cdots\!69}$, $\frac{94\!\cdots\!26}{16\!\cdots\!25}a^{20}+\frac{47\!\cdots\!23}{65\!\cdots\!90}a^{19}+\frac{16\!\cdots\!41}{16\!\cdots\!25}a^{18}+\frac{14\!\cdots\!14}{16\!\cdots\!25}a^{17}+\frac{15\!\cdots\!28}{32\!\cdots\!45}a^{16}+\frac{21\!\cdots\!73}{32\!\cdots\!50}a^{15}-\frac{23\!\cdots\!39}{32\!\cdots\!50}a^{14}+\frac{16\!\cdots\!31}{65\!\cdots\!90}a^{13}-\frac{30\!\cdots\!01}{32\!\cdots\!50}a^{12}+\frac{45\!\cdots\!44}{16\!\cdots\!25}a^{11}-\frac{23\!\cdots\!11}{16\!\cdots\!25}a^{10}-\frac{21\!\cdots\!38}{16\!\cdots\!25}a^{9}+\frac{65\!\cdots\!47}{32\!\cdots\!45}a^{8}-\frac{61\!\cdots\!18}{16\!\cdots\!25}a^{7}+\frac{93\!\cdots\!47}{16\!\cdots\!25}a^{6}-\frac{42\!\cdots\!93}{16\!\cdots\!25}a^{5}+\frac{58\!\cdots\!42}{16\!\cdots\!25}a^{4}-\frac{75\!\cdots\!61}{16\!\cdots\!25}a^{3}+\frac{90\!\cdots\!34}{16\!\cdots\!25}a^{2}-\frac{52\!\cdots\!22}{32\!\cdots\!45}a-\frac{10\!\cdots\!03}{65\!\cdots\!69}$, $\frac{59\!\cdots\!37}{32\!\cdots\!50}a^{20}+\frac{42\!\cdots\!92}{32\!\cdots\!45}a^{19}+\frac{12\!\cdots\!67}{32\!\cdots\!50}a^{18}+\frac{31\!\cdots\!49}{16\!\cdots\!25}a^{17}+\frac{12\!\cdots\!42}{65\!\cdots\!69}a^{16}+\frac{23\!\cdots\!63}{32\!\cdots\!50}a^{15}+\frac{12\!\cdots\!11}{32\!\cdots\!50}a^{14}-\frac{13\!\cdots\!41}{65\!\cdots\!90}a^{13}+\frac{20\!\cdots\!22}{16\!\cdots\!25}a^{12}-\frac{49\!\cdots\!87}{32\!\cdots\!50}a^{11}+\frac{73\!\cdots\!19}{16\!\cdots\!25}a^{10}-\frac{19\!\cdots\!68}{16\!\cdots\!25}a^{9}-\frac{42\!\cdots\!22}{32\!\cdots\!45}a^{8}+\frac{78\!\cdots\!92}{16\!\cdots\!25}a^{7}-\frac{65\!\cdots\!38}{16\!\cdots\!25}a^{6}+\frac{98\!\cdots\!22}{16\!\cdots\!25}a^{5}-\frac{35\!\cdots\!18}{16\!\cdots\!25}a^{4}+\frac{14\!\cdots\!64}{16\!\cdots\!25}a^{3}-\frac{14\!\cdots\!86}{16\!\cdots\!25}a^{2}-\frac{25\!\cdots\!60}{65\!\cdots\!69}a-\frac{18\!\cdots\!69}{65\!\cdots\!69}$, $\frac{28\!\cdots\!81}{32\!\cdots\!50}a^{20}+\frac{93\!\cdots\!03}{65\!\cdots\!90}a^{19}+\frac{44\!\cdots\!21}{32\!\cdots\!50}a^{18}-\frac{47\!\cdots\!73}{16\!\cdots\!25}a^{17}+\frac{15\!\cdots\!82}{32\!\cdots\!45}a^{16}+\frac{10\!\cdots\!19}{32\!\cdots\!50}a^{15}-\frac{39\!\cdots\!06}{16\!\cdots\!25}a^{14}+\frac{33\!\cdots\!31}{65\!\cdots\!69}a^{13}-\frac{53\!\cdots\!03}{32\!\cdots\!50}a^{12}+\frac{52\!\cdots\!79}{32\!\cdots\!50}a^{11}-\frac{27\!\cdots\!18}{16\!\cdots\!25}a^{10}-\frac{18\!\cdots\!49}{16\!\cdots\!25}a^{9}+\frac{38\!\cdots\!28}{65\!\cdots\!69}a^{8}-\frac{14\!\cdots\!54}{16\!\cdots\!25}a^{7}+\frac{17\!\cdots\!86}{16\!\cdots\!25}a^{6}-\frac{11\!\cdots\!09}{16\!\cdots\!25}a^{5}+\frac{63\!\cdots\!71}{16\!\cdots\!25}a^{4}-\frac{11\!\cdots\!88}{16\!\cdots\!25}a^{3}-\frac{38\!\cdots\!33}{16\!\cdots\!25}a^{2}+\frac{90\!\cdots\!12}{32\!\cdots\!45}a-\frac{28\!\cdots\!63}{65\!\cdots\!69}$, $\frac{74\!\cdots\!37}{16\!\cdots\!25}a^{20}+\frac{75\!\cdots\!34}{32\!\cdots\!45}a^{19}+\frac{12\!\cdots\!92}{16\!\cdots\!25}a^{18}+\frac{51\!\cdots\!98}{16\!\cdots\!25}a^{17}+\frac{10\!\cdots\!25}{65\!\cdots\!69}a^{16}+\frac{35\!\cdots\!51}{32\!\cdots\!50}a^{15}-\frac{90\!\cdots\!64}{16\!\cdots\!25}a^{14}-\frac{27\!\cdots\!57}{65\!\cdots\!90}a^{13}+\frac{10\!\cdots\!13}{32\!\cdots\!50}a^{12}-\frac{94\!\cdots\!49}{32\!\cdots\!50}a^{11}+\frac{43\!\cdots\!38}{16\!\cdots\!25}a^{10}-\frac{38\!\cdots\!61}{16\!\cdots\!25}a^{9}-\frac{17\!\cdots\!79}{32\!\cdots\!45}a^{8}+\frac{18\!\cdots\!84}{16\!\cdots\!25}a^{7}-\frac{21\!\cdots\!51}{16\!\cdots\!25}a^{6}+\frac{22\!\cdots\!19}{16\!\cdots\!25}a^{5}-\frac{15\!\cdots\!36}{16\!\cdots\!25}a^{4}+\frac{48\!\cdots\!28}{16\!\cdots\!25}a^{3}-\frac{14\!\cdots\!72}{16\!\cdots\!25}a^{2}-\frac{20\!\cdots\!60}{65\!\cdots\!69}a+\frac{76\!\cdots\!31}{65\!\cdots\!69}$, $\frac{13\!\cdots\!07}{32\!\cdots\!50}a^{20}+\frac{11\!\cdots\!26}{32\!\cdots\!45}a^{19}+\frac{11\!\cdots\!81}{16\!\cdots\!25}a^{18}+\frac{10\!\cdots\!43}{32\!\cdots\!50}a^{17}+\frac{16\!\cdots\!27}{65\!\cdots\!90}a^{16}+\frac{10\!\cdots\!43}{32\!\cdots\!50}a^{15}-\frac{27\!\cdots\!19}{32\!\cdots\!50}a^{14}+\frac{58\!\cdots\!04}{32\!\cdots\!45}a^{13}-\frac{21\!\cdots\!91}{32\!\cdots\!50}a^{12}+\frac{11\!\cdots\!23}{32\!\cdots\!50}a^{11}-\frac{11\!\cdots\!86}{16\!\cdots\!25}a^{10}-\frac{12\!\cdots\!68}{16\!\cdots\!25}a^{9}+\frac{67\!\cdots\!96}{32\!\cdots\!45}a^{8}-\frac{45\!\cdots\!63}{16\!\cdots\!25}a^{7}+\frac{62\!\cdots\!22}{16\!\cdots\!25}a^{6}-\frac{31\!\cdots\!68}{16\!\cdots\!25}a^{5}+\frac{24\!\cdots\!17}{16\!\cdots\!25}a^{4}-\frac{43\!\cdots\!06}{16\!\cdots\!25}a^{3}-\frac{32\!\cdots\!41}{16\!\cdots\!25}a^{2}+\frac{72\!\cdots\!21}{32\!\cdots\!45}a-\frac{11\!\cdots\!59}{65\!\cdots\!69}$
| 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: | \( 11785083997.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{7}\cdot 11785083997.5 \cdot 1}{2\cdot\sqrt{467975407174598390024135662204092416}}\cr\approx \mathstrut & 0.426245904664 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 + 16*x^19 - 6*x^18 + 60*x^17 + 24*x^16 - 262*x^15 + 630*x^14 - 2088*x^13 + 2404*x^12 - 2916*x^11 - 68*x^10 + 6420*x^9 - 11568*x^8 + 16832*x^7 - 14508*x^6 + 9552*x^5 - 4776*x^4 + 204*x^3 + 360*x^2 - 600*x + 500)
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 + 16*x^19 - 6*x^18 + 60*x^17 + 24*x^16 - 262*x^15 + 630*x^14 - 2088*x^13 + 2404*x^12 - 2916*x^11 - 68*x^10 + 6420*x^9 - 11568*x^8 + 16832*x^7 - 14508*x^6 + 9552*x^5 - 4776*x^4 + 204*x^3 + 360*x^2 - 600*x + 500, 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 + 16*x^19 - 6*x^18 + 60*x^17 + 24*x^16 - 262*x^15 + 630*x^14 - 2088*x^13 + 2404*x^12 - 2916*x^11 - 68*x^10 + 6420*x^9 - 11568*x^8 + 16832*x^7 - 14508*x^6 + 9552*x^5 - 4776*x^4 + 204*x^3 + 360*x^2 - 600*x + 500);
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 + 16*x^19 - 6*x^18 + 60*x^17 + 24*x^16 - 262*x^15 + 630*x^14 - 2088*x^13 + 2404*x^12 - 2916*x^11 - 68*x^10 + 6420*x^9 - 11568*x^8 + 16832*x^7 - 14508*x^6 + 9552*x^5 - 4776*x^4 + 204*x^3 + 360*x^2 - 600*x + 500);
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
$C_{21}:C_6$ (as 21T11):
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 15 conjugacy class representatives for $C_{21}:C_6$ |
| Character table for $C_{21}:C_6$ |
Intermediate fields
| 3.1.44.1, 7.7.1817487424.1 |
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 | $21$ | ${\href{/padicField/5.3.0.1}{3} }^{7}$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.7.0.1}{7} }$ | R | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }^{7}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }^{7}$ | ${\href{/padicField/37.3.0.1}{3} }^{7}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.3.0.1}{3} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.21.20.1 | $x^{21} + 2$ | $21$ | $1$ | $20$ | 21T11 | $[\ ]_{21}^{6}$ |
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(73\)
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 73.6.4.1 | $x^{6} + 210 x^{5} + 14715 x^{4} + 345246 x^{3} + 88905 x^{2} + 1076160 x + 24967804$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 73.6.4.1 | $x^{6} + 210 x^{5} + 14715 x^{4} + 345246 x^{3} + 88905 x^{2} + 1076160 x + 24967804$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |