Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 17*x^19 + 29*x^18 + 208*x^17 - 102*x^16 - 1409*x^15 - x^14 + 3951*x^13 + 2695*x^12 - 3516*x^11 - 6803*x^10 - 5950*x^9 - 1331*x^8 + 8064*x^7 + 12913*x^6 + 7673*x^5 + 1263*x^4 - 612*x^3 - 375*x^2 - 90*x - 9)
gp: K = bnfinit(y^21 - 3*y^20 - 17*y^19 + 29*y^18 + 208*y^17 - 102*y^16 - 1409*y^15 - y^14 + 3951*y^13 + 2695*y^12 - 3516*y^11 - 6803*y^10 - 5950*y^9 - 1331*y^8 + 8064*y^7 + 12913*y^6 + 7673*y^5 + 1263*y^4 - 612*y^3 - 375*y^2 - 90*y - 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 3*x^20 - 17*x^19 + 29*x^18 + 208*x^17 - 102*x^16 - 1409*x^15 - x^14 + 3951*x^13 + 2695*x^12 - 3516*x^11 - 6803*x^10 - 5950*x^9 - 1331*x^8 + 8064*x^7 + 12913*x^6 + 7673*x^5 + 1263*x^4 - 612*x^3 - 375*x^2 - 90*x - 9);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 3*x^20 - 17*x^19 + 29*x^18 + 208*x^17 - 102*x^16 - 1409*x^15 - x^14 + 3951*x^13 + 2695*x^12 - 3516*x^11 - 6803*x^10 - 5950*x^9 - 1331*x^8 + 8064*x^7 + 12913*x^6 + 7673*x^5 + 1263*x^4 - 612*x^3 - 375*x^2 - 90*x - 9)
\( x^{21} - 3 x^{20} - 17 x^{19} + 29 x^{18} + 208 x^{17} - 102 x^{16} - 1409 x^{15} - x^{14} + 3951 x^{13} + \cdots - 9 \)
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: |
\(-6400890533463290121080023755169887\)
\(\medspace = -\,3^{21}\cdot 29^{9}\cdot 59^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(40.72\) | 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: | $3^{11/6}29^{1/2}59^{2/3}\approx 611.6307435196422$ | ||
| Ramified primes: |
\(3\), \(29\), \(59\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-87}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{14}-\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{18}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{10}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}$, $\frac{1}{9}a^{19}+\frac{1}{9}a^{18}+\frac{1}{9}a^{17}-\frac{1}{9}a^{16}-\frac{1}{9}a^{15}+\frac{1}{3}a^{14}-\frac{4}{9}a^{13}+\frac{1}{9}a^{12}+\frac{2}{9}a^{11}+\frac{2}{9}a^{10}-\frac{4}{9}a^{8}+\frac{4}{9}a^{7}-\frac{1}{3}a^{6}+\frac{2}{9}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{24\!\cdots\!21}a^{20}+\frac{11\!\cdots\!99}{24\!\cdots\!21}a^{19}+\frac{52\!\cdots\!68}{24\!\cdots\!21}a^{18}+\frac{71\!\cdots\!86}{82\!\cdots\!07}a^{17}-\frac{10\!\cdots\!68}{24\!\cdots\!21}a^{16}+\frac{11\!\cdots\!22}{24\!\cdots\!21}a^{15}+\frac{79\!\cdots\!11}{24\!\cdots\!21}a^{14}-\frac{39\!\cdots\!90}{82\!\cdots\!07}a^{13}-\frac{45\!\cdots\!98}{91\!\cdots\!23}a^{12}+\frac{45\!\cdots\!49}{24\!\cdots\!21}a^{11}-\frac{27\!\cdots\!14}{24\!\cdots\!21}a^{10}+\frac{11\!\cdots\!92}{24\!\cdots\!21}a^{9}-\frac{33\!\cdots\!48}{11\!\cdots\!03}a^{8}-\frac{11\!\cdots\!80}{24\!\cdots\!21}a^{7}-\frac{11\!\cdots\!98}{24\!\cdots\!21}a^{6}-\frac{23\!\cdots\!31}{24\!\cdots\!21}a^{5}+\frac{33\!\cdots\!41}{82\!\cdots\!07}a^{4}-\frac{11\!\cdots\!89}{82\!\cdots\!07}a^{3}-\frac{29\!\cdots\!46}{82\!\cdots\!07}a^{2}+\frac{43\!\cdots\!76}{39\!\cdots\!01}a-\frac{10\!\cdots\!66}{27\!\cdots\!69}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{10\!\cdots\!35}{27\!\cdots\!17}a^{20}-\frac{29\!\cdots\!97}{27\!\cdots\!17}a^{19}-\frac{19\!\cdots\!34}{27\!\cdots\!17}a^{18}+\frac{94\!\cdots\!43}{93\!\cdots\!39}a^{17}+\frac{23\!\cdots\!35}{27\!\cdots\!17}a^{16}-\frac{75\!\cdots\!10}{27\!\cdots\!17}a^{15}-\frac{16\!\cdots\!78}{27\!\cdots\!17}a^{14}-\frac{79\!\cdots\!52}{93\!\cdots\!39}a^{13}+\frac{54\!\cdots\!75}{31\!\cdots\!13}a^{12}+\frac{33\!\cdots\!34}{27\!\cdots\!17}a^{11}-\frac{48\!\cdots\!94}{27\!\cdots\!17}a^{10}-\frac{81\!\cdots\!94}{27\!\cdots\!17}a^{9}-\frac{20\!\cdots\!20}{93\!\cdots\!39}a^{8}-\frac{88\!\cdots\!44}{27\!\cdots\!17}a^{7}+\frac{96\!\cdots\!28}{27\!\cdots\!17}a^{6}+\frac{15\!\cdots\!19}{27\!\cdots\!17}a^{5}+\frac{24\!\cdots\!64}{93\!\cdots\!39}a^{4}-\frac{15\!\cdots\!29}{93\!\cdots\!39}a^{3}-\frac{39\!\cdots\!87}{93\!\cdots\!39}a^{2}-\frac{26\!\cdots\!88}{31\!\cdots\!13}a-\frac{21\!\cdots\!41}{31\!\cdots\!13}$, $\frac{12\!\cdots\!24}{27\!\cdots\!69}a^{20}-\frac{12\!\cdots\!97}{82\!\cdots\!07}a^{19}-\frac{59\!\cdots\!75}{82\!\cdots\!07}a^{18}+\frac{12\!\cdots\!46}{82\!\cdots\!07}a^{17}+\frac{73\!\cdots\!88}{82\!\cdots\!07}a^{16}-\frac{58\!\cdots\!86}{82\!\cdots\!07}a^{15}-\frac{16\!\cdots\!98}{27\!\cdots\!69}a^{14}+\frac{13\!\cdots\!76}{82\!\cdots\!07}a^{13}+\frac{14\!\cdots\!75}{82\!\cdots\!07}a^{12}+\frac{59\!\cdots\!30}{82\!\cdots\!07}a^{11}-\frac{14\!\cdots\!59}{82\!\cdots\!07}a^{10}-\frac{69\!\cdots\!52}{27\!\cdots\!69}a^{9}-\frac{68\!\cdots\!52}{35\!\cdots\!09}a^{8}-\frac{33\!\cdots\!60}{82\!\cdots\!07}a^{7}+\frac{99\!\cdots\!39}{27\!\cdots\!69}a^{6}+\frac{39\!\cdots\!40}{82\!\cdots\!07}a^{5}+\frac{56\!\cdots\!36}{27\!\cdots\!69}a^{4}-\frac{14\!\cdots\!41}{27\!\cdots\!69}a^{3}-\frac{69\!\cdots\!47}{27\!\cdots\!69}a^{2}-\frac{34\!\cdots\!07}{39\!\cdots\!01}a-\frac{12\!\cdots\!91}{91\!\cdots\!23}$, $\frac{92\!\cdots\!01}{27\!\cdots\!69}a^{20}-\frac{30\!\cdots\!23}{27\!\cdots\!69}a^{19}-\frac{14\!\cdots\!74}{27\!\cdots\!69}a^{18}+\frac{31\!\cdots\!94}{27\!\cdots\!69}a^{17}+\frac{18\!\cdots\!08}{27\!\cdots\!69}a^{16}-\frac{15\!\cdots\!61}{27\!\cdots\!69}a^{15}-\frac{12\!\cdots\!19}{27\!\cdots\!69}a^{14}+\frac{39\!\cdots\!51}{27\!\cdots\!69}a^{13}+\frac{11\!\cdots\!23}{91\!\cdots\!23}a^{12}+\frac{13\!\cdots\!21}{27\!\cdots\!69}a^{11}-\frac{12\!\cdots\!15}{91\!\cdots\!23}a^{10}-\frac{50\!\cdots\!35}{27\!\cdots\!69}a^{9}-\frac{55\!\cdots\!05}{39\!\cdots\!01}a^{8}+\frac{33\!\cdots\!35}{91\!\cdots\!23}a^{7}+\frac{24\!\cdots\!03}{91\!\cdots\!23}a^{6}+\frac{31\!\cdots\!36}{91\!\cdots\!23}a^{5}+\frac{39\!\cdots\!08}{27\!\cdots\!69}a^{4}-\frac{12\!\cdots\!55}{27\!\cdots\!69}a^{3}-\frac{16\!\cdots\!33}{91\!\cdots\!23}a^{2}-\frac{24\!\cdots\!88}{39\!\cdots\!01}a-\frac{89\!\cdots\!12}{91\!\cdots\!23}$, $\frac{37\!\cdots\!54}{27\!\cdots\!69}a^{20}-\frac{37\!\cdots\!88}{82\!\cdots\!07}a^{19}-\frac{17\!\cdots\!49}{82\!\cdots\!07}a^{18}+\frac{38\!\cdots\!69}{82\!\cdots\!07}a^{17}+\frac{22\!\cdots\!51}{82\!\cdots\!07}a^{16}-\frac{18\!\cdots\!30}{82\!\cdots\!07}a^{15}-\frac{50\!\cdots\!29}{27\!\cdots\!69}a^{14}+\frac{51\!\cdots\!81}{82\!\cdots\!07}a^{13}+\frac{42\!\cdots\!49}{82\!\cdots\!07}a^{12}+\frac{16\!\cdots\!39}{82\!\cdots\!07}a^{11}-\frac{45\!\cdots\!71}{82\!\cdots\!07}a^{10}-\frac{68\!\cdots\!02}{91\!\cdots\!23}a^{9}-\frac{20\!\cdots\!27}{35\!\cdots\!09}a^{8}+\frac{54\!\cdots\!26}{82\!\cdots\!07}a^{7}+\frac{30\!\cdots\!02}{27\!\cdots\!69}a^{6}+\frac{11\!\cdots\!65}{82\!\cdots\!07}a^{5}+\frac{53\!\cdots\!02}{91\!\cdots\!23}a^{4}-\frac{20\!\cdots\!39}{91\!\cdots\!23}a^{3}-\frac{20\!\cdots\!69}{27\!\cdots\!69}a^{2}-\frac{10\!\cdots\!53}{39\!\cdots\!01}a-\frac{37\!\cdots\!20}{91\!\cdots\!23}$, $\frac{49\!\cdots\!64}{48\!\cdots\!17}a^{20}-\frac{14\!\cdots\!92}{43\!\cdots\!53}a^{19}-\frac{70\!\cdots\!27}{43\!\cdots\!53}a^{18}+\frac{15\!\cdots\!58}{43\!\cdots\!53}a^{17}+\frac{87\!\cdots\!66}{43\!\cdots\!53}a^{16}-\frac{73\!\cdots\!13}{43\!\cdots\!53}a^{15}-\frac{20\!\cdots\!07}{14\!\cdots\!51}a^{14}+\frac{19\!\cdots\!65}{43\!\cdots\!53}a^{13}+\frac{16\!\cdots\!66}{43\!\cdots\!53}a^{12}+\frac{65\!\cdots\!57}{43\!\cdots\!53}a^{11}-\frac{17\!\cdots\!47}{43\!\cdots\!53}a^{10}-\frac{81\!\cdots\!59}{14\!\cdots\!51}a^{9}-\frac{80\!\cdots\!24}{18\!\cdots\!11}a^{8}-\frac{76\!\cdots\!47}{43\!\cdots\!53}a^{7}+\frac{39\!\cdots\!38}{48\!\cdots\!17}a^{6}+\frac{45\!\cdots\!56}{43\!\cdots\!53}a^{5}+\frac{64\!\cdots\!32}{14\!\cdots\!51}a^{4}-\frac{18\!\cdots\!67}{14\!\cdots\!51}a^{3}-\frac{79\!\cdots\!68}{14\!\cdots\!51}a^{2}-\frac{39\!\cdots\!18}{20\!\cdots\!79}a-\frac{15\!\cdots\!20}{48\!\cdots\!17}$, $\frac{49\!\cdots\!79}{27\!\cdots\!69}a^{20}-\frac{48\!\cdots\!34}{82\!\cdots\!07}a^{19}-\frac{23\!\cdots\!40}{82\!\cdots\!07}a^{18}+\frac{50\!\cdots\!57}{82\!\cdots\!07}a^{17}+\frac{29\!\cdots\!26}{82\!\cdots\!07}a^{16}-\frac{24\!\cdots\!83}{82\!\cdots\!07}a^{15}-\frac{22\!\cdots\!84}{91\!\cdots\!23}a^{14}+\frac{63\!\cdots\!79}{82\!\cdots\!07}a^{13}+\frac{55\!\cdots\!72}{82\!\cdots\!07}a^{12}+\frac{21\!\cdots\!74}{82\!\cdots\!07}a^{11}-\frac{58\!\cdots\!00}{82\!\cdots\!07}a^{10}-\frac{26\!\cdots\!35}{27\!\cdots\!69}a^{9}-\frac{26\!\cdots\!21}{35\!\cdots\!09}a^{8}-\frac{12\!\cdots\!15}{82\!\cdots\!07}a^{7}+\frac{13\!\cdots\!47}{91\!\cdots\!23}a^{6}+\frac{15\!\cdots\!27}{82\!\cdots\!07}a^{5}+\frac{21\!\cdots\!19}{27\!\cdots\!69}a^{4}+\frac{52\!\cdots\!22}{91\!\cdots\!23}a^{3}-\frac{22\!\cdots\!71}{27\!\cdots\!69}a^{2}-\frac{11\!\cdots\!25}{39\!\cdots\!01}a-\frac{45\!\cdots\!92}{91\!\cdots\!23}$, $\frac{48\!\cdots\!71}{35\!\cdots\!09}a^{20}-\frac{16\!\cdots\!33}{35\!\cdots\!09}a^{19}-\frac{76\!\cdots\!59}{35\!\cdots\!09}a^{18}+\frac{55\!\cdots\!55}{11\!\cdots\!03}a^{17}+\frac{94\!\cdots\!20}{35\!\cdots\!09}a^{16}-\frac{80\!\cdots\!50}{35\!\cdots\!09}a^{15}-\frac{65\!\cdots\!45}{35\!\cdots\!09}a^{14}+\frac{71\!\cdots\!19}{11\!\cdots\!03}a^{13}+\frac{20\!\cdots\!17}{39\!\cdots\!01}a^{12}+\frac{69\!\cdots\!88}{35\!\cdots\!09}a^{11}-\frac{19\!\cdots\!33}{35\!\cdots\!09}a^{10}-\frac{26\!\cdots\!01}{35\!\cdots\!09}a^{9}-\frac{67\!\cdots\!23}{11\!\cdots\!03}a^{8}-\frac{25\!\cdots\!40}{35\!\cdots\!09}a^{7}+\frac{38\!\cdots\!27}{35\!\cdots\!09}a^{6}+\frac{49\!\cdots\!67}{35\!\cdots\!09}a^{5}+\frac{70\!\cdots\!84}{11\!\cdots\!03}a^{4}-\frac{11\!\cdots\!87}{11\!\cdots\!03}a^{3}-\frac{88\!\cdots\!59}{11\!\cdots\!03}a^{2}-\frac{10\!\cdots\!47}{39\!\cdots\!01}a-\frac{16\!\cdots\!33}{39\!\cdots\!01}$, $\frac{44\!\cdots\!37}{27\!\cdots\!69}a^{20}-\frac{49\!\cdots\!00}{91\!\cdots\!23}a^{19}-\frac{69\!\cdots\!44}{27\!\cdots\!69}a^{18}+\frac{50\!\cdots\!08}{91\!\cdots\!23}a^{17}+\frac{28\!\cdots\!76}{91\!\cdots\!23}a^{16}-\frac{25\!\cdots\!90}{91\!\cdots\!23}a^{15}-\frac{59\!\cdots\!46}{27\!\cdots\!69}a^{14}+\frac{70\!\cdots\!56}{91\!\cdots\!23}a^{13}+\frac{16\!\cdots\!92}{27\!\cdots\!69}a^{12}+\frac{19\!\cdots\!92}{91\!\cdots\!23}a^{11}-\frac{17\!\cdots\!75}{27\!\cdots\!69}a^{10}-\frac{23\!\cdots\!74}{27\!\cdots\!69}a^{9}-\frac{77\!\cdots\!97}{11\!\cdots\!03}a^{8}+\frac{48\!\cdots\!40}{27\!\cdots\!69}a^{7}+\frac{11\!\cdots\!36}{91\!\cdots\!23}a^{6}+\frac{44\!\cdots\!50}{27\!\cdots\!69}a^{5}+\frac{18\!\cdots\!76}{27\!\cdots\!69}a^{4}-\frac{87\!\cdots\!75}{27\!\cdots\!69}a^{3}-\frac{79\!\cdots\!66}{91\!\cdots\!23}a^{2}-\frac{11\!\cdots\!45}{39\!\cdots\!01}a-\frac{37\!\cdots\!89}{91\!\cdots\!23}$, $\frac{40\!\cdots\!19}{24\!\cdots\!21}a^{20}-\frac{12\!\cdots\!41}{24\!\cdots\!21}a^{19}-\frac{69\!\cdots\!28}{24\!\cdots\!21}a^{18}+\frac{40\!\cdots\!89}{82\!\cdots\!07}a^{17}+\frac{85\!\cdots\!05}{24\!\cdots\!21}a^{16}-\frac{45\!\cdots\!42}{24\!\cdots\!21}a^{15}-\frac{58\!\cdots\!25}{24\!\cdots\!21}a^{14}+\frac{80\!\cdots\!29}{82\!\cdots\!07}a^{13}+\frac{18\!\cdots\!41}{27\!\cdots\!69}a^{12}+\frac{99\!\cdots\!99}{24\!\cdots\!21}a^{11}-\frac{16\!\cdots\!84}{24\!\cdots\!21}a^{10}-\frac{27\!\cdots\!83}{24\!\cdots\!21}a^{9}-\frac{10\!\cdots\!72}{11\!\cdots\!03}a^{8}-\frac{26\!\cdots\!62}{24\!\cdots\!21}a^{7}+\frac{34\!\cdots\!06}{24\!\cdots\!21}a^{6}+\frac{51\!\cdots\!89}{24\!\cdots\!21}a^{5}+\frac{84\!\cdots\!91}{82\!\cdots\!07}a^{4}+\frac{74\!\cdots\!77}{82\!\cdots\!07}a^{3}-\frac{11\!\cdots\!63}{82\!\cdots\!07}a^{2}-\frac{19\!\cdots\!72}{39\!\cdots\!01}a-\frac{22\!\cdots\!87}{27\!\cdots\!69}$, $\frac{82\!\cdots\!52}{82\!\cdots\!07}a^{20}-\frac{27\!\cdots\!34}{82\!\cdots\!07}a^{19}-\frac{12\!\cdots\!13}{82\!\cdots\!07}a^{18}+\frac{94\!\cdots\!08}{27\!\cdots\!69}a^{17}+\frac{16\!\cdots\!96}{82\!\cdots\!07}a^{16}-\frac{14\!\cdots\!45}{82\!\cdots\!07}a^{15}-\frac{11\!\cdots\!13}{82\!\cdots\!07}a^{14}+\frac{13\!\cdots\!91}{27\!\cdots\!69}a^{13}+\frac{10\!\cdots\!45}{27\!\cdots\!69}a^{12}+\frac{10\!\cdots\!94}{82\!\cdots\!07}a^{11}-\frac{32\!\cdots\!77}{82\!\cdots\!07}a^{10}-\frac{43\!\cdots\!79}{82\!\cdots\!07}a^{9}-\frac{47\!\cdots\!52}{11\!\cdots\!03}a^{8}+\frac{11\!\cdots\!10}{82\!\cdots\!07}a^{7}+\frac{65\!\cdots\!44}{82\!\cdots\!07}a^{6}+\frac{81\!\cdots\!90}{82\!\cdots\!07}a^{5}+\frac{36\!\cdots\!24}{91\!\cdots\!23}a^{4}-\frac{60\!\cdots\!22}{27\!\cdots\!69}a^{3}-\frac{14\!\cdots\!14}{27\!\cdots\!69}a^{2}-\frac{70\!\cdots\!71}{39\!\cdots\!01}a-\frac{21\!\cdots\!82}{91\!\cdots\!23}$, $\frac{67\!\cdots\!48}{24\!\cdots\!21}a^{20}-\frac{22\!\cdots\!87}{24\!\cdots\!21}a^{19}-\frac{10\!\cdots\!63}{24\!\cdots\!21}a^{18}+\frac{77\!\cdots\!51}{82\!\cdots\!07}a^{17}+\frac{13\!\cdots\!32}{24\!\cdots\!21}a^{16}-\frac{11\!\cdots\!92}{24\!\cdots\!21}a^{15}-\frac{90\!\cdots\!92}{24\!\cdots\!21}a^{14}+\frac{10\!\cdots\!96}{82\!\cdots\!07}a^{13}+\frac{94\!\cdots\!26}{91\!\cdots\!23}a^{12}+\frac{90\!\cdots\!36}{24\!\cdots\!21}a^{11}-\frac{26\!\cdots\!92}{24\!\cdots\!21}a^{10}-\frac{36\!\cdots\!37}{24\!\cdots\!21}a^{9}-\frac{13\!\cdots\!76}{11\!\cdots\!03}a^{8}+\frac{60\!\cdots\!78}{24\!\cdots\!21}a^{7}+\frac{53\!\cdots\!08}{24\!\cdots\!21}a^{6}+\frac{67\!\cdots\!01}{24\!\cdots\!21}a^{5}+\frac{92\!\cdots\!68}{82\!\cdots\!07}a^{4}-\frac{36\!\cdots\!92}{82\!\cdots\!07}a^{3}-\frac{11\!\cdots\!41}{82\!\cdots\!07}a^{2}-\frac{16\!\cdots\!00}{39\!\cdots\!01}a-\frac{14\!\cdots\!12}{27\!\cdots\!69}$, $\frac{29\!\cdots\!59}{82\!\cdots\!07}a^{20}-\frac{99\!\cdots\!09}{82\!\cdots\!07}a^{19}-\frac{47\!\cdots\!76}{82\!\cdots\!07}a^{18}+\frac{10\!\cdots\!32}{82\!\cdots\!07}a^{17}+\frac{58\!\cdots\!96}{82\!\cdots\!07}a^{16}-\frac{16\!\cdots\!45}{27\!\cdots\!69}a^{15}-\frac{40\!\cdots\!59}{82\!\cdots\!07}a^{14}+\frac{12\!\cdots\!07}{82\!\cdots\!07}a^{13}+\frac{11\!\cdots\!56}{82\!\cdots\!07}a^{12}+\frac{45\!\cdots\!25}{82\!\cdots\!07}a^{11}-\frac{39\!\cdots\!24}{27\!\cdots\!69}a^{10}-\frac{16\!\cdots\!79}{82\!\cdots\!07}a^{9}-\frac{55\!\cdots\!37}{35\!\cdots\!09}a^{8}-\frac{33\!\cdots\!27}{27\!\cdots\!69}a^{7}+\frac{24\!\cdots\!78}{82\!\cdots\!07}a^{6}+\frac{34\!\cdots\!65}{91\!\cdots\!23}a^{5}+\frac{44\!\cdots\!98}{27\!\cdots\!69}a^{4}-\frac{29\!\cdots\!98}{91\!\cdots\!23}a^{3}-\frac{19\!\cdots\!01}{91\!\cdots\!23}a^{2}-\frac{28\!\cdots\!70}{39\!\cdots\!01}a-\frac{99\!\cdots\!60}{91\!\cdots\!23}$, $\frac{40\!\cdots\!34}{27\!\cdots\!69}a^{20}-\frac{41\!\cdots\!78}{82\!\cdots\!07}a^{19}-\frac{19\!\cdots\!91}{82\!\cdots\!07}a^{18}+\frac{42\!\cdots\!61}{82\!\cdots\!07}a^{17}+\frac{24\!\cdots\!23}{82\!\cdots\!07}a^{16}-\frac{21\!\cdots\!59}{82\!\cdots\!07}a^{15}-\frac{18\!\cdots\!57}{91\!\cdots\!23}a^{14}+\frac{58\!\cdots\!22}{82\!\cdots\!07}a^{13}+\frac{46\!\cdots\!02}{82\!\cdots\!07}a^{12}+\frac{16\!\cdots\!34}{82\!\cdots\!07}a^{11}-\frac{49\!\cdots\!08}{82\!\cdots\!07}a^{10}-\frac{22\!\cdots\!41}{27\!\cdots\!69}a^{9}-\frac{21\!\cdots\!55}{35\!\cdots\!09}a^{8}+\frac{18\!\cdots\!69}{82\!\cdots\!07}a^{7}+\frac{32\!\cdots\!44}{27\!\cdots\!69}a^{6}+\frac{12\!\cdots\!37}{82\!\cdots\!07}a^{5}+\frac{16\!\cdots\!56}{27\!\cdots\!69}a^{4}-\frac{11\!\cdots\!51}{27\!\cdots\!69}a^{3}-\frac{22\!\cdots\!53}{27\!\cdots\!69}a^{2}-\frac{98\!\cdots\!04}{39\!\cdots\!01}a-\frac{31\!\cdots\!08}{91\!\cdots\!23}$
| 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: | \( 5132288610.41 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{7}\cdot 5132288610.41 \cdot 1}{2\cdot\sqrt{6400890533463290121080023755169887}}\cr\approx \mathstrut & 1.58719367111 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 17*x^19 + 29*x^18 + 208*x^17 - 102*x^16 - 1409*x^15 - x^14 + 3951*x^13 + 2695*x^12 - 3516*x^11 - 6803*x^10 - 5950*x^9 - 1331*x^8 + 8064*x^7 + 12913*x^6 + 7673*x^5 + 1263*x^4 - 612*x^3 - 375*x^2 - 90*x - 9)
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 - 3*x^20 - 17*x^19 + 29*x^18 + 208*x^17 - 102*x^16 - 1409*x^15 - x^14 + 3951*x^13 + 2695*x^12 - 3516*x^11 - 6803*x^10 - 5950*x^9 - 1331*x^8 + 8064*x^7 + 12913*x^6 + 7673*x^5 + 1263*x^4 - 612*x^3 - 375*x^2 - 90*x - 9, 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 - 3*x^20 - 17*x^19 + 29*x^18 + 208*x^17 - 102*x^16 - 1409*x^15 - x^14 + 3951*x^13 + 2695*x^12 - 3516*x^11 - 6803*x^10 - 5950*x^9 - 1331*x^8 + 8064*x^7 + 12913*x^6 + 7673*x^5 + 1263*x^4 - 612*x^3 - 375*x^2 - 90*x - 9);
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 - 3*x^20 - 17*x^19 + 29*x^18 + 208*x^17 - 102*x^16 - 1409*x^15 - x^14 + 3951*x^13 + 2695*x^12 - 3516*x^11 - 6803*x^10 - 5950*x^9 - 1331*x^8 + 8064*x^7 + 12913*x^6 + 7673*x^5 + 1263*x^4 - 612*x^3 - 375*x^2 - 90*x - 9);
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
| 3.1.87.1, 7.7.2134162809.1 |
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 | $21$ | R | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }$ | $21$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | $21$ | ${\href{/padicField/17.3.0.1}{3} }^{7}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.3.0.1}{3} }^{7}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.3.5.3 | $x^{3} + 9 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.6.11.2 | $x^{6} + 9 x^{2} + 24$ | $6$ | $1$ | $11$ | $D_{6}$ | $[5/2]_{2}^{2}$ | |
|
\(29\)
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.2 | $x^{4} - 696 x^{2} + 1682$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 29.8.4.1 | $x^{8} + 2784 x^{7} + 2906616 x^{6} + 1348864734 x^{5} + 234834277018 x^{4} + 41857830864 x^{3} + 492109772617 x^{2} + 3561769809750 x + 616658760166$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(59\)
| 59.3.2.1 | $x^{3} + 59$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 59.4.0.1 | $x^{4} + 2 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 59.4.0.1 | $x^{4} + 2 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 59.4.0.1 | $x^{4} + 2 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 59.6.4.1 | $x^{6} + 174 x^{5} + 10098 x^{4} + 195926 x^{3} + 30462 x^{2} + 595416 x + 11494565$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |