Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 9*x^19 + 10*x^18 + 28*x^17 - 211*x^16 + 324*x^15 + 384*x^14 - 1692*x^13 + 1991*x^12 + 2315*x^11 - 1214*x^10 - 9164*x^9 - 5328*x^8 + 34796*x^7 - 21283*x^6 - 16215*x^5 + 19798*x^4 - 2184*x^3 - 3057*x^2 + 711*x - 41)
gp: K = bnfinit(y^21 - y^20 - 9*y^19 + 10*y^18 + 28*y^17 - 211*y^16 + 324*y^15 + 384*y^14 - 1692*y^13 + 1991*y^12 + 2315*y^11 - 1214*y^10 - 9164*y^9 - 5328*y^8 + 34796*y^7 - 21283*y^6 - 16215*y^5 + 19798*y^4 - 2184*y^3 - 3057*y^2 + 711*y - 41, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - x^20 - 9*x^19 + 10*x^18 + 28*x^17 - 211*x^16 + 324*x^15 + 384*x^14 - 1692*x^13 + 1991*x^12 + 2315*x^11 - 1214*x^10 - 9164*x^9 - 5328*x^8 + 34796*x^7 - 21283*x^6 - 16215*x^5 + 19798*x^4 - 2184*x^3 - 3057*x^2 + 711*x - 41);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - x^20 - 9*x^19 + 10*x^18 + 28*x^17 - 211*x^16 + 324*x^15 + 384*x^14 - 1692*x^13 + 1991*x^12 + 2315*x^11 - 1214*x^10 - 9164*x^9 - 5328*x^8 + 34796*x^7 - 21283*x^6 - 16215*x^5 + 19798*x^4 - 2184*x^3 - 3057*x^2 + 711*x - 41)
\( x^{21} - x^{20} - 9 x^{19} + 10 x^{18} + 28 x^{17} - 211 x^{16} + 324 x^{15} + 384 x^{14} - 1692 x^{13} + \cdots - 41 \)
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: |
\(-13347832346292311387708944226103\)
\(\medspace = -\,3^{7}\cdot 29^{19}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.35\) | 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: | $3^{1/2}29^{13/14}\approx 39.49131773649383$ | ||
| Ramified primes: |
\(3\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-87}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{41}a^{15}-\frac{14}{41}a^{14}+\frac{20}{41}a^{13}-\frac{2}{41}a^{12}-\frac{2}{41}a^{11}-\frac{18}{41}a^{10}+\frac{13}{41}a^{9}-\frac{11}{41}a^{8}+\frac{15}{41}a^{7}-\frac{13}{41}a^{6}+\frac{15}{41}a^{5}+\frac{12}{41}a^{4}+\frac{2}{41}a^{3}-\frac{18}{41}a$, $\frac{1}{41}a^{16}-\frac{12}{41}a^{14}-\frac{9}{41}a^{13}+\frac{11}{41}a^{12}-\frac{5}{41}a^{11}+\frac{7}{41}a^{10}+\frac{7}{41}a^{9}-\frac{16}{41}a^{8}-\frac{8}{41}a^{7}-\frac{3}{41}a^{6}+\frac{17}{41}a^{5}+\frac{6}{41}a^{4}-\frac{13}{41}a^{3}-\frac{18}{41}a^{2}-\frac{6}{41}a$, $\frac{1}{41}a^{17}-\frac{13}{41}a^{14}+\frac{5}{41}a^{13}+\frac{12}{41}a^{12}-\frac{17}{41}a^{11}-\frac{4}{41}a^{10}+\frac{17}{41}a^{9}-\frac{17}{41}a^{8}+\frac{13}{41}a^{7}-\frac{16}{41}a^{6}-\frac{19}{41}a^{5}+\frac{8}{41}a^{4}+\frac{6}{41}a^{3}-\frac{6}{41}a^{2}-\frac{11}{41}a$, $\frac{1}{41}a^{18}-\frac{13}{41}a^{14}-\frac{15}{41}a^{13}-\frac{2}{41}a^{12}+\frac{11}{41}a^{11}-\frac{12}{41}a^{10}-\frac{12}{41}a^{9}-\frac{7}{41}a^{8}+\frac{15}{41}a^{7}+\frac{17}{41}a^{6}-\frac{2}{41}a^{5}-\frac{2}{41}a^{4}+\frac{20}{41}a^{3}-\frac{11}{41}a^{2}+\frac{12}{41}a$, $\frac{1}{41}a^{19}+\frac{8}{41}a^{14}+\frac{12}{41}a^{13}-\frac{15}{41}a^{12}+\frac{3}{41}a^{11}-\frac{2}{41}a^{9}-\frac{5}{41}a^{8}+\frac{7}{41}a^{7}-\frac{7}{41}a^{6}-\frac{12}{41}a^{5}+\frac{12}{41}a^{4}+\frac{15}{41}a^{3}+\frac{12}{41}a^{2}+\frac{12}{41}a$, $\frac{1}{36\!\cdots\!99}a^{20}-\frac{17\!\cdots\!32}{36\!\cdots\!99}a^{19}+\frac{42\!\cdots\!52}{36\!\cdots\!99}a^{18}+\frac{28\!\cdots\!26}{36\!\cdots\!99}a^{17}-\frac{12\!\cdots\!04}{36\!\cdots\!99}a^{16}+\frac{22\!\cdots\!31}{36\!\cdots\!99}a^{15}+\frac{52\!\cdots\!60}{36\!\cdots\!99}a^{14}-\frac{13\!\cdots\!28}{36\!\cdots\!99}a^{13}-\frac{16\!\cdots\!14}{36\!\cdots\!99}a^{12}+\frac{25\!\cdots\!55}{36\!\cdots\!99}a^{11}+\frac{49\!\cdots\!80}{36\!\cdots\!99}a^{10}-\frac{25\!\cdots\!38}{36\!\cdots\!99}a^{9}+\frac{15\!\cdots\!81}{36\!\cdots\!99}a^{8}+\frac{13\!\cdots\!37}{36\!\cdots\!99}a^{7}+\frac{78\!\cdots\!81}{36\!\cdots\!99}a^{6}-\frac{64\!\cdots\!37}{36\!\cdots\!99}a^{5}+\frac{10\!\cdots\!55}{36\!\cdots\!99}a^{4}+\frac{18\!\cdots\!81}{36\!\cdots\!99}a^{3}+\frac{13\!\cdots\!23}{36\!\cdots\!99}a^{2}+\frac{25\!\cdots\!82}{36\!\cdots\!99}a-\frac{21\!\cdots\!24}{89\!\cdots\!39}$
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: | $13$ | 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: |
$\frac{29\!\cdots\!19}{36\!\cdots\!99}a^{20}-\frac{21\!\cdots\!36}{36\!\cdots\!99}a^{19}-\frac{26\!\cdots\!18}{36\!\cdots\!99}a^{18}+\frac{21\!\cdots\!07}{36\!\cdots\!99}a^{17}+\frac{85\!\cdots\!34}{36\!\cdots\!99}a^{16}-\frac{59\!\cdots\!42}{36\!\cdots\!99}a^{15}+\frac{79\!\cdots\!10}{36\!\cdots\!99}a^{14}+\frac{12\!\cdots\!19}{36\!\cdots\!99}a^{13}-\frac{45\!\cdots\!82}{36\!\cdots\!99}a^{12}+\frac{47\!\cdots\!95}{36\!\cdots\!99}a^{11}+\frac{77\!\cdots\!23}{36\!\cdots\!99}a^{10}-\frac{12\!\cdots\!97}{36\!\cdots\!99}a^{9}-\frac{26\!\cdots\!23}{36\!\cdots\!99}a^{8}-\frac{22\!\cdots\!51}{36\!\cdots\!99}a^{7}+\frac{93\!\cdots\!85}{36\!\cdots\!99}a^{6}-\frac{39\!\cdots\!97}{36\!\cdots\!99}a^{5}-\frac{52\!\cdots\!59}{36\!\cdots\!99}a^{4}+\frac{43\!\cdots\!82}{36\!\cdots\!99}a^{3}+\frac{14\!\cdots\!84}{36\!\cdots\!99}a^{2}-\frac{74\!\cdots\!61}{36\!\cdots\!99}a+\frac{19\!\cdots\!97}{89\!\cdots\!39}$, $\frac{50\!\cdots\!87}{36\!\cdots\!99}a^{20}-\frac{44\!\cdots\!36}{36\!\cdots\!99}a^{19}-\frac{45\!\cdots\!53}{36\!\cdots\!99}a^{18}+\frac{45\!\cdots\!16}{36\!\cdots\!99}a^{17}+\frac{14\!\cdots\!80}{36\!\cdots\!99}a^{16}-\frac{10\!\cdots\!14}{36\!\cdots\!99}a^{15}+\frac{15\!\cdots\!27}{36\!\cdots\!99}a^{14}+\frac{21\!\cdots\!83}{36\!\cdots\!99}a^{13}-\frac{82\!\cdots\!29}{36\!\cdots\!99}a^{12}+\frac{90\!\cdots\!83}{36\!\cdots\!99}a^{11}+\frac{12\!\cdots\!66}{36\!\cdots\!99}a^{10}-\frac{46\!\cdots\!00}{36\!\cdots\!99}a^{9}-\frac{46\!\cdots\!08}{36\!\cdots\!99}a^{8}-\frac{32\!\cdots\!27}{36\!\cdots\!99}a^{7}+\frac{17\!\cdots\!67}{36\!\cdots\!99}a^{6}-\frac{87\!\cdots\!68}{36\!\cdots\!99}a^{5}-\frac{91\!\cdots\!57}{36\!\cdots\!99}a^{4}+\frac{89\!\cdots\!88}{36\!\cdots\!99}a^{3}-\frac{76\!\cdots\!88}{36\!\cdots\!99}a^{2}-\frac{15\!\cdots\!00}{36\!\cdots\!99}a+\frac{44\!\cdots\!67}{89\!\cdots\!39}$, $\frac{19\!\cdots\!30}{36\!\cdots\!99}a^{20}-\frac{15\!\cdots\!84}{36\!\cdots\!99}a^{19}-\frac{17\!\cdots\!39}{36\!\cdots\!99}a^{18}+\frac{15\!\cdots\!16}{36\!\cdots\!99}a^{17}+\frac{57\!\cdots\!63}{36\!\cdots\!99}a^{16}-\frac{40\!\cdots\!93}{36\!\cdots\!99}a^{15}+\frac{55\!\cdots\!00}{36\!\cdots\!99}a^{14}+\frac{84\!\cdots\!73}{36\!\cdots\!99}a^{13}-\frac{31\!\cdots\!97}{36\!\cdots\!99}a^{12}+\frac{33\!\cdots\!80}{36\!\cdots\!99}a^{11}+\frac{51\!\cdots\!45}{36\!\cdots\!99}a^{10}-\frac{12\!\cdots\!92}{36\!\cdots\!99}a^{9}-\frac{17\!\cdots\!84}{36\!\cdots\!99}a^{8}-\frac{13\!\cdots\!24}{36\!\cdots\!99}a^{7}+\frac{64\!\cdots\!98}{36\!\cdots\!99}a^{6}-\frac{29\!\cdots\!89}{36\!\cdots\!99}a^{5}-\frac{35\!\cdots\!04}{36\!\cdots\!99}a^{4}+\frac{32\!\cdots\!59}{36\!\cdots\!99}a^{3}+\frac{66\!\cdots\!46}{36\!\cdots\!99}a^{2}-\frac{58\!\cdots\!47}{36\!\cdots\!99}a+\frac{16\!\cdots\!78}{89\!\cdots\!39}$, $\frac{92\!\cdots\!24}{36\!\cdots\!99}a^{20}-\frac{78\!\cdots\!42}{36\!\cdots\!99}a^{19}-\frac{84\!\cdots\!29}{36\!\cdots\!99}a^{18}+\frac{79\!\cdots\!78}{36\!\cdots\!99}a^{17}+\frac{27\!\cdots\!82}{36\!\cdots\!99}a^{16}-\frac{19\!\cdots\!51}{36\!\cdots\!99}a^{15}+\frac{27\!\cdots\!26}{36\!\cdots\!99}a^{14}+\frac{39\!\cdots\!13}{36\!\cdots\!99}a^{13}-\frac{15\!\cdots\!37}{36\!\cdots\!99}a^{12}+\frac{16\!\cdots\!99}{36\!\cdots\!99}a^{11}+\frac{23\!\cdots\!59}{36\!\cdots\!99}a^{10}-\frac{75\!\cdots\!31}{36\!\cdots\!99}a^{9}-\frac{86\!\cdots\!71}{36\!\cdots\!99}a^{8}-\frac{63\!\cdots\!61}{36\!\cdots\!99}a^{7}+\frac{31\!\cdots\!68}{36\!\cdots\!99}a^{6}-\frac{14\!\cdots\!98}{36\!\cdots\!99}a^{5}-\frac{17\!\cdots\!59}{36\!\cdots\!99}a^{4}+\frac{15\!\cdots\!01}{36\!\cdots\!99}a^{3}+\frac{34\!\cdots\!90}{36\!\cdots\!99}a^{2}-\frac{26\!\cdots\!31}{36\!\cdots\!99}a+\frac{50\!\cdots\!20}{89\!\cdots\!39}$, $\frac{11\!\cdots\!84}{36\!\cdots\!99}a^{20}-\frac{19\!\cdots\!53}{36\!\cdots\!99}a^{19}-\frac{10\!\cdots\!77}{36\!\cdots\!99}a^{18}+\frac{18\!\cdots\!48}{36\!\cdots\!99}a^{17}+\frac{31\!\cdots\!03}{36\!\cdots\!99}a^{16}-\frac{26\!\cdots\!88}{36\!\cdots\!99}a^{15}+\frac{50\!\cdots\!59}{36\!\cdots\!99}a^{14}+\frac{33\!\cdots\!32}{36\!\cdots\!99}a^{13}-\frac{23\!\cdots\!31}{36\!\cdots\!99}a^{12}+\frac{31\!\cdots\!47}{36\!\cdots\!99}a^{11}+\frac{20\!\cdots\!57}{36\!\cdots\!99}a^{10}-\frac{41\!\cdots\!48}{36\!\cdots\!99}a^{9}-\frac{12\!\cdots\!66}{36\!\cdots\!99}a^{8}+\frac{26\!\cdots\!75}{36\!\cdots\!99}a^{7}+\frac{50\!\cdots\!96}{36\!\cdots\!99}a^{6}-\frac{41\!\cdots\!77}{36\!\cdots\!99}a^{5}-\frac{22\!\cdots\!32}{36\!\cdots\!99}a^{4}+\frac{35\!\cdots\!00}{36\!\cdots\!99}a^{3}-\frac{35\!\cdots\!22}{36\!\cdots\!99}a^{2}-\frac{64\!\cdots\!08}{36\!\cdots\!99}a+\frac{18\!\cdots\!71}{89\!\cdots\!39}$, $\frac{11\!\cdots\!66}{36\!\cdots\!99}a^{20}-\frac{10\!\cdots\!48}{36\!\cdots\!99}a^{19}-\frac{10\!\cdots\!16}{36\!\cdots\!99}a^{18}+\frac{10\!\cdots\!27}{36\!\cdots\!99}a^{17}+\frac{33\!\cdots\!51}{36\!\cdots\!99}a^{16}-\frac{24\!\cdots\!17}{36\!\cdots\!99}a^{15}+\frac{35\!\cdots\!23}{36\!\cdots\!99}a^{14}+\frac{48\!\cdots\!38}{36\!\cdots\!99}a^{13}-\frac{19\!\cdots\!37}{36\!\cdots\!99}a^{12}+\frac{21\!\cdots\!28}{36\!\cdots\!99}a^{11}+\frac{29\!\cdots\!84}{36\!\cdots\!99}a^{10}-\frac{10\!\cdots\!88}{36\!\cdots\!99}a^{9}-\frac{10\!\cdots\!33}{36\!\cdots\!99}a^{8}-\frac{73\!\cdots\!34}{36\!\cdots\!99}a^{7}+\frac{39\!\cdots\!63}{36\!\cdots\!99}a^{6}-\frac{21\!\cdots\!84}{36\!\cdots\!99}a^{5}-\frac{20\!\cdots\!32}{36\!\cdots\!99}a^{4}+\frac{21\!\cdots\!46}{36\!\cdots\!99}a^{3}-\frac{59\!\cdots\!83}{36\!\cdots\!99}a^{2}-\frac{34\!\cdots\!61}{36\!\cdots\!99}a+\frac{93\!\cdots\!05}{89\!\cdots\!39}$, $\frac{11\!\cdots\!15}{36\!\cdots\!99}a^{20}-\frac{10\!\cdots\!45}{36\!\cdots\!99}a^{19}-\frac{10\!\cdots\!52}{36\!\cdots\!99}a^{18}+\frac{10\!\cdots\!94}{36\!\cdots\!99}a^{17}+\frac{33\!\cdots\!83}{36\!\cdots\!99}a^{16}-\frac{23\!\cdots\!20}{36\!\cdots\!99}a^{15}+\frac{34\!\cdots\!23}{36\!\cdots\!99}a^{14}+\frac{48\!\cdots\!06}{36\!\cdots\!99}a^{13}-\frac{18\!\cdots\!24}{36\!\cdots\!99}a^{12}+\frac{20\!\cdots\!61}{36\!\cdots\!99}a^{11}+\frac{28\!\cdots\!63}{36\!\cdots\!99}a^{10}-\frac{10\!\cdots\!14}{36\!\cdots\!99}a^{9}-\frac{10\!\cdots\!71}{36\!\cdots\!99}a^{8}-\frac{74\!\cdots\!22}{36\!\cdots\!99}a^{7}+\frac{39\!\cdots\!23}{36\!\cdots\!99}a^{6}-\frac{19\!\cdots\!22}{36\!\cdots\!99}a^{5}-\frac{20\!\cdots\!38}{36\!\cdots\!99}a^{4}+\frac{19\!\cdots\!06}{36\!\cdots\!99}a^{3}-\frac{21\!\cdots\!94}{36\!\cdots\!99}a^{2}-\frac{34\!\cdots\!65}{36\!\cdots\!99}a+\frac{93\!\cdots\!69}{89\!\cdots\!39}$, $\frac{17\!\cdots\!38}{36\!\cdots\!99}a^{20}-\frac{35\!\cdots\!38}{36\!\cdots\!99}a^{19}-\frac{16\!\cdots\!23}{36\!\cdots\!99}a^{18}+\frac{34\!\cdots\!06}{36\!\cdots\!99}a^{17}+\frac{52\!\cdots\!57}{36\!\cdots\!99}a^{16}-\frac{42\!\cdots\!41}{36\!\cdots\!99}a^{15}+\frac{88\!\cdots\!80}{36\!\cdots\!99}a^{14}+\frac{52\!\cdots\!15}{36\!\cdots\!99}a^{13}-\frac{40\!\cdots\!16}{36\!\cdots\!99}a^{12}+\frac{53\!\cdots\!81}{36\!\cdots\!99}a^{11}+\frac{32\!\cdots\!49}{36\!\cdots\!99}a^{10}-\frac{84\!\cdots\!55}{36\!\cdots\!99}a^{9}-\frac{21\!\cdots\!17}{36\!\cdots\!99}a^{8}+\frac{30\!\cdots\!68}{36\!\cdots\!99}a^{7}+\frac{88\!\cdots\!28}{36\!\cdots\!99}a^{6}-\frac{71\!\cdots\!43}{36\!\cdots\!99}a^{5}-\frac{44\!\cdots\!23}{36\!\cdots\!99}a^{4}+\frac{63\!\cdots\!45}{36\!\cdots\!99}a^{3}-\frac{41\!\cdots\!62}{36\!\cdots\!99}a^{2}-\frac{11\!\cdots\!06}{36\!\cdots\!99}a+\frac{19\!\cdots\!96}{89\!\cdots\!39}$, $\frac{58\!\cdots\!09}{36\!\cdots\!99}a^{20}-\frac{25\!\cdots\!21}{36\!\cdots\!99}a^{19}-\frac{53\!\cdots\!83}{36\!\cdots\!99}a^{18}+\frac{28\!\cdots\!05}{36\!\cdots\!99}a^{17}+\frac{17\!\cdots\!63}{36\!\cdots\!99}a^{16}-\frac{11\!\cdots\!70}{36\!\cdots\!99}a^{15}+\frac{12\!\cdots\!46}{36\!\cdots\!99}a^{14}+\frac{28\!\cdots\!99}{36\!\cdots\!99}a^{13}-\frac{81\!\cdots\!53}{36\!\cdots\!99}a^{12}+\frac{73\!\cdots\!23}{36\!\cdots\!99}a^{11}+\frac{16\!\cdots\!20}{36\!\cdots\!99}a^{10}+\frac{29\!\cdots\!22}{36\!\cdots\!99}a^{9}-\frac{49\!\cdots\!61}{36\!\cdots\!99}a^{8}-\frac{57\!\cdots\!05}{36\!\cdots\!99}a^{7}+\frac{16\!\cdots\!70}{36\!\cdots\!99}a^{6}-\frac{40\!\cdots\!18}{36\!\cdots\!99}a^{5}-\frac{10\!\cdots\!61}{36\!\cdots\!99}a^{4}+\frac{60\!\cdots\!98}{36\!\cdots\!99}a^{3}+\frac{10\!\cdots\!89}{36\!\cdots\!99}a^{2}-\frac{10\!\cdots\!54}{36\!\cdots\!99}a+\frac{15\!\cdots\!95}{89\!\cdots\!39}$, $\frac{11\!\cdots\!93}{36\!\cdots\!99}a^{20}-\frac{10\!\cdots\!84}{36\!\cdots\!99}a^{19}-\frac{10\!\cdots\!56}{36\!\cdots\!99}a^{18}+\frac{10\!\cdots\!72}{36\!\cdots\!99}a^{17}+\frac{32\!\cdots\!72}{36\!\cdots\!99}a^{16}-\frac{23\!\cdots\!38}{36\!\cdots\!99}a^{15}+\frac{33\!\cdots\!30}{36\!\cdots\!99}a^{14}+\frac{46\!\cdots\!23}{36\!\cdots\!99}a^{13}-\frac{18\!\cdots\!89}{36\!\cdots\!99}a^{12}+\frac{20\!\cdots\!23}{36\!\cdots\!99}a^{11}+\frac{28\!\cdots\!40}{36\!\cdots\!99}a^{10}-\frac{11\!\cdots\!14}{36\!\cdots\!99}a^{9}-\frac{10\!\cdots\!75}{36\!\cdots\!99}a^{8}-\frac{69\!\cdots\!88}{36\!\cdots\!99}a^{7}+\frac{38\!\cdots\!75}{36\!\cdots\!99}a^{6}-\frac{19\!\cdots\!23}{36\!\cdots\!99}a^{5}-\frac{20\!\cdots\!94}{36\!\cdots\!99}a^{4}+\frac{20\!\cdots\!75}{36\!\cdots\!99}a^{3}-\frac{30\!\cdots\!97}{36\!\cdots\!99}a^{2}-\frac{36\!\cdots\!02}{36\!\cdots\!99}a+\frac{94\!\cdots\!54}{89\!\cdots\!39}$, $\frac{21\!\cdots\!96}{36\!\cdots\!99}a^{20}-\frac{12\!\cdots\!86}{36\!\cdots\!99}a^{19}-\frac{19\!\cdots\!88}{36\!\cdots\!99}a^{18}+\frac{13\!\cdots\!22}{36\!\cdots\!99}a^{17}+\frac{63\!\cdots\!21}{36\!\cdots\!99}a^{16}-\frac{43\!\cdots\!30}{36\!\cdots\!99}a^{15}+\frac{53\!\cdots\!63}{36\!\cdots\!99}a^{14}+\frac{98\!\cdots\!89}{36\!\cdots\!99}a^{13}-\frac{32\!\cdots\!05}{36\!\cdots\!99}a^{12}+\frac{31\!\cdots\!24}{36\!\cdots\!99}a^{11}+\frac{58\!\cdots\!17}{36\!\cdots\!99}a^{10}+\frac{10\!\cdots\!82}{36\!\cdots\!99}a^{9}-\frac{18\!\cdots\!45}{36\!\cdots\!99}a^{8}-\frac{18\!\cdots\!29}{36\!\cdots\!99}a^{7}+\frac{65\!\cdots\!11}{36\!\cdots\!99}a^{6}-\frac{23\!\cdots\!02}{36\!\cdots\!99}a^{5}-\frac{36\!\cdots\!39}{36\!\cdots\!99}a^{4}+\frac{27\!\cdots\!50}{36\!\cdots\!99}a^{3}+\frac{13\!\cdots\!31}{36\!\cdots\!99}a^{2}-\frac{40\!\cdots\!75}{36\!\cdots\!99}a+\frac{10\!\cdots\!64}{89\!\cdots\!39}$, $\frac{39\!\cdots\!60}{36\!\cdots\!99}a^{20}-\frac{44\!\cdots\!85}{36\!\cdots\!99}a^{19}-\frac{36\!\cdots\!84}{36\!\cdots\!99}a^{18}+\frac{44\!\cdots\!52}{36\!\cdots\!99}a^{17}+\frac{11\!\cdots\!11}{36\!\cdots\!99}a^{16}-\frac{85\!\cdots\!80}{36\!\cdots\!99}a^{15}+\frac{13\!\cdots\!83}{36\!\cdots\!99}a^{14}+\frac{15\!\cdots\!66}{36\!\cdots\!99}a^{13}-\frac{70\!\cdots\!24}{36\!\cdots\!99}a^{12}+\frac{82\!\cdots\!70}{36\!\cdots\!99}a^{11}+\frac{94\!\cdots\!10}{36\!\cdots\!99}a^{10}-\frac{68\!\cdots\!36}{36\!\cdots\!99}a^{9}-\frac{39\!\cdots\!76}{36\!\cdots\!99}a^{8}-\frac{18\!\cdots\!07}{36\!\cdots\!99}a^{7}+\frac{14\!\cdots\!30}{36\!\cdots\!99}a^{6}-\frac{89\!\cdots\!79}{36\!\cdots\!99}a^{5}-\frac{76\!\cdots\!59}{36\!\cdots\!99}a^{4}+\frac{86\!\cdots\!53}{36\!\cdots\!99}a^{3}-\frac{36\!\cdots\!06}{36\!\cdots\!99}a^{2}-\frac{15\!\cdots\!06}{36\!\cdots\!99}a+\frac{47\!\cdots\!59}{89\!\cdots\!39}$, $\frac{47\!\cdots\!71}{56\!\cdots\!27}a^{20}+\frac{96\!\cdots\!09}{56\!\cdots\!27}a^{19}-\frac{42\!\cdots\!12}{56\!\cdots\!27}a^{18}+\frac{43\!\cdots\!86}{56\!\cdots\!27}a^{17}+\frac{13\!\cdots\!70}{56\!\cdots\!27}a^{16}-\frac{86\!\cdots\!10}{56\!\cdots\!27}a^{15}+\frac{67\!\cdots\!11}{56\!\cdots\!27}a^{14}+\frac{25\!\cdots\!58}{56\!\cdots\!27}a^{13}-\frac{55\!\cdots\!10}{56\!\cdots\!27}a^{12}+\frac{38\!\cdots\!20}{56\!\cdots\!27}a^{11}+\frac{14\!\cdots\!42}{56\!\cdots\!27}a^{10}+\frac{91\!\cdots\!16}{56\!\cdots\!27}a^{9}-\frac{34\!\cdots\!56}{56\!\cdots\!27}a^{8}-\frac{60\!\cdots\!68}{56\!\cdots\!27}a^{7}+\frac{10\!\cdots\!97}{56\!\cdots\!27}a^{6}+\frac{64\!\cdots\!47}{56\!\cdots\!27}a^{5}-\frac{71\!\cdots\!34}{56\!\cdots\!27}a^{4}+\frac{20\!\cdots\!34}{56\!\cdots\!27}a^{3}+\frac{11\!\cdots\!21}{56\!\cdots\!27}a^{2}-\frac{31\!\cdots\!59}{56\!\cdots\!27}a+\frac{10\!\cdots\!40}{13\!\cdots\!47}$
| 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: | \( 35353252.3989 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{7}\cdot 35353252.3989 \cdot 1}{2\cdot\sqrt{13347832346292311387708944226103}}\cr\approx \mathstrut & 0.239421448944 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 9*x^19 + 10*x^18 + 28*x^17 - 211*x^16 + 324*x^15 + 384*x^14 - 1692*x^13 + 1991*x^12 + 2315*x^11 - 1214*x^10 - 9164*x^9 - 5328*x^8 + 34796*x^7 - 21283*x^6 - 16215*x^5 + 19798*x^4 - 2184*x^3 - 3057*x^2 + 711*x - 41)
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 - 9*x^19 + 10*x^18 + 28*x^17 - 211*x^16 + 324*x^15 + 384*x^14 - 1692*x^13 + 1991*x^12 + 2315*x^11 - 1214*x^10 - 9164*x^9 - 5328*x^8 + 34796*x^7 - 21283*x^6 - 16215*x^5 + 19798*x^4 - 2184*x^3 - 3057*x^2 + 711*x - 41, 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 - 9*x^19 + 10*x^18 + 28*x^17 - 211*x^16 + 324*x^15 + 384*x^14 - 1692*x^13 + 1991*x^12 + 2315*x^11 - 1214*x^10 - 9164*x^9 - 5328*x^8 + 34796*x^7 - 21283*x^6 - 16215*x^5 + 19798*x^4 - 2184*x^3 - 3057*x^2 + 711*x - 41);
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 - 9*x^19 + 10*x^18 + 28*x^17 - 211*x^16 + 324*x^15 + 384*x^14 - 1692*x^13 + 1991*x^12 + 2315*x^11 - 1214*x^10 - 9164*x^9 - 5328*x^8 + 34796*x^7 - 21283*x^6 - 16215*x^5 + 19798*x^4 - 2184*x^3 - 3057*x^2 + 711*x - 41);
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
$S_3\times C_7$ (as 21T6):
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 42 |
| The 21 conjugacy class representatives for $S_3\times C_7$ |
| Character table for $S_3\times C_7$ |
Intermediate fields
| 3.1.87.1, 7.7.594823321.1 |
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
| Galois closure: | 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 | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }$ | $21$ | $21$ | $21$ | ${\href{/padicField/17.3.0.1}{3} }^{7}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.7.0.1}{7} }$ | R | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.1.0.1}{1} }^{21}$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }$ | $21$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.7.0.1 | $x^{7} + 2 x^{2} + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 3.14.7.1 | $x^{14} - 486 x^{4} - 2187$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ | |
|
\(29\)
| 29.7.6.2 | $x^{7} + 29$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 29.14.13.11 | $x^{14} + 348$ | $14$ | $1$ | $13$ | $C_{14}$ | $[\ ]_{14}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.87.2t1.a.a | $1$ | $ 3 \cdot 29 $ | \(\Q(\sqrt{-87}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.87.14t1.a.a | $1$ | $ 3 \cdot 29 $ | 14.0.22439994995240462987343.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| * | 1.29.7t1.a.a | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| 1.87.14t1.a.b | $1$ | $ 3 \cdot 29 $ | 14.0.22439994995240462987343.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.87.14t1.a.c | $1$ | $ 3 \cdot 29 $ | 14.0.22439994995240462987343.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| * | 1.29.7t1.a.b | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.29.7t1.a.c | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| 1.87.14t1.a.d | $1$ | $ 3 \cdot 29 $ | 14.0.22439994995240462987343.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| * | 1.29.7t1.a.d | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.29.7t1.a.e | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| 1.87.14t1.a.e | $1$ | $ 3 \cdot 29 $ | 14.0.22439994995240462987343.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.87.14t1.a.f | $1$ | $ 3 \cdot 29 $ | 14.0.22439994995240462987343.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| * | 1.29.7t1.a.f | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 2.87.3t2.a.a | $2$ | $ 3 \cdot 29 $ | 3.1.87.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.2523.21t6.a.a | $2$ | $ 3 \cdot 29^{2}$ | 21.7.13347832346292311387708944226103.1 | $S_3\times C_7$ (as 21T6) | $0$ | $0$ |
| * | 2.2523.21t6.a.b | $2$ | $ 3 \cdot 29^{2}$ | 21.7.13347832346292311387708944226103.1 | $S_3\times C_7$ (as 21T6) | $0$ | $0$ |
| * | 2.2523.21t6.a.c | $2$ | $ 3 \cdot 29^{2}$ | 21.7.13347832346292311387708944226103.1 | $S_3\times C_7$ (as 21T6) | $0$ | $0$ |
| * | 2.2523.21t6.a.d | $2$ | $ 3 \cdot 29^{2}$ | 21.7.13347832346292311387708944226103.1 | $S_3\times C_7$ (as 21T6) | $0$ | $0$ |
| * | 2.2523.21t6.a.e | $2$ | $ 3 \cdot 29^{2}$ | 21.7.13347832346292311387708944226103.1 | $S_3\times C_7$ (as 21T6) | $0$ | $0$ |
| * | 2.2523.21t6.a.f | $2$ | $ 3 \cdot 29^{2}$ | 21.7.13347832346292311387708944226103.1 | $S_3\times C_7$ (as 21T6) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.