Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 217*x^19 + 19544*x^17 - 959637*x^15 - 33706*x^14 + 28285005*x^13 + 3493224*x^12 - 517708667*x^11 - 143018778*x^10 + 5859384083*x^9 + 2942499210*x^8 - 39262594646*x^7 - 31779062112*x^6 + 138562884152*x^5 + 168986405644*x^4 - 168971552743*x^3 - 345735927544*x^2 - 146449603860*x - 6727656872)
gp: K = bnfinit(y^21 - 217*y^19 + 19544*y^17 - 959637*y^15 - 33706*y^14 + 28285005*y^13 + 3493224*y^12 - 517708667*y^11 - 143018778*y^10 + 5859384083*y^9 + 2942499210*y^8 - 39262594646*y^7 - 31779062112*y^6 + 138562884152*y^5 + 168986405644*y^4 - 168971552743*y^3 - 345735927544*y^2 - 146449603860*y - 6727656872, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 217*x^19 + 19544*x^17 - 959637*x^15 - 33706*x^14 + 28285005*x^13 + 3493224*x^12 - 517708667*x^11 - 143018778*x^10 + 5859384083*x^9 + 2942499210*x^8 - 39262594646*x^7 - 31779062112*x^6 + 138562884152*x^5 + 168986405644*x^4 - 168971552743*x^3 - 345735927544*x^2 - 146449603860*x - 6727656872);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 217*x^19 + 19544*x^17 - 959637*x^15 - 33706*x^14 + 28285005*x^13 + 3493224*x^12 - 517708667*x^11 - 143018778*x^10 + 5859384083*x^9 + 2942499210*x^8 - 39262594646*x^7 - 31779062112*x^6 + 138562884152*x^5 + 168986405644*x^4 - 168971552743*x^3 - 345735927544*x^2 - 146449603860*x - 6727656872)
\( x^{21} - 217 x^{19} + 19544 x^{17} - 959637 x^{15} - 33706 x^{14} + 28285005 x^{13} + 3493224 x^{12} + \cdots - 6727656872 \)
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: | $[21, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(35135593769377589230177713282906868218673215743721472\)
\(\medspace = 2^{18}\cdot 7^{35}\cdot 29^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(317.82\) | 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: | $2\cdot 7^{509/294}29^{6/7}\approx 1041.419982801981$ | ||
| Ramified primes: |
\(2\), \(7\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{203}a^{11}+\frac{1}{7}a^{10}+\frac{73}{203}a^{9}-\frac{3}{7}a^{8}-\frac{89}{203}a^{7}-\frac{8}{29}a^{5}+\frac{79}{203}a^{4}+\frac{2}{7}a^{3}-\frac{1}{7}a^{2}+\frac{1}{7}a-\frac{3}{7}$, $\frac{1}{609}a^{12}+\frac{1}{609}a^{11}-\frac{130}{609}a^{10}-\frac{304}{609}a^{9}+\frac{38}{203}a^{8}+\frac{8}{87}a^{7}-\frac{37}{87}a^{6}+\frac{23}{609}a^{5}+\frac{79}{609}a^{4}+\frac{2}{7}a^{3}+\frac{8}{21}a^{2}-\frac{1}{7}a+\frac{1}{3}$, $\frac{1}{1218}a^{13}+\frac{1}{1218}a^{11}-\frac{299}{1218}a^{9}-\frac{10}{21}a^{8}+\frac{39}{406}a^{7}+\frac{47}{203}a^{6}+\frac{83}{174}a^{5}+\frac{85}{609}a^{4}-\frac{1}{6}a^{3}+\frac{2}{21}a^{2}-\frac{5}{42}a-\frac{2}{21}$, $\frac{1}{35322}a^{14}+\frac{5}{11774}a^{12}-\frac{1}{609}a^{11}+\frac{1487}{5046}a^{10}+\frac{82}{203}a^{9}+\frac{3471}{11774}a^{8}-\frac{4180}{17661}a^{7}-\frac{15}{58}a^{6}-\frac{100}{203}a^{5}+\frac{15}{406}a^{4}+\frac{1}{3}a^{3}-\frac{17}{42}a^{2}-\frac{10}{21}a-\frac{4}{21}$, $\frac{1}{35322}a^{15}-\frac{1}{2523}a^{13}-\frac{1}{17661}a^{11}-\frac{8}{21}a^{10}+\frac{2736}{5887}a^{9}+\frac{830}{5887}a^{8}+\frac{26}{609}a^{7}-\frac{13}{87}a^{6}+\frac{13}{87}a^{5}-\frac{16}{609}a^{4}-\frac{2}{21}a^{3}+\frac{8}{21}a^{2}+\frac{3}{14}a+\frac{1}{7}$, $\frac{1}{35322}a^{16}-\frac{4}{5887}a^{12}-\frac{1}{609}a^{11}+\frac{5323}{17661}a^{10}-\frac{6355}{17661}a^{9}-\frac{2654}{17661}a^{8}-\frac{3884}{17661}a^{7}+\frac{20}{87}a^{6}+\frac{6}{203}a^{5}+\frac{124}{609}a^{4}-\frac{8}{21}a^{3}+\frac{1}{6}a^{2}-\frac{2}{21}a+\frac{3}{7}$, $\frac{1}{1024338}a^{17}-\frac{1}{73167}a^{15}-\frac{59}{512169}a^{13}+\frac{610}{512169}a^{11}+\frac{56314}{512169}a^{10}+\frac{2582}{17661}a^{9}+\frac{286}{17661}a^{8}-\frac{10}{29}a^{7}-\frac{181}{609}a^{6}+\frac{1}{29}a^{5}+\frac{128}{609}a^{4}-\frac{449}{1218}a^{3}-\frac{3}{7}a^{2}+\frac{5}{21}a$, $\frac{1}{1024338}a^{18}-\frac{1}{73167}a^{16}-\frac{1}{512169}a^{14}-\frac{202}{512169}a^{12}+\frac{808}{512169}a^{11}-\frac{1859}{17661}a^{10}+\frac{7739}{17661}a^{9}+\frac{185}{5887}a^{8}+\frac{6016}{17661}a^{7}-\frac{13}{87}a^{6}-\frac{194}{609}a^{5}-\frac{107}{406}a^{4}-\frac{8}{21}a^{3}-\frac{2}{7}a^{2}-\frac{10}{21}a+\frac{1}{7}$, $\frac{1}{29705802}a^{19}-\frac{1}{2121843}a^{17}-\frac{59}{14852901}a^{15}+\frac{3974}{14852901}a^{13}+\frac{2512}{4950967}a^{12}-\frac{724}{512169}a^{11}-\frac{55691}{170723}a^{10}+\frac{5147}{17661}a^{9}-\frac{848}{17661}a^{8}+\frac{47}{609}a^{7}-\frac{1679}{5887}a^{6}-\frac{10019}{35322}a^{5}-\frac{139}{609}a^{4}-\frac{176}{609}a^{3}+\frac{4}{21}a^{2}-\frac{4}{21}a-\frac{4}{21}$, $\frac{1}{17\!\cdots\!12}a^{20}+\frac{18\!\cdots\!13}{44\!\cdots\!53}a^{19}-\frac{22\!\cdots\!87}{17\!\cdots\!12}a^{18}-\frac{11\!\cdots\!09}{29\!\cdots\!02}a^{17}+\frac{82\!\cdots\!86}{63\!\cdots\!79}a^{16}-\frac{12\!\cdots\!65}{88\!\cdots\!06}a^{15}-\frac{21\!\cdots\!77}{25\!\cdots\!16}a^{14}-\frac{32\!\cdots\!68}{21\!\cdots\!93}a^{13}-\frac{16\!\cdots\!55}{25\!\cdots\!16}a^{12}-\frac{19\!\cdots\!23}{10\!\cdots\!38}a^{11}-\frac{31\!\cdots\!15}{20\!\cdots\!76}a^{10}+\frac{14\!\cdots\!25}{52\!\cdots\!33}a^{9}+\frac{24\!\cdots\!35}{70\!\cdots\!44}a^{8}+\frac{10\!\cdots\!41}{52\!\cdots\!33}a^{7}-\frac{10\!\cdots\!89}{52\!\cdots\!33}a^{6}+\frac{13\!\cdots\!39}{35\!\cdots\!22}a^{5}+\frac{80\!\cdots\!42}{18\!\cdots\!77}a^{4}-\frac{32\!\cdots\!50}{60\!\cdots\!59}a^{3}-\frac{50\!\cdots\!97}{25\!\cdots\!52}a^{2}-\frac{23\!\cdots\!20}{62\!\cdots\!13}a-\frac{39\!\cdots\!69}{62\!\cdots\!13}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $20$ | 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: |
$\frac{84\!\cdots\!17}{29\!\cdots\!68}a^{20}-\frac{21\!\cdots\!85}{14\!\cdots\!34}a^{19}-\frac{16\!\cdots\!77}{29\!\cdots\!68}a^{18}+\frac{41\!\cdots\!01}{14\!\cdots\!34}a^{17}+\frac{60\!\cdots\!81}{14\!\cdots\!34}a^{16}-\frac{32\!\cdots\!69}{14\!\cdots\!34}a^{15}-\frac{33\!\cdots\!15}{20\!\cdots\!76}a^{14}+\frac{45\!\cdots\!17}{50\!\cdots\!46}a^{13}+\frac{10\!\cdots\!81}{29\!\cdots\!68}a^{12}-\frac{17\!\cdots\!15}{86\!\cdots\!37}a^{11}-\frac{43\!\cdots\!03}{10\!\cdots\!92}a^{10}+\frac{43\!\cdots\!89}{17\!\cdots\!74}a^{9}+\frac{11\!\cdots\!91}{34\!\cdots\!48}a^{8}-\frac{61\!\cdots\!97}{35\!\cdots\!22}a^{7}-\frac{94\!\cdots\!99}{59\!\cdots\!06}a^{6}+\frac{10\!\cdots\!47}{17\!\cdots\!74}a^{5}+\frac{16\!\cdots\!13}{29\!\cdots\!53}a^{4}-\frac{46\!\cdots\!01}{59\!\cdots\!06}a^{3}-\frac{11\!\cdots\!69}{11\!\cdots\!12}a^{2}-\frac{15\!\cdots\!95}{59\!\cdots\!06}a-\frac{11\!\cdots\!24}{20\!\cdots\!71}$, $\frac{84\!\cdots\!17}{29\!\cdots\!68}a^{20}-\frac{21\!\cdots\!85}{14\!\cdots\!34}a^{19}-\frac{16\!\cdots\!77}{29\!\cdots\!68}a^{18}+\frac{41\!\cdots\!01}{14\!\cdots\!34}a^{17}+\frac{60\!\cdots\!81}{14\!\cdots\!34}a^{16}-\frac{32\!\cdots\!69}{14\!\cdots\!34}a^{15}-\frac{33\!\cdots\!15}{20\!\cdots\!76}a^{14}+\frac{45\!\cdots\!17}{50\!\cdots\!46}a^{13}+\frac{10\!\cdots\!81}{29\!\cdots\!68}a^{12}-\frac{17\!\cdots\!15}{86\!\cdots\!37}a^{11}-\frac{43\!\cdots\!03}{10\!\cdots\!92}a^{10}+\frac{43\!\cdots\!89}{17\!\cdots\!74}a^{9}+\frac{11\!\cdots\!91}{34\!\cdots\!48}a^{8}-\frac{61\!\cdots\!97}{35\!\cdots\!22}a^{7}-\frac{94\!\cdots\!99}{59\!\cdots\!06}a^{6}+\frac{10\!\cdots\!47}{17\!\cdots\!74}a^{5}+\frac{16\!\cdots\!13}{29\!\cdots\!53}a^{4}-\frac{46\!\cdots\!01}{59\!\cdots\!06}a^{3}-\frac{11\!\cdots\!69}{11\!\cdots\!12}a^{2}-\frac{15\!\cdots\!95}{59\!\cdots\!06}a-\frac{32\!\cdots\!95}{20\!\cdots\!71}$, $\frac{13\!\cdots\!01}{59\!\cdots\!04}a^{20}-\frac{51\!\cdots\!75}{29\!\cdots\!02}a^{19}-\frac{24\!\cdots\!27}{59\!\cdots\!04}a^{18}+\frac{10\!\cdots\!67}{29\!\cdots\!02}a^{17}+\frac{12\!\cdots\!71}{44\!\cdots\!53}a^{16}-\frac{23\!\cdots\!77}{88\!\cdots\!06}a^{15}-\frac{17\!\cdots\!01}{17\!\cdots\!12}a^{14}+\frac{44\!\cdots\!33}{42\!\cdots\!86}a^{13}+\frac{30\!\cdots\!53}{17\!\cdots\!12}a^{12}-\frac{35\!\cdots\!00}{15\!\cdots\!57}a^{11}-\frac{23\!\cdots\!57}{20\!\cdots\!76}a^{10}+\frac{30\!\cdots\!63}{10\!\cdots\!66}a^{9}-\frac{79\!\cdots\!51}{70\!\cdots\!44}a^{8}-\frac{21\!\cdots\!81}{10\!\cdots\!66}a^{7}+\frac{18\!\cdots\!99}{52\!\cdots\!33}a^{6}+\frac{35\!\cdots\!65}{50\!\cdots\!46}a^{5}-\frac{73\!\cdots\!62}{60\!\cdots\!59}a^{4}-\frac{54\!\cdots\!83}{52\!\cdots\!22}a^{3}-\frac{94\!\cdots\!03}{25\!\cdots\!52}a^{2}+\frac{16\!\cdots\!19}{12\!\cdots\!26}a+\frac{40\!\cdots\!08}{62\!\cdots\!13}$, $\frac{13\!\cdots\!91}{17\!\cdots\!12}a^{20}-\frac{72\!\cdots\!07}{29\!\cdots\!02}a^{19}-\frac{88\!\cdots\!31}{59\!\cdots\!04}a^{18}+\frac{30\!\cdots\!75}{63\!\cdots\!79}a^{17}+\frac{54\!\cdots\!44}{44\!\cdots\!53}a^{16}-\frac{34\!\cdots\!29}{88\!\cdots\!06}a^{15}-\frac{31\!\cdots\!29}{59\!\cdots\!04}a^{14}+\frac{14\!\cdots\!33}{88\!\cdots\!06}a^{13}+\frac{24\!\cdots\!79}{17\!\cdots\!12}a^{12}-\frac{58\!\cdots\!60}{15\!\cdots\!57}a^{11}-\frac{44\!\cdots\!39}{20\!\cdots\!76}a^{10}+\frac{54\!\cdots\!81}{10\!\cdots\!66}a^{9}+\frac{15\!\cdots\!95}{70\!\cdots\!44}a^{8}-\frac{39\!\cdots\!51}{10\!\cdots\!66}a^{7}-\frac{20\!\cdots\!19}{15\!\cdots\!38}a^{6}+\frac{45\!\cdots\!23}{35\!\cdots\!22}a^{5}+\frac{17\!\cdots\!37}{36\!\cdots\!54}a^{4}-\frac{13\!\cdots\!82}{18\!\cdots\!77}a^{3}-\frac{19\!\cdots\!29}{25\!\cdots\!52}a^{2}-\frac{73\!\cdots\!23}{17\!\cdots\!18}a-\frac{40\!\cdots\!23}{20\!\cdots\!71}$, $\frac{14\!\cdots\!67}{59\!\cdots\!04}a^{20}-\frac{49\!\cdots\!71}{44\!\cdots\!53}a^{19}-\frac{28\!\cdots\!47}{59\!\cdots\!04}a^{18}+\frac{19\!\cdots\!87}{88\!\cdots\!06}a^{17}+\frac{16\!\cdots\!92}{44\!\cdots\!53}a^{16}-\frac{10\!\cdots\!12}{63\!\cdots\!79}a^{15}-\frac{27\!\cdots\!91}{17\!\cdots\!12}a^{14}+\frac{61\!\cdots\!65}{88\!\cdots\!06}a^{13}+\frac{65\!\cdots\!59}{17\!\cdots\!12}a^{12}-\frac{34\!\cdots\!07}{21\!\cdots\!51}a^{11}-\frac{15\!\cdots\!13}{29\!\cdots\!68}a^{10}+\frac{21\!\cdots\!01}{10\!\cdots\!66}a^{9}+\frac{32\!\cdots\!43}{70\!\cdots\!44}a^{8}-\frac{15\!\cdots\!13}{10\!\cdots\!66}a^{7}-\frac{13\!\cdots\!51}{52\!\cdots\!33}a^{6}+\frac{12\!\cdots\!28}{25\!\cdots\!73}a^{5}+\frac{32\!\cdots\!67}{36\!\cdots\!54}a^{4}-\frac{18\!\cdots\!03}{36\!\cdots\!54}a^{3}-\frac{34\!\cdots\!33}{25\!\cdots\!52}a^{2}-\frac{12\!\cdots\!18}{20\!\cdots\!71}a-\frac{23\!\cdots\!73}{62\!\cdots\!13}$, $\frac{12\!\cdots\!47}{17\!\cdots\!12}a^{20}-\frac{22\!\cdots\!35}{88\!\cdots\!06}a^{19}-\frac{26\!\cdots\!89}{17\!\cdots\!12}a^{18}+\frac{23\!\cdots\!71}{44\!\cdots\!53}a^{17}+\frac{11\!\cdots\!39}{88\!\cdots\!06}a^{16}-\frac{38\!\cdots\!71}{88\!\cdots\!06}a^{15}-\frac{10\!\cdots\!01}{17\!\cdots\!12}a^{14}+\frac{85\!\cdots\!71}{44\!\cdots\!53}a^{13}+\frac{25\!\cdots\!53}{17\!\cdots\!12}a^{12}-\frac{15\!\cdots\!29}{30\!\cdots\!14}a^{11}-\frac{43\!\cdots\!65}{20\!\cdots\!76}a^{10}+\frac{40\!\cdots\!79}{52\!\cdots\!33}a^{9}+\frac{13\!\cdots\!01}{70\!\cdots\!44}a^{8}-\frac{37\!\cdots\!06}{52\!\cdots\!33}a^{7}-\frac{23\!\cdots\!38}{17\!\cdots\!11}a^{6}+\frac{65\!\cdots\!62}{17\!\cdots\!11}a^{5}+\frac{21\!\cdots\!01}{26\!\cdots\!11}a^{4}-\frac{36\!\cdots\!13}{36\!\cdots\!54}a^{3}-\frac{82\!\cdots\!49}{25\!\cdots\!52}a^{2}+\frac{63\!\cdots\!86}{62\!\cdots\!13}a+\frac{15\!\cdots\!94}{29\!\cdots\!53}$, $\frac{27\!\cdots\!19}{21\!\cdots\!93}a^{20}-\frac{81\!\cdots\!19}{88\!\cdots\!06}a^{19}-\frac{41\!\cdots\!18}{14\!\cdots\!51}a^{18}+\frac{29\!\cdots\!57}{14\!\cdots\!51}a^{17}+\frac{15\!\cdots\!78}{63\!\cdots\!79}a^{16}-\frac{26\!\cdots\!74}{14\!\cdots\!51}a^{15}-\frac{54\!\cdots\!98}{44\!\cdots\!53}a^{14}+\frac{12\!\cdots\!23}{14\!\cdots\!51}a^{13}+\frac{15\!\cdots\!25}{44\!\cdots\!53}a^{12}-\frac{32\!\cdots\!43}{15\!\cdots\!57}a^{11}-\frac{98\!\cdots\!65}{15\!\cdots\!57}a^{10}+\frac{14\!\cdots\!82}{52\!\cdots\!33}a^{9}+\frac{12\!\cdots\!66}{17\!\cdots\!11}a^{8}-\frac{79\!\cdots\!77}{52\!\cdots\!33}a^{7}-\frac{25\!\cdots\!11}{52\!\cdots\!33}a^{6}-\frac{48\!\cdots\!41}{10\!\cdots\!66}a^{5}+\frac{32\!\cdots\!42}{18\!\cdots\!77}a^{4}+\frac{20\!\cdots\!91}{26\!\cdots\!11}a^{3}-\frac{16\!\cdots\!83}{62\!\cdots\!13}a^{2}-\frac{47\!\cdots\!32}{20\!\cdots\!71}a-\frac{23\!\cdots\!13}{20\!\cdots\!71}$, $\frac{19\!\cdots\!75}{88\!\cdots\!06}a^{20}-\frac{76\!\cdots\!97}{88\!\cdots\!06}a^{19}-\frac{38\!\cdots\!11}{88\!\cdots\!06}a^{18}+\frac{75\!\cdots\!71}{44\!\cdots\!53}a^{17}+\frac{51\!\cdots\!01}{14\!\cdots\!51}a^{16}-\frac{59\!\cdots\!90}{44\!\cdots\!53}a^{15}-\frac{44\!\cdots\!01}{29\!\cdots\!02}a^{14}+\frac{49\!\cdots\!15}{88\!\cdots\!06}a^{13}+\frac{10\!\cdots\!11}{29\!\cdots\!02}a^{12}-\frac{13\!\cdots\!39}{10\!\cdots\!38}a^{11}-\frac{17\!\cdots\!13}{30\!\cdots\!14}a^{10}+\frac{17\!\cdots\!73}{10\!\cdots\!66}a^{9}+\frac{56\!\cdots\!63}{10\!\cdots\!66}a^{8}-\frac{42\!\cdots\!13}{35\!\cdots\!22}a^{7}-\frac{16\!\cdots\!20}{52\!\cdots\!33}a^{6}+\frac{10\!\cdots\!87}{25\!\cdots\!73}a^{5}+\frac{67\!\cdots\!46}{60\!\cdots\!59}a^{4}-\frac{58\!\cdots\!03}{17\!\cdots\!74}a^{3}-\frac{30\!\cdots\!79}{17\!\cdots\!18}a^{2}-\frac{15\!\cdots\!75}{17\!\cdots\!18}a-\frac{24\!\cdots\!60}{62\!\cdots\!13}$, $\frac{98\!\cdots\!39}{17\!\cdots\!12}a^{20}+\frac{76\!\cdots\!09}{14\!\cdots\!51}a^{19}-\frac{38\!\cdots\!49}{25\!\cdots\!16}a^{18}-\frac{14\!\cdots\!68}{14\!\cdots\!51}a^{17}+\frac{15\!\cdots\!37}{88\!\cdots\!06}a^{16}+\frac{20\!\cdots\!75}{29\!\cdots\!02}a^{15}-\frac{18\!\cdots\!57}{17\!\cdots\!12}a^{14}-\frac{36\!\cdots\!11}{14\!\cdots\!51}a^{13}+\frac{89\!\cdots\!07}{25\!\cdots\!16}a^{12}+\frac{14\!\cdots\!91}{30\!\cdots\!14}a^{11}-\frac{44\!\cdots\!59}{61\!\cdots\!28}a^{10}-\frac{24\!\cdots\!35}{52\!\cdots\!33}a^{9}+\frac{61\!\cdots\!61}{70\!\cdots\!44}a^{8}+\frac{42\!\cdots\!80}{17\!\cdots\!11}a^{7}-\frac{31\!\cdots\!88}{52\!\cdots\!33}a^{6}-\frac{13\!\cdots\!45}{10\!\cdots\!66}a^{5}+\frac{77\!\cdots\!67}{36\!\cdots\!54}a^{4}+\frac{28\!\cdots\!29}{36\!\cdots\!54}a^{3}-\frac{76\!\cdots\!01}{25\!\cdots\!52}a^{2}-\frac{13\!\cdots\!45}{62\!\cdots\!13}a-\frac{10\!\cdots\!70}{62\!\cdots\!13}$, $\frac{90\!\cdots\!25}{14\!\cdots\!51}a^{20}-\frac{22\!\cdots\!65}{12\!\cdots\!58}a^{19}-\frac{18\!\cdots\!54}{14\!\cdots\!51}a^{18}+\frac{16\!\cdots\!91}{44\!\cdots\!53}a^{17}+\frac{46\!\cdots\!38}{44\!\cdots\!53}a^{16}-\frac{13\!\cdots\!23}{44\!\cdots\!53}a^{15}-\frac{41\!\cdots\!97}{88\!\cdots\!06}a^{14}+\frac{58\!\cdots\!29}{44\!\cdots\!53}a^{13}+\frac{10\!\cdots\!29}{88\!\cdots\!06}a^{12}-\frac{16\!\cdots\!40}{52\!\cdots\!33}a^{11}-\frac{59\!\cdots\!69}{30\!\cdots\!14}a^{10}+\frac{23\!\cdots\!90}{52\!\cdots\!33}a^{9}+\frac{28\!\cdots\!39}{15\!\cdots\!38}a^{8}-\frac{17\!\cdots\!70}{52\!\cdots\!33}a^{7}-\frac{11\!\cdots\!67}{10\!\cdots\!66}a^{6}+\frac{12\!\cdots\!19}{10\!\cdots\!66}a^{5}+\frac{13\!\cdots\!01}{36\!\cdots\!54}a^{4}-\frac{14\!\cdots\!08}{18\!\cdots\!77}a^{3}-\frac{23\!\cdots\!41}{41\!\cdots\!42}a^{2}-\frac{18\!\cdots\!54}{62\!\cdots\!13}a-\frac{12\!\cdots\!65}{89\!\cdots\!59}$, $\frac{27\!\cdots\!73}{17\!\cdots\!12}a^{20}-\frac{19\!\cdots\!14}{44\!\cdots\!53}a^{19}-\frac{56\!\cdots\!49}{17\!\cdots\!12}a^{18}+\frac{38\!\cdots\!36}{44\!\cdots\!53}a^{17}+\frac{24\!\cdots\!03}{88\!\cdots\!06}a^{16}-\frac{10\!\cdots\!27}{14\!\cdots\!51}a^{15}-\frac{72\!\cdots\!17}{59\!\cdots\!04}a^{14}+\frac{89\!\cdots\!05}{29\!\cdots\!02}a^{13}+\frac{59\!\cdots\!07}{17\!\cdots\!12}a^{12}-\frac{37\!\cdots\!20}{51\!\cdots\!19}a^{11}-\frac{34\!\cdots\!13}{61\!\cdots\!28}a^{10}+\frac{10\!\cdots\!47}{10\!\cdots\!66}a^{9}+\frac{41\!\cdots\!95}{70\!\cdots\!44}a^{8}-\frac{79\!\cdots\!89}{10\!\cdots\!66}a^{7}-\frac{66\!\cdots\!75}{17\!\cdots\!11}a^{6}+\frac{19\!\cdots\!08}{75\!\cdots\!19}a^{5}+\frac{49\!\cdots\!37}{36\!\cdots\!54}a^{4}-\frac{69\!\cdots\!53}{18\!\cdots\!77}a^{3}-\frac{53\!\cdots\!87}{25\!\cdots\!52}a^{2}-\frac{11\!\cdots\!71}{89\!\cdots\!59}a-\frac{37\!\cdots\!09}{62\!\cdots\!13}$, $\frac{91\!\cdots\!95}{59\!\cdots\!04}a^{20}-\frac{62\!\cdots\!65}{88\!\cdots\!06}a^{19}-\frac{54\!\cdots\!87}{17\!\cdots\!12}a^{18}+\frac{40\!\cdots\!69}{29\!\cdots\!02}a^{17}+\frac{21\!\cdots\!53}{88\!\cdots\!06}a^{16}-\frac{96\!\cdots\!67}{88\!\cdots\!06}a^{15}-\frac{17\!\cdots\!93}{17\!\cdots\!12}a^{14}+\frac{19\!\cdots\!07}{44\!\cdots\!53}a^{13}+\frac{41\!\cdots\!09}{17\!\cdots\!12}a^{12}-\frac{10\!\cdots\!13}{10\!\cdots\!66}a^{11}-\frac{69\!\cdots\!13}{20\!\cdots\!76}a^{10}+\frac{69\!\cdots\!30}{52\!\cdots\!33}a^{9}+\frac{65\!\cdots\!35}{21\!\cdots\!32}a^{8}-\frac{49\!\cdots\!86}{52\!\cdots\!33}a^{7}-\frac{19\!\cdots\!33}{10\!\cdots\!66}a^{6}+\frac{17\!\cdots\!59}{52\!\cdots\!33}a^{5}+\frac{23\!\cdots\!81}{36\!\cdots\!54}a^{4}-\frac{61\!\cdots\!05}{18\!\cdots\!77}a^{3}-\frac{27\!\cdots\!01}{25\!\cdots\!52}a^{2}-\frac{31\!\cdots\!75}{62\!\cdots\!13}a-\frac{48\!\cdots\!18}{20\!\cdots\!71}$, $\frac{60\!\cdots\!77}{88\!\cdots\!06}a^{20}+\frac{36\!\cdots\!09}{63\!\cdots\!79}a^{19}-\frac{12\!\cdots\!01}{88\!\cdots\!06}a^{18}-\frac{34\!\cdots\!63}{29\!\cdots\!02}a^{17}+\frac{37\!\cdots\!45}{29\!\cdots\!02}a^{16}+\frac{86\!\cdots\!33}{88\!\cdots\!06}a^{15}-\frac{53\!\cdots\!07}{88\!\cdots\!06}a^{14}-\frac{19\!\cdots\!44}{44\!\cdots\!53}a^{13}+\frac{15\!\cdots\!27}{88\!\cdots\!06}a^{12}+\frac{19\!\cdots\!06}{15\!\cdots\!57}a^{11}-\frac{29\!\cdots\!61}{10\!\cdots\!38}a^{10}-\frac{11\!\cdots\!88}{52\!\cdots\!33}a^{9}+\frac{31\!\cdots\!15}{10\!\cdots\!66}a^{8}+\frac{13\!\cdots\!29}{52\!\cdots\!33}a^{7}-\frac{94\!\cdots\!65}{52\!\cdots\!33}a^{6}-\frac{87\!\cdots\!01}{52\!\cdots\!33}a^{5}+\frac{32\!\cdots\!82}{60\!\cdots\!59}a^{4}+\frac{67\!\cdots\!79}{12\!\cdots\!18}a^{3}-\frac{39\!\cdots\!33}{62\!\cdots\!13}a^{2}-\frac{91\!\cdots\!25}{12\!\cdots\!26}a-\frac{22\!\cdots\!26}{62\!\cdots\!13}$, $\frac{35\!\cdots\!99}{17\!\cdots\!12}a^{20}-\frac{60\!\cdots\!30}{63\!\cdots\!79}a^{19}-\frac{69\!\cdots\!21}{17\!\cdots\!12}a^{18}+\frac{82\!\cdots\!24}{44\!\cdots\!53}a^{17}+\frac{90\!\cdots\!23}{29\!\cdots\!02}a^{16}-\frac{13\!\cdots\!61}{88\!\cdots\!06}a^{15}-\frac{22\!\cdots\!17}{17\!\cdots\!12}a^{14}+\frac{26\!\cdots\!33}{44\!\cdots\!53}a^{13}+\frac{52\!\cdots\!85}{17\!\cdots\!12}a^{12}-\frac{14\!\cdots\!97}{10\!\cdots\!38}a^{11}-\frac{85\!\cdots\!05}{20\!\cdots\!76}a^{10}+\frac{94\!\cdots\!94}{52\!\cdots\!33}a^{9}+\frac{26\!\cdots\!25}{70\!\cdots\!44}a^{8}-\frac{67\!\cdots\!70}{52\!\cdots\!33}a^{7}-\frac{11\!\cdots\!10}{52\!\cdots\!33}a^{6}+\frac{48\!\cdots\!17}{10\!\cdots\!66}a^{5}+\frac{15\!\cdots\!01}{18\!\cdots\!77}a^{4}-\frac{18\!\cdots\!59}{36\!\cdots\!54}a^{3}-\frac{50\!\cdots\!35}{35\!\cdots\!36}a^{2}-\frac{57\!\cdots\!16}{89\!\cdots\!59}a-\frac{63\!\cdots\!10}{20\!\cdots\!71}$, $\frac{37\!\cdots\!12}{44\!\cdots\!53}a^{20}-\frac{29\!\cdots\!11}{88\!\cdots\!06}a^{19}-\frac{20\!\cdots\!93}{12\!\cdots\!58}a^{18}+\frac{94\!\cdots\!20}{14\!\cdots\!51}a^{17}+\frac{19\!\cdots\!28}{14\!\cdots\!51}a^{16}-\frac{22\!\cdots\!18}{44\!\cdots\!53}a^{15}-\frac{15\!\cdots\!77}{29\!\cdots\!02}a^{14}+\frac{58\!\cdots\!39}{29\!\cdots\!02}a^{13}+\frac{36\!\cdots\!03}{29\!\cdots\!02}a^{12}-\frac{13\!\cdots\!33}{30\!\cdots\!14}a^{11}-\frac{51\!\cdots\!27}{30\!\cdots\!14}a^{10}+\frac{81\!\cdots\!97}{15\!\cdots\!38}a^{9}+\frac{15\!\cdots\!97}{10\!\cdots\!66}a^{8}-\frac{37\!\cdots\!45}{10\!\cdots\!66}a^{7}-\frac{82\!\cdots\!23}{10\!\cdots\!66}a^{6}+\frac{59\!\cdots\!80}{52\!\cdots\!33}a^{5}+\frac{46\!\cdots\!45}{18\!\cdots\!77}a^{4}-\frac{11\!\cdots\!89}{12\!\cdots\!18}a^{3}-\frac{15\!\cdots\!49}{41\!\cdots\!42}a^{2}-\frac{22\!\cdots\!75}{12\!\cdots\!26}a-\frac{17\!\cdots\!77}{20\!\cdots\!71}$, $\frac{16\!\cdots\!25}{29\!\cdots\!02}a^{20}-\frac{11\!\cdots\!04}{44\!\cdots\!53}a^{19}-\frac{48\!\cdots\!48}{44\!\cdots\!53}a^{18}+\frac{65\!\cdots\!29}{12\!\cdots\!58}a^{17}+\frac{12\!\cdots\!25}{14\!\cdots\!51}a^{16}-\frac{17\!\cdots\!79}{42\!\cdots\!86}a^{15}-\frac{15\!\cdots\!69}{44\!\cdots\!53}a^{14}+\frac{14\!\cdots\!07}{88\!\cdots\!06}a^{13}+\frac{37\!\cdots\!98}{44\!\cdots\!53}a^{12}-\frac{11\!\cdots\!07}{30\!\cdots\!14}a^{11}-\frac{18\!\cdots\!16}{15\!\cdots\!57}a^{10}+\frac{51\!\cdots\!63}{10\!\cdots\!66}a^{9}+\frac{56\!\cdots\!68}{52\!\cdots\!33}a^{8}-\frac{17\!\cdots\!65}{50\!\cdots\!46}a^{7}-\frac{66\!\cdots\!79}{10\!\cdots\!66}a^{6}+\frac{12\!\cdots\!67}{10\!\cdots\!66}a^{5}+\frac{41\!\cdots\!61}{18\!\cdots\!77}a^{4}-\frac{23\!\cdots\!46}{18\!\cdots\!77}a^{3}-\frac{78\!\cdots\!67}{20\!\cdots\!71}a^{2}-\frac{10\!\cdots\!58}{62\!\cdots\!13}a-\frac{16\!\cdots\!35}{20\!\cdots\!71}$, $\frac{68\!\cdots\!25}{44\!\cdots\!53}a^{20}-\frac{61\!\cdots\!51}{88\!\cdots\!06}a^{19}-\frac{10\!\cdots\!41}{29\!\cdots\!02}a^{18}+\frac{81\!\cdots\!13}{44\!\cdots\!53}a^{17}+\frac{27\!\cdots\!99}{88\!\cdots\!06}a^{16}-\frac{12\!\cdots\!72}{63\!\cdots\!79}a^{15}-\frac{13\!\cdots\!91}{88\!\cdots\!06}a^{14}+\frac{32\!\cdots\!85}{29\!\cdots\!02}a^{13}+\frac{40\!\cdots\!45}{88\!\cdots\!06}a^{12}-\frac{49\!\cdots\!45}{14\!\cdots\!34}a^{11}-\frac{25\!\cdots\!77}{30\!\cdots\!14}a^{10}+\frac{20\!\cdots\!75}{35\!\cdots\!22}a^{9}+\frac{96\!\cdots\!41}{10\!\cdots\!66}a^{8}-\frac{18\!\cdots\!59}{35\!\cdots\!22}a^{7}-\frac{62\!\cdots\!37}{10\!\cdots\!66}a^{6}+\frac{44\!\cdots\!35}{25\!\cdots\!73}a^{5}+\frac{37\!\cdots\!98}{18\!\cdots\!77}a^{4}+\frac{72\!\cdots\!57}{36\!\cdots\!54}a^{3}-\frac{18\!\cdots\!15}{62\!\cdots\!13}a^{2}-\frac{74\!\cdots\!33}{41\!\cdots\!42}a-\frac{52\!\cdots\!57}{62\!\cdots\!13}$, $\frac{84\!\cdots\!13}{29\!\cdots\!02}a^{20}-\frac{26\!\cdots\!88}{14\!\cdots\!51}a^{19}-\frac{55\!\cdots\!77}{88\!\cdots\!06}a^{18}+\frac{65\!\cdots\!77}{14\!\cdots\!51}a^{17}+\frac{36\!\cdots\!42}{63\!\cdots\!79}a^{16}-\frac{40\!\cdots\!95}{88\!\cdots\!06}a^{15}-\frac{24\!\cdots\!67}{88\!\cdots\!06}a^{14}+\frac{35\!\cdots\!77}{14\!\cdots\!51}a^{13}+\frac{24\!\cdots\!41}{29\!\cdots\!02}a^{12}-\frac{10\!\cdots\!07}{15\!\cdots\!57}a^{11}-\frac{45\!\cdots\!45}{30\!\cdots\!14}a^{10}+\frac{62\!\cdots\!57}{52\!\cdots\!33}a^{9}+\frac{17\!\cdots\!03}{10\!\cdots\!66}a^{8}-\frac{17\!\cdots\!38}{17\!\cdots\!11}a^{7}-\frac{80\!\cdots\!35}{75\!\cdots\!19}a^{6}+\frac{17\!\cdots\!11}{52\!\cdots\!33}a^{5}+\frac{22\!\cdots\!84}{60\!\cdots\!59}a^{4}+\frac{75\!\cdots\!21}{18\!\cdots\!77}a^{3}-\frac{68\!\cdots\!43}{12\!\cdots\!26}a^{2}-\frac{62\!\cdots\!39}{17\!\cdots\!18}a-\frac{61\!\cdots\!28}{20\!\cdots\!71}$, $\frac{27\!\cdots\!89}{88\!\cdots\!06}a^{20}-\frac{40\!\cdots\!50}{44\!\cdots\!53}a^{19}-\frac{27\!\cdots\!19}{44\!\cdots\!53}a^{18}+\frac{78\!\cdots\!09}{44\!\cdots\!53}a^{17}+\frac{14\!\cdots\!07}{29\!\cdots\!02}a^{16}-\frac{40\!\cdots\!25}{29\!\cdots\!02}a^{15}-\frac{30\!\cdots\!23}{14\!\cdots\!51}a^{14}+\frac{34\!\cdots\!74}{63\!\cdots\!79}a^{13}+\frac{22\!\cdots\!35}{44\!\cdots\!53}a^{12}-\frac{17\!\cdots\!27}{15\!\cdots\!57}a^{11}-\frac{16\!\cdots\!24}{21\!\cdots\!51}a^{10}+\frac{99\!\cdots\!44}{75\!\cdots\!19}a^{9}+\frac{11\!\cdots\!56}{17\!\cdots\!11}a^{8}-\frac{38\!\cdots\!19}{52\!\cdots\!33}a^{7}-\frac{35\!\cdots\!93}{10\!\cdots\!66}a^{6}+\frac{22\!\cdots\!60}{17\!\cdots\!11}a^{5}+\frac{51\!\cdots\!72}{60\!\cdots\!59}a^{4}+\frac{40\!\cdots\!64}{18\!\cdots\!77}a^{3}-\frac{85\!\cdots\!41}{12\!\cdots\!26}a^{2}-\frac{49\!\cdots\!49}{12\!\cdots\!26}a-\frac{17\!\cdots\!58}{89\!\cdots\!59}$, $\frac{26\!\cdots\!15}{59\!\cdots\!04}a^{20}-\frac{78\!\cdots\!82}{63\!\cdots\!79}a^{19}-\frac{22\!\cdots\!83}{25\!\cdots\!16}a^{18}+\frac{21\!\cdots\!47}{88\!\cdots\!06}a^{17}+\frac{21\!\cdots\!29}{29\!\cdots\!02}a^{16}-\frac{27\!\cdots\!42}{14\!\cdots\!51}a^{15}-\frac{55\!\cdots\!29}{17\!\cdots\!12}a^{14}+\frac{10\!\cdots\!10}{14\!\cdots\!51}a^{13}+\frac{46\!\cdots\!59}{59\!\cdots\!04}a^{12}-\frac{46\!\cdots\!83}{30\!\cdots\!14}a^{11}-\frac{73\!\cdots\!23}{61\!\cdots\!28}a^{10}+\frac{30\!\cdots\!10}{17\!\cdots\!11}a^{9}+\frac{80\!\cdots\!01}{70\!\cdots\!44}a^{8}-\frac{67\!\cdots\!82}{75\!\cdots\!19}a^{7}-\frac{67\!\cdots\!33}{10\!\cdots\!66}a^{6}+\frac{10\!\cdots\!91}{10\!\cdots\!66}a^{5}+\frac{33\!\cdots\!60}{18\!\cdots\!77}a^{4}+\frac{12\!\cdots\!60}{18\!\cdots\!77}a^{3}-\frac{45\!\cdots\!69}{25\!\cdots\!52}a^{2}-\frac{50\!\cdots\!05}{41\!\cdots\!42}a-\frac{36\!\cdots\!74}{62\!\cdots\!13}$
| 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: | \( 100927220322000000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 100927220322000000000 \cdot 1}{2\cdot\sqrt{35135593769377589230177713282906868218673215743721472}}\cr\approx \mathstrut & 0.564591813950889 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 217*x^19 + 19544*x^17 - 959637*x^15 - 33706*x^14 + 28285005*x^13 + 3493224*x^12 - 517708667*x^11 - 143018778*x^10 + 5859384083*x^9 + 2942499210*x^8 - 39262594646*x^7 - 31779062112*x^6 + 138562884152*x^5 + 168986405644*x^4 - 168971552743*x^3 - 345735927544*x^2 - 146449603860*x - 6727656872)
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 - 217*x^19 + 19544*x^17 - 959637*x^15 - 33706*x^14 + 28285005*x^13 + 3493224*x^12 - 517708667*x^11 - 143018778*x^10 + 5859384083*x^9 + 2942499210*x^8 - 39262594646*x^7 - 31779062112*x^6 + 138562884152*x^5 + 168986405644*x^4 - 168971552743*x^3 - 345735927544*x^2 - 146449603860*x - 6727656872, 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 - 217*x^19 + 19544*x^17 - 959637*x^15 - 33706*x^14 + 28285005*x^13 + 3493224*x^12 - 517708667*x^11 - 143018778*x^10 + 5859384083*x^9 + 2942499210*x^8 - 39262594646*x^7 - 31779062112*x^6 + 138562884152*x^5 + 168986405644*x^4 - 168971552743*x^3 - 345735927544*x^2 - 146449603860*x - 6727656872);
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 - 217*x^19 + 19544*x^17 - 959637*x^15 - 33706*x^14 + 28285005*x^13 + 3493224*x^12 - 517708667*x^11 - 143018778*x^10 + 5859384083*x^9 + 2942499210*x^8 - 39262594646*x^7 - 31779062112*x^6 + 138562884152*x^5 + 168986405644*x^4 - 168971552743*x^3 - 345735927544*x^2 - 146449603860*x - 6727656872);
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
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 294 |
| The 14 conjugacy class representatives for $C_7:F_7$ |
| Character table for $C_7:F_7$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\) |
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 21 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{7}$ | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.2.0.1}{2} }^{9}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{7}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.3.0.1}{3} }$ | R | ${\href{/padicField/31.3.0.1}{3} }^{7}$ | ${\href{/padicField/37.3.0.1}{3} }^{7}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{9}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{7}$ | ${\href{/padicField/53.3.0.1}{3} }^{7}$ | ${\href{/padicField/59.3.0.1}{3} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
|
\(7\)
| Deg $21$ | $21$ | $1$ | $35$ | |||
|
\(29\)
| 29.7.0.1 | $x^{7} + 2 x + 27$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 29.7.6.3 | $x^{7} + 87$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ | |
| 29.7.6.4 | $x^{7} + 174$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |