Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 - 92*x^20 + 978*x^19 - 23031*x^18 + 208542*x^17 + 3327324*x^16 - 36397800*x^15 - 19963545*x^14 - 406653528*x^13 - 16539946488*x^12 + 256034067942*x^11 + 682357346031*x^10 - 13494358927986*x^9 + 9822827363220*x^8 + 261815058820176*x^7 - 813912422758992*x^6 - 647257611403968*x^5 + 8751031478070272*x^4 - 24165278101734848*x^3 + 30990808995877088*x^2 - 13115399746724928*x - 2634468582334464)
gp: K = bnfinit(y^22 - 4*y^21 - 92*y^20 + 978*y^19 - 23031*y^18 + 208542*y^17 + 3327324*y^16 - 36397800*y^15 - 19963545*y^14 - 406653528*y^13 - 16539946488*y^12 + 256034067942*y^11 + 682357346031*y^10 - 13494358927986*y^9 + 9822827363220*y^8 + 261815058820176*y^7 - 813912422758992*y^6 - 647257611403968*y^5 + 8751031478070272*y^4 - 24165278101734848*y^3 + 30990808995877088*y^2 - 13115399746724928*y - 2634468582334464, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 4*x^21 - 92*x^20 + 978*x^19 - 23031*x^18 + 208542*x^17 + 3327324*x^16 - 36397800*x^15 - 19963545*x^14 - 406653528*x^13 - 16539946488*x^12 + 256034067942*x^11 + 682357346031*x^10 - 13494358927986*x^9 + 9822827363220*x^8 + 261815058820176*x^7 - 813912422758992*x^6 - 647257611403968*x^5 + 8751031478070272*x^4 - 24165278101734848*x^3 + 30990808995877088*x^2 - 13115399746724928*x - 2634468582334464);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 4*x^21 - 92*x^20 + 978*x^19 - 23031*x^18 + 208542*x^17 + 3327324*x^16 - 36397800*x^15 - 19963545*x^14 - 406653528*x^13 - 16539946488*x^12 + 256034067942*x^11 + 682357346031*x^10 - 13494358927986*x^9 + 9822827363220*x^8 + 261815058820176*x^7 - 813912422758992*x^6 - 647257611403968*x^5 + 8751031478070272*x^4 - 24165278101734848*x^3 + 30990808995877088*x^2 - 13115399746724928*x - 2634468582334464)
\( x^{22} - 4 x^{21} - 92 x^{20} + 978 x^{19} - 23031 x^{18} + 208542 x^{17} + 3327324 x^{16} + \cdots - 26\!\cdots\!64 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[10, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(254171926510889578368436897006059571070510600002237220914603229184\)
\(\medspace = 2^{30}\cdot 3^{28}\cdot 7^{4}\cdot 23^{4}\cdot 137^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(939.64\) | 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: | $2^{103/48}3^{85/54}7^{1/2}23^{1/2}137^{4/5}\approx 16209.655612939698$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(23\), \(137\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{11}+\frac{1}{6}a^{9}+\frac{1}{6}a^{7}-\frac{1}{6}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{12}a^{12}-\frac{1}{6}a^{10}-\frac{1}{6}a^{8}-\frac{1}{2}a^{7}-\frac{1}{3}a^{6}-\frac{1}{2}a^{5}+\frac{5}{12}a^{4}+\frac{1}{6}a^{2}$, $\frac{1}{12}a^{13}-\frac{1}{6}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{3}a$, $\frac{1}{12}a^{14}-\frac{1}{6}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{12}a^{15}-\frac{1}{6}a^{9}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{3}a^{3}$, $\frac{1}{12}a^{16}-\frac{1}{6}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{6}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{24}a^{17}-\frac{1}{24}a^{13}-\frac{1}{24}a^{9}+\frac{1}{6}a^{7}-\frac{1}{2}a^{6}-\frac{7}{24}a^{5}-\frac{1}{2}a^{4}-\frac{1}{3}a^{3}$, $\frac{1}{24}a^{18}-\frac{1}{24}a^{14}-\frac{1}{24}a^{10}+\frac{1}{6}a^{8}-\frac{1}{2}a^{7}-\frac{7}{24}a^{6}-\frac{1}{2}a^{5}-\frac{1}{3}a^{4}$, $\frac{1}{48}a^{19}-\frac{1}{24}a^{16}+\frac{1}{48}a^{15}-\frac{1}{24}a^{14}-\frac{1}{48}a^{11}+\frac{1}{12}a^{10}-\frac{1}{4}a^{9}+\frac{5}{24}a^{8}+\frac{23}{48}a^{7}-\frac{3}{8}a^{6}-\frac{1}{6}a^{5}-\frac{1}{6}a^{4}+\frac{1}{6}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{2016}a^{20}-\frac{1}{168}a^{18}+\frac{1}{336}a^{17}+\frac{19}{672}a^{16}-\frac{13}{336}a^{15}-\frac{1}{56}a^{14}-\frac{1}{168}a^{13}-\frac{1}{224}a^{12}-\frac{3}{56}a^{11}+\frac{19}{84}a^{10}-\frac{53}{336}a^{9}-\frac{41}{224}a^{8}+\frac{3}{16}a^{7}-\frac{5}{168}a^{6}-\frac{17}{168}a^{5}+\frac{1}{7}a^{4}+\frac{3}{7}a^{3}-\frac{7}{18}a^{2}-\frac{4}{21}a-\frac{2}{7}$, $\frac{1}{26\!\cdots\!80}a^{21}+\frac{25\!\cdots\!71}{13\!\cdots\!40}a^{20}+\frac{24\!\cdots\!29}{43\!\cdots\!88}a^{19}+\frac{25\!\cdots\!63}{43\!\cdots\!80}a^{18}+\frac{44\!\cdots\!19}{87\!\cdots\!60}a^{17}+\frac{10\!\cdots\!13}{54\!\cdots\!60}a^{16}-\frac{56\!\cdots\!61}{36\!\cdots\!40}a^{15}+\frac{84\!\cdots\!19}{36\!\cdots\!40}a^{14}-\frac{33\!\cdots\!79}{87\!\cdots\!60}a^{13}+\frac{27\!\cdots\!65}{87\!\cdots\!76}a^{12}-\frac{10\!\cdots\!61}{15\!\cdots\!60}a^{11}+\frac{30\!\cdots\!89}{43\!\cdots\!80}a^{10}+\frac{35\!\cdots\!73}{17\!\cdots\!52}a^{9}+\frac{36\!\cdots\!77}{21\!\cdots\!40}a^{8}+\frac{73\!\cdots\!33}{54\!\cdots\!60}a^{7}+\frac{17\!\cdots\!77}{42\!\cdots\!08}a^{6}-\frac{10\!\cdots\!63}{10\!\cdots\!20}a^{5}+\frac{14\!\cdots\!15}{10\!\cdots\!72}a^{4}-\frac{16\!\cdots\!27}{40\!\cdots\!45}a^{3}-\frac{14\!\cdots\!64}{40\!\cdots\!45}a^{2}-\frac{41\!\cdots\!81}{90\!\cdots\!10}a+\frac{11\!\cdots\!58}{15\!\cdots\!35}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
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: | $15$ | 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: |
$\frac{65\!\cdots\!37}{18\!\cdots\!40}a^{21}-\frac{12\!\cdots\!77}{37\!\cdots\!80}a^{20}-\frac{39\!\cdots\!57}{30\!\cdots\!54}a^{19}+\frac{13\!\cdots\!11}{30\!\cdots\!40}a^{18}-\frac{81\!\cdots\!04}{77\!\cdots\!35}a^{17}+\frac{16\!\cdots\!77}{12\!\cdots\!60}a^{16}+\frac{82\!\cdots\!59}{20\!\cdots\!60}a^{15}-\frac{78\!\cdots\!29}{51\!\cdots\!90}a^{14}+\frac{50\!\cdots\!77}{61\!\cdots\!80}a^{13}-\frac{15\!\cdots\!41}{24\!\cdots\!32}a^{12}-\frac{14\!\cdots\!19}{63\!\cdots\!60}a^{11}+\frac{80\!\cdots\!47}{77\!\cdots\!35}a^{10}-\frac{11\!\cdots\!91}{30\!\cdots\!54}a^{9}-\frac{33\!\cdots\!73}{12\!\cdots\!60}a^{8}+\frac{11\!\cdots\!81}{61\!\cdots\!80}a^{7}-\frac{10\!\cdots\!59}{59\!\cdots\!53}a^{6}-\frac{29\!\cdots\!97}{15\!\cdots\!70}a^{5}+\frac{26\!\cdots\!79}{30\!\cdots\!54}a^{4}-\frac{42\!\cdots\!42}{23\!\cdots\!05}a^{3}+\frac{95\!\cdots\!27}{46\!\cdots\!10}a^{2}-\frac{20\!\cdots\!73}{25\!\cdots\!45}a-\frac{13\!\cdots\!47}{85\!\cdots\!15}$, $\frac{21\!\cdots\!23}{30\!\cdots\!54}a^{21}-\frac{45\!\cdots\!81}{61\!\cdots\!08}a^{20}-\frac{54\!\cdots\!65}{41\!\cdots\!72}a^{19}+\frac{15\!\cdots\!51}{20\!\cdots\!36}a^{18}-\frac{10\!\cdots\!62}{51\!\cdots\!09}a^{17}+\frac{29\!\cdots\!03}{10\!\cdots\!18}a^{16}+\frac{16\!\cdots\!19}{45\!\cdots\!08}a^{15}-\frac{95\!\cdots\!31}{34\!\cdots\!06}a^{14}+\frac{90\!\cdots\!61}{51\!\cdots\!09}a^{13}-\frac{29\!\cdots\!73}{20\!\cdots\!36}a^{12}-\frac{10\!\cdots\!97}{58\!\cdots\!96}a^{11}+\frac{39\!\cdots\!51}{20\!\cdots\!36}a^{10}-\frac{85\!\cdots\!25}{10\!\cdots\!18}a^{9}-\frac{20\!\cdots\!04}{51\!\cdots\!09}a^{8}+\frac{14\!\cdots\!05}{41\!\cdots\!72}a^{7}-\frac{19\!\cdots\!05}{39\!\cdots\!02}a^{6}-\frac{28\!\cdots\!65}{10\!\cdots\!18}a^{5}+\frac{14\!\cdots\!47}{10\!\cdots\!18}a^{4}-\frac{50\!\cdots\!66}{15\!\cdots\!27}a^{3}+\frac{11\!\cdots\!39}{30\!\cdots\!54}a^{2}-\frac{26\!\cdots\!85}{17\!\cdots\!03}a-\frac{17\!\cdots\!81}{57\!\cdots\!01}$, $\frac{73\!\cdots\!91}{13\!\cdots\!40}a^{21}-\frac{16\!\cdots\!67}{32\!\cdots\!60}a^{20}+\frac{10\!\cdots\!33}{43\!\cdots\!88}a^{19}+\frac{93\!\cdots\!43}{21\!\cdots\!40}a^{18}-\frac{89\!\cdots\!51}{43\!\cdots\!80}a^{17}+\frac{56\!\cdots\!09}{21\!\cdots\!40}a^{16}+\frac{29\!\cdots\!11}{72\!\cdots\!80}a^{15}-\frac{97\!\cdots\!79}{45\!\cdots\!05}a^{14}+\frac{94\!\cdots\!11}{43\!\cdots\!80}a^{13}-\frac{12\!\cdots\!79}{54\!\cdots\!86}a^{12}-\frac{46\!\cdots\!67}{44\!\cdots\!60}a^{11}+\frac{61\!\cdots\!49}{21\!\cdots\!40}a^{10}-\frac{40\!\cdots\!93}{87\!\cdots\!76}a^{9}-\frac{18\!\cdots\!31}{21\!\cdots\!40}a^{8}+\frac{80\!\cdots\!99}{21\!\cdots\!40}a^{7}+\frac{49\!\cdots\!45}{21\!\cdots\!54}a^{6}-\frac{54\!\cdots\!41}{10\!\cdots\!20}a^{5}+\frac{16\!\cdots\!73}{10\!\cdots\!72}a^{4}-\frac{19\!\cdots\!53}{81\!\cdots\!90}a^{3}+\frac{50\!\cdots\!17}{40\!\cdots\!45}a^{2}+\frac{26\!\cdots\!09}{45\!\cdots\!05}a+\frac{77\!\cdots\!91}{15\!\cdots\!35}$, $\frac{60\!\cdots\!53}{13\!\cdots\!40}a^{21}-\frac{12\!\cdots\!67}{65\!\cdots\!20}a^{20}-\frac{19\!\cdots\!07}{43\!\cdots\!88}a^{19}+\frac{10\!\cdots\!79}{21\!\cdots\!40}a^{18}-\frac{45\!\cdots\!13}{43\!\cdots\!80}a^{17}+\frac{52\!\cdots\!33}{54\!\cdots\!60}a^{16}+\frac{11\!\cdots\!53}{72\!\cdots\!80}a^{15}-\frac{63\!\cdots\!81}{36\!\cdots\!40}a^{14}-\frac{65\!\cdots\!07}{43\!\cdots\!80}a^{13}-\frac{49\!\cdots\!65}{43\!\cdots\!88}a^{12}-\frac{23\!\cdots\!97}{31\!\cdots\!20}a^{11}+\frac{26\!\cdots\!57}{21\!\cdots\!40}a^{10}+\frac{29\!\cdots\!89}{87\!\cdots\!76}a^{9}-\frac{74\!\cdots\!49}{10\!\cdots\!20}a^{8}+\frac{63\!\cdots\!77}{21\!\cdots\!40}a^{7}+\frac{12\!\cdots\!25}{84\!\cdots\!16}a^{6}-\frac{38\!\cdots\!53}{10\!\cdots\!20}a^{5}-\frac{49\!\cdots\!69}{54\!\cdots\!86}a^{4}+\frac{38\!\cdots\!91}{81\!\cdots\!90}a^{3}-\frac{25\!\cdots\!29}{40\!\cdots\!45}a^{2}+\frac{10\!\cdots\!77}{45\!\cdots\!05}a+\frac{74\!\cdots\!93}{15\!\cdots\!35}$, $\frac{23\!\cdots\!37}{65\!\cdots\!20}a^{21}-\frac{33\!\cdots\!51}{65\!\cdots\!20}a^{20}+\frac{37\!\cdots\!81}{21\!\cdots\!44}a^{19}+\frac{20\!\cdots\!33}{54\!\cdots\!60}a^{18}-\frac{29\!\cdots\!47}{21\!\cdots\!40}a^{17}+\frac{43\!\cdots\!61}{21\!\cdots\!40}a^{16}-\frac{10\!\cdots\!99}{18\!\cdots\!20}a^{15}-\frac{46\!\cdots\!53}{36\!\cdots\!40}a^{14}+\frac{39\!\cdots\!57}{21\!\cdots\!40}a^{13}-\frac{68\!\cdots\!27}{43\!\cdots\!88}a^{12}+\frac{30\!\cdots\!97}{15\!\cdots\!60}a^{11}+\frac{27\!\cdots\!47}{27\!\cdots\!30}a^{10}-\frac{43\!\cdots\!01}{43\!\cdots\!88}a^{9}+\frac{65\!\cdots\!11}{21\!\cdots\!40}a^{8}+\frac{37\!\cdots\!91}{13\!\cdots\!15}a^{7}-\frac{21\!\cdots\!47}{84\!\cdots\!16}a^{6}+\frac{21\!\cdots\!63}{54\!\cdots\!60}a^{5}+\frac{17\!\cdots\!01}{54\!\cdots\!86}a^{4}-\frac{60\!\cdots\!16}{40\!\cdots\!45}a^{3}+\frac{91\!\cdots\!68}{40\!\cdots\!45}a^{2}-\frac{44\!\cdots\!89}{45\!\cdots\!05}a-\frac{29\!\cdots\!31}{15\!\cdots\!35}$, $\frac{21\!\cdots\!89}{25\!\cdots\!60}a^{21}-\frac{21\!\cdots\!13}{10\!\cdots\!40}a^{20}-\frac{13\!\cdots\!65}{16\!\cdots\!84}a^{19}+\frac{11\!\cdots\!03}{16\!\cdots\!40}a^{18}-\frac{31\!\cdots\!13}{16\!\cdots\!40}a^{17}+\frac{49\!\cdots\!93}{33\!\cdots\!80}a^{16}+\frac{17\!\cdots\!91}{55\!\cdots\!80}a^{15}-\frac{14\!\cdots\!99}{55\!\cdots\!80}a^{14}-\frac{12\!\cdots\!34}{20\!\cdots\!55}a^{13}-\frac{28\!\cdots\!73}{66\!\cdots\!36}a^{12}-\frac{88\!\cdots\!81}{59\!\cdots\!30}a^{11}+\frac{32\!\cdots\!99}{16\!\cdots\!40}a^{10}+\frac{29\!\cdots\!21}{33\!\cdots\!68}a^{9}-\frac{34\!\cdots\!77}{33\!\cdots\!80}a^{8}-\frac{12\!\cdots\!51}{16\!\cdots\!40}a^{7}+\frac{10\!\cdots\!31}{47\!\cdots\!24}a^{6}-\frac{30\!\cdots\!31}{83\!\cdots\!20}a^{5}-\frac{19\!\cdots\!11}{16\!\cdots\!84}a^{4}+\frac{36\!\cdots\!47}{62\!\cdots\!65}a^{3}-\frac{14\!\cdots\!07}{12\!\cdots\!30}a^{2}+\frac{50\!\cdots\!83}{69\!\cdots\!85}a+\frac{80\!\cdots\!47}{23\!\cdots\!95}$, $\frac{30\!\cdots\!89}{65\!\cdots\!20}a^{21}-\frac{32\!\cdots\!37}{65\!\cdots\!20}a^{20}-\frac{55\!\cdots\!13}{54\!\cdots\!86}a^{19}+\frac{14\!\cdots\!83}{27\!\cdots\!30}a^{18}-\frac{31\!\cdots\!29}{21\!\cdots\!40}a^{17}+\frac{41\!\cdots\!07}{21\!\cdots\!40}a^{16}+\frac{10\!\cdots\!19}{36\!\cdots\!40}a^{15}-\frac{69\!\cdots\!31}{36\!\cdots\!40}a^{14}+\frac{25\!\cdots\!39}{21\!\cdots\!40}a^{13}-\frac{41\!\cdots\!57}{43\!\cdots\!88}a^{12}-\frac{55\!\cdots\!89}{38\!\cdots\!90}a^{11}+\frac{70\!\cdots\!43}{54\!\cdots\!60}a^{10}-\frac{23\!\cdots\!99}{43\!\cdots\!88}a^{9}-\frac{60\!\cdots\!83}{21\!\cdots\!40}a^{8}+\frac{25\!\cdots\!61}{10\!\cdots\!20}a^{7}-\frac{35\!\cdots\!07}{12\!\cdots\!88}a^{6}-\frac{20\!\cdots\!13}{10\!\cdots\!20}a^{5}+\frac{10\!\cdots\!51}{10\!\cdots\!72}a^{4}-\frac{84\!\cdots\!32}{40\!\cdots\!45}a^{3}+\frac{19\!\cdots\!67}{81\!\cdots\!90}a^{2}-\frac{41\!\cdots\!88}{45\!\cdots\!05}a-\frac{28\!\cdots\!57}{15\!\cdots\!35}$, $\frac{68\!\cdots\!69}{32\!\cdots\!88}a^{21}-\frac{42\!\cdots\!73}{48\!\cdots\!32}a^{20}-\frac{46\!\cdots\!45}{24\!\cdots\!16}a^{19}+\frac{10\!\cdots\!05}{48\!\cdots\!32}a^{18}-\frac{47\!\cdots\!17}{96\!\cdots\!64}a^{17}+\frac{10\!\cdots\!75}{24\!\cdots\!16}a^{16}+\frac{70\!\cdots\!08}{10\!\cdots\!09}a^{15}-\frac{18\!\cdots\!05}{24\!\cdots\!16}a^{14}-\frac{29\!\cdots\!11}{96\!\cdots\!64}a^{13}-\frac{41\!\cdots\!13}{48\!\cdots\!32}a^{12}-\frac{21\!\cdots\!16}{61\!\cdots\!23}a^{11}+\frac{88\!\cdots\!65}{16\!\cdots\!44}a^{10}+\frac{44\!\cdots\!47}{32\!\cdots\!88}a^{9}-\frac{34\!\cdots\!73}{12\!\cdots\!08}a^{8}+\frac{30\!\cdots\!49}{12\!\cdots\!08}a^{7}+\frac{17\!\cdots\!95}{31\!\cdots\!08}a^{6}-\frac{10\!\cdots\!45}{60\!\cdots\!54}a^{5}-\frac{33\!\cdots\!75}{30\!\cdots\!27}a^{4}+\frac{37\!\cdots\!63}{20\!\cdots\!18}a^{3}-\frac{16\!\cdots\!33}{30\!\cdots\!27}a^{2}+\frac{22\!\cdots\!34}{30\!\cdots\!27}a-\frac{38\!\cdots\!13}{10\!\cdots\!09}$, $\frac{55\!\cdots\!35}{18\!\cdots\!52}a^{21}-\frac{23\!\cdots\!47}{18\!\cdots\!52}a^{20}-\frac{40\!\cdots\!67}{31\!\cdots\!92}a^{19}+\frac{14\!\cdots\!67}{31\!\cdots\!92}a^{18}-\frac{55\!\cdots\!01}{62\!\cdots\!84}a^{17}+\frac{21\!\cdots\!45}{62\!\cdots\!84}a^{16}+\frac{14\!\cdots\!83}{51\!\cdots\!32}a^{15}-\frac{46\!\cdots\!22}{12\!\cdots\!83}a^{14}-\frac{25\!\cdots\!45}{62\!\cdots\!84}a^{13}-\frac{31\!\cdots\!83}{62\!\cdots\!84}a^{12}-\frac{22\!\cdots\!45}{31\!\cdots\!92}a^{11}+\frac{11\!\cdots\!21}{44\!\cdots\!56}a^{10}+\frac{15\!\cdots\!19}{88\!\cdots\!12}a^{9}-\frac{91\!\cdots\!45}{62\!\cdots\!84}a^{8}+\frac{48\!\cdots\!67}{38\!\cdots\!49}a^{7}+\frac{30\!\cdots\!26}{10\!\cdots\!77}a^{6}-\frac{11\!\cdots\!47}{15\!\cdots\!96}a^{5}-\frac{49\!\cdots\!90}{38\!\cdots\!49}a^{4}+\frac{92\!\cdots\!90}{11\!\cdots\!47}a^{3}-\frac{44\!\cdots\!13}{23\!\cdots\!94}a^{2}+\frac{17\!\cdots\!65}{12\!\cdots\!83}a+\frac{10\!\cdots\!43}{43\!\cdots\!61}$, $\frac{11\!\cdots\!05}{43\!\cdots\!88}a^{21}-\frac{10\!\cdots\!29}{43\!\cdots\!88}a^{20}-\frac{70\!\cdots\!79}{72\!\cdots\!48}a^{19}+\frac{22\!\cdots\!59}{72\!\cdots\!48}a^{18}-\frac{11\!\cdots\!91}{14\!\cdots\!96}a^{17}+\frac{14\!\cdots\!07}{14\!\cdots\!96}a^{16}+\frac{18\!\cdots\!07}{60\!\cdots\!54}a^{15}-\frac{13\!\cdots\!83}{12\!\cdots\!08}a^{14}+\frac{84\!\cdots\!25}{14\!\cdots\!96}a^{13}-\frac{63\!\cdots\!29}{14\!\cdots\!96}a^{12}-\frac{19\!\cdots\!71}{10\!\cdots\!64}a^{11}+\frac{56\!\cdots\!95}{72\!\cdots\!48}a^{10}-\frac{37\!\cdots\!77}{14\!\cdots\!96}a^{9}-\frac{29\!\cdots\!71}{14\!\cdots\!96}a^{8}+\frac{51\!\cdots\!89}{36\!\cdots\!24}a^{7}-\frac{16\!\cdots\!51}{14\!\cdots\!36}a^{6}-\frac{52\!\cdots\!99}{36\!\cdots\!24}a^{5}+\frac{11\!\cdots\!53}{18\!\cdots\!62}a^{4}-\frac{37\!\cdots\!04}{27\!\cdots\!43}a^{3}+\frac{83\!\cdots\!95}{54\!\cdots\!86}a^{2}-\frac{17\!\cdots\!19}{30\!\cdots\!27}a-\frac{12\!\cdots\!49}{10\!\cdots\!09}$, $\frac{79\!\cdots\!37}{13\!\cdots\!40}a^{21}-\frac{44\!\cdots\!13}{65\!\cdots\!20}a^{20}-\frac{26\!\cdots\!05}{27\!\cdots\!43}a^{19}+\frac{14\!\cdots\!01}{21\!\cdots\!40}a^{18}-\frac{81\!\cdots\!17}{43\!\cdots\!80}a^{17}+\frac{13\!\cdots\!77}{54\!\cdots\!60}a^{16}+\frac{88\!\cdots\!91}{36\!\cdots\!40}a^{15}-\frac{87\!\cdots\!09}{36\!\cdots\!40}a^{14}+\frac{67\!\cdots\!57}{43\!\cdots\!80}a^{13}-\frac{55\!\cdots\!99}{43\!\cdots\!88}a^{12}-\frac{15\!\cdots\!59}{15\!\cdots\!60}a^{11}+\frac{34\!\cdots\!03}{21\!\cdots\!40}a^{10}-\frac{61\!\cdots\!75}{87\!\cdots\!76}a^{9}-\frac{34\!\cdots\!61}{10\!\cdots\!20}a^{8}+\frac{30\!\cdots\!49}{10\!\cdots\!20}a^{7}-\frac{47\!\cdots\!99}{12\!\cdots\!88}a^{6}-\frac{58\!\cdots\!43}{27\!\cdots\!30}a^{5}+\frac{12\!\cdots\!79}{10\!\cdots\!72}a^{4}-\frac{20\!\cdots\!91}{81\!\cdots\!90}a^{3}+\frac{11\!\cdots\!44}{40\!\cdots\!45}a^{2}-\frac{49\!\cdots\!32}{45\!\cdots\!05}a-\frac{34\!\cdots\!53}{15\!\cdots\!35}$, $\frac{36\!\cdots\!67}{14\!\cdots\!60}a^{21}+\frac{28\!\cdots\!67}{72\!\cdots\!80}a^{20}-\frac{27\!\cdots\!91}{12\!\cdots\!08}a^{19}+\frac{36\!\cdots\!81}{24\!\cdots\!60}a^{18}-\frac{24\!\cdots\!47}{48\!\cdots\!20}a^{17}+\frac{12\!\cdots\!03}{40\!\cdots\!60}a^{16}+\frac{38\!\cdots\!31}{40\!\cdots\!60}a^{15}-\frac{30\!\cdots\!71}{60\!\cdots\!40}a^{14}-\frac{12\!\cdots\!13}{48\!\cdots\!20}a^{13}-\frac{10\!\cdots\!51}{48\!\cdots\!32}a^{12}-\frac{28\!\cdots\!73}{57\!\cdots\!80}a^{11}+\frac{10\!\cdots\!23}{24\!\cdots\!60}a^{10}+\frac{11\!\cdots\!61}{32\!\cdots\!88}a^{9}-\frac{38\!\cdots\!11}{20\!\cdots\!80}a^{8}-\frac{22\!\cdots\!97}{40\!\cdots\!60}a^{7}+\frac{33\!\cdots\!93}{77\!\cdots\!02}a^{6}-\frac{32\!\cdots\!87}{12\!\cdots\!80}a^{5}-\frac{16\!\cdots\!87}{60\!\cdots\!54}a^{4}+\frac{47\!\cdots\!52}{45\!\cdots\!05}a^{3}-\frac{15\!\cdots\!47}{90\!\cdots\!10}a^{2}+\frac{41\!\cdots\!58}{50\!\cdots\!45}a+\frac{79\!\cdots\!91}{50\!\cdots\!45}$, $\frac{44\!\cdots\!67}{13\!\cdots\!40}a^{21}-\frac{10\!\cdots\!27}{16\!\cdots\!80}a^{20}-\frac{57\!\cdots\!01}{27\!\cdots\!43}a^{19}+\frac{18\!\cdots\!81}{21\!\cdots\!40}a^{18}-\frac{51\!\cdots\!87}{43\!\cdots\!80}a^{17}+\frac{39\!\cdots\!33}{21\!\cdots\!40}a^{16}+\frac{52\!\cdots\!23}{18\!\cdots\!20}a^{15}-\frac{14\!\cdots\!28}{45\!\cdots\!05}a^{14}+\frac{64\!\cdots\!67}{43\!\cdots\!80}a^{13}+\frac{85\!\cdots\!25}{21\!\cdots\!44}a^{12}-\frac{50\!\cdots\!79}{15\!\cdots\!60}a^{11}+\frac{38\!\cdots\!83}{21\!\cdots\!40}a^{10}-\frac{81\!\cdots\!33}{87\!\cdots\!76}a^{9}-\frac{24\!\cdots\!47}{21\!\cdots\!40}a^{8}+\frac{36\!\cdots\!97}{54\!\cdots\!60}a^{7}+\frac{85\!\cdots\!05}{42\!\cdots\!08}a^{6}-\frac{19\!\cdots\!97}{10\!\cdots\!20}a^{5}+\frac{49\!\cdots\!85}{10\!\cdots\!72}a^{4}+\frac{65\!\cdots\!37}{40\!\cdots\!45}a^{3}-\frac{27\!\cdots\!27}{81\!\cdots\!90}a^{2}+\frac{77\!\cdots\!18}{45\!\cdots\!05}a+\frac{49\!\cdots\!27}{15\!\cdots\!35}$, $\frac{79\!\cdots\!71}{13\!\cdots\!40}a^{21}-\frac{10\!\cdots\!44}{40\!\cdots\!45}a^{20}-\frac{24\!\cdots\!93}{43\!\cdots\!88}a^{19}+\frac{13\!\cdots\!33}{21\!\cdots\!40}a^{18}-\frac{61\!\cdots\!31}{43\!\cdots\!80}a^{17}+\frac{28\!\cdots\!99}{21\!\cdots\!40}a^{16}+\frac{14\!\cdots\!01}{72\!\cdots\!80}a^{15}-\frac{81\!\cdots\!87}{36\!\cdots\!40}a^{14}-\frac{38\!\cdots\!89}{43\!\cdots\!80}a^{13}-\frac{53\!\cdots\!97}{21\!\cdots\!44}a^{12}-\frac{31\!\cdots\!59}{31\!\cdots\!20}a^{11}+\frac{34\!\cdots\!19}{21\!\cdots\!40}a^{10}+\frac{34\!\cdots\!75}{87\!\cdots\!76}a^{9}-\frac{17\!\cdots\!61}{21\!\cdots\!40}a^{8}+\frac{15\!\cdots\!89}{21\!\cdots\!40}a^{7}+\frac{13\!\cdots\!55}{84\!\cdots\!16}a^{6}-\frac{28\!\cdots\!83}{54\!\cdots\!60}a^{5}-\frac{34\!\cdots\!35}{10\!\cdots\!72}a^{4}+\frac{43\!\cdots\!87}{81\!\cdots\!90}a^{3}-\frac{12\!\cdots\!81}{81\!\cdots\!90}a^{2}+\frac{95\!\cdots\!49}{45\!\cdots\!05}a-\frac{16\!\cdots\!49}{15\!\cdots\!35}$, $\frac{18\!\cdots\!59}{54\!\cdots\!60}a^{21}-\frac{64\!\cdots\!23}{21\!\cdots\!40}a^{20}-\frac{96\!\cdots\!73}{72\!\cdots\!48}a^{19}+\frac{63\!\cdots\!89}{18\!\cdots\!20}a^{18}-\frac{34\!\cdots\!43}{36\!\cdots\!40}a^{17}+\frac{88\!\cdots\!63}{72\!\cdots\!80}a^{16}+\frac{26\!\cdots\!53}{60\!\cdots\!40}a^{15}-\frac{19\!\cdots\!43}{15\!\cdots\!35}a^{14}+\frac{49\!\cdots\!57}{90\!\cdots\!10}a^{13}-\frac{84\!\cdots\!67}{14\!\cdots\!96}a^{12}-\frac{99\!\cdots\!59}{51\!\cdots\!20}a^{11}+\frac{79\!\cdots\!81}{90\!\cdots\!10}a^{10}-\frac{13\!\cdots\!09}{72\!\cdots\!48}a^{9}-\frac{18\!\cdots\!67}{72\!\cdots\!80}a^{8}+\frac{91\!\cdots\!71}{90\!\cdots\!10}a^{7}+\frac{38\!\cdots\!49}{35\!\cdots\!59}a^{6}-\frac{20\!\cdots\!71}{18\!\cdots\!20}a^{5}+\frac{86\!\cdots\!37}{36\!\cdots\!24}a^{4}-\frac{67\!\cdots\!28}{13\!\cdots\!15}a^{3}+\frac{11\!\cdots\!99}{13\!\cdots\!15}a^{2}-\frac{24\!\cdots\!34}{50\!\cdots\!45}a-\frac{45\!\cdots\!83}{50\!\cdots\!45}$
| 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: | \( 739930020231000000000000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 739930020231000000000000000 \cdot 1}{2\cdot\sqrt{254171926510889578368436897006059571070510600002237220914603229184}}\cr\approx \mathstrut & 46.2355500355209 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 - 92*x^20 + 978*x^19 - 23031*x^18 + 208542*x^17 + 3327324*x^16 - 36397800*x^15 - 19963545*x^14 - 406653528*x^13 - 16539946488*x^12 + 256034067942*x^11 + 682357346031*x^10 - 13494358927986*x^9 + 9822827363220*x^8 + 261815058820176*x^7 - 813912422758992*x^6 - 647257611403968*x^5 + 8751031478070272*x^4 - 24165278101734848*x^3 + 30990808995877088*x^2 - 13115399746724928*x - 2634468582334464)
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^22 - 4*x^21 - 92*x^20 + 978*x^19 - 23031*x^18 + 208542*x^17 + 3327324*x^16 - 36397800*x^15 - 19963545*x^14 - 406653528*x^13 - 16539946488*x^12 + 256034067942*x^11 + 682357346031*x^10 - 13494358927986*x^9 + 9822827363220*x^8 + 261815058820176*x^7 - 813912422758992*x^6 - 647257611403968*x^5 + 8751031478070272*x^4 - 24165278101734848*x^3 + 30990808995877088*x^2 - 13115399746724928*x - 2634468582334464, 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^22 - 4*x^21 - 92*x^20 + 978*x^19 - 23031*x^18 + 208542*x^17 + 3327324*x^16 - 36397800*x^15 - 19963545*x^14 - 406653528*x^13 - 16539946488*x^12 + 256034067942*x^11 + 682357346031*x^10 - 13494358927986*x^9 + 9822827363220*x^8 + 261815058820176*x^7 - 813912422758992*x^6 - 647257611403968*x^5 + 8751031478070272*x^4 - 24165278101734848*x^3 + 30990808995877088*x^2 - 13115399746724928*x - 2634468582334464);
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^22 - 4*x^21 - 92*x^20 + 978*x^19 - 23031*x^18 + 208542*x^17 + 3327324*x^16 - 36397800*x^15 - 19963545*x^14 - 406653528*x^13 - 16539946488*x^12 + 256034067942*x^11 + 682357346031*x^10 - 13494358927986*x^9 + 9822827363220*x^8 + 261815058820176*x^7 - 813912422758992*x^6 - 647257611403968*x^5 + 8751031478070272*x^4 - 24165278101734848*x^3 + 30990808995877088*x^2 - 13115399746724928*x - 2634468582334464);
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
$C_2^{10}.A_{11}$ (as 22T49):
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 20437401600 |
| The 200 conjugacy class representatives for $C_2^{10}.A_{11}$ |
| Character table for $C_2^{10}.A_{11}$ |
Intermediate fields
| 11.7.63019333158425674204677255696384.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 2.8.16.69 | $x^{8} + 2 x^{6} + 4 x^{4} + 4 x + 2$ | $8$ | $1$ | $16$ | $C_4\wr C_2$ | $[2, 2, 5/2]^{4}$ | |
| 2.12.14.2 | $x^{12} + 2 x^{4} + 2 x^{3} + 2$ | $12$ | $1$ | $14$ | 12T27 | $[4/3, 4/3]_{3}^{4}$ | |
|
\(3\)
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.9.13.2 | $x^{9} + 6 x^{5} + 3 x^{3} + 3$ | $9$ | $1$ | $13$ | $C_3^2 : D_{6} $ | $[3/2, 3/2, 5/3]_{2}^{2}$ | |
| 3.9.13.2 | $x^{9} + 6 x^{5} + 3 x^{3} + 3$ | $9$ | $1$ | $13$ | $C_3^2 : D_{6} $ | $[3/2, 3/2, 5/3]_{2}^{2}$ | |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
|
\(23\)
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.7.0.1 | $x^{7} + 21 x + 18$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 23.7.0.1 | $x^{7} + 21 x + 18$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
|
\(137\)
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 137.5.4.1 | $x^{5} + 137$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 137.5.4.1 | $x^{5} + 137$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 137.10.8.1 | $x^{10} + 655 x^{9} + 171625 x^{8} + 22488770 x^{7} + 1474044185 x^{6} + 38714410755 x^{5} + 4422222290 x^{4} + 225901280 x^{3} + 3082903315 x^{2} + 201591447850 x + 5280771081809$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |