Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 175*x^36 + 9273*x^32 + 161966*x^28 + 849466*x^24 + 1781941*x^20 + 1558024*x^16 + 496503*x^12 + 42966*x^8 + 1085*x^4 + 1)
gp: K = bnfinit(y^40 + 175*y^36 + 9273*y^32 + 161966*y^28 + 849466*y^24 + 1781941*y^20 + 1558024*y^16 + 496503*y^12 + 42966*y^8 + 1085*y^4 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 + 175*x^36 + 9273*x^32 + 161966*x^28 + 849466*x^24 + 1781941*x^20 + 1558024*x^16 + 496503*x^12 + 42966*x^8 + 1085*x^4 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 + 175*x^36 + 9273*x^32 + 161966*x^28 + 849466*x^24 + 1781941*x^20 + 1558024*x^16 + 496503*x^12 + 42966*x^8 + 1085*x^4 + 1)
\( x^{40} + 175 x^{36} + 9273 x^{32} + 161966 x^{28} + 849466 x^{24} + 1781941 x^{20} + 1558024 x^{16} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $40$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 20]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(243425272531849218182083207491041757811794582804889600000000000000000000\)
\(\medspace = 2^{80}\cdot 5^{20}\cdot 11^{32}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(60.91\) | 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: | $2^{2}5^{1/2}11^{4/5}\approx 60.905868659110936$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $40$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(440=2^{3}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(389,·)$, $\chi_{440}(9,·)$, $\chi_{440}(269,·)$, $\chi_{440}(399,·)$, $\chi_{440}(401,·)$, $\chi_{440}(279,·)$, $\chi_{440}(411,·)$, $\chi_{440}(31,·)$, $\chi_{440}(289,·)$, $\chi_{440}(291,·)$, $\chi_{440}(421,·)$, $\chi_{440}(169,·)$, $\chi_{440}(71,·)$, $\chi_{440}(301,·)$, $\chi_{440}(49,·)$, $\chi_{440}(179,·)$, $\chi_{440}(309,·)$, $\chi_{440}(311,·)$, $\chi_{440}(159,·)$, $\chi_{440}(181,·)$, $\chi_{440}(69,·)$, $\chi_{440}(199,·)$, $\chi_{440}(201,·)$, $\chi_{440}(331,·)$, $\chi_{440}(141,·)$, $\chi_{440}(81,·)$, $\chi_{440}(339,·)$, $\chi_{440}(89,·)$, $\chi_{440}(91,·)$, $\chi_{440}(221,·)$, $\chi_{440}(419,·)$, $\chi_{440}(59,·)$, $\chi_{440}(229,·)$, $\chi_{440}(379,·)$, $\chi_{440}(361,·)$, $\chi_{440}(191,·)$, $\chi_{440}(111,·)$, $\chi_{440}(119,·)$, $\chi_{440}(251,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{524288}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{3}a^{20}-\frac{1}{3}a^{16}+\frac{1}{3}a^{12}-\frac{1}{3}a^{8}-\frac{1}{3}a^{4}-\frac{1}{3}$, $\frac{1}{3}a^{21}-\frac{1}{3}a^{17}+\frac{1}{3}a^{13}-\frac{1}{3}a^{9}-\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{3}a^{22}-\frac{1}{3}a^{18}+\frac{1}{3}a^{14}-\frac{1}{3}a^{10}-\frac{1}{3}a^{6}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{23}-\frac{1}{3}a^{19}+\frac{1}{3}a^{15}-\frac{1}{3}a^{11}-\frac{1}{3}a^{7}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{24}+\frac{1}{3}a^{8}+\frac{1}{3}a^{4}-\frac{1}{3}$, $\frac{1}{3}a^{25}+\frac{1}{3}a^{9}+\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{3}a^{26}+\frac{1}{3}a^{10}+\frac{1}{3}a^{6}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{27}+\frac{1}{3}a^{11}+\frac{1}{3}a^{7}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{28}+\frac{1}{3}a^{12}+\frac{1}{3}a^{8}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{29}+\frac{1}{3}a^{13}+\frac{1}{3}a^{9}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{30}+\frac{1}{3}a^{14}+\frac{1}{3}a^{10}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{31}+\frac{1}{3}a^{15}+\frac{1}{3}a^{11}-\frac{1}{3}a^{7}$, $\frac{1}{269679}a^{32}-\frac{2840}{89893}a^{28}-\frac{36257}{269679}a^{24}-\frac{1414}{89893}a^{20}-\frac{115250}{269679}a^{16}-\frac{130310}{269679}a^{12}-\frac{7664}{89893}a^{8}+\frac{17023}{269679}a^{4}-\frac{56974}{269679}$, $\frac{1}{269679}a^{33}-\frac{2840}{89893}a^{29}-\frac{36257}{269679}a^{25}-\frac{1414}{89893}a^{21}-\frac{115250}{269679}a^{17}-\frac{130310}{269679}a^{13}-\frac{7664}{89893}a^{9}+\frac{17023}{269679}a^{5}-\frac{56974}{269679}a$, $\frac{1}{269679}a^{34}-\frac{2840}{89893}a^{30}-\frac{36257}{269679}a^{26}-\frac{1414}{89893}a^{22}-\frac{115250}{269679}a^{18}-\frac{130310}{269679}a^{14}-\frac{7664}{89893}a^{10}+\frac{17023}{269679}a^{6}-\frac{56974}{269679}a^{2}$, $\frac{1}{269679}a^{35}-\frac{2840}{89893}a^{31}-\frac{36257}{269679}a^{27}-\frac{1414}{89893}a^{23}-\frac{115250}{269679}a^{19}-\frac{130310}{269679}a^{15}-\frac{7664}{89893}a^{11}+\frac{17023}{269679}a^{7}-\frac{56974}{269679}a^{3}$, $\frac{1}{12\!\cdots\!81}a^{36}+\frac{16\!\cdots\!92}{12\!\cdots\!81}a^{32}-\frac{92\!\cdots\!16}{12\!\cdots\!81}a^{28}+\frac{17\!\cdots\!64}{12\!\cdots\!81}a^{24}+\frac{12\!\cdots\!20}{12\!\cdots\!81}a^{20}-\frac{39\!\cdots\!06}{12\!\cdots\!81}a^{16}+\frac{15\!\cdots\!87}{12\!\cdots\!81}a^{12}+\frac{57\!\cdots\!74}{12\!\cdots\!81}a^{8}+\frac{86\!\cdots\!61}{41\!\cdots\!27}a^{4}+\frac{28\!\cdots\!36}{12\!\cdots\!81}$, $\frac{1}{12\!\cdots\!81}a^{37}+\frac{16\!\cdots\!92}{12\!\cdots\!81}a^{33}-\frac{92\!\cdots\!16}{12\!\cdots\!81}a^{29}+\frac{17\!\cdots\!64}{12\!\cdots\!81}a^{25}+\frac{12\!\cdots\!20}{12\!\cdots\!81}a^{21}-\frac{39\!\cdots\!06}{12\!\cdots\!81}a^{17}+\frac{15\!\cdots\!87}{12\!\cdots\!81}a^{13}+\frac{57\!\cdots\!74}{12\!\cdots\!81}a^{9}+\frac{86\!\cdots\!61}{41\!\cdots\!27}a^{5}+\frac{28\!\cdots\!36}{12\!\cdots\!81}a$, $\frac{1}{12\!\cdots\!81}a^{38}+\frac{16\!\cdots\!92}{12\!\cdots\!81}a^{34}-\frac{92\!\cdots\!16}{12\!\cdots\!81}a^{30}+\frac{17\!\cdots\!64}{12\!\cdots\!81}a^{26}+\frac{12\!\cdots\!20}{12\!\cdots\!81}a^{22}-\frac{39\!\cdots\!06}{12\!\cdots\!81}a^{18}+\frac{15\!\cdots\!87}{12\!\cdots\!81}a^{14}+\frac{57\!\cdots\!74}{12\!\cdots\!81}a^{10}+\frac{86\!\cdots\!61}{41\!\cdots\!27}a^{6}+\frac{28\!\cdots\!36}{12\!\cdots\!81}a^{2}$, $\frac{1}{12\!\cdots\!81}a^{39}+\frac{16\!\cdots\!92}{12\!\cdots\!81}a^{35}-\frac{92\!\cdots\!16}{12\!\cdots\!81}a^{31}+\frac{17\!\cdots\!64}{12\!\cdots\!81}a^{27}+\frac{12\!\cdots\!20}{12\!\cdots\!81}a^{23}-\frac{39\!\cdots\!06}{12\!\cdots\!81}a^{19}+\frac{15\!\cdots\!87}{12\!\cdots\!81}a^{15}+\frac{57\!\cdots\!74}{12\!\cdots\!81}a^{11}+\frac{86\!\cdots\!61}{41\!\cdots\!27}a^{7}+\frac{28\!\cdots\!36}{12\!\cdots\!81}a^{3}$
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
$C_{155}\times C_{155}$, which has order $24025$ (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: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{305129630405970959162}{37260142057183116579951} a^{37} + \frac{17799580877135952392828}{12420047352394372193317} a^{33} + \frac{2829650962622683518719896}{37260142057183116579951} a^{29} + \frac{16476744417685704651858369}{12420047352394372193317} a^{25} + \frac{86452717278717868054754281}{12420047352394372193317} a^{21} + \frac{544445021764559724516187729}{37260142057183116579951} a^{17} + \frac{158812680314314296283165743}{12420047352394372193317} a^{13} + \frac{50549700223384721143289054}{12420047352394372193317} a^{9} + \frac{12680131192389703024317364}{37260142057183116579951} a^{5} + \frac{223816153561166017405595}{37260142057183116579951} a \)
(order $8$)
| 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: |
$\frac{37\!\cdots\!96}{12\!\cdots\!81}a^{37}+\frac{21\!\cdots\!36}{41\!\cdots\!27}a^{33}+\frac{34\!\cdots\!77}{12\!\cdots\!81}a^{29}+\frac{19\!\cdots\!44}{41\!\cdots\!27}a^{25}+\frac{10\!\cdots\!52}{41\!\cdots\!27}a^{21}+\frac{58\!\cdots\!84}{12\!\cdots\!81}a^{17}+\frac{13\!\cdots\!04}{41\!\cdots\!27}a^{13}+\frac{17\!\cdots\!64}{41\!\cdots\!27}a^{9}-\frac{20\!\cdots\!25}{12\!\cdots\!81}a^{5}-\frac{11\!\cdots\!68}{12\!\cdots\!81}a$, $\frac{38\!\cdots\!12}{12\!\cdots\!81}a^{39}+\frac{22\!\cdots\!02}{41\!\cdots\!27}a^{35}+\frac{35\!\cdots\!84}{12\!\cdots\!81}a^{31}+\frac{62\!\cdots\!70}{12\!\cdots\!81}a^{27}+\frac{32\!\cdots\!02}{12\!\cdots\!81}a^{23}+\frac{68\!\cdots\!82}{12\!\cdots\!81}a^{19}+\frac{19\!\cdots\!75}{41\!\cdots\!27}a^{15}+\frac{61\!\cdots\!42}{41\!\cdots\!27}a^{11}+\frac{14\!\cdots\!28}{12\!\cdots\!81}a^{7}+\frac{31\!\cdots\!31}{12\!\cdots\!81}a^{3}$, $\frac{88\!\cdots\!27}{41\!\cdots\!27}a^{39}+\frac{46\!\cdots\!88}{12\!\cdots\!81}a^{35}+\frac{24\!\cdots\!78}{12\!\cdots\!81}a^{31}+\frac{42\!\cdots\!76}{12\!\cdots\!81}a^{27}+\frac{22\!\cdots\!46}{12\!\cdots\!81}a^{23}+\frac{15\!\cdots\!80}{41\!\cdots\!27}a^{19}+\frac{40\!\cdots\!24}{12\!\cdots\!81}a^{15}+\frac{12\!\cdots\!22}{12\!\cdots\!81}a^{11}+\frac{94\!\cdots\!30}{12\!\cdots\!81}a^{7}+\frac{65\!\cdots\!40}{41\!\cdots\!27}a^{3}$, $\frac{11\!\cdots\!99}{41\!\cdots\!27}a^{37}+\frac{20\!\cdots\!93}{41\!\cdots\!27}a^{33}+\frac{10\!\cdots\!27}{41\!\cdots\!27}a^{29}+\frac{55\!\cdots\!24}{12\!\cdots\!81}a^{25}+\frac{28\!\cdots\!48}{12\!\cdots\!81}a^{21}+\frac{60\!\cdots\!29}{12\!\cdots\!81}a^{17}+\frac{51\!\cdots\!04}{12\!\cdots\!81}a^{13}+\frac{51\!\cdots\!99}{41\!\cdots\!27}a^{9}+\frac{33\!\cdots\!48}{41\!\cdots\!27}a^{5}+\frac{13\!\cdots\!05}{12\!\cdots\!81}a$, $\frac{11\!\cdots\!33}{41\!\cdots\!27}a^{36}+\frac{19\!\cdots\!00}{41\!\cdots\!27}a^{32}+\frac{10\!\cdots\!19}{41\!\cdots\!27}a^{28}+\frac{18\!\cdots\!34}{41\!\cdots\!27}a^{24}+\frac{94\!\cdots\!71}{41\!\cdots\!27}a^{20}+\frac{19\!\cdots\!01}{41\!\cdots\!27}a^{16}+\frac{16\!\cdots\!93}{41\!\cdots\!27}a^{12}+\frac{43\!\cdots\!10}{41\!\cdots\!27}a^{8}+\frac{17\!\cdots\!13}{41\!\cdots\!27}a^{4}-\frac{10\!\cdots\!15}{41\!\cdots\!27}$, $\frac{37\!\cdots\!48}{41\!\cdots\!27}a^{38}+\frac{19\!\cdots\!50}{12\!\cdots\!81}a^{34}+\frac{10\!\cdots\!45}{12\!\cdots\!81}a^{30}+\frac{18\!\cdots\!20}{12\!\cdots\!81}a^{26}+\frac{94\!\cdots\!22}{12\!\cdots\!81}a^{22}+\frac{19\!\cdots\!84}{12\!\cdots\!81}a^{18}+\frac{16\!\cdots\!30}{12\!\cdots\!81}a^{14}+\frac{15\!\cdots\!92}{41\!\cdots\!27}a^{10}+\frac{20\!\cdots\!58}{12\!\cdots\!81}a^{6}-\frac{14\!\cdots\!63}{12\!\cdots\!81}a^{2}$, $\frac{92\!\cdots\!79}{12\!\cdots\!81}a^{39}+\frac{16\!\cdots\!92}{12\!\cdots\!81}a^{35}+\frac{28\!\cdots\!59}{41\!\cdots\!27}a^{31}+\frac{49\!\cdots\!00}{41\!\cdots\!27}a^{27}+\frac{78\!\cdots\!34}{12\!\cdots\!81}a^{23}+\frac{54\!\cdots\!73}{41\!\cdots\!27}a^{19}+\frac{14\!\cdots\!77}{12\!\cdots\!81}a^{15}+\frac{44\!\cdots\!58}{12\!\cdots\!81}a^{11}+\frac{11\!\cdots\!06}{41\!\cdots\!27}a^{7}+\frac{24\!\cdots\!69}{41\!\cdots\!27}a^{3}$, $\frac{25\!\cdots\!88}{41\!\cdots\!27}a^{36}+\frac{13\!\cdots\!96}{12\!\cdots\!81}a^{32}+\frac{69\!\cdots\!75}{12\!\cdots\!81}a^{28}+\frac{12\!\cdots\!25}{12\!\cdots\!81}a^{24}+\frac{63\!\cdots\!28}{12\!\cdots\!81}a^{20}+\frac{43\!\cdots\!32}{41\!\cdots\!27}a^{16}+\frac{11\!\cdots\!62}{12\!\cdots\!81}a^{12}+\frac{32\!\cdots\!21}{12\!\cdots\!81}a^{8}+\frac{19\!\cdots\!90}{12\!\cdots\!81}a^{4}+\frac{11\!\cdots\!10}{41\!\cdots\!27}$, $\frac{84\!\cdots\!67}{12\!\cdots\!81}a^{39}+\frac{49\!\cdots\!44}{41\!\cdots\!27}a^{35}+\frac{78\!\cdots\!99}{12\!\cdots\!81}a^{31}+\frac{13\!\cdots\!43}{12\!\cdots\!81}a^{27}+\frac{70\!\cdots\!30}{12\!\cdots\!81}a^{23}+\frac{14\!\cdots\!24}{12\!\cdots\!81}a^{19}+\frac{38\!\cdots\!14}{41\!\cdots\!27}a^{15}+\frac{97\!\cdots\!04}{41\!\cdots\!27}a^{11}+\frac{86\!\cdots\!28}{12\!\cdots\!81}a^{7}-\frac{69\!\cdots\!33}{12\!\cdots\!81}a^{3}$, $\frac{24\!\cdots\!69}{12\!\cdots\!17}a^{39}-\frac{25\!\cdots\!64}{99\!\cdots\!17}a^{38}+\frac{13\!\cdots\!96}{37\!\cdots\!51}a^{35}-\frac{13\!\cdots\!46}{29\!\cdots\!51}a^{34}+\frac{68\!\cdots\!76}{37\!\cdots\!51}a^{31}-\frac{23\!\cdots\!53}{99\!\cdots\!17}a^{30}+\frac{12\!\cdots\!31}{37\!\cdots\!51}a^{27}-\frac{12\!\cdots\!74}{29\!\cdots\!51}a^{26}+\frac{63\!\cdots\!30}{37\!\cdots\!51}a^{23}-\frac{64\!\cdots\!62}{29\!\cdots\!51}a^{22}+\frac{44\!\cdots\!06}{12\!\cdots\!17}a^{19}-\frac{13\!\cdots\!33}{29\!\cdots\!51}a^{18}+\frac{38\!\cdots\!27}{12\!\cdots\!17}a^{15}-\frac{39\!\cdots\!68}{99\!\cdots\!17}a^{14}+\frac{36\!\cdots\!16}{37\!\cdots\!51}a^{11}-\frac{37\!\cdots\!66}{29\!\cdots\!51}a^{10}+\frac{10\!\cdots\!98}{12\!\cdots\!17}a^{7}-\frac{11\!\cdots\!60}{99\!\cdots\!17}a^{6}+\frac{23\!\cdots\!16}{12\!\cdots\!17}a^{3}-\frac{99\!\cdots\!84}{29\!\cdots\!51}a^{2}+1$, $\frac{24\!\cdots\!69}{12\!\cdots\!17}a^{39}-\frac{30\!\cdots\!62}{37\!\cdots\!51}a^{37}-\frac{11\!\cdots\!33}{41\!\cdots\!27}a^{36}+\frac{13\!\cdots\!96}{37\!\cdots\!51}a^{35}-\frac{17\!\cdots\!28}{12\!\cdots\!17}a^{33}-\frac{19\!\cdots\!00}{41\!\cdots\!27}a^{32}+\frac{68\!\cdots\!76}{37\!\cdots\!51}a^{31}-\frac{28\!\cdots\!96}{37\!\cdots\!51}a^{29}-\frac{10\!\cdots\!19}{41\!\cdots\!27}a^{28}+\frac{12\!\cdots\!31}{37\!\cdots\!51}a^{27}-\frac{16\!\cdots\!69}{12\!\cdots\!17}a^{25}-\frac{18\!\cdots\!34}{41\!\cdots\!27}a^{24}+\frac{63\!\cdots\!30}{37\!\cdots\!51}a^{23}-\frac{86\!\cdots\!81}{12\!\cdots\!17}a^{21}-\frac{94\!\cdots\!71}{41\!\cdots\!27}a^{20}+\frac{44\!\cdots\!06}{12\!\cdots\!17}a^{19}-\frac{54\!\cdots\!29}{37\!\cdots\!51}a^{17}-\frac{19\!\cdots\!01}{41\!\cdots\!27}a^{16}+\frac{38\!\cdots\!27}{12\!\cdots\!17}a^{15}-\frac{15\!\cdots\!43}{12\!\cdots\!17}a^{13}-\frac{16\!\cdots\!93}{41\!\cdots\!27}a^{12}+\frac{36\!\cdots\!16}{37\!\cdots\!51}a^{11}-\frac{50\!\cdots\!54}{12\!\cdots\!17}a^{9}-\frac{43\!\cdots\!10}{41\!\cdots\!27}a^{8}+\frac{10\!\cdots\!98}{12\!\cdots\!17}a^{7}-\frac{12\!\cdots\!64}{37\!\cdots\!51}a^{5}-\frac{17\!\cdots\!13}{41\!\cdots\!27}a^{4}+\frac{23\!\cdots\!16}{12\!\cdots\!17}a^{3}-\frac{22\!\cdots\!95}{37\!\cdots\!51}a+\frac{10\!\cdots\!15}{41\!\cdots\!27}$, $\frac{88\!\cdots\!27}{41\!\cdots\!27}a^{39}-\frac{25\!\cdots\!64}{99\!\cdots\!17}a^{38}+\frac{46\!\cdots\!88}{12\!\cdots\!81}a^{35}-\frac{13\!\cdots\!46}{29\!\cdots\!51}a^{34}+\frac{24\!\cdots\!78}{12\!\cdots\!81}a^{31}-\frac{23\!\cdots\!53}{99\!\cdots\!17}a^{30}+\frac{42\!\cdots\!76}{12\!\cdots\!81}a^{27}-\frac{12\!\cdots\!74}{29\!\cdots\!51}a^{26}+\frac{22\!\cdots\!46}{12\!\cdots\!81}a^{23}-\frac{64\!\cdots\!62}{29\!\cdots\!51}a^{22}+\frac{15\!\cdots\!80}{41\!\cdots\!27}a^{19}-\frac{13\!\cdots\!33}{29\!\cdots\!51}a^{18}+\frac{40\!\cdots\!24}{12\!\cdots\!81}a^{15}-\frac{39\!\cdots\!68}{99\!\cdots\!17}a^{14}+\frac{12\!\cdots\!22}{12\!\cdots\!81}a^{11}-\frac{37\!\cdots\!66}{29\!\cdots\!51}a^{10}+\frac{94\!\cdots\!30}{12\!\cdots\!81}a^{7}-\frac{11\!\cdots\!60}{99\!\cdots\!17}a^{6}+\frac{65\!\cdots\!40}{41\!\cdots\!27}a^{3}-\frac{99\!\cdots\!84}{29\!\cdots\!51}a^{2}+1$, $\frac{25\!\cdots\!64}{99\!\cdots\!17}a^{38}-\frac{23\!\cdots\!44}{12\!\cdots\!81}a^{37}+\frac{13\!\cdots\!46}{29\!\cdots\!51}a^{34}-\frac{40\!\cdots\!23}{12\!\cdots\!81}a^{33}+\frac{23\!\cdots\!53}{99\!\cdots\!17}a^{30}-\frac{21\!\cdots\!48}{12\!\cdots\!81}a^{29}+\frac{12\!\cdots\!74}{29\!\cdots\!51}a^{26}-\frac{37\!\cdots\!44}{12\!\cdots\!81}a^{25}+\frac{64\!\cdots\!62}{29\!\cdots\!51}a^{22}-\frac{65\!\cdots\!80}{41\!\cdots\!27}a^{21}+\frac{13\!\cdots\!33}{29\!\cdots\!51}a^{18}-\frac{40\!\cdots\!04}{12\!\cdots\!81}a^{17}+\frac{39\!\cdots\!68}{99\!\cdots\!17}a^{14}-\frac{35\!\cdots\!22}{12\!\cdots\!81}a^{13}+\frac{37\!\cdots\!66}{29\!\cdots\!51}a^{10}-\frac{36\!\cdots\!22}{41\!\cdots\!27}a^{9}+\frac{11\!\cdots\!60}{99\!\cdots\!17}a^{6}-\frac{85\!\cdots\!50}{12\!\cdots\!81}a^{5}+\frac{99\!\cdots\!84}{29\!\cdots\!51}a^{2}-\frac{14\!\cdots\!28}{12\!\cdots\!81}a+1$, $\frac{24\!\cdots\!69}{12\!\cdots\!17}a^{39}-\frac{30\!\cdots\!65}{12\!\cdots\!81}a^{38}+\frac{30\!\cdots\!62}{37\!\cdots\!51}a^{37}+\frac{13\!\cdots\!96}{37\!\cdots\!51}a^{35}-\frac{54\!\cdots\!81}{12\!\cdots\!81}a^{34}+\frac{17\!\cdots\!28}{12\!\cdots\!17}a^{33}+\frac{68\!\cdots\!76}{37\!\cdots\!51}a^{31}-\frac{95\!\cdots\!26}{41\!\cdots\!27}a^{30}+\frac{28\!\cdots\!96}{37\!\cdots\!51}a^{29}+\frac{12\!\cdots\!31}{37\!\cdots\!51}a^{27}-\frac{50\!\cdots\!56}{12\!\cdots\!81}a^{26}+\frac{16\!\cdots\!69}{12\!\cdots\!17}a^{25}+\frac{63\!\cdots\!30}{37\!\cdots\!51}a^{23}-\frac{26\!\cdots\!01}{12\!\cdots\!81}a^{22}+\frac{86\!\cdots\!81}{12\!\cdots\!17}a^{21}+\frac{44\!\cdots\!06}{12\!\cdots\!17}a^{19}-\frac{56\!\cdots\!20}{12\!\cdots\!81}a^{18}+\frac{54\!\cdots\!29}{37\!\cdots\!51}a^{17}+\frac{38\!\cdots\!27}{12\!\cdots\!17}a^{15}-\frac{50\!\cdots\!81}{12\!\cdots\!81}a^{14}+\frac{15\!\cdots\!43}{12\!\cdots\!17}a^{13}+\frac{36\!\cdots\!16}{37\!\cdots\!51}a^{11}-\frac{17\!\cdots\!96}{12\!\cdots\!81}a^{10}+\frac{50\!\cdots\!54}{12\!\cdots\!17}a^{9}+\frac{10\!\cdots\!98}{12\!\cdots\!17}a^{7}-\frac{67\!\cdots\!70}{41\!\cdots\!27}a^{6}+\frac{12\!\cdots\!64}{37\!\cdots\!51}a^{5}+\frac{23\!\cdots\!16}{12\!\cdots\!17}a^{3}-\frac{68\!\cdots\!75}{12\!\cdots\!81}a^{2}+\frac{22\!\cdots\!95}{37\!\cdots\!51}a$, $\frac{24\!\cdots\!69}{12\!\cdots\!17}a^{39}-\frac{30\!\cdots\!62}{37\!\cdots\!51}a^{37}-\frac{34\!\cdots\!87}{45\!\cdots\!39}a^{36}+\frac{13\!\cdots\!96}{37\!\cdots\!51}a^{35}-\frac{17\!\cdots\!28}{12\!\cdots\!17}a^{33}-\frac{18\!\cdots\!12}{13\!\cdots\!17}a^{32}+\frac{68\!\cdots\!76}{37\!\cdots\!51}a^{31}-\frac{28\!\cdots\!96}{37\!\cdots\!51}a^{29}-\frac{95\!\cdots\!69}{13\!\cdots\!17}a^{28}+\frac{12\!\cdots\!31}{37\!\cdots\!51}a^{27}-\frac{16\!\cdots\!69}{12\!\cdots\!17}a^{25}-\frac{16\!\cdots\!79}{13\!\cdots\!17}a^{24}+\frac{63\!\cdots\!30}{37\!\cdots\!51}a^{23}-\frac{86\!\cdots\!81}{12\!\cdots\!17}a^{21}-\frac{86\!\cdots\!50}{13\!\cdots\!17}a^{20}+\frac{44\!\cdots\!06}{12\!\cdots\!17}a^{19}-\frac{54\!\cdots\!29}{37\!\cdots\!51}a^{17}-\frac{59\!\cdots\!66}{45\!\cdots\!39}a^{16}+\frac{38\!\cdots\!27}{12\!\cdots\!17}a^{15}-\frac{15\!\cdots\!43}{12\!\cdots\!17}a^{13}-\frac{15\!\cdots\!47}{13\!\cdots\!17}a^{12}+\frac{36\!\cdots\!16}{37\!\cdots\!51}a^{11}-\frac{50\!\cdots\!54}{12\!\cdots\!17}a^{9}-\frac{44\!\cdots\!17}{13\!\cdots\!17}a^{8}+\frac{10\!\cdots\!98}{12\!\cdots\!17}a^{7}-\frac{12\!\cdots\!64}{37\!\cdots\!51}a^{5}-\frac{24\!\cdots\!48}{13\!\cdots\!17}a^{4}+\frac{23\!\cdots\!16}{12\!\cdots\!17}a^{3}-\frac{22\!\cdots\!95}{37\!\cdots\!51}a-\frac{35\!\cdots\!51}{45\!\cdots\!39}$, $\frac{16\!\cdots\!20}{41\!\cdots\!27}a^{39}-\frac{23\!\cdots\!72}{12\!\cdots\!81}a^{37}+\frac{42\!\cdots\!24}{41\!\cdots\!27}a^{36}+\frac{29\!\cdots\!83}{41\!\cdots\!27}a^{35}-\frac{40\!\cdots\!11}{12\!\cdots\!81}a^{33}+\frac{22\!\cdots\!16}{12\!\cdots\!81}a^{32}+\frac{15\!\cdots\!44}{41\!\cdots\!27}a^{31}-\frac{21\!\cdots\!13}{12\!\cdots\!81}a^{29}+\frac{11\!\cdots\!74}{12\!\cdots\!81}a^{28}+\frac{27\!\cdots\!87}{41\!\cdots\!27}a^{27}-\frac{37\!\cdots\!99}{12\!\cdots\!81}a^{25}+\frac{20\!\cdots\!49}{12\!\cdots\!81}a^{24}+\frac{14\!\cdots\!73}{41\!\cdots\!27}a^{23}-\frac{19\!\cdots\!97}{12\!\cdots\!81}a^{21}+\frac{10\!\cdots\!63}{12\!\cdots\!81}a^{20}+\frac{29\!\cdots\!19}{41\!\cdots\!27}a^{19}-\frac{40\!\cdots\!68}{12\!\cdots\!81}a^{17}+\frac{73\!\cdots\!39}{41\!\cdots\!27}a^{16}+\frac{25\!\cdots\!69}{41\!\cdots\!27}a^{15}-\frac{11\!\cdots\!58}{41\!\cdots\!27}a^{13}+\frac{18\!\cdots\!50}{12\!\cdots\!81}a^{12}+\frac{81\!\cdots\!93}{41\!\cdots\!27}a^{11}-\frac{34\!\cdots\!99}{41\!\cdots\!27}a^{9}+\frac{52\!\cdots\!11}{12\!\cdots\!81}a^{8}+\frac{68\!\cdots\!38}{41\!\cdots\!27}a^{7}-\frac{23\!\cdots\!15}{41\!\cdots\!27}a^{5}+\frac{26\!\cdots\!03}{12\!\cdots\!81}a^{4}+\frac{16\!\cdots\!09}{41\!\cdots\!27}a^{3}-\frac{51\!\cdots\!28}{41\!\cdots\!27}a+\frac{10\!\cdots\!82}{41\!\cdots\!27}$, $\frac{29\!\cdots\!19}{12\!\cdots\!81}a^{39}+\frac{30\!\cdots\!62}{37\!\cdots\!51}a^{38}-\frac{16\!\cdots\!29}{37\!\cdots\!51}a^{36}+\frac{17\!\cdots\!26}{41\!\cdots\!27}a^{35}+\frac{17\!\cdots\!28}{12\!\cdots\!17}a^{34}-\frac{28\!\cdots\!35}{37\!\cdots\!51}a^{32}+\frac{27\!\cdots\!86}{12\!\cdots\!81}a^{31}+\frac{28\!\cdots\!96}{37\!\cdots\!51}a^{30}-\frac{15\!\cdots\!31}{37\!\cdots\!51}a^{28}+\frac{16\!\cdots\!14}{41\!\cdots\!27}a^{27}+\frac{16\!\cdots\!69}{12\!\cdots\!17}a^{26}-\frac{26\!\cdots\!32}{37\!\cdots\!51}a^{24}+\frac{84\!\cdots\!65}{41\!\cdots\!27}a^{23}+\frac{86\!\cdots\!81}{12\!\cdots\!17}a^{22}-\frac{46\!\cdots\!23}{12\!\cdots\!17}a^{20}+\frac{53\!\cdots\!43}{12\!\cdots\!81}a^{19}+\frac{54\!\cdots\!29}{37\!\cdots\!51}a^{18}-\frac{10\!\cdots\!29}{12\!\cdots\!17}a^{16}+\frac{15\!\cdots\!32}{41\!\cdots\!27}a^{15}+\frac{15\!\cdots\!43}{12\!\cdots\!17}a^{14}-\frac{27\!\cdots\!05}{37\!\cdots\!51}a^{12}+\frac{50\!\cdots\!28}{41\!\cdots\!27}a^{11}+\frac{50\!\cdots\!54}{12\!\cdots\!17}a^{10}-\frac{31\!\cdots\!56}{12\!\cdots\!17}a^{8}+\frac{13\!\cdots\!91}{12\!\cdots\!81}a^{7}+\frac{12\!\cdots\!64}{37\!\cdots\!51}a^{6}-\frac{30\!\cdots\!49}{12\!\cdots\!17}a^{4}+\frac{36\!\cdots\!51}{12\!\cdots\!81}a^{3}+\frac{22\!\cdots\!95}{37\!\cdots\!51}a^{2}-\frac{24\!\cdots\!69}{12\!\cdots\!17}$, $\frac{11\!\cdots\!52}{12\!\cdots\!81}a^{39}-\frac{68\!\cdots\!40}{12\!\cdots\!81}a^{38}-\frac{80\!\cdots\!79}{41\!\cdots\!27}a^{37}+\frac{20\!\cdots\!04}{12\!\cdots\!81}a^{35}-\frac{11\!\cdots\!07}{12\!\cdots\!81}a^{34}-\frac{14\!\cdots\!08}{41\!\cdots\!27}a^{33}+\frac{36\!\cdots\!45}{41\!\cdots\!27}a^{31}-\frac{18\!\cdots\!56}{41\!\cdots\!27}a^{30}-\frac{74\!\cdots\!83}{41\!\cdots\!27}a^{29}+\frac{19\!\cdots\!64}{12\!\cdots\!81}a^{27}-\frac{77\!\cdots\!46}{12\!\cdots\!81}a^{26}-\frac{13\!\cdots\!32}{41\!\cdots\!27}a^{25}+\frac{10\!\cdots\!40}{12\!\cdots\!81}a^{23}-\frac{28\!\cdots\!37}{12\!\cdots\!81}a^{22}-\frac{67\!\cdots\!55}{41\!\cdots\!27}a^{21}+\frac{21\!\cdots\!84}{12\!\cdots\!81}a^{19}+\frac{17\!\cdots\!10}{12\!\cdots\!81}a^{18}-\frac{13\!\cdots\!21}{41\!\cdots\!27}a^{17}+\frac{65\!\cdots\!52}{41\!\cdots\!27}a^{15}+\frac{51\!\cdots\!07}{12\!\cdots\!81}a^{14}-\frac{11\!\cdots\!81}{41\!\cdots\!27}a^{13}+\frac{68\!\cdots\!28}{12\!\cdots\!81}a^{11}+\frac{49\!\cdots\!60}{12\!\cdots\!81}a^{10}-\frac{32\!\cdots\!66}{41\!\cdots\!27}a^{9}+\frac{79\!\cdots\!47}{12\!\cdots\!81}a^{7}+\frac{50\!\cdots\!28}{41\!\cdots\!27}a^{6}-\frac{12\!\cdots\!93}{41\!\cdots\!27}a^{5}+\frac{27\!\cdots\!84}{12\!\cdots\!81}a^{3}+\frac{83\!\cdots\!76}{12\!\cdots\!81}a^{2}+\frac{26\!\cdots\!67}{41\!\cdots\!27}a$, $\frac{28\!\cdots\!61}{41\!\cdots\!27}a^{39}-\frac{39\!\cdots\!98}{12\!\cdots\!81}a^{38}-\frac{57\!\cdots\!03}{41\!\cdots\!27}a^{37}+\frac{14\!\cdots\!62}{12\!\cdots\!81}a^{35}-\frac{22\!\cdots\!86}{41\!\cdots\!27}a^{34}-\frac{10\!\cdots\!40}{41\!\cdots\!27}a^{33}+\frac{78\!\cdots\!70}{12\!\cdots\!81}a^{31}-\frac{36\!\cdots\!11}{12\!\cdots\!81}a^{30}-\frac{53\!\cdots\!14}{41\!\cdots\!27}a^{29}+\frac{13\!\cdots\!35}{12\!\cdots\!81}a^{27}-\frac{21\!\cdots\!53}{41\!\cdots\!27}a^{26}-\frac{92\!\cdots\!74}{41\!\cdots\!27}a^{25}+\frac{72\!\cdots\!16}{12\!\cdots\!81}a^{23}-\frac{33\!\cdots\!42}{12\!\cdots\!81}a^{22}-\frac{48\!\cdots\!74}{41\!\cdots\!27}a^{21}+\frac{50\!\cdots\!85}{41\!\cdots\!27}a^{19}-\frac{23\!\cdots\!54}{41\!\cdots\!27}a^{18}-\frac{99\!\cdots\!06}{41\!\cdots\!27}a^{17}+\frac{13\!\cdots\!01}{12\!\cdots\!81}a^{15}-\frac{61\!\cdots\!81}{12\!\cdots\!81}a^{14}-\frac{83\!\cdots\!03}{41\!\cdots\!27}a^{13}+\frac{43\!\cdots\!57}{12\!\cdots\!81}a^{11}-\frac{82\!\cdots\!35}{51\!\cdots\!41}a^{10}-\frac{23\!\cdots\!20}{41\!\cdots\!27}a^{9}+\frac{39\!\cdots\!76}{12\!\cdots\!81}a^{7}-\frac{60\!\cdots\!97}{41\!\cdots\!27}a^{6}-\frac{91\!\cdots\!58}{41\!\cdots\!27}a^{5}+\frac{40\!\cdots\!07}{41\!\cdots\!27}a^{3}-\frac{38\!\cdots\!90}{12\!\cdots\!81}a^{2}+\frac{11\!\cdots\!94}{41\!\cdots\!27}a$
| 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: | \( 3147005109105615.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 3147005109105615.0 \cdot 24025}{8\cdot\sqrt{243425272531849218182083207491041757811794582804889600000000000000000000}}\cr\approx \mathstrut & 0.176151113184114 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^40 + 175*x^36 + 9273*x^32 + 161966*x^28 + 849466*x^24 + 1781941*x^20 + 1558024*x^16 + 496503*x^12 + 42966*x^8 + 1085*x^4 + 1)
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^40 + 175*x^36 + 9273*x^32 + 161966*x^28 + 849466*x^24 + 1781941*x^20 + 1558024*x^16 + 496503*x^12 + 42966*x^8 + 1085*x^4 + 1, 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^40 + 175*x^36 + 9273*x^32 + 161966*x^28 + 849466*x^24 + 1781941*x^20 + 1558024*x^16 + 496503*x^12 + 42966*x^8 + 1085*x^4 + 1);
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^40 + 175*x^36 + 9273*x^32 + 161966*x^28 + 849466*x^24 + 1781941*x^20 + 1558024*x^16 + 496503*x^12 + 42966*x^8 + 1085*x^4 + 1);
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
$C_2^2\times C_{10}$ (as 40T7):
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)
| An abelian group of order 40 |
| The 40 conjugacy class representatives for $C_2^2\times C_{10}$ |
| Character table for $C_2^2\times C_{10}$ is not computed |
Intermediate fields
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 | ${\href{/padicField/3.10.0.1}{10} }^{4}$ | R | ${\href{/padicField/7.10.0.1}{10} }^{4}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{20}$ | ${\href{/padicField/29.10.0.1}{10} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{4}$ | ${\href{/padicField/41.5.0.1}{5} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{20}$ | ${\href{/padicField/47.10.0.1}{10} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $40$ | $4$ | $10$ | $80$ | |||
|
\(5\)
| Deg $20$ | $2$ | $10$ | $10$ | |||
| Deg $20$ | $2$ | $10$ | $10$ | ||||
|
\(11\)
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |