Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 81*x^34 + 2814*x^32 + 56015*x^30 + 716880*x^28 + 6252714*x^26 + 38356241*x^24 + 168143160*x^22 + 529714350*x^20 + 1196774341*x^18 + 1922710668*x^16 + 2166050013*x^14 + 1679154720*x^12 + 874409358*x^10 + 295502889*x^8 + 61327725*x^6 + 7101906*x^4 + 378984*x^2 + 5041)
gp: K = bnfinit(y^36 + 81*y^34 + 2814*y^32 + 56015*y^30 + 716880*y^28 + 6252714*y^26 + 38356241*y^24 + 168143160*y^22 + 529714350*y^20 + 1196774341*y^18 + 1922710668*y^16 + 2166050013*y^14 + 1679154720*y^12 + 874409358*y^10 + 295502889*y^8 + 61327725*y^6 + 7101906*y^4 + 378984*y^2 + 5041, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 + 81*x^34 + 2814*x^32 + 56015*x^30 + 716880*x^28 + 6252714*x^26 + 38356241*x^24 + 168143160*x^22 + 529714350*x^20 + 1196774341*x^18 + 1922710668*x^16 + 2166050013*x^14 + 1679154720*x^12 + 874409358*x^10 + 295502889*x^8 + 61327725*x^6 + 7101906*x^4 + 378984*x^2 + 5041);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 + 81*x^34 + 2814*x^32 + 56015*x^30 + 716880*x^28 + 6252714*x^26 + 38356241*x^24 + 168143160*x^22 + 529714350*x^20 + 1196774341*x^18 + 1922710668*x^16 + 2166050013*x^14 + 1679154720*x^12 + 874409358*x^10 + 295502889*x^8 + 61327725*x^6 + 7101906*x^4 + 378984*x^2 + 5041)
\( x^{36} + 81 x^{34} + 2814 x^{32} + 56015 x^{30} + 716880 x^{28} + 6252714 x^{26} + 38356241 x^{24} + \cdots + 5041 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $36$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4006019960016756356145265577845273975236004675584000000000000000000\)
\(\medspace = 2^{36}\cdot 3^{48}\cdot 5^{18}\cdot 7^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(70.81\) | 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: | $2\cdot 3^{4/3}5^{1/2}7^{2/3}\approx 70.80686465272596$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1260=2^{2}\cdot 3^{2}\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1260}(1,·)$, $\chi_{1260}(1159,·)$, $\chi_{1260}(919,·)$, $\chi_{1260}(781,·)$, $\chi_{1260}(529,·)$, $\chi_{1260}(1171,·)$, $\chi_{1260}(151,·)$, $\chi_{1260}(1051,·)$, $\chi_{1260}(541,·)$, $\chi_{1260}(799,·)$, $\chi_{1260}(289,·)$, $\chi_{1260}(421,·)$, $\chi_{1260}(169,·)$, $\chi_{1260}(1201,·)$, $\chi_{1260}(949,·)$, $\chi_{1260}(571,·)$, $\chi_{1260}(319,·)$, $\chi_{1260}(961,·)$, $\chi_{1260}(1219,·)$, $\chi_{1260}(709,·)$, $\chi_{1260}(841,·)$, $\chi_{1260}(331,·)$, $\chi_{1260}(589,·)$, $\chi_{1260}(79,·)$, $\chi_{1260}(211,·)$, $\chi_{1260}(361,·)$, $\chi_{1260}(991,·)$, $\chi_{1260}(739,·)$, $\chi_{1260}(1129,·)$, $\chi_{1260}(109,·)$, $\chi_{1260}(751,·)$, $\chi_{1260}(1009,·)$, $\chi_{1260}(499,·)$, $\chi_{1260}(631,·)$, $\chi_{1260}(121,·)$, $\chi_{1260}(379,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{71}a^{29}+\frac{17}{71}a^{27}-\frac{8}{71}a^{25}-\frac{10}{71}a^{23}+\frac{27}{71}a^{21}+\frac{13}{71}a^{19}-\frac{18}{71}a^{17}+\frac{15}{71}a^{15}+\frac{8}{71}a^{13}+\frac{8}{71}a^{11}-\frac{24}{71}a^{9}+\frac{1}{71}a^{7}-\frac{13}{71}a^{5}-\frac{17}{71}a^{3}+\frac{16}{71}a$, $\frac{1}{71}a^{30}+\frac{17}{71}a^{28}-\frac{8}{71}a^{26}-\frac{10}{71}a^{24}+\frac{27}{71}a^{22}+\frac{13}{71}a^{20}-\frac{18}{71}a^{18}+\frac{15}{71}a^{16}+\frac{8}{71}a^{14}+\frac{8}{71}a^{12}-\frac{24}{71}a^{10}+\frac{1}{71}a^{8}-\frac{13}{71}a^{6}-\frac{17}{71}a^{4}+\frac{16}{71}a^{2}$, $\frac{1}{71}a^{31}-\frac{13}{71}a^{27}-\frac{16}{71}a^{25}-\frac{16}{71}a^{23}-\frac{20}{71}a^{21}-\frac{26}{71}a^{19}-\frac{34}{71}a^{17}-\frac{34}{71}a^{15}+\frac{14}{71}a^{13}-\frac{18}{71}a^{11}-\frac{17}{71}a^{9}-\frac{30}{71}a^{7}-\frac{9}{71}a^{5}+\frac{21}{71}a^{3}+\frac{12}{71}a$, $\frac{1}{5770099}a^{32}-\frac{28577}{5770099}a^{30}-\frac{2814267}{5770099}a^{28}+\frac{2605893}{5770099}a^{26}+\frac{811509}{5770099}a^{24}+\frac{2246043}{5770099}a^{22}-\frac{2178264}{5770099}a^{20}-\frac{315141}{5770099}a^{18}-\frac{1656208}{5770099}a^{16}+\frac{2706964}{5770099}a^{14}-\frac{1412701}{5770099}a^{12}-\frac{38298}{5770099}a^{10}+\frac{2476131}{5770099}a^{8}-\frac{2710405}{5770099}a^{6}+\frac{765144}{5770099}a^{4}+\frac{456621}{5770099}a^{2}+\frac{17077}{81269}$, $\frac{1}{5770099}a^{33}-\frac{28577}{5770099}a^{31}+\frac{30148}{5770099}a^{29}-\frac{969943}{5770099}a^{27}+\frac{1136585}{5770099}a^{25}+\frac{2652388}{5770099}a^{23}-\frac{390346}{5770099}a^{21}+\frac{2041660}{5770099}a^{19}-\frac{924787}{5770099}a^{17}-\frac{11093}{81269}a^{15}-\frac{1737777}{5770099}a^{13}-\frac{363374}{5770099}a^{11}-\frac{2318740}{5770099}a^{9}+\frac{134010}{5770099}a^{7}-\frac{1591657}{5770099}a^{5}-\frac{1737642}{5770099}a^{3}+\frac{562315}{5770099}a$, $\frac{1}{12\!\cdots\!21}a^{34}+\frac{68\!\cdots\!78}{12\!\cdots\!21}a^{32}-\frac{23\!\cdots\!43}{12\!\cdots\!21}a^{30}+\frac{26\!\cdots\!81}{12\!\cdots\!21}a^{28}+\frac{36\!\cdots\!45}{12\!\cdots\!21}a^{26}+\frac{17\!\cdots\!31}{12\!\cdots\!21}a^{24}+\frac{10\!\cdots\!67}{12\!\cdots\!21}a^{22}-\frac{22\!\cdots\!08}{12\!\cdots\!21}a^{20}-\frac{51\!\cdots\!86}{12\!\cdots\!21}a^{18}+\frac{22\!\cdots\!77}{12\!\cdots\!21}a^{16}+\frac{10\!\cdots\!44}{12\!\cdots\!21}a^{14}-\frac{59\!\cdots\!59}{12\!\cdots\!21}a^{12}-\frac{10\!\cdots\!20}{12\!\cdots\!21}a^{10}-\frac{49\!\cdots\!87}{12\!\cdots\!21}a^{8}+\frac{44\!\cdots\!77}{12\!\cdots\!21}a^{6}-\frac{16\!\cdots\!54}{12\!\cdots\!21}a^{4}-\frac{12\!\cdots\!28}{12\!\cdots\!21}a^{2}+\frac{79\!\cdots\!47}{17\!\cdots\!51}$, $\frac{1}{12\!\cdots\!21}a^{35}+\frac{68\!\cdots\!78}{12\!\cdots\!21}a^{33}-\frac{23\!\cdots\!43}{12\!\cdots\!21}a^{31}-\frac{80\!\cdots\!21}{12\!\cdots\!21}a^{29}-\frac{22\!\cdots\!89}{12\!\cdots\!21}a^{27}+\frac{45\!\cdots\!47}{12\!\cdots\!21}a^{25}+\frac{45\!\cdots\!87}{12\!\cdots\!21}a^{23}+\frac{67\!\cdots\!59}{12\!\cdots\!21}a^{21}+\frac{25\!\cdots\!09}{12\!\cdots\!21}a^{19}-\frac{38\!\cdots\!08}{12\!\cdots\!21}a^{17}-\frac{50\!\cdots\!86}{12\!\cdots\!21}a^{15}+\frac{35\!\cdots\!46}{12\!\cdots\!21}a^{13}-\frac{37\!\cdots\!36}{12\!\cdots\!21}a^{11}+\frac{33\!\cdots\!61}{12\!\cdots\!21}a^{9}+\frac{40\!\cdots\!75}{12\!\cdots\!21}a^{7}+\frac{43\!\cdots\!72}{12\!\cdots\!21}a^{5}+\frac{45\!\cdots\!06}{12\!\cdots\!21}a^{3}+\frac{98\!\cdots\!05}{12\!\cdots\!21}a$
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
$C_{3}\times C_{6}\times C_{6}\times C_{12}\times C_{252}$, which has order $326592$ (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: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{11668155635671881393}{1002736895565255980039} a^{35} + \frac{938276035513231596047}{1002736895565255980039} a^{33} + \frac{32283807410423355517893}{1002736895565255980039} a^{31} + \frac{634654985108940857242377}{1002736895565255980039} a^{29} + \frac{7992411865836211603767424}{1002736895565255980039} a^{27} + \frac{68269950421701000071832360}{1002736895565255980039} a^{25} + \frac{5739546355554253770546201}{14123054867116281409} a^{23} + \frac{1722948118930681427303238362}{1002736895565255980039} a^{21} + \frac{5170537122883937374236160608}{1002736895565255980039} a^{19} + \frac{10932935199281709383236891581}{1002736895565255980039} a^{17} + \frac{16026747093101357680474632097}{1002736895565255980039} a^{15} + \frac{15884373575992061212839665505}{1002736895565255980039} a^{13} + \frac{10292704807702966549762934568}{1002736895565255980039} a^{11} + \frac{4183467902858993711269190544}{1002736895565255980039} a^{9} + \frac{1006607747918671501487159052}{1002736895565255980039} a^{7} + \frac{130616866199337710650813773}{1002736895565255980039} a^{5} + \frac{7645229074990333655777267}{1002736895565255980039} a^{3} + \frac{112176035263440902615685}{1002736895565255980039} a \)
(order $4$)
| 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: |
$\frac{26\!\cdots\!28}{90\!\cdots\!09}a^{34}+\frac{21\!\cdots\!37}{90\!\cdots\!09}a^{32}+\frac{73\!\cdots\!30}{90\!\cdots\!09}a^{30}+\frac{14\!\cdots\!04}{90\!\cdots\!09}a^{28}+\frac{17\!\cdots\!52}{90\!\cdots\!09}a^{26}+\frac{14\!\cdots\!72}{90\!\cdots\!09}a^{24}+\frac{48\!\cdots\!00}{50\!\cdots\!89}a^{22}+\frac{35\!\cdots\!36}{90\!\cdots\!09}a^{20}+\frac{10\!\cdots\!80}{90\!\cdots\!09}a^{18}+\frac{20\!\cdots\!58}{90\!\cdots\!09}a^{16}+\frac{27\!\cdots\!30}{90\!\cdots\!09}a^{14}+\frac{23\!\cdots\!98}{90\!\cdots\!09}a^{12}+\frac{11\!\cdots\!00}{90\!\cdots\!09}a^{10}+\frac{27\!\cdots\!45}{90\!\cdots\!09}a^{8}-\frac{64\!\cdots\!22}{90\!\cdots\!09}a^{6}-\frac{13\!\cdots\!88}{90\!\cdots\!09}a^{4}-\frac{15\!\cdots\!45}{90\!\cdots\!09}a^{2}-\frac{13\!\cdots\!59}{12\!\cdots\!79}$, $\frac{55\!\cdots\!77}{13\!\cdots\!19}a^{35}+\frac{44\!\cdots\!59}{13\!\cdots\!19}a^{33}+\frac{15\!\cdots\!80}{13\!\cdots\!19}a^{31}+\frac{30\!\cdots\!39}{13\!\cdots\!19}a^{29}+\frac{38\!\cdots\!18}{13\!\cdots\!19}a^{27}+\frac{32\!\cdots\!38}{13\!\cdots\!19}a^{25}+\frac{19\!\cdots\!22}{13\!\cdots\!19}a^{23}+\frac{82\!\cdots\!80}{13\!\cdots\!19}a^{21}+\frac{24\!\cdots\!40}{13\!\cdots\!19}a^{19}+\frac{51\!\cdots\!15}{13\!\cdots\!19}a^{17}+\frac{75\!\cdots\!29}{13\!\cdots\!19}a^{15}+\frac{74\!\cdots\!07}{13\!\cdots\!19}a^{13}+\frac{47\!\cdots\!14}{13\!\cdots\!19}a^{11}+\frac{18\!\cdots\!77}{13\!\cdots\!19}a^{9}+\frac{43\!\cdots\!85}{13\!\cdots\!19}a^{7}+\frac{52\!\cdots\!07}{13\!\cdots\!19}a^{5}+\frac{26\!\cdots\!66}{13\!\cdots\!19}a^{3}+\frac{30\!\cdots\!21}{13\!\cdots\!19}a$, $\frac{20\!\cdots\!30}{13\!\cdots\!19}a^{35}+\frac{16\!\cdots\!46}{13\!\cdots\!19}a^{33}+\frac{57\!\cdots\!33}{13\!\cdots\!19}a^{31}+\frac{11\!\cdots\!56}{13\!\cdots\!19}a^{29}+\frac{14\!\cdots\!22}{13\!\cdots\!19}a^{27}+\frac{12\!\cdots\!98}{13\!\cdots\!19}a^{25}+\frac{72\!\cdots\!13}{13\!\cdots\!19}a^{23}+\frac{30\!\cdots\!82}{13\!\cdots\!19}a^{21}+\frac{92\!\cdots\!08}{13\!\cdots\!19}a^{19}+\frac{19\!\cdots\!16}{13\!\cdots\!19}a^{17}+\frac{28\!\cdots\!66}{13\!\cdots\!19}a^{15}+\frac{28\!\cdots\!12}{13\!\cdots\!19}a^{13}+\frac{18\!\cdots\!42}{13\!\cdots\!19}a^{11}+\frac{73\!\cdots\!01}{13\!\cdots\!19}a^{9}+\frac{17\!\cdots\!77}{13\!\cdots\!19}a^{7}+\frac{22\!\cdots\!40}{13\!\cdots\!19}a^{5}+\frac{12\!\cdots\!73}{13\!\cdots\!19}a^{3}+\frac{17\!\cdots\!06}{13\!\cdots\!19}a$, $\frac{16\!\cdots\!16}{12\!\cdots\!21}a^{35}+\frac{13\!\cdots\!74}{12\!\cdots\!21}a^{33}+\frac{46\!\cdots\!54}{12\!\cdots\!21}a^{31}+\frac{90\!\cdots\!85}{12\!\cdots\!21}a^{29}+\frac{11\!\cdots\!01}{12\!\cdots\!21}a^{27}+\frac{97\!\cdots\!51}{12\!\cdots\!21}a^{25}+\frac{58\!\cdots\!10}{12\!\cdots\!21}a^{23}+\frac{24\!\cdots\!01}{12\!\cdots\!21}a^{21}+\frac{73\!\cdots\!06}{12\!\cdots\!21}a^{19}+\frac{15\!\cdots\!91}{12\!\cdots\!21}a^{17}+\frac{22\!\cdots\!40}{12\!\cdots\!21}a^{15}+\frac{22\!\cdots\!92}{12\!\cdots\!21}a^{13}+\frac{14\!\cdots\!12}{12\!\cdots\!21}a^{11}+\frac{60\!\cdots\!41}{12\!\cdots\!21}a^{9}+\frac{14\!\cdots\!37}{12\!\cdots\!21}a^{7}+\frac{19\!\cdots\!53}{12\!\cdots\!21}a^{5}+\frac{11\!\cdots\!28}{12\!\cdots\!21}a^{3}+\frac{18\!\cdots\!92}{12\!\cdots\!21}a$, $\frac{24\!\cdots\!89}{12\!\cdots\!21}a^{35}+\frac{19\!\cdots\!41}{12\!\cdots\!21}a^{33}+\frac{67\!\cdots\!27}{12\!\cdots\!21}a^{31}+\frac{13\!\cdots\!82}{12\!\cdots\!21}a^{29}+\frac{16\!\cdots\!65}{12\!\cdots\!21}a^{27}+\frac{14\!\cdots\!11}{12\!\cdots\!21}a^{25}+\frac{85\!\cdots\!41}{12\!\cdots\!21}a^{23}+\frac{36\!\cdots\!83}{12\!\cdots\!21}a^{21}+\frac{10\!\cdots\!94}{12\!\cdots\!21}a^{19}+\frac{23\!\cdots\!32}{12\!\cdots\!21}a^{17}+\frac{34\!\cdots\!57}{12\!\cdots\!21}a^{15}+\frac{33\!\cdots\!97}{12\!\cdots\!21}a^{13}+\frac{22\!\cdots\!60}{12\!\cdots\!21}a^{11}+\frac{91\!\cdots\!25}{12\!\cdots\!21}a^{9}+\frac{22\!\cdots\!09}{12\!\cdots\!21}a^{7}+\frac{31\!\cdots\!06}{12\!\cdots\!21}a^{5}+\frac{20\!\cdots\!15}{12\!\cdots\!21}a^{3}+\frac{45\!\cdots\!77}{12\!\cdots\!21}a$, $\frac{72\!\cdots\!58}{12\!\cdots\!21}a^{35}+\frac{58\!\cdots\!32}{12\!\cdots\!21}a^{33}+\frac{19\!\cdots\!60}{12\!\cdots\!21}a^{31}+\frac{39\!\cdots\!56}{12\!\cdots\!21}a^{29}+\frac{49\!\cdots\!86}{12\!\cdots\!21}a^{27}+\frac{42\!\cdots\!86}{12\!\cdots\!21}a^{25}+\frac{25\!\cdots\!48}{12\!\cdots\!21}a^{23}+\frac{10\!\cdots\!10}{12\!\cdots\!21}a^{21}+\frac{32\!\cdots\!23}{12\!\cdots\!21}a^{19}+\frac{68\!\cdots\!37}{12\!\cdots\!21}a^{17}+\frac{10\!\cdots\!11}{12\!\cdots\!21}a^{15}+\frac{10\!\cdots\!63}{12\!\cdots\!21}a^{13}+\frac{65\!\cdots\!42}{12\!\cdots\!21}a^{11}+\frac{27\!\cdots\!76}{12\!\cdots\!21}a^{9}+\frac{67\!\cdots\!79}{12\!\cdots\!21}a^{7}+\frac{90\!\cdots\!88}{12\!\cdots\!21}a^{5}+\frac{54\!\cdots\!58}{12\!\cdots\!21}a^{3}+\frac{81\!\cdots\!43}{12\!\cdots\!21}a$, $\frac{27\!\cdots\!54}{12\!\cdots\!21}a^{35}+\frac{22\!\cdots\!40}{12\!\cdots\!21}a^{33}+\frac{76\!\cdots\!93}{12\!\cdots\!21}a^{31}+\frac{15\!\cdots\!19}{12\!\cdots\!21}a^{29}+\frac{18\!\cdots\!97}{12\!\cdots\!21}a^{27}+\frac{16\!\cdots\!31}{12\!\cdots\!21}a^{25}+\frac{95\!\cdots\!57}{12\!\cdots\!21}a^{23}+\frac{40\!\cdots\!37}{12\!\cdots\!21}a^{21}+\frac{12\!\cdots\!66}{12\!\cdots\!21}a^{19}+\frac{25\!\cdots\!53}{12\!\cdots\!21}a^{17}+\frac{36\!\cdots\!54}{12\!\cdots\!21}a^{15}+\frac{35\!\cdots\!16}{12\!\cdots\!21}a^{13}+\frac{22\!\cdots\!66}{12\!\cdots\!21}a^{11}+\frac{85\!\cdots\!18}{12\!\cdots\!21}a^{9}+\frac{18\!\cdots\!06}{12\!\cdots\!21}a^{7}+\frac{17\!\cdots\!07}{12\!\cdots\!21}a^{5}+\frac{45\!\cdots\!79}{12\!\cdots\!21}a^{3}-\frac{17\!\cdots\!38}{12\!\cdots\!21}a$, $\frac{11\!\cdots\!19}{12\!\cdots\!21}a^{35}+\frac{96\!\cdots\!57}{12\!\cdots\!21}a^{33}+\frac{33\!\cdots\!01}{12\!\cdots\!21}a^{31}+\frac{65\!\cdots\!37}{12\!\cdots\!21}a^{29}+\frac{82\!\cdots\!15}{12\!\cdots\!21}a^{27}+\frac{70\!\cdots\!07}{12\!\cdots\!21}a^{25}+\frac{41\!\cdots\!47}{12\!\cdots\!21}a^{23}+\frac{17\!\cdots\!41}{12\!\cdots\!21}a^{21}+\frac{52\!\cdots\!60}{12\!\cdots\!21}a^{19}+\frac{11\!\cdots\!89}{12\!\cdots\!21}a^{17}+\frac{16\!\cdots\!17}{12\!\cdots\!21}a^{15}+\frac{16\!\cdots\!87}{12\!\cdots\!21}a^{13}+\frac{10\!\cdots\!57}{12\!\cdots\!21}a^{11}+\frac{41\!\cdots\!33}{12\!\cdots\!21}a^{9}+\frac{96\!\cdots\!62}{12\!\cdots\!21}a^{7}+\frac{11\!\cdots\!39}{12\!\cdots\!21}a^{5}+\frac{62\!\cdots\!11}{12\!\cdots\!21}a^{3}+\frac{57\!\cdots\!57}{12\!\cdots\!21}a$, $\frac{66\!\cdots\!00}{10\!\cdots\!91}a^{34}+\frac{53\!\cdots\!40}{10\!\cdots\!91}a^{32}+\frac{18\!\cdots\!90}{10\!\cdots\!91}a^{30}+\frac{36\!\cdots\!48}{10\!\cdots\!91}a^{28}+\frac{45\!\cdots\!76}{10\!\cdots\!91}a^{26}+\frac{39\!\cdots\!00}{10\!\cdots\!91}a^{24}+\frac{12\!\cdots\!88}{57\!\cdots\!11}a^{22}+\frac{98\!\cdots\!68}{10\!\cdots\!91}a^{20}+\frac{29\!\cdots\!06}{10\!\cdots\!91}a^{18}+\frac{62\!\cdots\!02}{10\!\cdots\!91}a^{16}+\frac{92\!\cdots\!66}{10\!\cdots\!91}a^{14}+\frac{92\!\cdots\!06}{10\!\cdots\!91}a^{12}+\frac{60\!\cdots\!60}{10\!\cdots\!91}a^{10}+\frac{24\!\cdots\!95}{10\!\cdots\!91}a^{8}+\frac{59\!\cdots\!06}{10\!\cdots\!91}a^{6}+\frac{78\!\cdots\!61}{10\!\cdots\!91}a^{4}+\frac{45\!\cdots\!33}{10\!\cdots\!91}a^{2}+\frac{91\!\cdots\!84}{14\!\cdots\!21}$, $\frac{26\!\cdots\!40}{12\!\cdots\!21}a^{34}+\frac{21\!\cdots\!61}{12\!\cdots\!21}a^{32}+\frac{72\!\cdots\!50}{12\!\cdots\!21}a^{30}+\frac{14\!\cdots\!37}{12\!\cdots\!21}a^{28}+\frac{17\!\cdots\!63}{12\!\cdots\!21}a^{26}+\frac{15\!\cdots\!59}{12\!\cdots\!21}a^{24}+\frac{91\!\cdots\!83}{12\!\cdots\!21}a^{22}+\frac{38\!\cdots\!02}{12\!\cdots\!21}a^{20}+\frac{11\!\cdots\!42}{12\!\cdots\!21}a^{18}+\frac{24\!\cdots\!02}{12\!\cdots\!21}a^{16}+\frac{36\!\cdots\!76}{12\!\cdots\!21}a^{14}+\frac{36\!\cdots\!82}{12\!\cdots\!21}a^{12}+\frac{23\!\cdots\!15}{12\!\cdots\!21}a^{10}+\frac{95\!\cdots\!56}{12\!\cdots\!21}a^{8}+\frac{23\!\cdots\!02}{12\!\cdots\!21}a^{6}+\frac{29\!\cdots\!11}{12\!\cdots\!21}a^{4}+\frac{17\!\cdots\!13}{12\!\cdots\!21}a^{2}+\frac{33\!\cdots\!18}{17\!\cdots\!51}$, $\frac{69\!\cdots\!30}{12\!\cdots\!21}a^{35}+\frac{55\!\cdots\!87}{12\!\cdots\!21}a^{33}+\frac{19\!\cdots\!92}{12\!\cdots\!21}a^{31}+\frac{37\!\cdots\!19}{12\!\cdots\!21}a^{29}+\frac{47\!\cdots\!00}{12\!\cdots\!21}a^{27}+\frac{40\!\cdots\!86}{12\!\cdots\!21}a^{25}+\frac{24\!\cdots\!22}{12\!\cdots\!21}a^{23}+\frac{10\!\cdots\!13}{12\!\cdots\!21}a^{21}+\frac{30\!\cdots\!31}{12\!\cdots\!21}a^{19}+\frac{64\!\cdots\!30}{12\!\cdots\!21}a^{17}+\frac{94\!\cdots\!44}{12\!\cdots\!21}a^{15}+\frac{93\!\cdots\!43}{12\!\cdots\!21}a^{13}+\frac{60\!\cdots\!21}{12\!\cdots\!21}a^{11}+\frac{24\!\cdots\!68}{12\!\cdots\!21}a^{9}+\frac{56\!\cdots\!55}{12\!\cdots\!21}a^{7}+\frac{70\!\cdots\!81}{12\!\cdots\!21}a^{5}+\frac{38\!\cdots\!94}{12\!\cdots\!21}a^{3}+\frac{60\!\cdots\!86}{12\!\cdots\!21}a$, $\frac{19\!\cdots\!88}{12\!\cdots\!21}a^{35}+\frac{21\!\cdots\!98}{17\!\cdots\!51}a^{33}+\frac{52\!\cdots\!68}{12\!\cdots\!21}a^{31}+\frac{10\!\cdots\!84}{12\!\cdots\!21}a^{29}+\frac{13\!\cdots\!65}{12\!\cdots\!21}a^{27}+\frac{11\!\cdots\!61}{12\!\cdots\!21}a^{25}+\frac{66\!\cdots\!68}{12\!\cdots\!21}a^{23}+\frac{27\!\cdots\!49}{12\!\cdots\!21}a^{21}+\frac{83\!\cdots\!49}{12\!\cdots\!21}a^{19}+\frac{17\!\cdots\!11}{12\!\cdots\!21}a^{17}+\frac{25\!\cdots\!89}{12\!\cdots\!21}a^{15}+\frac{25\!\cdots\!19}{12\!\cdots\!21}a^{13}+\frac{16\!\cdots\!67}{12\!\cdots\!21}a^{11}+\frac{65\!\cdots\!73}{12\!\cdots\!21}a^{9}+\frac{15\!\cdots\!11}{12\!\cdots\!21}a^{7}+\frac{18\!\cdots\!98}{12\!\cdots\!21}a^{5}+\frac{10\!\cdots\!66}{12\!\cdots\!21}a^{3}+\frac{11\!\cdots\!29}{12\!\cdots\!21}a$, $\frac{94\!\cdots\!52}{17\!\cdots\!21}a^{34}+\frac{75\!\cdots\!06}{17\!\cdots\!21}a^{32}+\frac{26\!\cdots\!22}{17\!\cdots\!21}a^{30}+\frac{51\!\cdots\!96}{17\!\cdots\!21}a^{28}+\frac{64\!\cdots\!04}{17\!\cdots\!21}a^{26}+\frac{54\!\cdots\!27}{17\!\cdots\!21}a^{24}+\frac{32\!\cdots\!16}{17\!\cdots\!21}a^{22}+\frac{13\!\cdots\!15}{17\!\cdots\!21}a^{20}+\frac{41\!\cdots\!67}{17\!\cdots\!21}a^{18}+\frac{87\!\cdots\!14}{17\!\cdots\!21}a^{16}+\frac{12\!\cdots\!48}{17\!\cdots\!21}a^{14}+\frac{12\!\cdots\!52}{17\!\cdots\!21}a^{12}+\frac{79\!\cdots\!83}{17\!\cdots\!21}a^{10}+\frac{31\!\cdots\!85}{17\!\cdots\!21}a^{8}+\frac{74\!\cdots\!97}{17\!\cdots\!21}a^{6}+\frac{93\!\cdots\!46}{17\!\cdots\!21}a^{4}+\frac{50\!\cdots\!90}{17\!\cdots\!21}a^{2}+\frac{93\!\cdots\!75}{24\!\cdots\!51}$, $\frac{14\!\cdots\!68}{12\!\cdots\!21}a^{34}+\frac{11\!\cdots\!60}{12\!\cdots\!21}a^{32}+\frac{39\!\cdots\!70}{12\!\cdots\!21}a^{30}+\frac{78\!\cdots\!28}{12\!\cdots\!21}a^{28}+\frac{10\!\cdots\!02}{12\!\cdots\!21}a^{26}+\frac{87\!\cdots\!28}{12\!\cdots\!21}a^{24}+\frac{53\!\cdots\!90}{12\!\cdots\!21}a^{22}+\frac{23\!\cdots\!68}{12\!\cdots\!21}a^{20}+\frac{71\!\cdots\!89}{12\!\cdots\!21}a^{18}+\frac{15\!\cdots\!11}{12\!\cdots\!21}a^{16}+\frac{24\!\cdots\!76}{12\!\cdots\!21}a^{14}+\frac{27\!\cdots\!02}{12\!\cdots\!21}a^{12}+\frac{19\!\cdots\!69}{12\!\cdots\!21}a^{10}+\frac{93\!\cdots\!83}{12\!\cdots\!21}a^{8}+\frac{27\!\cdots\!25}{12\!\cdots\!21}a^{6}+\frac{43\!\cdots\!67}{12\!\cdots\!21}a^{4}+\frac{30\!\cdots\!79}{12\!\cdots\!21}a^{2}+\frac{62\!\cdots\!21}{17\!\cdots\!51}$, $\frac{39\!\cdots\!39}{12\!\cdots\!21}a^{35}+\frac{31\!\cdots\!85}{12\!\cdots\!21}a^{33}+\frac{10\!\cdots\!53}{12\!\cdots\!21}a^{31}+\frac{21\!\cdots\!31}{12\!\cdots\!21}a^{29}+\frac{26\!\cdots\!21}{12\!\cdots\!21}a^{27}+\frac{32\!\cdots\!65}{17\!\cdots\!51}a^{25}+\frac{13\!\cdots\!25}{12\!\cdots\!21}a^{23}+\frac{58\!\cdots\!92}{12\!\cdots\!21}a^{21}+\frac{17\!\cdots\!28}{12\!\cdots\!21}a^{19}+\frac{36\!\cdots\!99}{12\!\cdots\!21}a^{17}+\frac{54\!\cdots\!84}{12\!\cdots\!21}a^{15}+\frac{53\!\cdots\!71}{12\!\cdots\!21}a^{13}+\frac{34\!\cdots\!35}{12\!\cdots\!21}a^{11}+\frac{14\!\cdots\!41}{12\!\cdots\!21}a^{9}+\frac{33\!\cdots\!17}{12\!\cdots\!21}a^{7}+\frac{43\!\cdots\!89}{12\!\cdots\!21}a^{5}+\frac{24\!\cdots\!38}{12\!\cdots\!21}a^{3}+\frac{27\!\cdots\!70}{12\!\cdots\!21}a$, $\frac{23\!\cdots\!41}{12\!\cdots\!21}a^{35}+\frac{18\!\cdots\!40}{12\!\cdots\!21}a^{33}+\frac{64\!\cdots\!61}{12\!\cdots\!21}a^{31}+\frac{12\!\cdots\!20}{12\!\cdots\!21}a^{29}+\frac{16\!\cdots\!34}{12\!\cdots\!21}a^{27}+\frac{13\!\cdots\!23}{12\!\cdots\!21}a^{25}+\frac{83\!\cdots\!95}{12\!\cdots\!21}a^{23}+\frac{35\!\cdots\!71}{12\!\cdots\!21}a^{21}+\frac{10\!\cdots\!66}{12\!\cdots\!21}a^{19}+\frac{23\!\cdots\!17}{12\!\cdots\!21}a^{17}+\frac{34\!\cdots\!28}{12\!\cdots\!21}a^{15}+\frac{35\!\cdots\!96}{12\!\cdots\!21}a^{13}+\frac{24\!\cdots\!68}{12\!\cdots\!21}a^{11}+\frac{10\!\cdots\!14}{12\!\cdots\!21}a^{9}+\frac{27\!\cdots\!41}{12\!\cdots\!21}a^{7}+\frac{39\!\cdots\!89}{12\!\cdots\!21}a^{5}+\frac{24\!\cdots\!70}{12\!\cdots\!21}a^{3}+\frac{34\!\cdots\!75}{12\!\cdots\!21}a$, $\frac{51\!\cdots\!04}{12\!\cdots\!21}a^{34}+\frac{41\!\cdots\!73}{12\!\cdots\!21}a^{32}+\frac{14\!\cdots\!58}{12\!\cdots\!21}a^{30}+\frac{27\!\cdots\!05}{12\!\cdots\!21}a^{28}+\frac{35\!\cdots\!20}{12\!\cdots\!21}a^{26}+\frac{30\!\cdots\!16}{12\!\cdots\!21}a^{24}+\frac{17\!\cdots\!39}{12\!\cdots\!21}a^{22}+\frac{75\!\cdots\!48}{12\!\cdots\!21}a^{20}+\frac{22\!\cdots\!02}{12\!\cdots\!21}a^{18}+\frac{48\!\cdots\!59}{12\!\cdots\!21}a^{16}+\frac{70\!\cdots\!24}{12\!\cdots\!21}a^{14}+\frac{70\!\cdots\!49}{12\!\cdots\!21}a^{12}+\frac{45\!\cdots\!78}{12\!\cdots\!21}a^{10}+\frac{26\!\cdots\!60}{17\!\cdots\!51}a^{8}+\frac{44\!\cdots\!58}{12\!\cdots\!21}a^{6}+\frac{57\!\cdots\!57}{12\!\cdots\!21}a^{4}+\frac{32\!\cdots\!95}{12\!\cdots\!21}a^{2}+\frac{62\!\cdots\!84}{17\!\cdots\!51}$
| 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: | \( 13624539961495.691 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 13624539961495.691 \cdot 326592}{4\cdot\sqrt{4006019960016756356145265577845273975236004675584000000000000000000}}\cr\approx \mathstrut & 0.129463836923254 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^36 + 81*x^34 + 2814*x^32 + 56015*x^30 + 716880*x^28 + 6252714*x^26 + 38356241*x^24 + 168143160*x^22 + 529714350*x^20 + 1196774341*x^18 + 1922710668*x^16 + 2166050013*x^14 + 1679154720*x^12 + 874409358*x^10 + 295502889*x^8 + 61327725*x^6 + 7101906*x^4 + 378984*x^2 + 5041)
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^36 + 81*x^34 + 2814*x^32 + 56015*x^30 + 716880*x^28 + 6252714*x^26 + 38356241*x^24 + 168143160*x^22 + 529714350*x^20 + 1196774341*x^18 + 1922710668*x^16 + 2166050013*x^14 + 1679154720*x^12 + 874409358*x^10 + 295502889*x^8 + 61327725*x^6 + 7101906*x^4 + 378984*x^2 + 5041, 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^36 + 81*x^34 + 2814*x^32 + 56015*x^30 + 716880*x^28 + 6252714*x^26 + 38356241*x^24 + 168143160*x^22 + 529714350*x^20 + 1196774341*x^18 + 1922710668*x^16 + 2166050013*x^14 + 1679154720*x^12 + 874409358*x^10 + 295502889*x^8 + 61327725*x^6 + 7101906*x^4 + 378984*x^2 + 5041);
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^36 + 81*x^34 + 2814*x^32 + 56015*x^30 + 716880*x^28 + 6252714*x^26 + 38356241*x^24 + 168143160*x^22 + 529714350*x^20 + 1196774341*x^18 + 1922710668*x^16 + 2166050013*x^14 + 1679154720*x^12 + 874409358*x^10 + 295502889*x^8 + 61327725*x^6 + 7101906*x^4 + 378984*x^2 + 5041);
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)
| An abelian group of order 36 |
| The 36 conjugacy class representatives for $C_6^2$ |
| Character table for $C_6^2$ |
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 | R | R | R | ${\href{/padicField/11.6.0.1}{6} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{12}$ | ${\href{/padicField/43.6.0.1}{6} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
|
\(3\)
| Deg $18$ | $3$ | $6$ | $24$ | |||
| Deg $18$ | $3$ | $6$ | $24$ | ||||
|
\(5\)
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(7\)
| 7.18.12.1 | $x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |
| 7.18.12.1 | $x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |