Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 9*x^16 + 15*x^15 + 17*x^14 - 71*x^13 + 30*x^12 + 89*x^11 - 142*x^10 + 61*x^9 + 128*x^8 - 265*x^7 + 227*x^6 - 66*x^5 - 73*x^4 + 95*x^3 - 47*x^2 + 11*x - 1)
gp: K = bnfinit(y^18 - y^17 - 9*y^16 + 15*y^15 + 17*y^14 - 71*y^13 + 30*y^12 + 89*y^11 - 142*y^10 + 61*y^9 + 128*y^8 - 265*y^7 + 227*y^6 - 66*y^5 - 73*y^4 + 95*y^3 - 47*y^2 + 11*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 - 9*x^16 + 15*x^15 + 17*x^14 - 71*x^13 + 30*x^12 + 89*x^11 - 142*x^10 + 61*x^9 + 128*x^8 - 265*x^7 + 227*x^6 - 66*x^5 - 73*x^4 + 95*x^3 - 47*x^2 + 11*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 - 9*x^16 + 15*x^15 + 17*x^14 - 71*x^13 + 30*x^12 + 89*x^11 - 142*x^10 + 61*x^9 + 128*x^8 - 265*x^7 + 227*x^6 - 66*x^5 - 73*x^4 + 95*x^3 - 47*x^2 + 11*x - 1)
\( x^{18} - x^{17} - 9 x^{16} + 15 x^{15} + 17 x^{14} - 71 x^{13} + 30 x^{12} + 89 x^{11} - 142 x^{10} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[10, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(140215359130103174768525\)
\(\medspace = 5^{2}\cdot 257^{6}\cdot 269^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.32\) | 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: | $5^{1/2}257^{1/2}269^{1/2}\approx 587.9328192914561$ | ||
| Ramified primes: |
\(5\), \(257\), \(269\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{269}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{193866599}a^{17}+\frac{75915647}{193866599}a^{16}-\frac{71222569}{193866599}a^{15}-\frac{19738532}{193866599}a^{14}-\frac{68768272}{193866599}a^{13}-\frac{38766691}{193866599}a^{12}+\frac{29503128}{193866599}a^{11}-\frac{74363724}{193866599}a^{10}+\frac{58935435}{193866599}a^{9}-\frac{68444506}{193866599}a^{8}+\frac{71769458}{193866599}a^{7}-\frac{48951491}{193866599}a^{6}+\frac{87082289}{193866599}a^{5}-\frac{11655098}{193866599}a^{4}-\frac{67336562}{193866599}a^{3}-\frac{22565206}{193866599}a^{2}-\frac{51262577}{193866599}a+\frac{23103954}{193866599}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{2968765153}{193866599}a^{17}-\frac{1230424054}{193866599}a^{16}-\frac{27576908880}{193866599}a^{15}+\frac{28359090569}{193866599}a^{14}+\frac{68332003004}{193866599}a^{13}-\frac{171348545160}{193866599}a^{12}-\frac{14738203465}{193866599}a^{11}+\frac{261538590015}{193866599}a^{10}-\frac{264318877645}{193866599}a^{9}+\frac{16618570041}{193866599}a^{8}+\frac{396629230747}{193866599}a^{7}-\frac{551006675158}{193866599}a^{6}+\frac{334943933776}{193866599}a^{5}+\frac{16233669241}{193866599}a^{4}-\frac{213594611902}{193866599}a^{3}+\frac{152287129296}{193866599}a^{2}-\frac{42921341621}{193866599}a+\frac{4531453626}{193866599}$, $a$, $\frac{4337587027}{193866599}a^{17}-\frac{1368821874}{193866599}a^{16}-\frac{40268707297}{193866599}a^{15}+\frac{37486896525}{193866599}a^{14}+\frac{102098070028}{193866599}a^{13}-\frac{239636675913}{193866599}a^{12}-\frac{41220934350}{193866599}a^{11}+\frac{371307041938}{193866599}a^{10}-\frac{354398767819}{193866599}a^{9}+\frac{273931002}{193866599}a^{8}+\frac{571829709497}{193866599}a^{7}-\frac{752831331408}{193866599}a^{6}+\frac{433625579971}{193866599}a^{5}+\frac{48663189994}{193866599}a^{4}-\frac{300410183730}{193866599}a^{3}+\frac{198476155663}{193866599}a^{2}-\frac{51579460973}{193866599}a+\frac{4792115676}{193866599}$, $\frac{4324210490}{193866599}a^{17}-\frac{4564505175}{193866599}a^{16}-\frac{39618178885}{193866599}a^{15}+\frac{67227200588}{193866599}a^{14}+\frac{78665985132}{193866599}a^{13}-\frac{318342321190}{193866599}a^{12}+\frac{123852886186}{193866599}a^{11}+\frac{427308297035}{193866599}a^{10}-\frac{621252727504}{193866599}a^{9}+\frac{218511571778}{193866599}a^{8}+\frac{607313538371}{193866599}a^{7}-\frac{1169130736006}{193866599}a^{6}+\frac{921800643698}{193866599}a^{5}-\frac{189941741163}{193866599}a^{4}-\frac{377753122604}{193866599}a^{3}+\frac{409237341967}{193866599}a^{2}-\frac{163148792499}{193866599}a+\frac{23007911742}{193866599}$, $\frac{3170497957}{193866599}a^{17}-\frac{2024246911}{193866599}a^{16}-\frac{29249104197}{193866599}a^{15}+\frac{36921515767}{193866599}a^{14}+\frac{67039478954}{193866599}a^{13}-\frac{200241283838}{193866599}a^{12}+\frac{23206976279}{193866599}a^{11}+\frac{288275260323}{193866599}a^{10}-\frac{344604895730}{193866599}a^{9}+\frac{72888642861}{193866599}a^{8}+\frac{428585769943}{193866599}a^{7}-\frac{683582958353}{193866599}a^{6}+\frac{477107587082}{193866599}a^{5}-\frac{45158872642}{193866599}a^{4}-\frac{242584168167}{193866599}a^{3}+\frac{213138223161}{193866599}a^{2}-\frac{75679986934}{193866599}a+\frac{10154960418}{193866599}$, $\frac{5973297393}{193866599}a^{17}-\frac{2537136691}{193866599}a^{16}-\frac{55386083348}{193866599}a^{15}+\frac{57737534422}{193866599}a^{14}+\frac{136302302861}{193866599}a^{13}-\frac{346657953211}{193866599}a^{12}-\frac{24397075491}{193866599}a^{11}+\frac{525554067021}{193866599}a^{10}-\frac{541936836784}{193866599}a^{9}+\frac{39593899475}{193866599}a^{8}+\frac{797436262102}{193866599}a^{7}-\frac{1120955699757}{193866599}a^{6}+\frac{689884488690}{193866599}a^{5}+\frac{25217678706}{193866599}a^{4}-\frac{431449456620}{193866599}a^{3}+\frac{314047975292}{193866599}a^{2}-\frac{89830849422}{193866599}a+\frac{9164633374}{193866599}$, $\frac{7456270394}{193866599}a^{17}-\frac{6252842222}{193866599}a^{16}-\frac{69113720580}{193866599}a^{15}+\frac{100504694141}{193866599}a^{14}+\frac{152047399598}{193866599}a^{13}-\frac{509025276358}{193866599}a^{12}+\frac{116851920033}{193866599}a^{11}+\frac{725376563622}{193866599}a^{10}-\frac{913302712526}{193866599}a^{9}+\frac{239138359262}{193866599}a^{8}+\frac{1045252171616}{193866599}a^{7}-\frac{1785436015640}{193866599}a^{6}+\frac{1286382670183}{193866599}a^{5}-\frac{166153891903}{193866599}a^{4}-\frac{621565021516}{193866599}a^{3}+\frac{577716906494}{193866599}a^{2}-\frac{204415260766}{193866599}a+\frac{25689610001}{193866599}$, $\frac{4827046347}{193866599}a^{17}-\frac{1822514107}{193866599}a^{16}-\frac{44750129760}{193866599}a^{15}+\frac{44596520737}{193866599}a^{14}+\frac{111438231752}{193866599}a^{13}-\frac{274749990780}{193866599}a^{12}-\frac{30498017641}{193866599}a^{11}+\frac{419948252857}{193866599}a^{10}-\frac{421425104268}{193866599}a^{9}+\frac{16831868028}{193866599}a^{8}+\frac{640837261850}{193866599}a^{7}-\frac{878360250600}{193866599}a^{6}+\frac{525790496170}{193866599}a^{5}+\frac{36355496012}{193866599}a^{4}-\frac{343390373866}{193866599}a^{3}+\frac{240714558247}{193866599}a^{2}-\frac{65110332313}{193866599}a+\frac{5994135417}{193866599}$, $\frac{16077491370}{193866599}a^{17}-\frac{8783046563}{193866599}a^{16}-\frac{149023413953}{193866599}a^{15}+\frac{173503331531}{193866599}a^{14}+\frac{355145966887}{193866599}a^{13}-\frac{981938865348}{193866599}a^{12}+\frac{28389566659}{193866599}a^{11}+\frac{1458826783643}{193866599}a^{10}-\frac{1611946287879}{193866599}a^{9}+\frac{225389504315}{193866599}a^{8}+\frac{2178742032897}{193866599}a^{7}-\frac{3265153778291}{193866599}a^{6}+\frac{2127345501916}{193866599}a^{5}-\frac{54096807533}{193866599}a^{4}-\frac{1216681895603}{193866599}a^{3}+\frac{965409564833}{193866599}a^{2}-\frac{299217732173}{193866599}a+\frac{32789419923}{193866599}$, $\frac{25461361093}{193866599}a^{17}-\frac{15420937611}{193866599}a^{16}-\frac{235235346674}{193866599}a^{15}+\frac{289193745598}{193866599}a^{14}+\frac{546929562033}{193866599}a^{13}-\frac{1592394707678}{193866599}a^{12}+\frac{135834179979}{193866599}a^{11}+\frac{2320525148297}{193866599}a^{10}-\frac{2701533953295}{193866599}a^{9}+\frac{486268382810}{193866599}a^{8}+\frac{3452493196116}{193866599}a^{7}-\frac{5387553584300}{193866599}a^{6}+\frac{3653457850637}{193866599}a^{5}-\frac{235485586461}{193866599}a^{4}-\frac{1954493151004}{193866599}a^{3}+\frac{1648570260459}{193866599}a^{2}-\frac{544602122463}{193866599}a+\frac{63764974039}{193866599}$, $\frac{7893927763}{193866599}a^{17}-\frac{4085979138}{193866599}a^{16}-\frac{72668882766}{193866599}a^{15}+\frac{83336291427}{193866599}a^{14}+\frac{171186642708}{193866599}a^{13}-\frac{475711934418}{193866599}a^{12}+\frac{15926965833}{193866599}a^{11}+\frac{693231051374}{193866599}a^{10}-\frac{793856348212}{193866599}a^{9}+\frac{125988171616}{193866599}a^{8}+\frac{1048972550854}{193866599}a^{7}-\frac{1591216471501}{193866599}a^{6}+\frac{1068349430398}{193866599}a^{5}-\frac{55157532433}{193866599}a^{4}-\frac{579291042624}{193866599}a^{3}+\frac{479918605422}{193866599}a^{2}-\frac{161145661901}{193866599}a+\frac{20233946251}{193866599}$, $\frac{13573465402}{193866599}a^{17}-\frac{8861119167}{193866599}a^{16}-\frac{125138541660}{193866599}a^{15}+\frac{160218579834}{193866599}a^{14}+\frac{285504369097}{193866599}a^{13}-\frac{864556249891}{193866599}a^{12}+\frac{109482091491}{193866599}a^{11}+\frac{1242840483700}{193866599}a^{10}-\frac{1500076243487}{193866599}a^{9}+\frac{312049592194}{193866599}a^{8}+\frac{1842499809318}{193866599}a^{7}-\frac{2960770843159}{193866599}a^{6}+\frac{2063168732048}{193866599}a^{5}-\frac{187157839633}{193866599}a^{4}-\frac{1053781265371}{193866599}a^{3}+\frac{927144534074}{193866599}a^{2}-\frac{319575450395}{193866599}a+\frac{39230722188}{193866599}$, $\frac{1037946459}{193866599}a^{17}+\frac{1364681744}{193866599}a^{16}-\frac{9324550532}{193866599}a^{15}-\frac{6461903896}{193866599}a^{14}+\frac{31444000493}{193866599}a^{13}-\frac{15068932493}{193866599}a^{12}-\frac{82376895606}{193866599}a^{11}+\frac{39573628375}{193866599}a^{10}+\frac{32012976764}{193866599}a^{9}-\frac{85730486091}{193866599}a^{8}+\frac{99183929920}{193866599}a^{7}+\frac{21420595304}{193866599}a^{6}-\frac{94347257565}{193866599}a^{5}+\frac{93027937464}{193866599}a^{4}-\frac{19298617493}{193866599}a^{3}-\frac{42392858466}{193866599}a^{2}+\frac{25121718039}{193866599}a-\frac{3506920151}{193866599}$
| 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: | \( 79194.0173874 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{4}\cdot 79194.0173874 \cdot 1}{2\cdot\sqrt{140215359130103174768525}}\cr\approx \mathstrut & 0.168765633774 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 9*x^16 + 15*x^15 + 17*x^14 - 71*x^13 + 30*x^12 + 89*x^11 - 142*x^10 + 61*x^9 + 128*x^8 - 265*x^7 + 227*x^6 - 66*x^5 - 73*x^4 + 95*x^3 - 47*x^2 + 11*x - 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^18 - x^17 - 9*x^16 + 15*x^15 + 17*x^14 - 71*x^13 + 30*x^12 + 89*x^11 - 142*x^10 + 61*x^9 + 128*x^8 - 265*x^7 + 227*x^6 - 66*x^5 - 73*x^4 + 95*x^3 - 47*x^2 + 11*x - 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^18 - x^17 - 9*x^16 + 15*x^15 + 17*x^14 - 71*x^13 + 30*x^12 + 89*x^11 - 142*x^10 + 61*x^9 + 128*x^8 - 265*x^7 + 227*x^6 - 66*x^5 - 73*x^4 + 95*x^3 - 47*x^2 + 11*x - 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^18 - x^17 - 9*x^16 + 15*x^15 + 17*x^14 - 71*x^13 + 30*x^12 + 89*x^11 - 142*x^10 + 61*x^9 + 128*x^8 - 265*x^7 + 227*x^6 - 66*x^5 - 73*x^4 + 95*x^3 - 47*x^2 + 11*x - 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
$D_6\wr S_3$ (as 18T556):
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 10368 |
| The 98 conjugacy class representatives for $D_6\wr S_3$ |
| Character table for $D_6\wr S_3$ |
Intermediate fields
| 3.3.257.1, 6.6.17767181.1, 9.5.22830827585.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 18 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 18.6.2606233441079984661125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $18$ | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.12.0.1 | $x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(257\)
| Deg $4$ | $2$ | $2$ | $2$ | |||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
|
\(269\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |