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Properties

Label 24.0.436...456.1
Degree $24$
Signature $[0, 12]$
Discriminant $4.368\times 10^{33}$
Root discriminant \(25.22\)
Ramified primes $2,7,239$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $C_2^3\times A_4$ (as 24T135)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 10*x^20 - 14*x^18 + 37*x^16 - 154*x^14 + 97*x^12 - 616*x^10 + 592*x^8 - 896*x^6 + 2560*x^4 + 4096)
 
gp: K = bnfinit(y^24 + 10*y^20 - 14*y^18 + 37*y^16 - 154*y^14 + 97*y^12 - 616*y^10 + 592*y^8 - 896*y^6 + 2560*y^4 + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 10*x^20 - 14*x^18 + 37*x^16 - 154*x^14 + 97*x^12 - 616*x^10 + 592*x^8 - 896*x^6 + 2560*x^4 + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 + 10*x^20 - 14*x^18 + 37*x^16 - 154*x^14 + 97*x^12 - 616*x^10 + 592*x^8 - 896*x^6 + 2560*x^4 + 4096)
 

\( x^{24} + 10 x^{20} - 14 x^{18} + 37 x^{16} - 154 x^{14} + 97 x^{12} - 616 x^{10} + 592 x^{8} + \cdots + 4096 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $24$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(4367896108464230086685082523795456\) \(\medspace = 2^{24}\cdot 7^{20}\cdot 239^{4}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(25.22\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2\cdot 7^{5/6}239^{1/2}\approx 156.48665697583843$
Ramified primes:   \(2\), \(7\), \(239\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{2048}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{10}+\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{12}+\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{7}-\frac{1}{4}a^{5}+\frac{3}{8}a^{3}$, $\frac{1}{16}a^{16}-\frac{1}{8}a^{12}-\frac{1}{8}a^{10}+\frac{1}{16}a^{8}-\frac{1}{8}a^{6}+\frac{5}{16}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{32}a^{17}+\frac{1}{16}a^{13}+\frac{1}{16}a^{11}-\frac{3}{32}a^{9}-\frac{1}{16}a^{7}+\frac{1}{32}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{256}a^{18}-\frac{1}{64}a^{16}-\frac{3}{128}a^{14}+\frac{5}{128}a^{12}-\frac{3}{256}a^{10}-\frac{7}{128}a^{8}-\frac{7}{256}a^{6}-\frac{3}{64}a^{4}-\frac{1}{8}a^{2}+\frac{1}{4}$, $\frac{1}{512}a^{19}-\frac{1}{128}a^{17}-\frac{3}{256}a^{15}-\frac{27}{256}a^{13}+\frac{61}{512}a^{11}+\frac{25}{256}a^{9}+\frac{121}{512}a^{7}-\frac{19}{128}a^{5}-\frac{1}{16}a^{3}-\frac{3}{8}a$, $\frac{1}{1024}a^{20}-\frac{11}{512}a^{16}+\frac{25}{512}a^{14}-\frac{27}{1024}a^{12}-\frac{45}{512}a^{10}+\frac{65}{1024}a^{8}-\frac{61}{128}a^{6}-\frac{21}{64}a^{4}-\frac{1}{16}a^{2}+\frac{1}{4}$, $\frac{1}{2048}a^{21}-\frac{11}{1024}a^{17}+\frac{25}{1024}a^{15}-\frac{27}{2048}a^{13}-\frac{45}{1024}a^{11}+\frac{65}{2048}a^{9}-\frac{61}{256}a^{7}-\frac{21}{128}a^{5}-\frac{1}{32}a^{3}-\frac{3}{8}a$, $\frac{1}{167936}a^{22}+\frac{5}{10496}a^{20}-\frac{75}{83968}a^{18}+\frac{553}{83968}a^{16}+\frac{12421}{167936}a^{14}-\frac{485}{83968}a^{12}+\frac{20897}{167936}a^{10}-\frac{371}{20992}a^{8}-\frac{3077}{10496}a^{6}-\frac{1161}{2624}a^{4}-\frac{291}{656}a^{2}-\frac{20}{41}$, $\frac{1}{335872}a^{23}+\frac{5}{20992}a^{21}-\frac{75}{167936}a^{19}+\frac{553}{167936}a^{17}+\frac{12421}{335872}a^{15}-\frac{485}{167936}a^{13}-\frac{21087}{335872}a^{11}+\frac{4877}{41984}a^{9}-\frac{453}{20992}a^{7}+\frac{151}{5248}a^{5}+\frac{201}{1312}a^{3}-\frac{10}{41}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{4}$, which has order $4$ (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:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{475}{335872} a^{23} + \frac{117}{41984} a^{21} + \frac{1439}{167936} a^{19} + \frac{2571}{167936} a^{17} - \frac{8617}{335872} a^{15} + \frac{373}{167936} a^{13} - \frac{122533}{335872} a^{11} + \frac{1493}{20992} a^{9} - \frac{10561}{10496} a^{7} + \frac{113}{82} a^{5} - \frac{629}{1312} a^{3} + \frac{1155}{328} a \)  (order $28$) Copy content Toggle raw display
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:   $\frac{13}{83968}a^{22}-\frac{13}{41984}a^{20}+\frac{501}{41984}a^{18}-\frac{109}{41984}a^{16}+\frac{6493}{83968}a^{14}-\frac{1697}{20992}a^{12}+\frac{12377}{83968}a^{10}-\frac{24745}{41984}a^{8}-\frac{503}{10496}a^{6}-\frac{401}{328}a^{4}-\frac{63}{656}a^{2}-\frac{28}{41}$, $\frac{219}{167936}a^{22}+\frac{29}{10496}a^{20}+\frac{303}{83968}a^{18}+\frac{403}{83968}a^{16}-\frac{12697}{167936}a^{14}-\frac{2567}{83968}a^{12}-\frac{76565}{167936}a^{10}+\frac{7639}{20992}a^{8}-\frac{1401}{2624}a^{6}+\frac{1779}{656}a^{4}+\frac{477}{656}a^{2}+\frac{643}{164}$, $\frac{221}{167936}a^{23}-\frac{221}{83968}a^{21}+\frac{973}{83968}a^{19}-\frac{3165}{83968}a^{17}+\frac{11981}{167936}a^{15}-\frac{8841}{41984}a^{13}+\frac{66745}{167936}a^{11}-\frac{52649}{83968}a^{9}+\frac{28267}{20992}a^{7}-\frac{3775}{2624}a^{5}+\frac{2373}{1312}a^{3}-\frac{583}{164}a$, $\frac{475}{335872}a^{23}+\frac{117}{41984}a^{21}+\frac{1439}{167936}a^{19}+\frac{2571}{167936}a^{17}-\frac{8617}{335872}a^{15}+\frac{373}{167936}a^{13}-\frac{122533}{335872}a^{11}+\frac{1493}{20992}a^{9}-\frac{10561}{10496}a^{7}+\frac{113}{82}a^{5}-\frac{629}{1312}a^{3}+\frac{1155}{328}a+1$, $\frac{221}{167936}a^{23}-\frac{221}{83968}a^{21}+\frac{973}{83968}a^{19}-\frac{3165}{83968}a^{17}+\frac{11981}{167936}a^{15}-\frac{8841}{41984}a^{13}+\frac{66745}{167936}a^{11}-\frac{52649}{83968}a^{9}+\frac{28267}{20992}a^{7}-\frac{3775}{2624}a^{5}+\frac{2373}{1312}a^{3}-\frac{419}{164}a-1$, $\frac{355}{335872}a^{23}+\frac{147}{41984}a^{21}+\frac{927}{167936}a^{19}+\frac{2467}{167936}a^{17}-\frac{11329}{335872}a^{15}-\frac{1451}{167936}a^{13}-\frac{99981}{335872}a^{11}+\frac{171}{10496}a^{9}-\frac{11903}{20992}a^{7}+\frac{4979}{5248}a^{5}-\frac{67}{1312}a^{3}+\frac{355}{164}a-1$, $\frac{1}{512}a^{23}-\frac{89}{41984}a^{22}+\frac{1}{512}a^{21}-\frac{109}{41984}a^{20}+\frac{9}{512}a^{19}-\frac{295}{20992}a^{18}-\frac{29}{10496}a^{16}+\frac{29}{512}a^{15}+\frac{465}{41984}a^{14}-\frac{63}{512}a^{13}+\frac{4289}{41984}a^{12}+\frac{5}{256}a^{11}+\frac{12309}{41984}a^{10}-\frac{313}{512}a^{9}+\frac{9747}{41984}a^{8}-\frac{17}{512}a^{7}+\frac{6001}{10496}a^{6}-\frac{153}{128}a^{5}-\frac{385}{328}a^{4}+\frac{5}{16}a^{3}-\frac{749}{656}a^{2}-\frac{5}{8}a-\frac{315}{82}$, $\frac{389}{167936}a^{23}+\frac{119}{167936}a^{22}-\frac{133}{41984}a^{21}+\frac{125}{41984}a^{20}+\frac{1165}{83968}a^{19}+\frac{259}{83968}a^{18}-\frac{4151}{83968}a^{17}+\frac{2339}{83968}a^{16}+\frac{9185}{167936}a^{15}-\frac{6757}{167936}a^{14}-\frac{24747}{83968}a^{13}+\frac{8951}{83968}a^{12}+\frac{47669}{167936}a^{11}-\frac{73297}{167936}a^{10}-\frac{26115}{41984}a^{9}+\frac{15719}{41984}a^{8}+\frac{29563}{20992}a^{7}-\frac{15121}{10496}a^{6}-\frac{1627}{5248}a^{5}+\frac{519}{328}a^{4}+\frac{2011}{656}a^{3}-\frac{689}{328}a^{2}+\frac{39}{328}a+\frac{525}{164}$, $\frac{345}{335872}a^{23}+\frac{9}{167936}a^{22}-\frac{111}{83968}a^{21}-\frac{115}{20992}a^{20}+\frac{1021}{167936}a^{19}+\frac{637}{83968}a^{18}-\frac{1915}{167936}a^{17}-\frac{3223}{83968}a^{16}-\frac{731}{335872}a^{15}+\frac{14045}{167936}a^{14}-\frac{13083}{167936}a^{13}-\frac{11089}{83968}a^{12}-\frac{21295}{335872}a^{11}+\frac{75897}{167936}a^{10}-\frac{5193}{83968}a^{9}-\frac{1345}{2624}a^{8}-\frac{157}{20992}a^{7}+\frac{10683}{10496}a^{6}+\frac{1267}{1312}a^{5}-\frac{3643}{2624}a^{4}+\frac{229}{328}a^{3}+\frac{579}{656}a^{2}+\frac{731}{328}a-\frac{155}{82}$, $\frac{231}{335872}a^{23}+\frac{195}{167936}a^{22}-\frac{17}{10496}a^{21}-\frac{241}{41984}a^{20}+\frac{59}{167936}a^{19}+\frac{1447}{83968}a^{18}-\frac{177}{167936}a^{17}-\frac{5161}{83968}a^{16}-\frac{11901}{335872}a^{15}+\frac{15559}{167936}a^{14}+\frac{14245}{167936}a^{13}-\frac{25121}{83968}a^{12}-\frac{93449}{335872}a^{11}+\frac{53307}{167936}a^{10}+\frac{25573}{41984}a^{9}-\frac{22551}{41984}a^{8}-\frac{10563}{10496}a^{7}+\frac{2301}{2624}a^{6}+\frac{5161}{2624}a^{5}+\frac{1483}{2624}a^{4}-\frac{2523}{1312}a^{3}+\frac{471}{328}a^{2}+\frac{1077}{328}a+\frac{159}{41}$, $\frac{93}{83968}a^{23}+\frac{13}{167936}a^{22}-\frac{93}{41984}a^{21}+\frac{7}{20992}a^{20}+\frac{405}{41984}a^{19}+\frac{665}{83968}a^{18}-\frac{1789}{41984}a^{17}+\frac{957}{83968}a^{16}+\frac{3053}{83968}a^{15}+\frac{8625}{167936}a^{14}-\frac{5025}{20992}a^{13}+\frac{2387}{83968}a^{12}+\frac{13961}{83968}a^{11}+\frac{14509}{167936}a^{10}-\frac{19481}{41984}a^{9}-\frac{2719}{10496}a^{8}+\frac{11149}{10496}a^{7}-\frac{193}{328}a^{6}+\frac{359}{656}a^{5}-\frac{1499}{1312}a^{4}+\frac{1675}{656}a^{3}-\frac{1159}{656}a^{2}+\frac{93}{41}a-\frac{97}{164}$ Copy content Toggle raw display (assuming GRH)
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:  \( 32831618.65116483 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 32831618.65116483 \cdot 4}{28\cdot\sqrt{4367896108464230086685082523795456}}\cr\approx \mathstrut & 0.268668360387794 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 + 10*x^20 - 14*x^18 + 37*x^16 - 154*x^14 + 97*x^12 - 616*x^10 + 592*x^8 - 896*x^6 + 2560*x^4 + 4096)
 
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^24 + 10*x^20 - 14*x^18 + 37*x^16 - 154*x^14 + 97*x^12 - 616*x^10 + 592*x^8 - 896*x^6 + 2560*x^4 + 4096, 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^24 + 10*x^20 - 14*x^18 + 37*x^16 - 154*x^14 + 97*x^12 - 616*x^10 + 592*x^8 - 896*x^6 + 2560*x^4 + 4096);
 
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^24 + 10*x^20 - 14*x^18 + 37*x^16 - 154*x^14 + 97*x^12 - 616*x^10 + 592*x^8 - 896*x^6 + 2560*x^4 + 4096);
 
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

$C_2^3\times A_4$ (as 24T135):

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 96
The 32 conjugacy class representatives for $C_2^3\times A_4$
Character table for $C_2^3\times A_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(i, \sqrt{7})\), 6.0.257079872.1, 6.6.4016873.1, 6.0.573839.1, 6.6.36725696.1, 6.0.153664.1, \(\Q(\zeta_{28})^+\), \(\Q(\zeta_{7})\), \(\Q(\zeta_{28})\), 12.0.66090060587536384.1, 12.0.1348776746684416.1, 12.0.66090060587536384.2, 12.12.66090060587536384.1, 12.0.16135268698129.1, 12.0.66090060587536384.3

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 24 siblings: data not computed
Degree 32 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{4}$ ${\href{/padicField/5.6.0.1}{6} }^{4}$ R ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{12}$ ${\href{/padicField/17.6.0.1}{6} }^{4}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{12}$ ${\href{/padicField/43.2.0.1}{2} }^{12}$ ${\href{/padicField/47.6.0.1}{6} }^{4}$ ${\href{/padicField/53.6.0.1}{6} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
\(7\) Copy content Toggle raw display 7.12.10.1$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
\(239\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$