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Properties

Label 24.0.394...336.1
Degree $24$
Signature $[0, 12]$
Discriminant $3.949\times 10^{33}$
Root discriminant \(25.11\)
Ramified primes $2,3,199$
Class number $3$ (GRH)
Class group [3] (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 + 6*x^22 + 9*x^20 - 21*x^18 - 72*x^16 + 33*x^14 + 341*x^12 + 132*x^10 - 1152*x^8 - 1344*x^6 + 2304*x^4 + 6144*x^2 + 4096)
 
gp: K = bnfinit(y^24 + 6*y^22 + 9*y^20 - 21*y^18 - 72*y^16 + 33*y^14 + 341*y^12 + 132*y^10 - 1152*y^8 - 1344*y^6 + 2304*y^4 + 6144*y^2 + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 6*x^22 + 9*x^20 - 21*x^18 - 72*x^16 + 33*x^14 + 341*x^12 + 132*x^10 - 1152*x^8 - 1344*x^6 + 2304*x^4 + 6144*x^2 + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 + 6*x^22 + 9*x^20 - 21*x^18 - 72*x^16 + 33*x^14 + 341*x^12 + 132*x^10 - 1152*x^8 - 1344*x^6 + 2304*x^4 + 6144*x^2 + 4096)
 

\( x^{24} + 6 x^{22} + 9 x^{20} - 21 x^{18} - 72 x^{16} + 33 x^{14} + 341 x^{12} + 132 x^{10} - 1152 x^{8} + \cdots + 4096 \) 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:   \(3949093091982271355222362303758336\) \(\medspace = 2^{24}\cdot 3^{36}\cdot 199^{4}\) 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:  \(25.11\)
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:  $2\cdot 3^{3/2}199^{1/2}\approx 146.6015006744474$
Ramified primes:   \(2\), \(3\), \(199\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{2}a^{12}+\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{15}-\frac{1}{4}a^{13}+\frac{1}{8}a^{11}+\frac{3}{8}a^{9}+\frac{1}{8}a^{5}-\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{16}-\frac{1}{8}a^{14}-\frac{7}{16}a^{12}+\frac{3}{16}a^{10}+\frac{1}{16}a^{6}-\frac{3}{16}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{17}-\frac{1}{16}a^{15}-\frac{7}{32}a^{13}+\frac{3}{32}a^{11}+\frac{1}{32}a^{7}+\frac{13}{32}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{1216}a^{18}+\frac{13}{608}a^{16}+\frac{33}{1216}a^{14}-\frac{545}{1216}a^{12}-\frac{1}{16}a^{10}-\frac{5}{64}a^{8}-\frac{599}{1216}a^{6}+\frac{1}{152}a^{4}-\frac{37}{76}a^{2}-\frac{8}{19}$, $\frac{1}{2432}a^{19}+\frac{13}{1216}a^{17}+\frac{33}{2432}a^{15}-\frac{545}{2432}a^{13}-\frac{1}{32}a^{11}+\frac{59}{128}a^{9}+\frac{617}{2432}a^{7}-\frac{151}{304}a^{5}+\frac{39}{152}a^{3}+\frac{11}{38}a$, $\frac{1}{4864}a^{20}-\frac{1}{2432}a^{18}-\frac{87}{4864}a^{16}-\frac{253}{4864}a^{14}+\frac{75}{304}a^{12}+\frac{11}{256}a^{10}+\frac{845}{4864}a^{8}-\frac{213}{1216}a^{6}+\frac{63}{304}a^{4}-\frac{17}{38}a^{2}-\frac{1}{19}$, $\frac{1}{9728}a^{21}-\frac{1}{4864}a^{19}-\frac{87}{9728}a^{17}-\frac{253}{9728}a^{15}+\frac{75}{608}a^{13}+\frac{11}{512}a^{11}-\frac{4019}{9728}a^{9}-\frac{213}{2432}a^{7}+\frac{63}{608}a^{5}-\frac{17}{76}a^{3}+\frac{9}{19}a$, $\frac{1}{19456}a^{22}-\frac{1}{9728}a^{20}-\frac{7}{19456}a^{18}-\frac{605}{19456}a^{16}-\frac{1}{19}a^{14}-\frac{6911}{19456}a^{12}+\frac{2061}{19456}a^{10}+\frac{319}{4864}a^{8}+\frac{141}{304}a^{6}-\frac{31}{152}a^{4}+\frac{23}{76}a^{2}-\frac{2}{19}$, $\frac{1}{38912}a^{23}-\frac{1}{19456}a^{21}-\frac{7}{38912}a^{19}-\frac{605}{38912}a^{17}-\frac{1}{38}a^{15}-\frac{6911}{38912}a^{13}-\frac{17395}{38912}a^{11}+\frac{319}{9728}a^{9}+\frac{141}{608}a^{7}-\frac{31}{304}a^{5}-\frac{53}{152}a^{3}-\frac{1}{19}a$ Copy content Toggle raw display

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

$C_{3}$, which has order $3$ (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:   \( \frac{21}{1216} a^{23} + \frac{403}{4864} a^{21} + \frac{37}{1216} a^{19} - \frac{2269}{4864} a^{17} - \frac{2825}{4864} a^{15} + \frac{4259}{2432} a^{13} + \frac{17647}{4864} a^{11} - \frac{21019}{4864} a^{9} - \frac{37199}{2432} a^{7} + \frac{1703}{608} a^{5} + \frac{6523}{152} a^{3} + \frac{1421}{38} a \)  (order $36$) 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:   $\frac{53}{2432}a^{22}-\frac{311}{4864}a^{20}-\frac{493}{1216}a^{18}+\frac{379}{4864}a^{16}+\frac{11651}{4864}a^{14}+\frac{2663}{2432}a^{12}-\frac{49377}{4864}a^{10}-\frac{43779}{4864}a^{8}+\frac{18837}{608}a^{6}+\frac{13311}{304}a^{4}-\frac{4681}{76}a^{2}-\frac{2282}{19}$, $\frac{21}{1216}a^{23}-\frac{5}{152}a^{22}+\frac{403}{4864}a^{21}-\frac{489}{4864}a^{20}+\frac{37}{1216}a^{19}+\frac{221}{2432}a^{18}-\frac{2269}{4864}a^{17}+\frac{3215}{4864}a^{16}-\frac{2825}{4864}a^{15}+\frac{301}{4864}a^{14}+\frac{4259}{2432}a^{13}-\frac{1781}{608}a^{12}+\frac{17647}{4864}a^{11}-\frac{9705}{4864}a^{10}-\frac{21019}{4864}a^{9}+\frac{45363}{4864}a^{8}-\frac{37199}{2432}a^{7}+\frac{14811}{1216}a^{6}+\frac{1703}{608}a^{5}-\frac{1413}{76}a^{4}+\frac{6523}{152}a^{3}-\frac{3235}{76}a^{2}+\frac{1421}{38}a-\frac{292}{19}$, $\frac{213}{9728}a^{22}+\frac{89}{608}a^{20}+\frac{1513}{9728}a^{18}-\frac{7287}{9728}a^{16}-\frac{395}{256}a^{14}+\frac{23621}{9728}a^{12}+\frac{81451}{9728}a^{10}-\frac{21081}{4864}a^{8}-\frac{39051}{1216}a^{6}-\frac{2295}{304}a^{4}+\frac{3159}{38}a^{2}+\frac{1722}{19}$, $\frac{1781}{19456}a^{22}+\frac{3131}{9728}a^{20}-\frac{2947}{19456}a^{18}-\frac{39297}{19456}a^{16}-\frac{1079}{1216}a^{14}+\frac{166341}{19456}a^{12}+\frac{174033}{19456}a^{10}-\frac{124993}{4864}a^{8}-\frac{55745}{1216}a^{6}+\frac{13239}{304}a^{4}+\frac{5557}{38}a^{2}+\frac{1495}{19}$, $\frac{403}{38912}a^{23}+\frac{663}{19456}a^{21}-\frac{1053}{38912}a^{19}-\frac{8595}{38912}a^{17}-\frac{381}{9728}a^{15}+\frac{37763}{38912}a^{13}+\frac{28555}{38912}a^{11}-\frac{15023}{4864}a^{9}-\frac{653}{152}a^{7}+\frac{1795}{304}a^{5}+\frac{280}{19}a^{3}+\frac{205}{38}a+1$, $\frac{297}{38912}a^{23}+\frac{213}{9728}a^{22}+\frac{89}{1024}a^{21}+\frac{89}{608}a^{20}+\frac{5841}{38912}a^{19}+\frac{1513}{9728}a^{18}-\frac{15645}{38912}a^{17}-\frac{7287}{9728}a^{16}-\frac{5869}{4864}a^{15}-\frac{395}{256}a^{14}+\frac{41385}{38912}a^{13}+\frac{23621}{9728}a^{12}+\frac{236829}{38912}a^{11}+\frac{81451}{9728}a^{10}-\frac{6511}{9728}a^{9}-\frac{21081}{4864}a^{8}-\frac{26763}{1216}a^{7}-\frac{39051}{1216}a^{6}-\frac{219}{19}a^{5}-\frac{2295}{304}a^{4}+\frac{429}{8}a^{3}+\frac{3159}{38}a^{2}+\frac{1299}{19}a+\frac{1741}{19}$, $\frac{397}{19456}a^{22}+\frac{129}{9728}a^{20}-\frac{3651}{19456}a^{18}-\frac{4189}{19456}a^{16}+\frac{4329}{4864}a^{14}+\frac{30013}{19456}a^{12}-\frac{61179}{19456}a^{10}-\frac{901}{128}a^{8}+\frac{4453}{608}a^{6}+\frac{3683}{152}a^{4}-8a^{2}-\frac{799}{19}$, $\frac{117}{4864}a^{22}-\frac{103}{1216}a^{20}-\frac{2391}{4864}a^{18}+\frac{669}{4864}a^{16}+\frac{7105}{2432}a^{14}+\frac{5773}{4864}a^{12}-\frac{60849}{4864}a^{10}-\frac{25715}{2432}a^{8}+\frac{11701}{304}a^{6}+\frac{16129}{304}a^{4}-\frac{5877}{76}a^{2}-\frac{2868}{19}$, $\frac{25}{38912}a^{23}+\frac{1337}{19456}a^{22}-\frac{555}{19456}a^{21}+\frac{3155}{9728}a^{20}-\frac{2999}{38912}a^{19}+\frac{2257}{19456}a^{18}+\frac{4487}{38912}a^{17}-\frac{35773}{19456}a^{16}+\frac{5265}{9728}a^{15}-\frac{5509}{2432}a^{14}-\frac{7639}{38912}a^{13}+\frac{133545}{19456}a^{12}-\frac{98271}{38912}a^{11}+\frac{276285}{19456}a^{10}-\frac{2449}{4864}a^{9}-\frac{83079}{4864}a^{8}+\frac{5199}{608}a^{7}-\frac{72641}{1216}a^{6}+\frac{3975}{608}a^{5}+\frac{215}{19}a^{4}-\frac{3055}{152}a^{3}+\frac{6371}{38}a^{2}-\frac{1037}{38}a+\frac{2803}{19}$, $\frac{1533}{9728}a^{23}+\frac{799}{19456}a^{22}+\frac{803}{1216}a^{21}-\frac{701}{9728}a^{20}+\frac{141}{9728}a^{19}-\frac{12545}{19456}a^{18}-\frac{37887}{9728}a^{17}-\frac{991}{19456}a^{16}-\frac{16903}{4864}a^{15}+\frac{17739}{4864}a^{14}+\frac{150001}{9728}a^{13}+\frac{48063}{19456}a^{12}+\frac{238795}{9728}a^{11}-\frac{293161}{19456}a^{10}-\frac{204855}{4864}a^{9}-\frac{2087}{128}a^{8}-\frac{267999}{2432}a^{7}+\frac{54001}{1216}a^{6}+\frac{30699}{608}a^{5}+\frac{11011}{152}a^{4}+\frac{24573}{76}a^{3}-\frac{6385}{76}a^{2}+\frac{9077}{38}a-\frac{3478}{19}$, $\frac{7281}{38912}a^{23}+\frac{937}{19456}a^{22}+\frac{16387}{19456}a^{21}+\frac{2669}{9728}a^{20}+\frac{6921}{38912}a^{19}+\frac{4073}{19456}a^{18}-\frac{188805}{38912}a^{17}-\frac{28505}{19456}a^{16}-\frac{25515}{4864}a^{15}-\frac{12039}{4864}a^{14}+\frac{38171}{2048}a^{13}+\frac{98377}{19456}a^{12}+\frac{1339365}{38912}a^{11}+\frac{275649}{19456}a^{10}-\frac{472003}{9728}a^{9}-\frac{25937}{2432}a^{8}-\frac{361495}{2432}a^{7}-\frac{68337}{1216}a^{6}+\frac{13715}{304}a^{5}-\frac{339}{76}a^{4}+\frac{32291}{76}a^{3}+\frac{5693}{38}a^{2}+\frac{689}{2}a+\frac{2909}{19}$ 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:  \( 43460476.10350948 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 43460476.10350948 \cdot 3}{36\cdot\sqrt{3949093091982271355222362303758336}}\cr\approx \mathstrut & 0.218184034503331 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 + 6*x^22 + 9*x^20 - 21*x^18 - 72*x^16 + 33*x^14 + 341*x^12 + 132*x^10 - 1152*x^8 - 1344*x^6 + 2304*x^4 + 6144*x^2 + 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 + 6*x^22 + 9*x^20 - 21*x^18 - 72*x^16 + 33*x^14 + 341*x^12 + 132*x^10 - 1152*x^8 - 1344*x^6 + 2304*x^4 + 6144*x^2 + 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 + 6*x^22 + 9*x^20 - 21*x^18 - 72*x^16 + 33*x^14 + 341*x^12 + 132*x^10 - 1152*x^8 - 1344*x^6 + 2304*x^4 + 6144*x^2 + 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 + 6*x^22 + 9*x^20 - 21*x^18 - 72*x^16 + 33*x^14 + 341*x^12 + 132*x^10 - 1152*x^8 - 1344*x^6 + 2304*x^4 + 6144*x^2 + 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 $C_2^3\times A_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{12})\), 6.6.83560896.1, \(\Q(\zeta_{9})\), 6.0.1305639.1, \(\Q(\zeta_{36})^+\), 6.0.250682688.4, 6.0.419904.1, 6.6.3916917.1, 12.0.62841810062905344.1, 12.0.62841810062905344.3, 12.0.62841810062905344.2, 12.0.6982423340322816.1, \(\Q(\zeta_{36})\), 12.12.62841810062905344.1, 12.0.15342238784889.1

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 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 R ${\href{/padicField/5.6.0.1}{6} }^{4}$ ${\href{/padicField/7.6.0.1}{6} }^{4}$ ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.3.0.1}{3} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{12}$ ${\href{/padicField/19.2.0.1}{2} }^{12}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{16}$ ${\href{/padicField/41.6.0.1}{6} }^{4}$ ${\href{/padicField/43.6.0.1}{6} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{12}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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 2.12.12.26$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}$
\(3\) Copy content Toggle raw display 3.12.18.82$x^{12} + 24 x^{11} + 252 x^{10} + 1558 x^{9} + 6450 x^{8} + 19068 x^{7} + 41627 x^{6} + 68094 x^{5} + 83298 x^{4} + 74306 x^{3} + 45618 x^{2} + 17400 x + 3277$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
3.12.18.82$x^{12} + 24 x^{11} + 252 x^{10} + 1558 x^{9} + 6450 x^{8} + 19068 x^{7} + 41627 x^{6} + 68094 x^{5} + 83298 x^{4} + 74306 x^{3} + 45618 x^{2} + 17400 x + 3277$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
\(199\) Copy content Toggle raw display 199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.4.2.1$x^{4} + 386 x^{3} + 37653 x^{2} + 77972 x + 7450967$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
199.4.2.1$x^{4} + 386 x^{3} + 37653 x^{2} + 77972 x + 7450967$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$