LMFDB - Number field 18.0.6131323620948659515767.1

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*x^2 - 3*x + 1)

gp:K = bnfinit(y^18 - 3*y^17 - 2*y^16 + 12*y^15 + 11*y^14 - 55*y^13 + 101*y^11 - 8*y^10 - 107*y^9 - 8*y^8 + 101*y^7 - 55*y^5 + 11*y^4 + 12*y^3 - 2*y^2 - 3*y + 1, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*x^2 - 3*x + 1);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 3*x^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*x^2 - 3*x + 1)

$ x^{18} - 3 x^{17} - 2 x^{16} + 12 x^{15} + 11 x^{14} - 55 x^{13} + 101 x^{11} - 8 x^{10} - 107 x^{9} + \cdots + 1 $ x^18 - 3*x^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*x^2 - 3*x + 1

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$18$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[0, 9]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$-6131323620948659515767$ -6131323620948659515767 $\medspace = -\,3^{6}\cdot 7^{15}\cdot 11^{6}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$16.23$	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:	$3^{1/2}7^{5/6}11^{1/2}\approx 29.074036854473896$
Ramified primes:	$3$, $7$, $11$ 3, 7, 11	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-7}) $
$\Aut(K/\Q)$:	$C_6$	comment:Autmorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:	$\Q(\zeta_{7})$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{4}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{6}+\frac{1}{4}a^{5}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{3}{8}a^{2}-\frac{3}{8}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{4}a^{8}-\frac{1}{8}a^{7}-\frac{1}{4}a^{6}+\frac{3}{8}a^{5}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}+\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{40}a^{14}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{10}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{40}$, $\frac{1}{40}a^{15}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{10}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{40}a$, $\frac{1}{200}a^{16}+\frac{1}{200}a^{15}+\frac{1}{200}a^{14}-\frac{1}{20}a^{13}-\frac{1}{40}a^{12}+\frac{1}{20}a^{11}+\frac{9}{40}a^{10}+\frac{23}{100}a^{9}+\frac{31}{200}a^{8}-\frac{1}{50}a^{7}+\frac{9}{40}a^{6}-\frac{1}{5}a^{5}+\frac{9}{40}a^{4}+\frac{9}{20}a^{3}+\frac{19}{50}a^{2}-\frac{49}{200}a-\frac{37}{100}$, $\frac{1}{200}a^{17}-\frac{1}{200}a^{14}+\frac{1}{40}a^{13}-\frac{1}{20}a^{12}+\frac{7}{40}a^{11}+\frac{13}{100}a^{10}-\frac{3}{40}a^{9}-\frac{1}{20}a^{8}-\frac{41}{200}a^{7}+\frac{1}{5}a^{6}-\frac{13}{40}a^{5}-\frac{3}{20}a^{4}-\frac{7}{100}a^{3}-\frac{1}{2}a^{2}+\frac{3}{8}a+\frac{59}{200}$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2*a - 1/2, 1/2*a^7 - 1/2, 1/2*a^8 - 1/2*a, 1/2*a^9 - 1/2*a^2, 1/2*a^10 - 1/2*a^3, 1/2*a^11 - 1/2*a^4, 1/8*a^12 - 1/8*a^10 - 1/8*a^8 - 1/4*a^7 - 1/8*a^6 + 1/4*a^5 - 1/8*a^4 - 1/2*a^3 + 3/8*a^2 - 3/8, 1/8*a^13 - 1/8*a^11 - 1/8*a^9 - 1/4*a^8 - 1/8*a^7 - 1/4*a^6 + 3/8*a^5 - 1/8*a^3 - 1/2*a^2 + 1/8*a - 1/2, 1/40*a^14 - 1/4*a^10 - 1/4*a^9 - 1/4*a^8 - 1/10*a^7 - 1/4*a^6 - 1/4*a^5 + 1/4*a^4 - 1/2*a^3 + 1/40, 1/40*a^15 - 1/4*a^11 - 1/4*a^10 - 1/4*a^9 - 1/10*a^8 - 1/4*a^7 - 1/4*a^6 + 1/4*a^5 - 1/2*a^4 + 1/40*a, 1/200*a^16 + 1/200*a^15 + 1/200*a^14 - 1/20*a^13 - 1/40*a^12 + 1/20*a^11 + 9/40*a^10 + 23/100*a^9 + 31/200*a^8 - 1/50*a^7 + 9/40*a^6 - 1/5*a^5 + 9/40*a^4 + 9/20*a^3 + 19/50*a^2 - 49/200*a - 37/100, 1/200*a^17 - 1/200*a^14 + 1/40*a^13 - 1/20*a^12 + 7/40*a^11 + 13/100*a^10 - 3/40*a^9 - 1/20*a^8 - 41/200*a^7 + 1/5*a^6 - 13/40*a^5 - 3/20*a^4 - 7/100*a^3 - 1/2*a^2 + 3/8*a + 59/200

comment:Integral basis

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

Ideal class group:		Trivial group, which has order $1$	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		Trivial group, which has order $1$	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$8$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -\frac{297}{200} a^{17} + \frac{693}{200} a^{16} + \frac{549}{100} a^{15} - \frac{297}{20} a^{14} - \frac{1053}{40} a^{13} + \frac{2651}{40} a^{12} + \frac{1827}{40} a^{11} - \frac{26037}{200} a^{10} - \frac{14067}{200} a^{9} + \frac{25083}{200} a^{8} + \frac{3609}{40} a^{7} - \frac{4059}{40} a^{6} - \frac{2583}{40} a^{5} + \frac{1989}{40} a^{4} + \frac{1089}{100} a^{3} - \frac{1341}{100} a^{2} - \frac{477}{200} a + \frac{117}{40} $ -(297)/(200)a^(17) + (693)/(200)a^(16) + (549)/(100)a^(15) - (297)/(20)a^(14) - (1053)/(40)a^(13) + (2651)/(40)a^(12) + (1827)/(40)a^(11) - (26037)/(200)a^(10) - (14067)/(200)a^(9) + (25083)/(200)a^(8) + (3609)/(40)a^(7) - (4059)/(40)a^(6) - (2583)/(40)a^(5) + (1989)/(40)a^(4) + (1089)/(100)a^(3) - (1341)/(100)a^(2) - (477)/(200)*a + (117)/(40) (order $14$)	comment:Generator for roots of unity 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:	$a^{17}-3a^{16}-2a^{15}+12a^{14}+11a^{13}-55a^{12}+101a^{10}-8a^{9}-107a^{8}-8a^{7}+101a^{6}-55a^{4}+11a^{3}+12a^{2}-2a-3$, $\frac{53}{40}a^{17}-\frac{287}{100}a^{16}-\frac{1049}{200}a^{15}+\frac{2391}{200}a^{14}+\frac{1013}{40}a^{13}-\frac{267}{5}a^{12}-\frac{1943}{40}a^{11}+\frac{2021}{20}a^{10}+\frac{16271}{200}a^{9}-\frac{4361}{50}a^{8}-\frac{19739}{200}a^{7}+\frac{641}{10}a^{6}+\frac{2797}{40}a^{5}-\frac{219}{10}a^{4}-\frac{327}{20}a^{3}+\frac{169}{50}a^{2}+\frac{463}{100}a-\frac{59}{200}$, $\frac{139}{100}a^{17}-\frac{79}{25}a^{16}-\frac{1077}{200}a^{15}+\frac{541}{40}a^{14}+\frac{1047}{40}a^{13}-\frac{303}{5}a^{12}-\frac{1973}{40}a^{11}+\frac{5947}{50}a^{10}+\frac{16733}{200}a^{9}-\frac{5773}{50}a^{8}-\frac{4361}{40}a^{7}+\frac{1783}{20}a^{6}+\frac{3317}{40}a^{5}-\frac{743}{20}a^{4}-\frac{4997}{200}a^{3}+\frac{171}{25}a^{2}+\frac{549}{100}a-\frac{83}{40}$, $\frac{467}{200}a^{17}-\frac{601}{100}a^{16}-\frac{184}{25}a^{15}+\frac{5061}{200}a^{14}+\frac{733}{20}a^{13}-\frac{2291}{20}a^{12}-\frac{199}{4}a^{11}+\frac{22251}{100}a^{10}+\frac{3707}{50}a^{9}-\frac{11663}{50}a^{8}-\frac{11167}{100}a^{7}+\frac{4089}{20}a^{6}+\frac{327}{4}a^{5}-\frac{439}{4}a^{4}-\frac{3293}{200}a^{3}+\frac{3099}{100}a^{2}+\frac{207}{50}a-\frac{1419}{200}$, $\frac{757}{200}a^{17}-\frac{1779}{200}a^{16}-\frac{2719}{200}a^{15}+\frac{3707}{100}a^{14}+\frac{267}{4}a^{13}-\frac{667}{4}a^{12}-\frac{2267}{20}a^{11}+\frac{31751}{100}a^{10}+\frac{9309}{50}a^{9}-\frac{30017}{100}a^{8}-\frac{24323}{100}a^{7}+\frac{951}{4}a^{6}+\frac{3473}{20}a^{5}-\frac{2089}{20}a^{4}-\frac{8183}{200}a^{3}+\frac{4171}{200}a^{2}+\frac{1981}{200}a-\frac{127}{25}$, $\frac{299}{100}a^{17}-\frac{351}{50}a^{16}-\frac{1067}{100}a^{15}+\frac{5823}{200}a^{14}+\frac{2101}{40}a^{13}-\frac{1313}{10}a^{12}-\frac{3537}{40}a^{11}+\frac{24909}{100}a^{10}+\frac{29121}{200}a^{9}-\frac{5923}{25}a^{8}-\frac{37777}{200}a^{7}+\frac{1927}{10}a^{6}+\frac{5393}{40}a^{5}-\frac{1717}{20}a^{4}-\frac{6007}{200}a^{3}+\frac{1049}{50}a^{2}+\frac{1891}{200}a-\frac{997}{200}$, $\frac{513}{200}a^{17}-\frac{1373}{200}a^{16}-\frac{1593}{200}a^{15}+\frac{1501}{50}a^{14}+\frac{1559}{40}a^{13}-\frac{5393}{40}a^{12}-\frac{407}{8}a^{11}+\frac{54653}{200}a^{10}+\frac{13647}{200}a^{9}-\frac{59913}{200}a^{8}-\frac{25001}{200}a^{7}+\frac{10667}{40}a^{6}+\frac{831}{8}a^{5}-\frac{1179}{8}a^{4}-\frac{719}{25}a^{3}+\frac{844}{25}a^{2}+\frac{433}{50}a-\frac{1141}{200}$, $\frac{269}{200}a^{17}-\frac{519}{200}a^{16}-\frac{1179}{200}a^{15}+\frac{2067}{200}a^{14}+\frac{144}{5}a^{13}-\frac{927}{20}a^{12}-62a^{11}+\frac{4091}{50}a^{10}+\frac{2827}{25}a^{9}-\frac{2791}{50}a^{8}-\frac{6337}{50}a^{7}+\frac{112}{5}a^{6}+\frac{343}{4}a^{5}+11a^{4}-\frac{4301}{200}a^{3}-\frac{1169}{200}a^{2}+\frac{471}{200}a+\frac{157}{200}$ a^17 - 3a^16 - 2a^15 + 12a^14 + 11a^13 - 55a^12 + 101a^10 - 8a^9 - 107a^8 - 8a^7 + 101a^6 - 55a^4 + 11a^3 + 12a^2 - 2a - 3, 53/40a^17 - 287/100a^16 - 1049/200a^15 + 2391/200a^14 + 1013/40a^13 - 267/5a^12 - 1943/40a^11 + 2021/20a^10 + 16271/200a^9 - 4361/50a^8 - 19739/200a^7 + 641/10a^6 + 2797/40a^5 - 219/10a^4 - 327/20a^3 + 169/50a^2 + 463/100a - 59/200, 139/100a^17 - 79/25a^16 - 1077/200a^15 + 541/40a^14 + 1047/40a^13 - 303/5a^12 - 1973/40a^11 + 5947/50a^10 + 16733/200a^9 - 5773/50a^8 - 4361/40a^7 + 1783/20a^6 + 3317/40a^5 - 743/20a^4 - 4997/200a^3 + 171/25a^2 + 549/100a - 83/40, 467/200a^17 - 601/100a^16 - 184/25a^15 + 5061/200a^14 + 733/20a^13 - 2291/20a^12 - 199/4a^11 + 22251/100a^10 + 3707/50a^9 - 11663/50a^8 - 11167/100a^7 + 4089/20a^6 + 327/4a^5 - 439/4a^4 - 3293/200a^3 + 3099/100a^2 + 207/50a - 1419/200, 757/200a^17 - 1779/200a^16 - 2719/200a^15 + 3707/100a^14 + 267/4a^13 - 667/4a^12 - 2267/20a^11 + 31751/100a^10 + 9309/50a^9 - 30017/100a^8 - 24323/100a^7 + 951/4a^6 + 3473/20a^5 - 2089/20a^4 - 8183/200a^3 + 4171/200a^2 + 1981/200a - 127/25, 299/100a^17 - 351/50a^16 - 1067/100a^15 + 5823/200a^14 + 2101/40a^13 - 1313/10a^12 - 3537/40a^11 + 24909/100a^10 + 29121/200a^9 - 5923/25a^8 - 37777/200a^7 + 1927/10a^6 + 5393/40a^5 - 1717/20a^4 - 6007/200a^3 + 1049/50a^2 + 1891/200a - 997/200, 513/200a^17 - 1373/200a^16 - 1593/200a^15 + 1501/50a^14 + 1559/40a^13 - 5393/40a^12 - 407/8a^11 + 54653/200a^10 + 13647/200a^9 - 59913/200a^8 - 25001/200a^7 + 10667/40a^6 + 831/8a^5 - 1179/8a^4 - 719/25a^3 + 844/25a^2 + 433/50a - 1141/200, 269/200a^17 - 519/200a^16 - 1179/200a^15 + 2067/200a^14 + 144/5a^13 - 927/20a^12 - 62a^11 + 4091/50a^10 + 2827/25a^9 - 2791/50a^8 - 6337/50a^7 + 112/5a^6 + 343/4a^5 + 11a^4 - 4301/200a^3 - 1169/200a^2 + 471/200*a + 157/200	comment:Fundamental units 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:	$ 27994.834823771416 $	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 27994.834823771416 \cdot 1}{14\cdot\sqrt{6131323620948659515767}}\cr\approx \mathstrut & 0.389754886856357 \end{aligned}\]

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*x^2 - 3*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))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^18 - 3*x^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*x^2 - 3*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)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*x^2 - 3*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)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*x^2 - 3*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

$C_6\times S_3$ (as 18T6):

comment: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)

A solvable group of order 36

The 18 conjugacy class representatives for $S_3 \times C_6$

Character table for $S_3 \times C_6$

Intermediate fields

$\Q(\sqrt{-7}) $, $\Q(\zeta_{7})^+$, 3.1.231.1, $\Q(\zeta_{7})$, 6.0.373527.1, 9.3.29595664791.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

comment:Intermediate fields

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

Galois closure:	data not computed
Degree 12 sibling:	12.0.364807736118799281.4
Degree 18 sibling:	deg 18
Minimal sibling:	12.0.364807736118799281.4

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.3.0.1}{3} }^{6}$	R	${\href{/padicField/5.6.0.1}{6} }^{3}$	R	R	${\href{/padicField/13.6.0.1}{6} }^{3}$	${\href{/padicField/17.6.0.1}{6} }^{3}$	${\href{/padicField/19.6.0.1}{6} }^{3}$	${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$	${\href{/padicField/29.3.0.1}{3} }^{6}$	${\href{/padicField/31.6.0.1}{6} }^{3}$	${\href{/padicField/37.3.0.1}{3} }^{6}$	${\href{/padicField/41.2.0.1}{2} }^{9}$	${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$	${\href{/padicField/47.6.0.1}{6} }^{3}$	${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$	${\href{/padicField/59.6.0.1}{6} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

comment:Frobenius cycle types

sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
$3$ 3	3.6.1.0a1.1	$x^{6} + 2 x^{4} + x^{2} + 2 x + 2$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
$3$ 3	3.6.2.6a1.2	$x^{12} + 4 x^{10} + 6 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} + 9 x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$	$2$	$6$	$6$	$C_6\times C_2$	$$[\ ]_{2}^{6}$$
$7$ 7	7.1.6.5a1.1	$x^{6} + 7$	$6$	$1$	$5$	$C_6$	$$[\ ]_{6}$$
$7$ 7	7.2.6.10a1.2	$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 192996 x^{6} + 266328 x^{5} + 234495 x^{4} + 131220 x^{3} + 45198 x^{2} + 8748 x + 736$	$6$	$2$	$10$	$C_6\times C_2$	$$[\ ]_{6}^{2}$$
$11$ 11	11.3.1.0a1.1	$x^{3} + 2 x + 9$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
	11.3.1.0a1.1	$x^{3} + 2 x + 9$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
	11.3.2.3a1.1	$x^{6} + 4 x^{4} + 18 x^{3} + 4 x^{2} + 47 x + 81$	$2$	$3$	$3$	$C_6$	$$[\ ]_{2}^{3}$$
	11.3.2.3a1.1	$x^{6} + 4 x^{4} + 18 x^{3} + 4 x^{2} + 47 x + 81$	$2$	$3$	$3$	$C_6$	$$[\ ]_{2}^{3}$$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*³⁶	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
	1.33.2t1.a.a	$1$	$ 3 \cdot 11 $	$\Q(\sqrt{33}) $	$C_2$ (as 2T1)	$1$	$1$
	1.231.2t1.a.a	$1$	$ 3 \cdot 7 \cdot 11 $	$\Q(\sqrt{-231}) $	$C_2$ (as 2T1)	$1$	$-1$
*³⁶	1.7.2t1.a.a	$1$	$ 7 $	$\Q(\sqrt{-7}) $	$C_2$ (as 2T1)	$1$	$-1$
	1.231.6t1.a.a	$1$	$ 3 \cdot 7 \cdot 11 $	6.0.603993159.1	$C_6$ (as 6T1)	$0$	$-1$
*³⁶	1.7.3t1.a.a	$1$	$ 7 $	$\Q(\zeta_{7})^+$	$C_3$ (as 3T1)	$0$	$1$
*³⁶	1.7.3t1.a.b	$1$	$ 7 $	$\Q(\zeta_{7})^+$	$C_3$ (as 3T1)	$0$	$1$
*³⁶	1.7.6t1.a.a	$1$	$ 7 $	$\Q(\zeta_{7})$	$C_6$ (as 6T1)	$0$	$-1$
	1.231.6t1.a.b	$1$	$ 3 \cdot 7 \cdot 11 $	6.0.603993159.1	$C_6$ (as 6T1)	$0$	$-1$
	1.231.6t1.b.a	$1$	$ 3 \cdot 7 \cdot 11 $	6.6.86284737.1	$C_6$ (as 6T1)	$0$	$1$
	1.231.6t1.b.b	$1$	$ 3 \cdot 7 \cdot 11 $	6.6.86284737.1	$C_6$ (as 6T1)	$0$	$1$
*³⁶	1.7.6t1.a.b	$1$	$ 7 $	$\Q(\zeta_{7})$	$C_6$ (as 6T1)	$0$	$-1$
*³⁶	2.231.3t2.a.a	$2$	$ 3 \cdot 7 \cdot 11 $	3.1.231.1	$S_3$ (as 3T2)	$1$	$0$
*³⁶	2.231.6t3.d.a	$2$	$ 3 \cdot 7 \cdot 11 $	6.2.1760913.1	$D_{6}$ (as 6T3)	$1$	$0$
*³⁶	2.1617.6t5.a.a	$2$	$ 3 \cdot 7^{2} \cdot 11 $	6.0.603993159.2	$S_3\times C_3$ (as 6T5)	$0$	$0$
*³⁶	2.1617.12t18.a.a	$2$	$ 3 \cdot 7^{2} \cdot 11 $	18.0.6131323620948659515767.1	$S_3 \times C_6$ (as 18T6)	$0$	$0$
*³⁶	2.1617.12t18.a.b	$2$	$ 3 \cdot 7^{2} \cdot 11 $	18.0.6131323620948659515767.1	$S_3 \times C_6$ (as 18T6)	$0$	$0$
*³⁶	2.1617.6t5.a.b	$2$	$ 3 \cdot 7^{2} \cdot 11 $	6.0.603993159.2	$S_3\times C_3$ (as 6T5)	$0$	$0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

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