LMFDB - Number field 18.0.5770142004982097067662109375.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 + 135*x^14 + 546*x^12 + 1287*x^10 - 76*x^9 + 1782*x^8 - 684*x^7 + 1386*x^6 - 2052*x^5 + 540*x^4 - 2280*x^3 + 81*x^2 - 684*x + 5779)

gp: K = bnfinit(y^18 + 18*y^16 + 135*y^14 + 546*y^12 + 1287*y^10 - 76*y^9 + 1782*y^8 - 684*y^7 + 1386*y^6 - 2052*y^5 + 540*y^4 - 2280*y^3 + 81*y^2 - 684*y + 5779, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 18*x^16 + 135*x^14 + 546*x^12 + 1287*x^10 - 76*x^9 + 1782*x^8 - 684*x^7 + 1386*x^6 - 2052*x^5 + 540*x^4 - 2280*x^3 + 81*x^2 - 684*x + 5779);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 18*x^16 + 135*x^14 + 546*x^12 + 1287*x^10 - 76*x^9 + 1782*x^8 - 684*x^7 + 1386*x^6 - 2052*x^5 + 540*x^4 - 2280*x^3 + 81*x^2 - 684*x + 5779)

$ x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} - 76 x^{9} + 1782 x^{8} - 684 x^{7} + 1386 x^{6} - 2052 x^{5} + 540 x^{4} - 2280 x^{3} + 81 x^{2} - 684 x + 5779 $ x^18 + 18*x^16 + 135*x^14 + 546*x^12 + 1287*x^10 - 76*x^9 + 1782*x^8 - 684*x^7 + 1386*x^6 - 2052*x^5 + 540*x^4 - 2280*x^3 + 81*x^2 - 684*x + 5779

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:	$[0, 9]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-5770142004982097067662109375$ -5770142004982097067662109375 $\medspace = -\,3^{45}\cdot 5^{9}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$34.86$	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))
Ramified primes:	$3$, $5$ 3, 5	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-15}) $
$\card{ \Gal(K/\Q) }$:	$18$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$135=3^{3}\cdot 5$
Dirichlet character group:	$\lbrace$$\chi_{135}(1,·)$, $\chi_{135}(134,·)$, $\chi_{135}(74,·)$, $\chi_{135}(76,·)$, $\chi_{135}(14,·)$, $\chi_{135}(16,·)$, $\chi_{135}(89,·)$, $\chi_{135}(91,·)$, $\chi_{135}(29,·)$, $\chi_{135}(31,·)$, $\chi_{135}(104,·)$, $\chi_{135}(106,·)$, $\chi_{135}(44,·)$, $\chi_{135}(46,·)$, $\chi_{135}(119,·)$, $\chi_{135}(121,·)$, $\chi_{135}(59,·)$, $\chi_{135}(61,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{256}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{34}a^{9}+\frac{9}{34}a^{7}-\frac{7}{34}a^{5}-\frac{2}{17}a^{3}+\frac{9}{34}a+\frac{13}{34}$, $\frac{1}{34}a^{10}+\frac{9}{34}a^{8}-\frac{7}{34}a^{6}-\frac{2}{17}a^{4}+\frac{9}{34}a^{2}+\frac{13}{34}a$, $\frac{1}{34}a^{11}+\frac{7}{17}a^{7}-\frac{9}{34}a^{5}+\frac{11}{34}a^{3}+\frac{13}{34}a^{2}-\frac{13}{34}a-\frac{15}{34}$, $\frac{1}{34}a^{12}+\frac{7}{17}a^{8}-\frac{9}{34}a^{6}+\frac{11}{34}a^{4}+\frac{13}{34}a^{3}-\frac{13}{34}a^{2}-\frac{15}{34}a$, $\frac{1}{34}a^{13}+\frac{1}{34}a^{7}+\frac{7}{34}a^{5}+\frac{13}{34}a^{4}+\frac{9}{34}a^{3}-\frac{15}{34}a^{2}+\frac{5}{17}a-\frac{6}{17}$, $\frac{1}{196418}a^{14}-\frac{38}{98209}a^{13}+\frac{7}{98209}a^{12}-\frac{494}{98209}a^{11}+\frac{77}{196418}a^{10}+\frac{837}{196418}a^{9}+\frac{105}{98209}a^{8}+\frac{2361}{11554}a^{7}+\frac{147}{98209}a^{6}+\frac{33523}{98209}a^{5}+\frac{5973}{196418}a^{4}-\frac{12693}{196418}a^{3}+\frac{23157}{196418}a^{2}+\frac{34168}{98209}a+\frac{86657}{196418}$, $\frac{1}{196418}a^{15}+\frac{15}{196418}a^{13}+\frac{38}{98209}a^{12}+\frac{45}{98209}a^{11}+\frac{456}{98209}a^{10}+\frac{275}{196418}a^{9}+\frac{2052}{98209}a^{8}+\frac{225}{98209}a^{7}-\frac{3917}{11554}a^{6}+\frac{189}{98209}a^{5}-\frac{24895}{98209}a^{4}-\frac{5637}{196418}a^{3}-\frac{83919}{196418}a^{2}-\frac{8658}{98209}a+\frac{23184}{98209}$, $\frac{1}{196418}a^{16}+\frac{608}{98209}a^{13}-\frac{60}{98209}a^{12}-\frac{1599}{196418}a^{11}-\frac{440}{98209}a^{10}-\frac{1337}{98209}a^{9}-\frac{1350}{98209}a^{8}-\frac{73613}{196418}a^{7}-\frac{2016}{98209}a^{6}+\frac{21075}{98209}a^{5}-\frac{47616}{98209}a^{4}+\frac{89145}{196418}a^{3}-\frac{360}{98209}a^{2}+\frac{42148}{98209}a+\frac{17301}{196418}$, $\frac{1}{196418}a^{17}-\frac{4}{5777}a^{13}-\frac{38}{5777}a^{12}-\frac{32}{5777}a^{11}+\frac{1903}{196418}a^{10}+\frac{2037}{196418}a^{9}+\frac{5935}{98209}a^{8}+\frac{45465}{196418}a^{7}+\frac{9856}{98209}a^{6}-\frac{46151}{196418}a^{5}+\frac{32279}{98209}a^{4}+\frac{72925}{196418}a^{3}-\frac{4975}{98209}a^{2}-\frac{11673}{98209}a-\frac{77533}{196418}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, 1/34*a^9 + 9/34*a^7 - 7/34*a^5 - 2/17*a^3 + 9/34*a + 13/34, 1/34*a^10 + 9/34*a^8 - 7/34*a^6 - 2/17*a^4 + 9/34*a^2 + 13/34*a, 1/34*a^11 + 7/17*a^7 - 9/34*a^5 + 11/34*a^3 + 13/34*a^2 - 13/34*a - 15/34, 1/34*a^12 + 7/17*a^8 - 9/34*a^6 + 11/34*a^4 + 13/34*a^3 - 13/34*a^2 - 15/34*a, 1/34*a^13 + 1/34*a^7 + 7/34*a^5 + 13/34*a^4 + 9/34*a^3 - 15/34*a^2 + 5/17*a - 6/17, 1/196418*a^14 - 38/98209*a^13 + 7/98209*a^12 - 494/98209*a^11 + 77/196418*a^10 + 837/196418*a^9 + 105/98209*a^8 + 2361/11554*a^7 + 147/98209*a^6 + 33523/98209*a^5 + 5973/196418*a^4 - 12693/196418*a^3 + 23157/196418*a^2 + 34168/98209*a + 86657/196418, 1/196418*a^15 + 15/196418*a^13 + 38/98209*a^12 + 45/98209*a^11 + 456/98209*a^10 + 275/196418*a^9 + 2052/98209*a^8 + 225/98209*a^7 - 3917/11554*a^6 + 189/98209*a^5 - 24895/98209*a^4 - 5637/196418*a^3 - 83919/196418*a^2 - 8658/98209*a + 23184/98209, 1/196418*a^16 + 608/98209*a^13 - 60/98209*a^12 - 1599/196418*a^11 - 440/98209*a^10 - 1337/98209*a^9 - 1350/98209*a^8 - 73613/196418*a^7 - 2016/98209*a^6 + 21075/98209*a^5 - 47616/98209*a^4 + 89145/196418*a^3 - 360/98209*a^2 + 42148/98209*a + 17301/196418, 1/196418*a^17 - 4/5777*a^13 - 38/5777*a^12 - 32/5777*a^11 + 1903/196418*a^10 + 2037/196418*a^9 + 5935/98209*a^8 + 45465/196418*a^7 + 9856/98209*a^6 - 46151/196418*a^5 + 32279/98209*a^4 + 72925/196418*a^3 - 4975/98209*a^2 - 11673/98209*a - 77533/196418

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

$C_{74}$, which has order $74$ (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:	$8$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -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:	$\frac{4}{98209}a^{15}+\frac{60}{98209}a^{13}+\frac{304}{98209}a^{12}+\frac{360}{98209}a^{11}+\frac{3648}{98209}a^{10}+\frac{1100}{98209}a^{9}+\frac{16416}{98209}a^{8}+\frac{1800}{98209}a^{7}+\frac{1663}{5777}a^{6}+\frac{1512}{98209}a^{5}-\frac{2742}{98209}a^{4}-\frac{22548}{98209}a^{3}-\frac{41049}{98209}a^{2}-\frac{69264}{98209}a-\frac{10946}{98209}$, $\frac{72}{98209}a^{15}+\frac{1080}{98209}a^{13}-\frac{305}{98209}a^{12}+\frac{6480}{98209}a^{11}-\frac{3660}{98209}a^{10}+\frac{19800}{98209}a^{9}-\frac{16470}{98209}a^{8}+\frac{32400}{98209}a^{7}-\frac{34160}{98209}a^{6}+\frac{27216}{98209}a^{5}-\frac{32025}{98209}a^{4}+\frac{10080}{98209}a^{3}-\frac{10980}{98209}a^{2}+\frac{1080}{98209}a-\frac{610}{98209}$, $\frac{114}{98209}a^{14}+\frac{3}{196418}a^{13}+\frac{1596}{98209}a^{12}+\frac{39}{196418}a^{11}+\frac{8778}{98209}a^{10}+\frac{195}{196418}a^{9}+\frac{23940}{98209}a^{8}+\frac{234}{98209}a^{7}+\frac{33516}{98209}a^{6}-\frac{16785}{196418}a^{5}+\frac{15803}{196418}a^{4}-\frac{43191}{98209}a^{3}-\frac{52184}{98209}a^{2}-\frac{43308}{98209}a-\frac{28657}{98209}$, $\frac{233}{196418}a^{14}-\frac{377}{196418}a^{13}+\frac{1631}{98209}a^{12}-\frac{4901}{196418}a^{11}+\frac{17941}{196418}a^{10}-\frac{24505}{196418}a^{9}+\frac{24465}{98209}a^{8}-\frac{29406}{98209}a^{7}+\frac{34251}{98209}a^{6}-\frac{34307}{98209}a^{5}+\frac{22834}{98209}a^{4}-\frac{34307}{196418}a^{3}+\frac{11417}{196418}a^{2}-\frac{4901}{196418}a+\frac{233}{98209}$, $\frac{21}{196418}a^{17}-\frac{38}{98209}a^{16}+\frac{349}{196418}a^{15}-\frac{494}{98209}a^{14}+\frac{1191}{98209}a^{13}-\frac{2660}{98209}a^{12}+\frac{4300}{98209}a^{11}-\frac{7448}{98209}a^{10}+\frac{17619}{196418}a^{9}-\frac{24929}{196418}a^{8}+\frac{1539}{11554}a^{7}-\frac{15469}{98209}a^{6}+\frac{2091}{11554}a^{5}-\frac{39989}{196418}a^{4}+\frac{43821}{196418}a^{3}-\frac{4565}{196418}a^{2}-\frac{270}{1853}a-\frac{45076}{98209}$, $\frac{21}{196418}a^{17}+\frac{21}{11554}a^{15}+\frac{147}{11554}a^{13}+\frac{273}{5777}a^{11}+\frac{798}{98209}a^{10}+\frac{1155}{11554}a^{9}+\frac{599}{11554}a^{8}+\frac{693}{5777}a^{7}+\frac{4822}{98209}a^{6}+\frac{441}{5777}a^{5}-\frac{17870}{98209}a^{4}+\frac{126}{5777}a^{3}-\frac{26266}{98209}a^{2}-\frac{60480}{98209}a-\frac{4181}{98209}$, $\frac{89}{196418}a^{16}+\frac{712}{98209}a^{14}+\frac{4628}{98209}a^{12}-\frac{987}{196418}a^{11}+\frac{15664}{98209}a^{10}-\frac{10857}{196418}a^{9}+\frac{29370}{98209}a^{8}-\frac{21714}{98209}a^{7}+\frac{29904}{98209}a^{6}-\frac{75999}{196418}a^{5}+\frac{14952}{98209}a^{4}-\frac{54285}{196418}a^{3}+\frac{2848}{98209}a^{2}-\frac{10857}{196418}a+\frac{89}{98209}$, $\frac{38}{98209}a^{16}+\frac{608}{98209}a^{14}+\frac{3952}{98209}a^{12}+\frac{1}{196418}a^{11}+\frac{13376}{98209}a^{10}+\frac{11}{196418}a^{9}+\frac{25080}{98209}a^{8}-\frac{5733}{196418}a^{7}+\frac{25536}{98209}a^{6}-\frac{20181}{98209}a^{5}+\frac{12768}{98209}a^{4}-\frac{80823}{196418}a^{3}-\frac{70237}{196418}a^{2}-\frac{20214}{98209}a-\frac{75025}{98209}$ 4/98209a^15 + 60/98209a^13 + 304/98209a^12 + 360/98209a^11 + 3648/98209a^10 + 1100/98209a^9 + 16416/98209a^8 + 1800/98209a^7 + 1663/5777a^6 + 1512/98209a^5 - 2742/98209a^4 - 22548/98209a^3 - 41049/98209a^2 - 69264/98209a - 10946/98209, 72/98209a^15 + 1080/98209a^13 - 305/98209a^12 + 6480/98209a^11 - 3660/98209a^10 + 19800/98209a^9 - 16470/98209a^8 + 32400/98209a^7 - 34160/98209a^6 + 27216/98209a^5 - 32025/98209a^4 + 10080/98209a^3 - 10980/98209a^2 + 1080/98209a - 610/98209, 114/98209a^14 + 3/196418a^13 + 1596/98209a^12 + 39/196418a^11 + 8778/98209a^10 + 195/196418a^9 + 23940/98209a^8 + 234/98209a^7 + 33516/98209a^6 - 16785/196418a^5 + 15803/196418a^4 - 43191/98209a^3 - 52184/98209a^2 - 43308/98209a - 28657/98209, 233/196418a^14 - 377/196418a^13 + 1631/98209a^12 - 4901/196418a^11 + 17941/196418a^10 - 24505/196418a^9 + 24465/98209a^8 - 29406/98209a^7 + 34251/98209a^6 - 34307/98209a^5 + 22834/98209a^4 - 34307/196418a^3 + 11417/196418a^2 - 4901/196418a + 233/98209, 21/196418a^17 - 38/98209a^16 + 349/196418a^15 - 494/98209a^14 + 1191/98209a^13 - 2660/98209a^12 + 4300/98209a^11 - 7448/98209a^10 + 17619/196418a^9 - 24929/196418a^8 + 1539/11554a^7 - 15469/98209a^6 + 2091/11554a^5 - 39989/196418a^4 + 43821/196418a^3 - 4565/196418a^2 - 270/1853a - 45076/98209, 21/196418a^17 + 21/11554a^15 + 147/11554a^13 + 273/5777a^11 + 798/98209a^10 + 1155/11554a^9 + 599/11554a^8 + 693/5777a^7 + 4822/98209a^6 + 441/5777a^5 - 17870/98209a^4 + 126/5777a^3 - 26266/98209a^2 - 60480/98209a - 4181/98209, 89/196418a^16 + 712/98209a^14 + 4628/98209a^12 - 987/196418a^11 + 15664/98209a^10 - 10857/196418a^9 + 29370/98209a^8 - 21714/98209a^7 + 29904/98209a^6 - 75999/196418a^5 + 14952/98209a^4 - 54285/196418a^3 + 2848/98209a^2 - 10857/196418a + 89/98209, 38/98209a^16 + 608/98209a^14 + 3952/98209a^12 + 1/196418a^11 + 13376/98209a^10 + 11/196418a^9 + 25080/98209a^8 - 5733/196418a^7 + 25536/98209a^6 - 20181/98209a^5 + 12768/98209a^4 - 80823/196418a^3 - 70237/196418a^2 - 20214/98209*a - 75025/98209 (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:	$ 40934.0329443 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 40934.0329443 \cdot 74}{2\cdot\sqrt{5770142004982097067662109375}}\cr\approx \mathstrut & 0.304306916445 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 + 135*x^14 + 546*x^12 + 1287*x^10 - 76*x^9 + 1782*x^8 - 684*x^7 + 1386*x^6 - 2052*x^5 + 540*x^4 - 2280*x^3 + 81*x^2 - 684*x + 5779)

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 + 18*x^16 + 135*x^14 + 546*x^12 + 1287*x^10 - 76*x^9 + 1782*x^8 - 684*x^7 + 1386*x^6 - 2052*x^5 + 540*x^4 - 2280*x^3 + 81*x^2 - 684*x + 5779, 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 + 18*x^16 + 135*x^14 + 546*x^12 + 1287*x^10 - 76*x^9 + 1782*x^8 - 684*x^7 + 1386*x^6 - 2052*x^5 + 540*x^4 - 2280*x^3 + 81*x^2 - 684*x + 5779);

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 + 18*x^16 + 135*x^14 + 546*x^12 + 1287*x^10 - 76*x^9 + 1782*x^8 - 684*x^7 + 1386*x^6 - 2052*x^5 + 540*x^4 - 2280*x^3 + 81*x^2 - 684*x + 5779);

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

$C_{18}$ (as 18T1):

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 cyclic group of order 18

The 18 conjugacy class representatives for $C_{18}$

Character table for $C_{18}$

Intermediate fields

$\Q(\sqrt{-15}) $, $\Q(\zeta_{9})^+$, 6.0.2460375.1, $\Q(\zeta_{27})^+$

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	${\href{/padicField/2.9.0.1}{9} }^{2}$	R	R	$18$	$18$	$18$	${\href{/padicField/17.3.0.1}{3} }^{6}$	${\href{/padicField/19.3.0.1}{3} }^{6}$	${\href{/padicField/23.9.0.1}{9} }^{2}$	$18$	${\href{/padicField/31.9.0.1}{9} }^{2}$	${\href{/padicField/37.6.0.1}{6} }^{3}$	$18$	$18$	${\href{/padicField/47.9.0.1}{9} }^{2}$	${\href{/padicField/53.1.0.1}{1} }^{18}$	$18$

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
$3$ 3		Deg $18$	$18$	$1$	$45$
$5$ 5	5.18.9.2	$x^{18} + 31250 x^{6} + 390625 x^{2} - 5859375$	$2$	$9$	$9$	$C_{18}$	$[\ ]_{2}^{9}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
*	1.15.2t1.a.a	$1$	$ 3 \cdot 5 $	$\Q(\sqrt{-15}) $	$C_2$ (as 2T1)	$1$	$-1$
*	1.27.9t1.a.a	$1$	$ 3^{3}$	$\Q(\zeta_{27})^+$	$C_9$ (as 9T1)	$0$	$1$
*	1.135.18t1.b.a	$1$	$ 3^{3} \cdot 5 $	18.0.5770142004982097067662109375.1	$C_{18}$ (as 18T1)	$0$	$-1$
*	1.27.9t1.a.b	$1$	$ 3^{3}$	$\Q(\zeta_{27})^+$	$C_9$ (as 9T1)	$0$	$1$
*	1.135.18t1.b.b	$1$	$ 3^{3} \cdot 5 $	18.0.5770142004982097067662109375.1	$C_{18}$ (as 18T1)	$0$	$-1$
*	1.9.3t1.a.a	$1$	$ 3^{2}$	$\Q(\zeta_{9})^+$	$C_3$ (as 3T1)	$0$	$1$
*	1.45.6t1.b.a	$1$	$ 3^{2} \cdot 5 $	6.0.2460375.1	$C_6$ (as 6T1)	$0$	$-1$
*	1.27.9t1.a.c	$1$	$ 3^{3}$	$\Q(\zeta_{27})^+$	$C_9$ (as 9T1)	$0$	$1$
*	1.135.18t1.b.c	$1$	$ 3^{3} \cdot 5 $	18.0.5770142004982097067662109375.1	$C_{18}$ (as 18T1)	$0$	$-1$
*	1.27.9t1.a.d	$1$	$ 3^{3}$	$\Q(\zeta_{27})^+$	$C_9$ (as 9T1)	$0$	$1$
*	1.135.18t1.b.d	$1$	$ 3^{3} \cdot 5 $	18.0.5770142004982097067662109375.1	$C_{18}$ (as 18T1)	$0$	$-1$
*	1.9.3t1.a.b	$1$	$ 3^{2}$	$\Q(\zeta_{9})^+$	$C_3$ (as 3T1)	$0$	$1$
*	1.45.6t1.b.b	$1$	$ 3^{2} \cdot 5 $	6.0.2460375.1	$C_6$ (as 6T1)	$0$	$-1$
*	1.27.9t1.a.e	$1$	$ 3^{3}$	$\Q(\zeta_{27})^+$	$C_9$ (as 9T1)	$0$	$1$
*	1.135.18t1.b.e	$1$	$ 3^{3} \cdot 5 $	18.0.5770142004982097067662109375.1	$C_{18}$ (as 18T1)	$0$	$-1$
*	1.27.9t1.a.f	$1$	$ 3^{3}$	$\Q(\zeta_{27})^+$	$C_9$ (as 9T1)	$0$	$1$
*	1.135.18t1.b.f	$1$	$ 3^{3} \cdot 5 $	18.0.5770142004982097067662109375.1	$C_{18}$ (as 18T1)	$0$	$-1$

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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