LMFDB - Number field 17.1.1863207356329472909377536.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 7*x^16 + 22*x^15 - 34*x^14 + 16*x^13 + 8*x^12 - 48*x^11 + 12*x^10 - 76*x^8 + 46*x^7 - 34*x^6 - 32*x^5 + 32*x^4 - 20*x^3 + 9*x - 3)

gp: K = bnfinit(y^17 - 7*y^16 + 22*y^15 - 34*y^14 + 16*y^13 + 8*y^12 - 48*y^11 + 12*y^10 - 76*y^8 + 46*y^7 - 34*y^6 - 32*y^5 + 32*y^4 - 20*y^3 + 9*y - 3, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 7*x^16 + 22*x^15 - 34*x^14 + 16*x^13 + 8*x^12 - 48*x^11 + 12*x^10 - 76*x^8 + 46*x^7 - 34*x^6 - 32*x^5 + 32*x^4 - 20*x^3 + 9*x - 3);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 7*x^16 + 22*x^15 - 34*x^14 + 16*x^13 + 8*x^12 - 48*x^11 + 12*x^10 - 76*x^8 + 46*x^7 - 34*x^6 - 32*x^5 + 32*x^4 - 20*x^3 + 9*x - 3)

$ x^{17} - 7 x^{16} + 22 x^{15} - 34 x^{14} + 16 x^{13} + 8 x^{12} - 48 x^{11} + 12 x^{10} - 76 x^{8} + \cdots - 3 $ x^17 - 7*x^16 + 22*x^15 - 34*x^14 + 16*x^13 + 8*x^12 - 48*x^11 + 12*x^10 - 76*x^8 + 46*x^7 - 34*x^6 - 32*x^5 + 32*x^4 - 20*x^3 + 9*x - 3

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$17$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[1, 8]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$1863207356329472909377536$ 1863207356329472909377536 $\medspace = 2^{41}\cdot 3^{25}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$26.77$	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:	not computed
Ramified primes:	$2$, $3$ 2, 3	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{6}) $
$\card{ \Aut(K/\Q) }$:	$1$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}$, $\frac{1}{9}a^{13}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{4}{9}a^{8}+\frac{4}{9}a^{7}+\frac{1}{9}a^{6}-\frac{2}{9}a^{5}-\frac{1}{3}a^{4}+\frac{1}{9}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{4}{9}a^{7}+\frac{4}{9}a^{6}-\frac{2}{9}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{12}-\frac{1}{9}a^{6}-\frac{1}{3}a^{4}-\frac{4}{9}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{8488134}a^{16}-\frac{106649}{4244067}a^{15}+\frac{1764}{471563}a^{14}+\frac{85970}{4244067}a^{13}-\frac{371570}{4244067}a^{12}-\frac{232907}{1414689}a^{11}+\frac{162694}{1414689}a^{10}+\frac{12476}{83217}a^{9}+\frac{3831}{27739}a^{8}-\frac{1545032}{4244067}a^{7}-\frac{1894081}{4244067}a^{6}-\frac{203957}{471563}a^{5}+\frac{779113}{4244067}a^{4}+\frac{1010333}{4244067}a^{3}-\frac{496954}{1414689}a^{2}+\frac{80763}{471563}a+\frac{109019}{2829378}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, 1/3*a^9 + 1/3*a^8 - 1/3*a^7 + 1/3*a^6 - 1/3*a^5 + 1/3*a^4 + 1/3*a^3, 1/3*a^10 + 1/3*a^8 - 1/3*a^7 + 1/3*a^6 - 1/3*a^5 - 1/3*a^3, 1/3*a^11 + 1/3*a^8 - 1/3*a^7 + 1/3*a^6 + 1/3*a^5 + 1/3*a^4 - 1/3*a^3, 1/3*a^12 + 1/3*a^8 - 1/3*a^7 - 1/3*a^5 + 1/3*a^4 - 1/3*a^3, 1/9*a^13 + 1/9*a^11 + 1/9*a^10 + 1/9*a^9 + 4/9*a^8 + 4/9*a^7 + 1/9*a^6 - 2/9*a^5 - 1/3*a^4 + 1/9*a^3 - 1/3*a^2 - 1/3*a - 1/3, 1/9*a^14 + 1/9*a^12 + 1/9*a^11 + 1/9*a^10 + 1/9*a^9 + 1/9*a^8 + 4/9*a^7 + 4/9*a^6 - 2/9*a^4 + 1/3*a^3 - 1/3*a^2 - 1/3*a, 1/9*a^15 + 1/9*a^12 - 1/9*a^6 - 1/3*a^4 - 4/9*a^3 + 1/3*a + 1/3, 1/8488134*a^16 - 106649/4244067*a^15 + 1764/471563*a^14 + 85970/4244067*a^13 - 371570/4244067*a^12 - 232907/1414689*a^11 + 162694/1414689*a^10 + 12476/83217*a^9 + 3831/27739*a^8 - 1545032/4244067*a^7 - 1894081/4244067*a^6 - 203957/471563*a^5 + 779113/4244067*a^4 + 1010333/4244067*a^3 - 496954/1414689*a^2 + 80763/471563*a + 109019/2829378

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:	$8$	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:	$\frac{71965}{1414689}a^{16}+\frac{1373197}{4244067}a^{15}-\frac{1253071}{1414689}a^{14}+\frac{3754001}{4244067}a^{13}+\frac{4103314}{4244067}a^{12}-\frac{10925566}{4244067}a^{11}+\frac{16156070}{4244067}a^{10}+\frac{398755}{249651}a^{9}-\frac{471239}{249651}a^{8}+\frac{34655582}{4244067}a^{7}+\frac{3149704}{4244067}a^{6}+\frac{3320315}{4244067}a^{5}+\frac{13151074}{1414689}a^{4}-\frac{5925392}{4244067}a^{3}+\frac{2097955}{1414689}a^{2}+\frac{4414559}{1414689}a-\frac{1419238}{1414689}$, $\frac{65611}{1414689}a^{16}-\frac{831344}{4244067}a^{15}+\frac{688540}{4244067}a^{14}+\frac{3633577}{4244067}a^{13}-\frac{9308347}{4244067}a^{12}-\frac{3286}{4244067}a^{11}-\frac{1299805}{4244067}a^{10}-\frac{908969}{249651}a^{9}+\frac{45751}{249651}a^{8}-\frac{1780534}{4244067}a^{7}-\frac{14022236}{4244067}a^{6}+\frac{19803958}{4244067}a^{5}+\frac{6666184}{4244067}a^{4}+\frac{269618}{471563}a^{3}+\frac{4391572}{1414689}a^{2}+\frac{251279}{1414689}a-\frac{293538}{471563}$, $\frac{233384}{4244067}a^{16}-\frac{2150152}{4244067}a^{15}+\frac{8284357}{4244067}a^{14}-\frac{16584793}{4244067}a^{13}+\frac{4753342}{1414689}a^{12}+\frac{1326769}{4244067}a^{11}-\frac{15076037}{4244067}a^{10}+\frac{1668410}{249651}a^{9}+\frac{663371}{249651}a^{8}-\frac{297853}{4244067}a^{7}+\frac{63004457}{4244067}a^{6}+\frac{4572089}{1414689}a^{5}+\frac{28640594}{4244067}a^{4}+\frac{48862516}{4244067}a^{3}+\frac{3387532}{1414689}a^{2}+\frac{1780890}{471563}a+\frac{3881135}{1414689}$, $\frac{8268277}{4244067}a^{16}-\frac{18232340}{1414689}a^{15}+\frac{160922171}{4244067}a^{14}-\frac{73037681}{1414689}a^{13}+\frac{45953729}{4244067}a^{12}+\frac{90976063}{4244067}a^{11}-\frac{370799195}{4244067}a^{10}-\frac{2813053}{249651}a^{9}-\frac{620119}{249651}a^{8}-\frac{217432249}{1414689}a^{7}+\frac{117058630}{4244067}a^{6}-\frac{230538256}{4244067}a^{5}-\frac{384470531}{4244067}a^{4}+\frac{109925465}{4244067}a^{3}-\frac{12871860}{471563}a^{2}-\frac{6757260}{471563}a+\frac{16518586}{1414689}$, $\frac{2171021}{4244067}a^{16}+\frac{5044279}{1414689}a^{15}-\frac{46449500}{4244067}a^{14}+\frac{66637939}{4244067}a^{13}-\frac{16125992}{4244067}a^{12}-\frac{38278901}{4244067}a^{11}+\frac{100869673}{4244067}a^{10}-\frac{495019}{249651}a^{9}-\frac{1744423}{249651}a^{8}+\frac{18176687}{471563}a^{7}-\frac{79880873}{4244067}a^{6}+\frac{3034328}{471563}a^{5}+\frac{31501829}{1414689}a^{4}-\frac{6568559}{471563}a^{3}+\frac{2420615}{471563}a^{2}+\frac{1958467}{471563}a-\frac{2789880}{471563}$, $\frac{1221494}{4244067}a^{16}+\frac{774517}{471563}a^{15}-\frac{15797006}{4244067}a^{14}+\frac{6709336}{4244067}a^{13}+\frac{34619653}{4244067}a^{12}-\frac{36811907}{4244067}a^{11}+\frac{49685320}{4244067}a^{10}+\frac{3405950}{249651}a^{9}-\frac{1144714}{249651}a^{8}+\frac{26620780}{1414689}a^{7}+\frac{49543450}{4244067}a^{6}-\frac{14043152}{1414689}a^{5}+\frac{24329734}{1414689}a^{4}-\frac{1615223}{471563}a^{3}-\frac{3444887}{471563}a^{2}+\frac{1748734}{471563}a-\frac{145230}{471563}$, $\frac{6419725}{4244067}a^{16}-\frac{45036521}{4244067}a^{15}+\frac{141340031}{4244067}a^{14}-\frac{217665352}{4244067}a^{13}+\frac{11564326}{471563}a^{12}+\frac{10495216}{1414689}a^{11}-\frac{27981178}{471563}a^{10}+\frac{741334}{83217}a^{9}+\frac{141248}{83217}a^{8}-\frac{449070811}{4244067}a^{7}+\frac{246828731}{4244067}a^{6}-\frac{236262023}{4244067}a^{5}-\frac{175880210}{4244067}a^{4}+\frac{25321577}{1414689}a^{3}-\frac{36193385}{1414689}a^{2}-\frac{6356473}{1414689}a+\frac{2634548}{471563}$, $\frac{44581}{249651}a^{16}+\frac{108938}{249651}a^{15}+\frac{345865}{249651}a^{14}-\frac{2330651}{249651}a^{13}+\frac{1512527}{83217}a^{12}-\frac{236341}{27739}a^{11}+\frac{847531}{83217}a^{10}+\frac{1847818}{83217}a^{9}+\frac{167381}{27739}a^{8}+\frac{4133281}{249651}a^{7}+\frac{9345406}{249651}a^{6}-\frac{3298429}{249651}a^{5}+\frac{6217232}{249651}a^{4}-\frac{154874}{83217}a^{3}-\frac{311272}{83217}a^{2}+\frac{272338}{83217}a-\frac{54848}{27739}$ -71965/1414689a^16 + 1373197/4244067a^15 - 1253071/1414689a^14 + 3754001/4244067a^13 + 4103314/4244067a^12 - 10925566/4244067a^11 + 16156070/4244067a^10 + 398755/249651a^9 - 471239/249651a^8 + 34655582/4244067a^7 + 3149704/4244067a^6 + 3320315/4244067a^5 + 13151074/1414689a^4 - 5925392/4244067a^3 + 2097955/1414689a^2 + 4414559/1414689a - 1419238/1414689, 65611/1414689a^16 - 831344/4244067a^15 + 688540/4244067a^14 + 3633577/4244067a^13 - 9308347/4244067a^12 - 3286/4244067a^11 - 1299805/4244067a^10 - 908969/249651a^9 + 45751/249651a^8 - 1780534/4244067a^7 - 14022236/4244067a^6 + 19803958/4244067a^5 + 6666184/4244067a^4 + 269618/471563a^3 + 4391572/1414689a^2 + 251279/1414689a - 293538/471563, 233384/4244067a^16 - 2150152/4244067a^15 + 8284357/4244067a^14 - 16584793/4244067a^13 + 4753342/1414689a^12 + 1326769/4244067a^11 - 15076037/4244067a^10 + 1668410/249651a^9 + 663371/249651a^8 - 297853/4244067a^7 + 63004457/4244067a^6 + 4572089/1414689a^5 + 28640594/4244067a^4 + 48862516/4244067a^3 + 3387532/1414689a^2 + 1780890/471563a + 3881135/1414689, 8268277/4244067a^16 - 18232340/1414689a^15 + 160922171/4244067a^14 - 73037681/1414689a^13 + 45953729/4244067a^12 + 90976063/4244067a^11 - 370799195/4244067a^10 - 2813053/249651a^9 - 620119/249651a^8 - 217432249/1414689a^7 + 117058630/4244067a^6 - 230538256/4244067a^5 - 384470531/4244067a^4 + 109925465/4244067a^3 - 12871860/471563a^2 - 6757260/471563a + 16518586/1414689, -2171021/4244067a^16 + 5044279/1414689a^15 - 46449500/4244067a^14 + 66637939/4244067a^13 - 16125992/4244067a^12 - 38278901/4244067a^11 + 100869673/4244067a^10 - 495019/249651a^9 - 1744423/249651a^8 + 18176687/471563a^7 - 79880873/4244067a^6 + 3034328/471563a^5 + 31501829/1414689a^4 - 6568559/471563a^3 + 2420615/471563a^2 + 1958467/471563a - 2789880/471563, -1221494/4244067a^16 + 774517/471563a^15 - 15797006/4244067a^14 + 6709336/4244067a^13 + 34619653/4244067a^12 - 36811907/4244067a^11 + 49685320/4244067a^10 + 3405950/249651a^9 - 1144714/249651a^8 + 26620780/1414689a^7 + 49543450/4244067a^6 - 14043152/1414689a^5 + 24329734/1414689a^4 - 1615223/471563a^3 - 3444887/471563a^2 + 1748734/471563a - 145230/471563, 6419725/4244067a^16 - 45036521/4244067a^15 + 141340031/4244067a^14 - 217665352/4244067a^13 + 11564326/471563a^12 + 10495216/1414689a^11 - 27981178/471563a^10 + 741334/83217a^9 + 141248/83217a^8 - 449070811/4244067a^7 + 246828731/4244067a^6 - 236262023/4244067a^5 - 175880210/4244067a^4 + 25321577/1414689a^3 - 36193385/1414689a^2 - 6356473/1414689a + 2634548/471563, -44581/249651a^16 + 108938/249651a^15 + 345865/249651a^14 - 2330651/249651a^13 + 1512527/83217a^12 - 236341/27739a^11 + 847531/83217a^10 + 1847818/83217a^9 + 167381/27739a^8 + 4133281/249651a^7 + 9345406/249651a^6 - 3298429/249651a^5 + 6217232/249651a^4 - 154874/83217a^3 - 311272/83217a^2 + 272338/83217a - 54848/27739 (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:	$ 1860535.6396997238 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 1860535.6396997238 \cdot 1}{2\cdot\sqrt{1863207356329472909377536}}\cr\approx \mathstrut & 3.31090214631167 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^17 - 7*x^16 + 22*x^15 - 34*x^14 + 16*x^13 + 8*x^12 - 48*x^11 + 12*x^10 - 76*x^8 + 46*x^7 - 34*x^6 - 32*x^5 + 32*x^4 - 20*x^3 + 9*x - 3)

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^17 - 7*x^16 + 22*x^15 - 34*x^14 + 16*x^13 + 8*x^12 - 48*x^11 + 12*x^10 - 76*x^8 + 46*x^7 - 34*x^6 - 32*x^5 + 32*x^4 - 20*x^3 + 9*x - 3, 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^17 - 7*x^16 + 22*x^15 - 34*x^14 + 16*x^13 + 8*x^12 - 48*x^11 + 12*x^10 - 76*x^8 + 46*x^7 - 34*x^6 - 32*x^5 + 32*x^4 - 20*x^3 + 9*x - 3);

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^17 - 7*x^16 + 22*x^15 - 34*x^14 + 16*x^13 + 8*x^12 - 48*x^11 + 12*x^10 - 76*x^8 + 46*x^7 - 34*x^6 - 32*x^5 + 32*x^4 - 20*x^3 + 9*x - 3);

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

$S_{17}$ (as 17T10):

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 non-solvable group of order 355687428096000

The 297 conjugacy class representatives for $S_{17}$

Character table for $S_{17}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

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	R	R	$17$	$16{,}\,{\href{/padicField/7.1.0.1}{1} }$	${\href{/padicField/11.7.0.1}{7} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$	${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$	${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$	${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }$	${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }$	${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$	$16{,}\,{\href{/padicField/31.1.0.1}{1} }$	${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.3.0.1}{3} }$	${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.3.0.1}{3} }$	$17$	${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$	${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
$2$ 2	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
$2$ 2		Deg $16$	$16$	$1$	$41$
$3$ 3	3.2.1.1	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.3.3.1	$x^{3} + 6 x + 3$	$3$	$1$	$3$	$S_3$	$[3/2]_{2}$
	3.12.21.8	$x^{12} + 3 x^{10} + 6 x^{9} + 3 x^{6} + 15 x^{3} + 24$	$12$	$1$	$21$	12T118	$[2, 9/4, 9/4]_{4}^{2}$

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