Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*x + 1)
gp: K = bnfinit(y^18 - 6*y^17 + 11*y^16 + 6*y^15 - 60*y^14 + 122*y^13 - 141*y^12 + 74*y^11 + 54*y^10 - 122*y^9 + 71*y^8 + 36*y^7 - 70*y^6 + 16*y^5 + 33*y^4 - 10*y^3 - 11*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*x + 1)
\( x^{18} - 6 x^{17} + 11 x^{16} + 6 x^{15} - 60 x^{14} + 122 x^{13} - 141 x^{12} + 74 x^{11} + 54 x^{10} + \cdots + 1 \)
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: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(19446011944726528000000\)
\(\medspace = 2^{18}\cdot 5^{6}\cdot 7^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.31\) | 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: | $2\cdot 5^{1/2}7^{5/6}\approx 22.634106993721137$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{86029642954203}a^{17}-\frac{869649721279}{6617664842631}a^{16}+\frac{7235668926211}{86029642954203}a^{15}-\frac{7506573131436}{28676547651401}a^{14}+\frac{34538291624156}{86029642954203}a^{13}-\frac{23411632094356}{86029642954203}a^{12}+\frac{3114905726745}{28676547651401}a^{11}-\frac{11118832043938}{86029642954203}a^{10}-\frac{14367304156966}{86029642954203}a^{9}+\frac{38301130535374}{86029642954203}a^{8}+\frac{14666286105091}{86029642954203}a^{7}+\frac{8375323339453}{86029642954203}a^{6}+\frac{5123296022740}{86029642954203}a^{5}-\frac{6937657893167}{28676547651401}a^{4}+\frac{5205403477032}{28676547651401}a^{3}-\frac{38217903043877}{86029642954203}a^{2}-\frac{14308576464381}{28676547651401}a+\frac{12557213140464}{28676547651401}$
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 $1$
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: |
\( -1 \)
(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: |
$\frac{3289542082228}{86029642954203}a^{17}-\frac{1799757701782}{6617664842631}a^{16}+\frac{55028069236597}{86029642954203}a^{15}-\frac{1500705654290}{28676547651401}a^{14}-\frac{237659493297832}{86029642954203}a^{13}+\frac{574520026251833}{86029642954203}a^{12}-\frac{249489167590787}{28676547651401}a^{11}+\frac{555080384696768}{86029642954203}a^{10}+\frac{32879807823350}{86029642954203}a^{9}-\frac{538733483242271}{86029642954203}a^{8}+\frac{470862843352528}{86029642954203}a^{7}-\frac{103227648218336}{86029642954203}a^{6}-\frac{72544003371644}{86029642954203}a^{5}-\frac{18642421980662}{28676547651401}a^{4}+\frac{57040437212715}{28676547651401}a^{3}+\frac{594280176319}{86029642954203}a^{2}-\frac{37901077484942}{28676547651401}a-\frac{13882749489042}{28676547651401}$, $\frac{10107444161534}{86029642954203}a^{17}-\frac{1310689001135}{2205888280877}a^{16}+\frac{17537457616095}{28676547651401}a^{15}+\frac{175103872114817}{86029642954203}a^{14}-\frac{569298995253299}{86029642954203}a^{13}+\frac{652011715389416}{86029642954203}a^{12}-\frac{151444481286266}{86029642954203}a^{11}-\frac{814379850117029}{86029642954203}a^{10}+\frac{14\!\cdots\!43}{86029642954203}a^{9}-\frac{253631069656090}{28676547651401}a^{8}-\frac{711567328958150}{86029642954203}a^{7}+\frac{14\!\cdots\!60}{86029642954203}a^{6}-\frac{535929923429204}{86029642954203}a^{5}-\frac{608595558301057}{86029642954203}a^{4}+\frac{240621300107278}{28676547651401}a^{3}+\frac{200603290735637}{86029642954203}a^{2}-\frac{332668600636666}{86029642954203}a-\frac{30017238356392}{28676547651401}$, $\frac{74650376}{9891875699}a^{17}-\frac{1777332580}{29675627097}a^{16}+\frac{7363630100}{29675627097}a^{15}-\frac{17336760668}{29675627097}a^{14}+\frac{10670882644}{29675627097}a^{13}+\frac{62280325111}{29675627097}a^{12}-\frac{72650552096}{9891875699}a^{11}+\frac{370306137974}{29675627097}a^{10}-\frac{391261400792}{29675627097}a^{9}+\frac{66568649411}{9891875699}a^{8}+\frac{112092967180}{29675627097}a^{7}-\frac{90341064076}{9891875699}a^{6}+\frac{50264912844}{9891875699}a^{5}+\frac{26212688363}{9891875699}a^{4}-\frac{158226548072}{29675627097}a^{3}+\frac{18799168058}{9891875699}a^{2}+\frac{61433386916}{29675627097}a-\frac{27683385680}{29675627097}$, $\frac{2088287592}{9891875699}a^{17}-\frac{34977325742}{29675627097}a^{16}+\frac{52470378560}{29675627097}a^{15}+\frac{71272482202}{29675627097}a^{14}-\frac{123602695158}{9891875699}a^{13}+\frac{606323429582}{29675627097}a^{12}-\frac{511906512268}{29675627097}a^{11}-\frac{9612171614}{9891875699}a^{10}+\frac{679569529268}{29675627097}a^{9}-\frac{686163569674}{29675627097}a^{8}+\frac{22721291710}{29675627097}a^{7}+\frac{634603152067}{29675627097}a^{6}-\frac{522999117230}{29675627097}a^{5}-\frac{84637216067}{29675627097}a^{4}+\frac{123505764802}{9891875699}a^{3}-\frac{16766685651}{9891875699}a^{2}-\frac{112958611532}{29675627097}a-\frac{6092800099}{9891875699}$, $\frac{8651079986890}{28676547651401}a^{17}-\frac{11694881464817}{6617664842631}a^{16}+\frac{89145352614793}{28676547651401}a^{15}+\frac{58126855775789}{28676547651401}a^{14}-\frac{15\!\cdots\!19}{86029642954203}a^{13}+\frac{30\!\cdots\!27}{86029642954203}a^{12}-\frac{34\!\cdots\!26}{86029642954203}a^{11}+\frac{17\!\cdots\!17}{86029642954203}a^{10}+\frac{14\!\cdots\!95}{86029642954203}a^{9}-\frac{32\!\cdots\!76}{86029642954203}a^{8}+\frac{689648349472216}{28676547651401}a^{7}+\frac{629147491123001}{86029642954203}a^{6}-\frac{16\!\cdots\!52}{86029642954203}a^{5}+\frac{636353941090256}{86029642954203}a^{4}+\frac{513662832723214}{86029642954203}a^{3}-\frac{64420431327545}{28676547651401}a^{2}-\frac{91941369888983}{86029642954203}a-\frac{180310182577403}{86029642954203}$, $\frac{906541454023}{28676547651401}a^{17}-\frac{1333271784442}{6617664842631}a^{16}+\frac{14175969202583}{28676547651401}a^{15}-\frac{9641921250957}{28676547651401}a^{14}-\frac{124313901835078}{86029642954203}a^{13}+\frac{469363496072683}{86029642954203}a^{12}-\frac{845823352099135}{86029642954203}a^{11}+\frac{881439220791175}{86029642954203}a^{10}-\frac{415755771133367}{86029642954203}a^{9}-\frac{335553376947262}{86029642954203}a^{8}+\frac{272503805942910}{28676547651401}a^{7}-\frac{546998419713986}{86029642954203}a^{6}-\frac{14051577243578}{86029642954203}a^{5}+\frac{312001250339572}{86029642954203}a^{4}-\frac{84521021113612}{86029642954203}a^{3}-\frac{41057870319303}{28676547651401}a^{2}+\frac{179017447191197}{86029642954203}a+\frac{37077460646351}{86029642954203}$, $\frac{916294797790}{28676547651401}a^{17}-\frac{241456815144}{2205888280877}a^{16}-\frac{5181770918704}{28676547651401}a^{15}+\frac{37991249766890}{28676547651401}a^{14}-\frac{52859138775899}{28676547651401}a^{13}-\frac{34164795129847}{28676547651401}a^{12}+\frac{216085338208427}{28676547651401}a^{11}-\frac{390778660702455}{28676547651401}a^{10}+\frac{397913170239529}{28676547651401}a^{9}-\frac{138683358538971}{28676547651401}a^{8}-\frac{158914298752714}{28676547651401}a^{7}+\frac{213832238008000}{28676547651401}a^{6}-\frac{9127710517037}{28676547651401}a^{5}-\frac{154209453726599}{28676547651401}a^{4}+\frac{69477622052192}{28676547651401}a^{3}+\frac{80225949934874}{28676547651401}a^{2}-\frac{34950514250917}{28676547651401}a-\frac{15849691165890}{28676547651401}$, $\frac{2462409851133}{28676547651401}a^{17}-\frac{765876205647}{2205888280877}a^{16}+\frac{3158278497347}{86029642954203}a^{15}+\frac{162829633093033}{86029642954203}a^{14}-\frac{322155527504248}{86029642954203}a^{13}+\frac{56916211352037}{28676547651401}a^{12}+\frac{330197699807980}{86029642954203}a^{11}-\frac{10\!\cdots\!08}{86029642954203}a^{10}+\frac{408323433552594}{28676547651401}a^{9}-\frac{400506332570348}{86029642954203}a^{8}-\frac{549387035025980}{86029642954203}a^{7}+\frac{931741342410524}{86029642954203}a^{6}-\frac{175124960863444}{86029642954203}a^{5}-\frac{468265886760979}{86029642954203}a^{4}+\frac{420668605741151}{86029642954203}a^{3}+\frac{62956954412114}{28676547651401}a^{2}-\frac{37380940708646}{28676547651401}a-\frac{49379204214196}{86029642954203}$, $\frac{2951818420362}{28676547651401}a^{17}-\frac{1579436660097}{2205888280877}a^{16}+\frac{154769277449699}{86029642954203}a^{15}-\frac{84363365764526}{86029642954203}a^{14}-\frac{481264572180898}{86029642954203}a^{13}+\frac{526670856791865}{28676547651401}a^{12}-\frac{27\!\cdots\!67}{86029642954203}a^{11}+\frac{29\!\cdots\!52}{86029642954203}a^{10}-\frac{540710591612818}{28676547651401}a^{9}-\frac{475953169958852}{86029642954203}a^{8}+\frac{18\!\cdots\!56}{86029642954203}a^{7}-\frac{15\!\cdots\!32}{86029642954203}a^{6}+\frac{255585881318399}{86029642954203}a^{5}+\frac{589601557249508}{86029642954203}a^{4}-\frac{491055860022328}{86029642954203}a^{3}+\frac{18790649378929}{28676547651401}a^{2}+\frac{28169930401567}{28676547651401}a-\frac{117963607151146}{86029642954203}$, $\frac{10204689914066}{86029642954203}a^{17}-\frac{5035829106185}{6617664842631}a^{16}+\frac{136453187319245}{86029642954203}a^{15}+\frac{6455215340188}{28676547651401}a^{14}-\frac{639471520822124}{86029642954203}a^{13}+\frac{14\!\cdots\!57}{86029642954203}a^{12}-\frac{637190176066562}{28676547651401}a^{11}+\frac{12\!\cdots\!57}{86029642954203}a^{10}+\frac{296466185733433}{86029642954203}a^{9}-\frac{13\!\cdots\!69}{86029642954203}a^{8}+\frac{961904634169472}{86029642954203}a^{7}+\frac{464202423936059}{86029642954203}a^{6}-\frac{12\!\cdots\!60}{86029642954203}a^{5}+\frac{196943704554768}{28676547651401}a^{4}+\frac{132606232663362}{28676547651401}a^{3}-\frac{458458619477317}{86029642954203}a^{2}+\frac{25798828980231}{28676547651401}a+\frac{25038610288890}{28676547651401}$, $\frac{1501062567904}{86029642954203}a^{17}-\frac{771607303784}{6617664842631}a^{16}+\frac{15606769062190}{86029642954203}a^{15}+\frac{12703287136105}{28676547651401}a^{14}-\frac{164732765071253}{86029642954203}a^{13}+\frac{65185296219122}{28676547651401}a^{12}+\frac{52843821157892}{86029642954203}a^{11}-\frac{174088246478180}{28676547651401}a^{10}+\frac{310258984982784}{28676547651401}a^{9}-\frac{245384771343460}{28676547651401}a^{8}-\frac{140489231310788}{86029642954203}a^{7}+\frac{836260291526558}{86029642954203}a^{6}-\frac{739466483441962}{86029642954203}a^{5}-\frac{372905951531}{86029642954203}a^{4}+\frac{504251724689654}{86029642954203}a^{3}-\frac{275531828235359}{86029642954203}a^{2}-\frac{191930130488392}{86029642954203}a+\frac{118307333344817}{86029642954203}$
| 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: | \( 13385.82469770262 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 13385.82469770262 \cdot 1}{2\cdot\sqrt{19446011944726528000000}}\cr\approx \mathstrut & 0.188998800336351 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*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))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^18 - 6*x^17 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*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)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^18 - 6*x^17 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*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)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*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):
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
| \(\Q(\sqrt{7}) \), 3.1.980.1, \(\Q(\zeta_{7})^+\), 6.2.26891200.1, \(\Q(\zeta_{28})^+\), 9.3.941192000.1 |
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
| Galois closure: | data not computed |
| Degree 12 sibling: | 12.0.368947264000000.7 |
| Degree 18 sibling: | 18.0.37980492079544000000000.2 |
| Minimal sibling: | 12.0.368947264000000.7 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{6}$ | R | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{9}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
|
\(5\)
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(7\)
| 7.18.15.1 | $x^{18} + 315 x^{12} + 9555 x^{6} + 76489$ | $6$ | $3$ | $15$ | $C_6 \times C_3$ | $[\ ]_{6}^{3}$ |