Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 11*x^15 + 45*x^14 + 12*x^13 - 111*x^12 + 30*x^11 + 48*x^10 - 60*x^9 + 195*x^8 + 105*x^7 - 285*x^6 - 90*x^5 + 165*x^4 + 29*x^3 - 42*x^2 + 1)
gp: K = bnfinit(y^18 - 3*y^17 - 11*y^15 + 45*y^14 + 12*y^13 - 111*y^12 + 30*y^11 + 48*y^10 - 60*y^9 + 195*y^8 + 105*y^7 - 285*y^6 - 90*y^5 + 165*y^4 + 29*y^3 - 42*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 11*x^15 + 45*x^14 + 12*x^13 - 111*x^12 + 30*x^11 + 48*x^10 - 60*x^9 + 195*x^8 + 105*x^7 - 285*x^6 - 90*x^5 + 165*x^4 + 29*x^3 - 42*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 11*x^15 + 45*x^14 + 12*x^13 - 111*x^12 + 30*x^11 + 48*x^10 - 60*x^9 + 195*x^8 + 105*x^7 - 285*x^6 - 90*x^5 + 165*x^4 + 29*x^3 - 42*x^2 + 1)
\( x^{18} - 3 x^{17} - 11 x^{15} + 45 x^{14} + 12 x^{13} - 111 x^{12} + 30 x^{11} + 48 x^{10} - 60 x^{9} + \cdots + 1 \)
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: |
\(435848050125000000000000\)
\(\medspace = 2^{12}\cdot 3^{20}\cdot 5^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.57\) | 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: | $2\cdot 3^{7/6}5^{5/6}\approx 27.55157706505336$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}-\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{11}-\frac{2}{5}a^{10}-\frac{1}{5}a^{8}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{3}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{14}-\frac{1}{5}a^{10}-\frac{1}{5}a^{9}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}$, $\frac{1}{5}a^{15}-\frac{1}{5}a^{11}-\frac{1}{5}a^{10}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a$, $\frac{1}{35}a^{16}-\frac{1}{35}a^{14}-\frac{3}{35}a^{12}-\frac{4}{35}a^{11}-\frac{3}{7}a^{10}-\frac{4}{35}a^{9}-\frac{1}{7}a^{8}+\frac{13}{35}a^{7}+\frac{1}{35}a^{6}-\frac{2}{7}a^{5}-\frac{8}{35}a^{4}+\frac{3}{7}a^{3}+\frac{1}{35}a^{2}-\frac{4}{35}a+\frac{3}{7}$, $\frac{1}{15590740940075}a^{17}+\frac{122159446329}{15590740940075}a^{16}-\frac{730066203812}{15590740940075}a^{15}+\frac{24666675231}{3118148188015}a^{14}-\frac{210312332491}{3118148188015}a^{13}+\frac{351261575187}{15590740940075}a^{12}+\frac{2894343158543}{15590740940075}a^{11}-\frac{4360413201264}{15590740940075}a^{10}+\frac{73497323828}{445449741145}a^{9}-\frac{112385146669}{445449741145}a^{8}+\frac{141505600101}{3118148188015}a^{7}-\frac{464108206601}{3118148188015}a^{6}+\frac{938540699519}{3118148188015}a^{5}-\frac{1190444889076}{3118148188015}a^{4}+\frac{83564506336}{623629637603}a^{3}-\frac{409396362238}{2227248705725}a^{2}+\frac{3697317323181}{15590740940075}a+\frac{5996826699112}{15590740940075}$
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: | $11$ | 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: |
$\frac{1018890478558}{15590740940075}a^{17}-\frac{4532807686913}{15590740940075}a^{16}+\frac{3932660615134}{15590740940075}a^{15}-\frac{400001818233}{623629637603}a^{14}+\frac{12520826660879}{3118148188015}a^{13}-\frac{47490709021124}{15590740940075}a^{12}-\frac{21623683545358}{2227248705725}a^{11}+\frac{181214948550858}{15590740940075}a^{10}+\frac{7328210476246}{3118148188015}a^{9}-\frac{19867360147006}{3118148188015}a^{8}+\frac{53469327658672}{3118148188015}a^{7}-\frac{36423254534801}{3118148188015}a^{6}-\frac{20478974673084}{623629637603}a^{5}+\frac{42843316184853}{3118148188015}a^{4}+\frac{72551918012501}{3118148188015}a^{3}-\frac{69551718630988}{15590740940075}a^{2}-\frac{104354525313687}{15590740940075}a+\frac{5071649085746}{15590740940075}$, $\frac{970199251171}{15590740940075}a^{17}-\frac{1622120498576}{15590740940075}a^{16}-\frac{4913661632407}{15590740940075}a^{15}-\frac{1247172839712}{3118148188015}a^{14}+\frac{4925170696209}{3118148188015}a^{13}+\frac{12113809077871}{2227248705725}a^{12}-\frac{155972044338692}{15590740940075}a^{11}-\frac{56564498661819}{15590740940075}a^{10}+\frac{33009339951654}{3118148188015}a^{9}-\frac{29914244213173}{3118148188015}a^{8}+\frac{5860003692291}{445449741145}a^{7}+\frac{74094399733434}{3118148188015}a^{6}-\frac{80694774765088}{3118148188015}a^{5}-\frac{1470925288552}{89089948229}a^{4}+\frac{9042725499260}{623629637603}a^{3}+\frac{49252934160099}{15590740940075}a^{2}-\frac{47589587034439}{15590740940075}a+\frac{10296306089212}{15590740940075}$, $\frac{890098485371}{15590740940075}a^{17}-\frac{2270837890891}{15590740940075}a^{16}-\frac{3089243754797}{15590740940075}a^{15}-\frac{1000861969343}{3118148188015}a^{14}+\frac{7403294872119}{3118148188015}a^{13}+\frac{7461278689356}{2227248705725}a^{12}-\frac{166138920437692}{15590740940075}a^{11}-\frac{66595820251659}{15590740940075}a^{10}+\frac{8491272554182}{623629637603}a^{9}-\frac{12094050381478}{3118148188015}a^{8}+\frac{4968670916149}{445449741145}a^{7}+\frac{51749679727113}{3118148188015}a^{6}-\frac{114106926473826}{3118148188015}a^{5}-\frac{13513684405039}{445449741145}a^{4}+\frac{11839259303762}{623629637603}a^{3}+\frac{221529643382434}{15590740940075}a^{2}-\frac{52839873315759}{15590740940075}a-\frac{16445017172503}{15590740940075}$, $\frac{423771732074}{2227248705725}a^{17}-\frac{7925363396908}{15590740940075}a^{16}-\frac{259533755418}{2227248705725}a^{15}-\frac{7000046823752}{3118148188015}a^{14}+\frac{3416750310016}{445449741145}a^{13}+\frac{69566262865726}{15590740940075}a^{12}-\frac{283124834584841}{15590740940075}a^{11}+\frac{45366291448848}{15590740940075}a^{10}+\frac{5138939416753}{3118148188015}a^{9}-\frac{26805176665289}{3118148188015}a^{8}+\frac{116914843414961}{3118148188015}a^{7}+\frac{78500654130993}{3118148188015}a^{6}-\frac{98268092352257}{3118148188015}a^{5}-\frac{58742200760428}{3118148188015}a^{4}+\frac{5498613518498}{623629637603}a^{3}+\frac{87505809220292}{15590740940075}a^{2}-\frac{5548906358727}{15590740940075}a-\frac{18326145733489}{15590740940075}$, $\frac{1606891121907}{15590740940075}a^{17}-\frac{1788426228622}{15590740940075}a^{16}-\frac{8980116162669}{15590740940075}a^{15}-\frac{618861031513}{623629637603}a^{14}+\frac{6116850458934}{3118148188015}a^{13}+\frac{158508461193989}{15590740940075}a^{12}-\frac{171262562678559}{15590740940075}a^{11}-\frac{161161558534948}{15590740940075}a^{10}+\frac{25862445135334}{3118148188015}a^{9}-\frac{29025552825582}{3118148188015}a^{8}+\frac{54645968232076}{3118148188015}a^{7}+\frac{27209875350341}{623629637603}a^{6}-\frac{4623503523998}{445449741145}a^{5}-\frac{99895288300643}{3118148188015}a^{4}+\frac{3167181440523}{3118148188015}a^{3}+\frac{146512521355128}{15590740940075}a^{2}-\frac{297735484984}{2227248705725}a-\frac{6786542409071}{15590740940075}$, $\frac{580180886321}{3118148188015}a^{17}-\frac{1532416324136}{3118148188015}a^{16}-\frac{115583092257}{623629637603}a^{15}-\frac{6375686541282}{3118148188015}a^{14}+\frac{23347894017606}{3118148188015}a^{13}+\frac{16015355845332}{3118148188015}a^{12}-\frac{61751695150672}{3118148188015}a^{11}+\frac{1984534672254}{3118148188015}a^{10}+\frac{27916103492677}{3118148188015}a^{9}-\frac{35124801618141}{3118148188015}a^{8}+\frac{21895542145740}{623629637603}a^{7}+\frac{19159464322807}{623629637603}a^{6}-\frac{19416295463702}{445449741145}a^{5}-\frac{79677516258901}{3118148188015}a^{4}+\frac{70117639758358}{3118148188015}a^{3}+\frac{23602447522729}{3118148188015}a^{2}-\frac{2586618757832}{445449741145}a-\frac{743035575063}{3118148188015}$, $\frac{472592702998}{2227248705725}a^{17}-\frac{10967447072326}{15590740940075}a^{16}+\frac{561638780984}{2227248705725}a^{15}-\frac{7943542596372}{3118148188015}a^{14}+\frac{4670921502934}{445449741145}a^{13}-\frac{17180306066803}{15590740940075}a^{12}-\frac{333909018211057}{15590740940075}a^{11}+\frac{180946963377911}{15590740940075}a^{10}+\frac{8182735586649}{3118148188015}a^{9}-\frac{21871459758817}{3118148188015}a^{8}+\frac{129478389537268}{3118148188015}a^{7}+\frac{21060203278847}{3118148188015}a^{6}-\frac{164074135061516}{3118148188015}a^{5}-\frac{7009961259708}{623629637603}a^{4}+\frac{90850913799007}{3118148188015}a^{3}+\frac{99803320645844}{15590740940075}a^{2}-\frac{100133447741609}{15590740940075}a-\frac{1962410799163}{15590740940075}$, $\frac{4894913508874}{15590740940075}a^{17}-\frac{13914968712599}{15590740940075}a^{16}-\frac{2502070024283}{15590740940075}a^{15}-\frac{10756068812109}{3118148188015}a^{14}+\frac{42496052528796}{3118148188015}a^{13}+\frac{13941505435789}{2227248705725}a^{12}-\frac{536077293129993}{15590740940075}a^{11}+\frac{46561498439394}{15590740940075}a^{10}+\frac{48210464505732}{3118148188015}a^{9}-\frac{46834099631867}{3118148188015}a^{8}+\frac{27376854423181}{445449741145}a^{7}+\frac{131144510951003}{3118148188015}a^{6}-\frac{262808587134428}{3118148188015}a^{5}-\frac{21715456348409}{445449741145}a^{4}+\frac{23368371514441}{623629637603}a^{3}+\frac{281663460664721}{15590740940075}a^{2}-\frac{85947673144716}{15590740940075}a-\frac{1546548852607}{15590740940075}$, $\frac{98692510687}{15590740940075}a^{17}+\frac{994825335088}{15590740940075}a^{16}-\frac{3115136949219}{15590740940075}a^{15}-\frac{107156691811}{623629637603}a^{14}-\frac{2330248246377}{3118148188015}a^{13}+\frac{50729363445934}{15590740940075}a^{12}+\frac{30474352602271}{15590740940075}a^{11}-\frac{102194100765268}{15590740940075}a^{10}-\frac{805042520188}{445449741145}a^{9}+\frac{418329051277}{445449741145}a^{8}-\frac{3200750070086}{3118148188015}a^{7}+\frac{44960834689368}{3118148188015}a^{6}+\frac{47671496781793}{3118148188015}a^{5}-\frac{32073857081128}{3118148188015}a^{4}-\frac{9225398607023}{623629637603}a^{3}+\frac{4359586860279}{2227248705725}a^{2}+\frac{52119729166182}{15590740940075}a-\frac{11030182686806}{15590740940075}$, $\frac{3125431347126}{15590740940075}a^{17}-\frac{8083369357081}{15590740940075}a^{16}-\frac{4994034764107}{15590740940075}a^{15}-\frac{5819632732342}{3118148188015}a^{14}+\frac{23793992520703}{3118148188015}a^{13}+\frac{114836418639617}{15590740940075}a^{12}-\frac{410223426923032}{15590740940075}a^{11}+\frac{47795280784666}{15590740940075}a^{10}+\frac{47150547210734}{3118148188015}a^{9}-\frac{57321696149047}{3118148188015}a^{8}+\frac{26418138196543}{623629637603}a^{7}+\frac{20891412491267}{623629637603}a^{6}-\frac{38666914999494}{623629637603}a^{5}-\frac{15172576635788}{623629637603}a^{4}+\frac{15997681836211}{445449741145}a^{3}+\frac{92399724080924}{15590740940075}a^{2}-\frac{132825621765999}{15590740940075}a+\frac{8266771429367}{15590740940075}$, $\frac{3692579973532}{15590740940075}a^{17}-\frac{9703392772212}{15590740940075}a^{16}-\frac{4391749753274}{15590740940075}a^{15}-\frac{7806121114291}{3118148188015}a^{14}+\frac{29683046332061}{3118148188015}a^{13}+\frac{110888983508579}{15590740940075}a^{12}-\frac{417501469758694}{15590740940075}a^{11}-\frac{3219088182508}{15590740940075}a^{10}+\frac{1310427098131}{89089948229}a^{9}-\frac{5697394628987}{445449741145}a^{8}+\frac{127512290741678}{3118148188015}a^{7}+\frac{134772741254591}{3118148188015}a^{6}-\frac{197919810568061}{3118148188015}a^{5}-\frac{122705947593471}{3118148188015}a^{4}+\frac{119015560344529}{3118148188015}a^{3}+\frac{36865014411689}{2227248705725}a^{2}-\frac{174302318736338}{15590740940075}a-\frac{11498902412241}{15590740940075}$
| 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: | \( 90961.5357762 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 90961.5357762 \cdot 1}{2\cdot\sqrt{435848050125000000000000}}\cr\approx \mathstrut & 0.271281025844 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 11*x^15 + 45*x^14 + 12*x^13 - 111*x^12 + 30*x^11 + 48*x^10 - 60*x^9 + 195*x^8 + 105*x^7 - 285*x^6 - 90*x^5 + 165*x^4 + 29*x^3 - 42*x^2 + 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 - 3*x^17 - 11*x^15 + 45*x^14 + 12*x^13 - 111*x^12 + 30*x^11 + 48*x^10 - 60*x^9 + 195*x^8 + 105*x^7 - 285*x^6 - 90*x^5 + 165*x^4 + 29*x^3 - 42*x^2 + 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 - 3*x^17 - 11*x^15 + 45*x^14 + 12*x^13 - 111*x^12 + 30*x^11 + 48*x^10 - 60*x^9 + 195*x^8 + 105*x^7 - 285*x^6 - 90*x^5 + 165*x^4 + 29*x^3 - 42*x^2 + 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 - 3*x^17 - 11*x^15 + 45*x^14 + 12*x^13 - 111*x^12 + 30*x^11 + 48*x^10 - 60*x^9 + 195*x^8 + 105*x^7 - 285*x^6 - 90*x^5 + 165*x^4 + 29*x^3 - 42*x^2 + 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
$S_3\times D_6$ (as 18T29):
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 72 |
| The 18 conjugacy class representatives for $S_3\times D_6$ |
| Character table for $S_3\times D_6$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 3.1.135.1, 3.3.2700.1, 6.2.91125.1, 6.6.36450000.1, 9.3.295245000000.1 |
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 12 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.262440000000000.4 |
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 | R | ${\href{/padicField/7.2.0.1}{2} }^{9}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{9}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
|
\(3\)
| 3.6.6.4 | $x^{6} + 48 x^{4} + 6 x^{3} + 36 x^{2} + 36 x + 9$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ |
| 3.12.14.11 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 792 x^{8} + 1728 x^{7} + 2918 x^{6} + 3684 x^{5} + 3156 x^{4} + 1376 x^{3} - 36 x^{2} - 168 x + 25$ | $6$ | $2$ | $14$ | $D_6$ | $[3/2]_{2}^{2}$ | |
|
\(5\)
| 5.6.5.1 | $x^{6} + 5$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |
| 5.12.10.1 | $x^{12} + 20 x^{7} + 10 x^{6} + 50 x^{2} + 100 x + 25$ | $6$ | $2$ | $10$ | $D_6$ | $[\ ]_{6}^{2}$ |