↑

Properties

Label 33.33.5964572044...6969.1
Degree $33$
Signature $[33, 0]$
Discriminant $3^{44}\cdot 67^{30}$
Root discriminant $197.80$
Ramified primes $3, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{33}$ (as 33T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2255257, 21340050, 1428300, -501812162, 963373038, 3064214298, -9149246964, -5020929429, 32585743542, -9740363856, -52500002157, 40721934162, 37619683596, -50183067855, -7076803698, 29700176281, -5392982019, -9398785623, 3707180205, 1580724570, -1026324261, -111316484, 158067366, -5289417, -14588595, 1676889, 813933, -139216, -26439, 5757, 450, -120, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^33 - 3*x^32 - 120*x^31 + 450*x^30 + 5757*x^29 - 26439*x^28 - 139216*x^27 + 813933*x^26 + 1676889*x^25 - 14588595*x^24 - 5289417*x^23 + 158067366*x^22 - 111316484*x^21 - 1026324261*x^20 + 1580724570*x^19 + 3707180205*x^18 - 9398785623*x^17 - 5392982019*x^16 + 29700176281*x^15 - 7076803698*x^14 - 50183067855*x^13 + 37619683596*x^12 + 40721934162*x^11 - 52500002157*x^10 - 9740363856*x^9 + 32585743542*x^8 - 5020929429*x^7 - 9149246964*x^6 + 3064214298*x^5 + 963373038*x^4 - 501812162*x^3 + 1428300*x^2 + 21340050*x - 2255257)
 
gp: K = bnfinit(x^33 - 3*x^32 - 120*x^31 + 450*x^30 + 5757*x^29 - 26439*x^28 - 139216*x^27 + 813933*x^26 + 1676889*x^25 - 14588595*x^24 - 5289417*x^23 + 158067366*x^22 - 111316484*x^21 - 1026324261*x^20 + 1580724570*x^19 + 3707180205*x^18 - 9398785623*x^17 - 5392982019*x^16 + 29700176281*x^15 - 7076803698*x^14 - 50183067855*x^13 + 37619683596*x^12 + 40721934162*x^11 - 52500002157*x^10 - 9740363856*x^9 + 32585743542*x^8 - 5020929429*x^7 - 9149246964*x^6 + 3064214298*x^5 + 963373038*x^4 - 501812162*x^3 + 1428300*x^2 + 21340050*x - 2255257, 1)
 

Normalized defining polynomial

\( x^{33} - 3 x^{32} - 120 x^{31} + 450 x^{30} + 5757 x^{29} - 26439 x^{28} - 139216 x^{27} + 813933 x^{26} + 1676889 x^{25} - 14588595 x^{24} - 5289417 x^{23} + 158067366 x^{22} - 111316484 x^{21} - 1026324261 x^{20} + 1580724570 x^{19} + 3707180205 x^{18} - 9398785623 x^{17} - 5392982019 x^{16} + 29700176281 x^{15} - 7076803698 x^{14} - 50183067855 x^{13} + 37619683596 x^{12} + 40721934162 x^{11} - 52500002157 x^{10} - 9740363856 x^{9} + 32585743542 x^{8} - 5020929429 x^{7} - 9149246964 x^{6} + 3064214298 x^{5} + 963373038 x^{4} - 501812162 x^{3} + 1428300 x^{2} + 21340050 x - 2255257 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $33$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[33, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5964572044631662930869897974580631723402610311315668047983759159277483196969=3^{44}\cdot 67^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $197.80$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(603=3^{2}\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{603}(1,·)$, $\chi_{603}(265,·)$, $\chi_{603}(394,·)$, $\chi_{603}(397,·)$, $\chi_{603}(403,·)$, $\chi_{603}(148,·)$, $\chi_{603}(277,·)$, $\chi_{603}(22,·)$, $\chi_{603}(25,·)$, $\chi_{603}(283,·)$, $\chi_{603}(292,·)$, $\chi_{603}(550,·)$, $\chi_{603}(40,·)$, $\chi_{603}(427,·)$, $\chi_{603}(466,·)$, $\chi_{603}(442,·)$, $\chi_{603}(64,·)$, $\chi_{603}(193,·)$, $\chi_{603}(424,·)$, $\chi_{603}(196,·)$, $\chi_{603}(202,·)$, $\chi_{603}(76,·)$, $\chi_{603}(82,·)$, $\chi_{603}(595,·)$, $\chi_{603}(598,·)$, $\chi_{603}(91,·)$, $\chi_{603}(349,·)$, $\chi_{603}(478,·)$, $\chi_{603}(223,·)$, $\chi_{603}(226,·)$, $\chi_{603}(484,·)$, $\chi_{603}(493,·)$, $\chi_{603}(241,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{37} a^{21} - \frac{18}{37} a^{20} + \frac{1}{37} a^{19} + \frac{7}{37} a^{18} + \frac{17}{37} a^{17} + \frac{16}{37} a^{16} - \frac{1}{37} a^{15} - \frac{14}{37} a^{14} + \frac{17}{37} a^{13} - \frac{5}{37} a^{12} - \frac{16}{37} a^{11} + \frac{17}{37} a^{10} - \frac{7}{37} a^{9} - \frac{9}{37} a^{8} + \frac{1}{37} a^{7} + \frac{11}{37} a^{6} + \frac{17}{37} a^{5} - \frac{2}{37} a^{4} - \frac{10}{37} a^{3} - \frac{9}{37} a^{2} + \frac{4}{37} a - \frac{7}{37}$, $\frac{1}{37} a^{22} + \frac{10}{37} a^{20} - \frac{12}{37} a^{19} - \frac{5}{37} a^{18} - \frac{11}{37} a^{17} - \frac{9}{37} a^{16} + \frac{5}{37} a^{15} - \frac{13}{37} a^{14} + \frac{5}{37} a^{13} + \frac{5}{37} a^{12} - \frac{12}{37} a^{11} + \frac{3}{37} a^{10} + \frac{13}{37} a^{9} - \frac{13}{37} a^{8} - \frac{8}{37} a^{7} - \frac{7}{37} a^{6} + \frac{8}{37} a^{5} - \frac{9}{37} a^{4} - \frac{4}{37} a^{3} - \frac{10}{37} a^{2} - \frac{9}{37} a - \frac{15}{37}$, $\frac{1}{37} a^{23} - \frac{17}{37} a^{20} - \frac{15}{37} a^{19} - \frac{7}{37} a^{18} + \frac{6}{37} a^{17} - \frac{7}{37} a^{16} - \frac{3}{37} a^{15} - \frac{3}{37} a^{14} - \frac{17}{37} a^{13} + \frac{1}{37} a^{12} + \frac{15}{37} a^{11} - \frac{9}{37} a^{10} - \frac{17}{37} a^{9} + \frac{8}{37} a^{8} - \frac{17}{37} a^{7} + \frac{9}{37} a^{6} + \frac{6}{37} a^{5} + \frac{16}{37} a^{4} + \frac{16}{37} a^{3} + \frac{7}{37} a^{2} - \frac{18}{37} a - \frac{4}{37}$, $\frac{1}{37} a^{24} + \frac{12}{37} a^{20} + \frac{10}{37} a^{19} + \frac{14}{37} a^{18} - \frac{14}{37} a^{17} + \frac{10}{37} a^{16} + \frac{17}{37} a^{15} + \frac{4}{37} a^{14} - \frac{6}{37} a^{13} + \frac{4}{37} a^{12} + \frac{15}{37} a^{11} + \frac{13}{37} a^{10} + \frac{15}{37} a^{8} - \frac{11}{37} a^{7} + \frac{8}{37} a^{6} + \frac{9}{37} a^{5} - \frac{18}{37} a^{4} - \frac{15}{37} a^{3} + \frac{14}{37} a^{2} - \frac{10}{37} a - \frac{8}{37}$, $\frac{1}{37} a^{25} + \frac{4}{37} a^{20} + \frac{2}{37} a^{19} + \frac{13}{37} a^{18} - \frac{9}{37} a^{17} + \frac{10}{37} a^{16} + \frac{16}{37} a^{15} + \frac{14}{37} a^{14} - \frac{15}{37} a^{13} + \frac{1}{37} a^{12} - \frac{17}{37} a^{11} + \frac{18}{37} a^{10} - \frac{12}{37} a^{9} - \frac{14}{37} a^{8} - \frac{4}{37} a^{7} - \frac{12}{37} a^{6} + \frac{9}{37} a^{4} - \frac{14}{37} a^{3} - \frac{13}{37} a^{2} + \frac{18}{37} a + \frac{10}{37}$, $\frac{1}{37} a^{26} + \frac{9}{37} a^{19} + \frac{16}{37} a^{17} - \frac{11}{37} a^{16} + \frac{18}{37} a^{15} + \frac{4}{37} a^{14} + \frac{7}{37} a^{13} + \frac{3}{37} a^{12} + \frac{8}{37} a^{11} - \frac{6}{37} a^{10} + \frac{14}{37} a^{9} - \frac{5}{37} a^{8} - \frac{16}{37} a^{7} - \frac{7}{37} a^{6} + \frac{15}{37} a^{5} - \frac{6}{37} a^{4} - \frac{10}{37} a^{3} + \frac{17}{37} a^{2} - \frac{6}{37} a - \frac{9}{37}$, $\frac{1}{37} a^{27} + \frac{9}{37} a^{20} + \frac{16}{37} a^{18} - \frac{11}{37} a^{17} + \frac{18}{37} a^{16} + \frac{4}{37} a^{15} + \frac{7}{37} a^{14} + \frac{3}{37} a^{13} + \frac{8}{37} a^{12} - \frac{6}{37} a^{11} + \frac{14}{37} a^{10} - \frac{5}{37} a^{9} - \frac{16}{37} a^{8} - \frac{7}{37} a^{7} + \frac{15}{37} a^{6} - \frac{6}{37} a^{5} - \frac{10}{37} a^{4} + \frac{17}{37} a^{3} - \frac{6}{37} a^{2} - \frac{9}{37} a$, $\frac{1}{37} a^{28} + \frac{14}{37} a^{20} + \frac{7}{37} a^{19} + \frac{13}{37} a^{17} + \frac{8}{37} a^{16} + \frac{16}{37} a^{15} + \frac{18}{37} a^{14} + \frac{3}{37} a^{13} + \frac{2}{37} a^{12} + \frac{10}{37} a^{11} - \frac{10}{37} a^{10} + \frac{10}{37} a^{9} + \frac{6}{37} a^{7} + \frac{6}{37} a^{6} - \frac{15}{37} a^{5} - \frac{2}{37} a^{4} + \frac{10}{37} a^{3} - \frac{2}{37} a^{2} + \frac{1}{37} a - \frac{11}{37}$, $\frac{1}{37} a^{29} - \frac{14}{37} a^{19} - \frac{11}{37} a^{18} - \frac{8}{37} a^{17} + \frac{14}{37} a^{16} - \frac{5}{37} a^{15} + \frac{14}{37} a^{14} - \frac{14}{37} a^{13} + \frac{6}{37} a^{12} - \frac{8}{37} a^{11} - \frac{6}{37} a^{10} - \frac{13}{37} a^{9} - \frac{16}{37} a^{8} - \frac{8}{37} a^{7} + \frac{16}{37} a^{6} - \frac{18}{37} a^{5} + \frac{1}{37} a^{4} - \frac{10}{37} a^{3} + \frac{16}{37} a^{2} + \frac{7}{37} a - \frac{13}{37}$, $\frac{1}{1546631857} a^{30} - \frac{7562821}{1546631857} a^{29} - \frac{4896456}{1546631857} a^{28} + \frac{20185048}{1546631857} a^{27} - \frac{6913244}{1546631857} a^{26} + \frac{3313359}{1546631857} a^{25} - \frac{16252646}{1546631857} a^{24} + \frac{17778467}{1546631857} a^{23} + \frac{10794678}{1546631857} a^{22} + \frac{7961201}{1546631857} a^{21} + \frac{324286745}{1546631857} a^{20} + \frac{7806}{174899} a^{19} - \frac{438113452}{1546631857} a^{18} - \frac{79225023}{1546631857} a^{17} - \frac{28931438}{1546631857} a^{16} + \frac{16534956}{1546631857} a^{15} + \frac{119655054}{1546631857} a^{14} - \frac{508590391}{1546631857} a^{13} - \frac{324766637}{1546631857} a^{12} - \frac{754912806}{1546631857} a^{11} - \frac{727366890}{1546631857} a^{10} - \frac{10316583}{41800861} a^{9} + \frac{464296109}{1546631857} a^{8} + \frac{731937116}{1546631857} a^{7} + \frac{497484}{6471263} a^{6} + \frac{688571048}{1546631857} a^{5} + \frac{423492216}{1546631857} a^{4} + \frac{115647973}{1546631857} a^{3} - \frac{419561467}{1546631857} a^{2} - \frac{39126189}{1546631857} a + \frac{565745468}{1546631857}$, $\frac{1}{416043969533} a^{31} - \frac{118}{416043969533} a^{30} + \frac{3168594619}{416043969533} a^{29} + \frac{1085866963}{416043969533} a^{28} - \frac{3855233415}{416043969533} a^{27} + \frac{3757355816}{416043969533} a^{26} + \frac{449471142}{416043969533} a^{25} - \frac{5298653514}{416043969533} a^{24} + \frac{18850529}{1546631857} a^{23} - \frac{5354984790}{416043969533} a^{22} - \frac{1897020823}{416043969533} a^{21} - \frac{29164745072}{416043969533} a^{20} - \frac{106517335415}{416043969533} a^{19} - \frac{32553990214}{416043969533} a^{18} - \frac{143074196345}{416043969533} a^{17} + \frac{77770145381}{416043969533} a^{16} + \frac{75583839287}{416043969533} a^{15} - \frac{86722461505}{416043969533} a^{14} + \frac{124638579543}{416043969533} a^{13} - \frac{165899656399}{416043969533} a^{12} + \frac{85406919405}{416043969533} a^{11} - \frac{116010149662}{416043969533} a^{10} + \frac{121589298046}{416043969533} a^{9} - \frac{50961566679}{416043969533} a^{8} + \frac{51928378743}{416043969533} a^{7} - \frac{22220341840}{416043969533} a^{6} + \frac{17886608861}{416043969533} a^{5} - \frac{144140579586}{416043969533} a^{4} - \frac{8291919840}{416043969533} a^{3} - \frac{1346550845}{416043969533} a^{2} - \frac{134513922795}{416043969533} a + \frac{55334985763}{416043969533}$, $\frac{1}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{32} + \frac{78898102346748383130806205027212491239971341120901950517952993451908606722775829088388932756612}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{31} - \frac{5571110383986394127633922495037964783242118393001825309882775252193247906050079577285329528525704}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{30} - \frac{962973700597999840965835846027967781359770607006226102834796379617224903530031791219448209809714203802374}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{29} + \frac{766066787542600991000737529279908155858816404643535840371620687450518706371689745765490556721343418079961}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{28} + \frac{236475120083611846328244423202616852376955820611511166240852643983825242627102364536311869047857033298690}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{27} - \frac{781615547491560159686080570881411657275796473346843866805801128139923761379026809673968570927217141338453}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{26} + \frac{809129888266593226763977363167458234907953972063430588181965776364789668261151332750364936076739569844803}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{25} - \frac{572625788273718282362365001426703477898074806962323717327848821109323620614451528065065395638039720876295}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{24} - \frac{147224266029240127989756191172901498287176087367091688926378619273387376567612495374563138847577678957589}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{23} - \frac{143266982308687212889057380085422657377565452716466276749146460128422926688292094105459210002445474847786}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{22} + \frac{760409435062755280317936010749150646393595777155571108567220153514936492795870546772798225302399320099165}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{21} + \frac{26222540494709416059512338852141996448595426945514862772493404520695105218646493026878017672191489877107075}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{20} - \frac{24326202641421960078031136928701133861894739214341483573238786975663121920764726837159050651994868650009681}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{19} + \frac{8973840198358688343896334348020192263808840443160197743957956133557684129351337361499345197134119653256760}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{18} - \frac{8650973522758928909397218806907095913584467488162800638019502285550663354876550254960905440183936327871703}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{17} + \frac{3737727790854191335912391051255885844028020331667746591648927419481840497215749283744473834057063264541779}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{16} - \frac{16885371085230638961940767850660054120160632681270260348575123154079639933284223419284080271958538516368254}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{15} - \frac{1006861225553649811083340986664899806372840656225882047538048765567205720953289370898570073583231334922653}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{14} - \frac{2775111145870777496768184216201938421540721718799593872166347237831550172147210632344491784545942201187037}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{13} + \frac{27453395380978152507460935424015426675846559164752822903614347480711547819015597629359418682590823886209690}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{12} + \frac{9050996795382296323219987700426356924666895554603813516043077609741194880415236075257837496957542397091794}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{11} + \frac{8234208725018140842879780725136621164407838429825448348061848543767705695208798117720300705148042938402196}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{10} - \frac{21234799110055257929304783526426600638670469824084378790086864496267544373441682914724755988697347462800616}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{9} + \frac{13691896500027545688929648049330665059372970640756199627005564089644059544305962844639202324458350099580741}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{8} - \frac{37814537744202221900647409251656678472806360096174541567799642021329303354794892632429571863420469653634418}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{7} - \frac{38669324063729685335037168205961025265946703742645737465262741853062905042273937117708342461817214797938653}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{6} + \frac{37162696572083154618014016223346316194174612427262804959122606795106451771065305619517414548466890353637602}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{5} + \frac{20204692963053877473900039649864639745126669902004184488275012087356455168385139419153295717196846096192650}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{4} - \frac{37926579220339952857825659920342710059227430135182059372397589331795452645340311375048082596065284209135183}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{3} + \frac{25598261337831816766071628324400796161494065688160449599540790627159568986233677784167900710248717632318294}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a^{2} - \frac{919792673761089792007356445368005770955054786001146353260820262979229815862935921407045446148358860524266}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449} a + \frac{27411226530064365063401752390203346584110630294783943636013431669760088677386411360014447264738360617539715}{82902972600072261025447329880212486090223211514534094621422134388806507065892909923236559945216287314007449}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $32$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5218493771929612000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{33}$ (as 33T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 33
The 33 conjugacy class representatives for $C_{33}$
Character table for $C_{33}$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 11.11.1822837804551761449.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $33$ R $33$ $33$ $33$ $33$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{3}$ $33$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{11}$ $33$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{33}$ $33$ $33$ $33$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{3}$ $33$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
67Data not computed