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Properties

Label 36.36.1914756742...0625.1
Degree $36$
Signature $[36, 0]$
Discriminant $5^{18}\cdot 7^{24}\cdot 13^{30}$
Root discriminant $69.37$
Ramified primes $5, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -118, 1167, 129082, -1337240, -10815600, 2585738, 91102932, 38527228, -344940108, -228251095, 754912148, 593149297, -1059071764, -914832149, 1004033756, 924513319, -662076108, -641419103, 308270172, 313099092, -101854202, -108821254, 23777282, 26998873, -3862424, -4751125, 423372, 583031, -29576, -48322, 1176, 2554, -20, -77, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 77*x^34 - 20*x^33 + 2554*x^32 + 1176*x^31 - 48322*x^30 - 29576*x^29 + 583031*x^28 + 423372*x^27 - 4751125*x^26 - 3862424*x^25 + 26998873*x^24 + 23777282*x^23 - 108821254*x^22 - 101854202*x^21 + 313099092*x^20 + 308270172*x^19 - 641419103*x^18 - 662076108*x^17 + 924513319*x^16 + 1004033756*x^15 - 914832149*x^14 - 1059071764*x^13 + 593149297*x^12 + 754912148*x^11 - 228251095*x^10 - 344940108*x^9 + 38527228*x^8 + 91102932*x^7 + 2585738*x^6 - 10815600*x^5 - 1337240*x^4 + 129082*x^3 + 1167*x^2 - 118*x + 1)
 
gp: K = bnfinit(x^36 - 77*x^34 - 20*x^33 + 2554*x^32 + 1176*x^31 - 48322*x^30 - 29576*x^29 + 583031*x^28 + 423372*x^27 - 4751125*x^26 - 3862424*x^25 + 26998873*x^24 + 23777282*x^23 - 108821254*x^22 - 101854202*x^21 + 313099092*x^20 + 308270172*x^19 - 641419103*x^18 - 662076108*x^17 + 924513319*x^16 + 1004033756*x^15 - 914832149*x^14 - 1059071764*x^13 + 593149297*x^12 + 754912148*x^11 - 228251095*x^10 - 344940108*x^9 + 38527228*x^8 + 91102932*x^7 + 2585738*x^6 - 10815600*x^5 - 1337240*x^4 + 129082*x^3 + 1167*x^2 - 118*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 77 x^{34} - 20 x^{33} + 2554 x^{32} + 1176 x^{31} - 48322 x^{30} - 29576 x^{29} + 583031 x^{28} + 423372 x^{27} - 4751125 x^{26} - 3862424 x^{25} + 26998873 x^{24} + 23777282 x^{23} - 108821254 x^{22} - 101854202 x^{21} + 313099092 x^{20} + 308270172 x^{19} - 641419103 x^{18} - 662076108 x^{17} + 924513319 x^{16} + 1004033756 x^{15} - 914832149 x^{14} - 1059071764 x^{13} + 593149297 x^{12} + 754912148 x^{11} - 228251095 x^{10} - 344940108 x^{9} + 38527228 x^{8} + 91102932 x^{7} + 2585738 x^{6} - 10815600 x^{5} - 1337240 x^{4} + 129082 x^{3} + 1167 x^{2} - 118 x + 1 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1914756742943481965797622489677413208586713574151377292633056640625=5^{18}\cdot 7^{24}\cdot 13^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.37$
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:  $5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(455=5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{455}(256,·)$, $\chi_{455}(1,·)$, $\chi_{455}(386,·)$, $\chi_{455}(4,·)$, $\chi_{455}(261,·)$, $\chi_{455}(134,·)$, $\chi_{455}(9,·)$, $\chi_{455}(394,·)$, $\chi_{455}(16,·)$, $\chi_{455}(274,·)$, $\chi_{455}(51,·)$, $\chi_{455}(29,·)$, $\chi_{455}(389,·)$, $\chi_{455}(289,·)$, $\chi_{455}(36,·)$, $\chi_{455}(296,·)$, $\chi_{455}(179,·)$, $\chi_{455}(309,·)$, $\chi_{455}(186,·)$, $\chi_{455}(316,·)$, $\chi_{455}(191,·)$, $\chi_{455}(64,·)$, $\chi_{455}(324,·)$, $\chi_{455}(326,·)$, $\chi_{455}(74,·)$, $\chi_{455}(204,·)$, $\chi_{455}(79,·)$, $\chi_{455}(81,·)$, $\chi_{455}(211,·)$, $\chi_{455}(144,·)$, $\chi_{455}(354,·)$, $\chi_{455}(361,·)$, $\chi_{455}(114,·)$, $\chi_{455}(116,·)$, $\chi_{455}(246,·)$, $\chi_{455}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{30} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{31} - \frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{32} - \frac{1}{4} a^{30} - \frac{1}{4} a^{28} - \frac{1}{4} a^{26} - \frac{1}{4} a^{20} - \frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{8} a^{33} - \frac{1}{8} a^{32} + \frac{1}{8} a^{31} - \frac{1}{8} a^{30} - \frac{1}{8} a^{29} + \frac{1}{8} a^{28} - \frac{1}{8} a^{27} + \frac{1}{8} a^{26} - \frac{1}{4} a^{25} - \frac{1}{4} a^{24} - \frac{1}{8} a^{21} - \frac{1}{8} a^{20} - \frac{1}{4} a^{19} - \frac{1}{4} a^{18} - \frac{1}{8} a^{17} + \frac{1}{8} a^{16} - \frac{3}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{8} a^{2} + \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{14576} a^{34} + \frac{153}{3644} a^{33} + \frac{311}{3644} a^{32} - \frac{200}{911} a^{31} - \frac{1373}{7288} a^{30} - \frac{203}{911} a^{29} - \frac{25}{3644} a^{28} - \frac{329}{1822} a^{27} + \frac{307}{14576} a^{26} + \frac{102}{911} a^{25} + \frac{1727}{7288} a^{24} + \frac{156}{911} a^{23} - \frac{2321}{14576} a^{22} - \frac{313}{7288} a^{21} - \frac{2071}{14576} a^{20} + \frac{305}{1822} a^{19} - \frac{19}{14576} a^{18} - \frac{85}{911} a^{17} + \frac{1795}{7288} a^{16} - \frac{1425}{3644} a^{15} - \frac{4291}{14576} a^{14} - \frac{841}{3644} a^{13} - \frac{485}{1822} a^{12} + \frac{231}{911} a^{11} - \frac{2271}{14576} a^{10} - \frac{237}{1822} a^{9} - \frac{2375}{7288} a^{8} - \frac{749}{3644} a^{7} - \frac{2721}{7288} a^{6} - \frac{257}{911} a^{5} + \frac{314}{911} a^{4} - \frac{34}{911} a^{3} + \frac{617}{1822} a^{2} + \frac{2853}{7288} a - \frac{2241}{14576}$, $\frac{1}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a^{35} + \frac{4328274678526859793905102702832041490670556779851892438844186412706313892718766510376311851}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a^{34} + \frac{1414042223650878980692267266778245948038232333588301779459467451116585292316638215786886775151}{23929727637688119167830473553598064993892544337025813819089185979497416127645428029186221380172} a^{33} - \frac{1747104540015465807850143861733166821867117954987789482882829844954675013176040686099287599187}{47859455275376238335660947107196129987785088674051627638178371958994832255290856058372442760344} a^{32} + \frac{19197814862183966598455976805786443573654644718838564887701613667636646997138761730712908365451}{95718910550752476671321894214392259975570177348103255276356743917989664510581712116744885520688} a^{31} + \frac{8813550665260073474442932994624538009677331213292732903607808532882333950318115543197355671013}{95718910550752476671321894214392259975570177348103255276356743917989664510581712116744885520688} a^{30} + \frac{1934627498875605684516948086579652837934979333694513514420194297270953009722241132582108145125}{47859455275376238335660947107196129987785088674051627638178371958994832255290856058372442760344} a^{29} + \frac{4123368520829595988762662262455335305425798067916105844753932170012217741990799167201306488221}{47859455275376238335660947107196129987785088674051627638178371958994832255290856058372442760344} a^{28} + \frac{39017785933311755975980133175354193000383535306029653006431617484552360703631493497228678923811}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a^{27} - \frac{13467634145908491772253387594643051787705759575737880360377840719984277486100926099121131215907}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a^{26} + \frac{6214264604455229133649114504902810061220136774362046866769544939127437206621972444770832068007}{95718910550752476671321894214392259975570177348103255276356743917989664510581712116744885520688} a^{25} + \frac{14706738577741635766121702689903206024969683994554023312603853864550308335076043597387698600929}{95718910550752476671321894214392259975570177348103255276356743917989664510581712116744885520688} a^{24} - \frac{24154006250941485029749941113830112940118883632664089188902912779928095995264995364943152608345}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a^{23} - \frac{11465256480683888907084230006688900765795963382908768157067820259115982775892289555671234840225}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a^{22} + \frac{656110531352890689415082731147159190803330173170743990665726048473628572654775862427501751795}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a^{21} - \frac{36251474845846459926865935008288441655859135319543874537568537081609388706651824652249903979649}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a^{20} - \frac{23113151604488730529181527624258537454851281732647588197189212329572801799661242078200668295147}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a^{19} + \frac{35169515755029624343839700958919027054473761635255366474525760911479944486856882108199604416691}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a^{18} - \frac{4074184025850443609437649779755253281347801866755426164178751174587431059245410566312783863785}{95718910550752476671321894214392259975570177348103255276356743917989664510581712116744885520688} a^{17} - \frac{2289253414858158022044764300167482221838843437222111792193100912901305269836846556228947624689}{95718910550752476671321894214392259975570177348103255276356743917989664510581712116744885520688} a^{16} - \frac{66925033419091720898662163701366572037008151952851644455378778691112327848348670278415666847095}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a^{15} + \frac{35355212719213825167870877989414118048410035951134590330154927590009230523090885861457964447391}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a^{14} - \frac{7110085862663919803693921903447797481715569080177356869719175679127026316676209571511160804445}{47859455275376238335660947107196129987785088674051627638178371958994832255290856058372442760344} a^{13} + \frac{2029098744200426415616109765015750165926954920928555891977190566113889158104357800984638427573}{5982431909422029791957618388399516248473136084256453454772296494874354031911357007296555345043} a^{12} - \frac{23579287209562786815934026766579832915397805449485834480255189865448508349779032448809177424783}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a^{11} + \frac{50896188340889074091283707551171154099126906006994429828863257514915605607586348658647813791247}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a^{10} + \frac{7633197925866065923305358451210191201988650492396128134468856967569154485381964792229945026293}{95718910550752476671321894214392259975570177348103255276356743917989664510581712116744885520688} a^{9} - \frac{28318780580956973793812736700430718770246281460044817188935301913462773096767850231043634394663}{95718910550752476671321894214392259975570177348103255276356743917989664510581712116744885520688} a^{8} + \frac{16849672562106884696292498591082723912366321818007726340854358263287427371760386009559094616853}{95718910550752476671321894214392259975570177348103255276356743917989664510581712116744885520688} a^{7} + \frac{33350346565356764129196221314662397120195733771833932364351760068907566666892783644720767840057}{95718910550752476671321894214392259975570177348103255276356743917989664510581712116744885520688} a^{6} + \frac{10109695696079395930227694303794478349234396365595892378056566551058010705128123450732141462191}{23929727637688119167830473553598064993892544337025813819089185979497416127645428029186221380172} a^{5} - \frac{4834323798222773008861053827349499938980754609636989680809166879002070700659659751052742528387}{23929727637688119167830473553598064993892544337025813819089185979497416127645428029186221380172} a^{4} + \frac{1952618661708908102071653403630388758303000812532810970781903857395512399773960091342122907387}{5982431909422029791957618388399516248473136084256453454772296494874354031911357007296555345043} a^{3} - \frac{35529076440787696522623935041005443919032120154297768193394294005461698033581234560157643762027}{95718910550752476671321894214392259975570177348103255276356743917989664510581712116744885520688} a^{2} + \frac{9215808886971525300083541190040069939078728839925937863275698834819900419573771362288643953277}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376} a + \frac{41485764893722151365448225548401061994396048142915058510751216294517699668801276267267680613073}{191437821101504953342643788428784519951140354696206510552713487835979329021163424233489771041376}$

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:  $35$
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:  \( 5028686096261389000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2$ (as 36T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), 3.3.169.1, 3.3.8281.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{5}, \sqrt{13})\), 6.6.46411625.1, 6.6.111434311625.2, 6.6.111434311625.1, 6.6.659374625.1, \(\Q(\zeta_{13})^+\), 6.6.3570125.1, 6.6.891474493.1, 6.6.8571870125.2, 6.6.891474493.2, 6.6.8571870125.1, 6.6.5274997.1, 6.6.300125.1, 9.9.567869252041.1, 12.12.2154038935140625.1, 12.12.12417605807337610140625.1, 12.12.12417605807337610140625.2, 12.12.434774896093890625.1, 18.18.1383747355171268959993166322265625.1, 18.18.708478645847689707516501157.1, 18.18.629834936354696841143908203125.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{6}$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed
$13$13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$