Properties

Label 33.33.236...601.1
Degree $33$
Signature $[33, 0]$
Discriminant $2.368\times 10^{73}$
Root discriminant $167.29$
Ramified primes $7, 67$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{33}$ (as 33T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^33 - 8*x^32 - 86*x^31 + 690*x^30 + 3527*x^29 - 25876*x^28 - 92162*x^27 + 549745*x^26 + 1676544*x^25 - 7221169*x^24 - 21518451*x^23 + 59984529*x^22 + 192193744*x^21 - 307177110*x^20 - 1170053321*x^19 + 864215343*x^18 + 4741699040*x^17 - 637264044*x^16 - 12404482875*x^15 - 3656463713*x^14 + 19997642137*x^13 + 12376322490*x^12 - 18399788371*x^11 - 16849661483*x^10 + 8289542908*x^9 + 11391691466*x^8 - 842642325*x^7 - 3707258605*x^6 - 522452271*x^5 + 463178377*x^4 + 128777001*x^3 - 2389174*x^2 - 1929880*x + 90889)
 
gp: K = bnfinit(x^33 - 8*x^32 - 86*x^31 + 690*x^30 + 3527*x^29 - 25876*x^28 - 92162*x^27 + 549745*x^26 + 1676544*x^25 - 7221169*x^24 - 21518451*x^23 + 59984529*x^22 + 192193744*x^21 - 307177110*x^20 - 1170053321*x^19 + 864215343*x^18 + 4741699040*x^17 - 637264044*x^16 - 12404482875*x^15 - 3656463713*x^14 + 19997642137*x^13 + 12376322490*x^12 - 18399788371*x^11 - 16849661483*x^10 + 8289542908*x^9 + 11391691466*x^8 - 842642325*x^7 - 3707258605*x^6 - 522452271*x^5 + 463178377*x^4 + 128777001*x^3 - 2389174*x^2 - 1929880*x + 90889, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![90889, -1929880, -2389174, 128777001, 463178377, -522452271, -3707258605, -842642325, 11391691466, 8289542908, -16849661483, -18399788371, 12376322490, 19997642137, -3656463713, -12404482875, -637264044, 4741699040, 864215343, -1170053321, -307177110, 192193744, 59984529, -21518451, -7221169, 1676544, 549745, -92162, -25876, 3527, 690, -86, -8, 1]);
 

\( x^{33} - 8 x^{32} - 86 x^{31} + 690 x^{30} + 3527 x^{29} - 25876 x^{28} - 92162 x^{27} + 549745 x^{26} + 1676544 x^{25} - 7221169 x^{24} - 21518451 x^{23} + 59984529 x^{22} + 192193744 x^{21} - 307177110 x^{20} - 1170053321 x^{19} + 864215343 x^{18} + 4741699040 x^{17} - 637264044 x^{16} - 12404482875 x^{15} - 3656463713 x^{14} + 19997642137 x^{13} + 12376322490 x^{12} - 18399788371 x^{11} - 16849661483 x^{10} + 8289542908 x^{9} + 11391691466 x^{8} - 842642325 x^{7} - 3707258605 x^{6} - 522452271 x^{5} + 463178377 x^{4} + 128777001 x^{3} - 2389174 x^{2} - 1929880 x + 90889 \)

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

Invariants

Degree:  $33$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[33, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(236\!\cdots\!601\)\(\medspace = 7^{22}\cdot 67^{30}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $167.29$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 67$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $33$
This field is Galois and abelian over $\Q$.
Conductor:  \(469=7\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{469}(1,·)$, $\chi_{469}(260,·)$, $\chi_{469}(263,·)$, $\chi_{469}(9,·)$, $\chi_{469}(394,·)$, $\chi_{469}(15,·)$, $\chi_{469}(403,·)$, $\chi_{469}(148,·)$, $\chi_{469}(277,·)$, $\chi_{469}(22,·)$, $\chi_{469}(25,·)$, $\chi_{469}(282,·)$, $\chi_{469}(156,·)$, $\chi_{469}(158,·)$, $\chi_{469}(417,·)$, $\chi_{469}(424,·)$, $\chi_{469}(135,·)$, $\chi_{469}(442,·)$, $\chi_{469}(64,·)$, $\chi_{469}(193,·)$, $\chi_{469}(198,·)$, $\chi_{469}(330,·)$, $\chi_{469}(464,·)$, $\chi_{469}(81,·)$, $\chi_{469}(466,·)$, $\chi_{469}(344,·)$, $\chi_{469}(92,·)$, $\chi_{469}(225,·)$, $\chi_{469}(226,·)$, $\chi_{469}(359,·)$, $\chi_{469}(107,·)$, $\chi_{469}(375,·)$, $\chi_{469}(149,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{29} a^{21} - \frac{8}{29} a^{20} - \frac{5}{29} a^{19} + \frac{10}{29} a^{18} - \frac{5}{29} a^{17} - \frac{2}{29} a^{16} + \frac{8}{29} a^{15} + \frac{4}{29} a^{13} + \frac{9}{29} a^{12} - \frac{3}{29} a^{11} + \frac{9}{29} a^{10} - \frac{5}{29} a^{9} - \frac{1}{29} a^{8} + \frac{6}{29} a^{7} + \frac{11}{29} a^{6} - \frac{13}{29} a^{5} + \frac{3}{29} a^{4} - \frac{13}{29} a^{3} - \frac{4}{29} a^{2} - \frac{4}{29} a + \frac{2}{29}$, $\frac{1}{29} a^{22} - \frac{11}{29} a^{20} - \frac{1}{29} a^{19} - \frac{12}{29} a^{18} - \frac{13}{29} a^{17} - \frac{8}{29} a^{16} + \frac{6}{29} a^{15} + \frac{4}{29} a^{14} + \frac{12}{29} a^{13} + \frac{11}{29} a^{12} + \frac{14}{29} a^{11} + \frac{9}{29} a^{10} - \frac{12}{29} a^{9} - \frac{2}{29} a^{8} + \frac{1}{29} a^{7} - \frac{12}{29} a^{6} - \frac{14}{29} a^{5} + \frac{11}{29} a^{4} + \frac{8}{29} a^{3} - \frac{7}{29} a^{2} - \frac{1}{29} a - \frac{13}{29}$, $\frac{1}{29} a^{23} - \frac{2}{29} a^{20} - \frac{9}{29} a^{19} + \frac{10}{29} a^{18} - \frac{5}{29} a^{17} + \frac{13}{29} a^{16} + \frac{5}{29} a^{15} + \frac{12}{29} a^{14} - \frac{3}{29} a^{13} - \frac{3}{29} a^{12} + \frac{5}{29} a^{11} + \frac{1}{29} a^{9} - \frac{10}{29} a^{8} - \frac{4}{29} a^{7} - \frac{9}{29} a^{6} + \frac{13}{29} a^{5} + \frac{12}{29} a^{4} - \frac{5}{29} a^{3} + \frac{13}{29} a^{2} + \frac{1}{29} a - \frac{7}{29}$, $\frac{1}{29} a^{24} + \frac{4}{29} a^{20} - \frac{14}{29} a^{18} + \frac{3}{29} a^{17} + \frac{1}{29} a^{16} - \frac{1}{29} a^{15} - \frac{3}{29} a^{14} + \frac{5}{29} a^{13} - \frac{6}{29} a^{12} - \frac{6}{29} a^{11} - \frac{10}{29} a^{10} + \frac{9}{29} a^{9} - \frac{6}{29} a^{8} + \frac{3}{29} a^{7} + \frac{6}{29} a^{6} - \frac{14}{29} a^{5} + \frac{1}{29} a^{4} - \frac{13}{29} a^{3} - \frac{7}{29} a^{2} + \frac{14}{29} a + \frac{4}{29}$, $\frac{1}{29} a^{25} + \frac{3}{29} a^{20} + \frac{6}{29} a^{19} - \frac{8}{29} a^{18} - \frac{8}{29} a^{17} + \frac{7}{29} a^{16} - \frac{6}{29} a^{15} + \frac{5}{29} a^{14} + \frac{7}{29} a^{13} - \frac{13}{29} a^{12} + \frac{2}{29} a^{11} + \frac{2}{29} a^{10} + \frac{14}{29} a^{9} + \frac{7}{29} a^{8} + \frac{11}{29} a^{7} - \frac{5}{29} a^{5} + \frac{4}{29} a^{4} - \frac{13}{29} a^{3} + \frac{1}{29} a^{2} - \frac{9}{29} a - \frac{8}{29}$, $\frac{1}{29} a^{26} + \frac{1}{29} a^{20} + \frac{7}{29} a^{19} - \frac{9}{29} a^{18} - \frac{7}{29} a^{17} + \frac{10}{29} a^{15} + \frac{7}{29} a^{14} + \frac{4}{29} a^{13} + \frac{4}{29} a^{12} + \frac{11}{29} a^{11} - \frac{13}{29} a^{10} - \frac{7}{29} a^{9} + \frac{14}{29} a^{8} + \frac{11}{29} a^{7} - \frac{9}{29} a^{6} + \frac{14}{29} a^{5} + \frac{7}{29} a^{4} + \frac{11}{29} a^{3} + \frac{3}{29} a^{2} + \frac{4}{29} a - \frac{6}{29}$, $\frac{1}{29} a^{27} - \frac{14}{29} a^{20} - \frac{4}{29} a^{19} + \frac{12}{29} a^{18} + \frac{5}{29} a^{17} + \frac{12}{29} a^{16} - \frac{1}{29} a^{15} + \frac{4}{29} a^{14} + \frac{2}{29} a^{12} - \frac{10}{29} a^{11} + \frac{13}{29} a^{10} - \frac{10}{29} a^{9} + \frac{12}{29} a^{8} + \frac{14}{29} a^{7} + \frac{3}{29} a^{6} - \frac{9}{29} a^{5} + \frac{8}{29} a^{4} - \frac{13}{29} a^{3} + \frac{8}{29} a^{2} - \frac{2}{29} a - \frac{2}{29}$, $\frac{1}{29} a^{28} - \frac{1}{29}$, $\frac{1}{29} a^{29} - \frac{1}{29} a$, $\frac{1}{7436963} a^{30} - \frac{40018}{7436963} a^{29} + \frac{19632}{7436963} a^{28} - \frac{110131}{7436963} a^{27} - \frac{12410}{7436963} a^{26} + \frac{71590}{7436963} a^{25} + \frac{76491}{7436963} a^{24} + \frac{85961}{7436963} a^{23} - \frac{3082}{256447} a^{22} - \frac{3264}{7436963} a^{21} + \frac{290295}{7436963} a^{20} - \frac{1060069}{7436963} a^{19} - \frac{2707658}{7436963} a^{18} + \frac{2148906}{7436963} a^{17} - \frac{3439636}{7436963} a^{16} + \frac{127496}{7436963} a^{15} + \frac{1234945}{7436963} a^{14} + \frac{5}{37} a^{13} + \frac{860104}{7436963} a^{12} + \frac{61644}{256447} a^{11} + \frac{3228921}{7436963} a^{10} - \frac{2983870}{7436963} a^{9} + \frac{3652868}{7436963} a^{8} - \frac{3629386}{7436963} a^{7} + \frac{1550764}{7436963} a^{6} - \frac{2858496}{7436963} a^{5} + \frac{2742473}{7436963} a^{4} + \frac{56530}{256447} a^{3} + \frac{462954}{7436963} a^{2} + \frac{2465172}{7436963} a + \frac{1889497}{7436963}$, $\frac{1}{721385411} a^{31} + \frac{46}{721385411} a^{30} + \frac{417177}{24875359} a^{29} + \frac{3237179}{721385411} a^{28} + \frac{10897062}{721385411} a^{27} - \frac{6795986}{721385411} a^{26} - \frac{9333536}{721385411} a^{25} + \frac{203154}{19496903} a^{24} - \frac{6385812}{721385411} a^{23} - \frac{4690041}{721385411} a^{22} - \frac{5588912}{721385411} a^{21} - \frac{44661523}{721385411} a^{20} - \frac{242902349}{721385411} a^{19} + \frac{283763070}{721385411} a^{18} - \frac{275120971}{721385411} a^{17} + \frac{172012437}{721385411} a^{16} + \frac{1402583}{3018349} a^{15} + \frac{84354146}{721385411} a^{14} - \frac{56951367}{721385411} a^{13} - \frac{327836347}{721385411} a^{12} + \frac{231630514}{721385411} a^{11} + \frac{236108763}{721385411} a^{10} - \frac{297850070}{721385411} a^{9} - \frac{268696651}{721385411} a^{8} - \frac{10273897}{24875359} a^{7} + \frac{275524258}{721385411} a^{6} + \frac{279484067}{721385411} a^{5} + \frac{50598316}{721385411} a^{4} + \frac{38726279}{721385411} a^{3} + \frac{270968515}{721385411} a^{2} - \frac{15305213}{721385411} a - \frac{2279339}{7436963}$, $\frac{1}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{32} - \frac{5695552672365422786650885919957576035254271146644755039895901266365073791533495614128545571660599}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{31} + \frac{1483291090227873719017308254505112377735370552572393456776554315693724890520438986089219885974333824}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{30} + \frac{402085909578855758887427829878690662739172006305792169422821510708325981858552862342819824908702409702067}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{29} + \frac{379373592008178180440034369127637972034987854873965843155630828266881937816884213697204490621670842721490}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{28} + \frac{63453323808487942844850179037033085104536937426336385382486899025158842748331209364848824666641463615332}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{27} + \frac{200708606174439824517248147530471977337010059832670142960377993060944020083391735793180019016037757135323}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{26} - \frac{205992970942112683570842078870065127208023594719349411661120433543796267560933265291595964010841255315748}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{25} + \frac{337717118661868277686332007262332576427960440427961715347307697822901846461228471647069396976262544743997}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{24} + \frac{423339439978826125060391863491164634128016954217760763746839734275240612991906152671609488730585680529805}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{23} - \frac{253665925182230560892815956259665894916411608484413623228617386681411917995260732004847138135909449710116}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{22} - \frac{43082532093950257470016120901315476128437612981252992639971360147613739608767786684555641619231189635269}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{21} + \frac{3094925776283896582559815656580337198775329134064028852866790505116809303631421280261063471817669564384716}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{20} + \frac{4397689509473345861856183722394204679124535199954212508399347788198112905747382116145158053587479011091560}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{19} + \frac{49734227059414946728586800703715167089541003992006191833655225515443599184235140527708127148488034282436}{783359082020907319038132243156721764473305187274938982386562452696856040657866956471696348426576000844221} a^{18} - \frac{12170398255672404682790474200754525168311302551373898524480830025953731438382031004764976747566654168076302}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{17} + \frac{5022475434010793797203735803929617897824618125920987765437177916070854794721758716441478455095662986361274}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{16} - \frac{3672368268092147014319291605219298048888090049897239901492330220686402357720887466955556213606457825054691}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{15} + \frac{10817593740583405537091630255439709052089297408600062642022502620866696742987679589234975949522043310000484}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{14} + \frac{11055843183565935019109588470318134009528572633708613289033465344112171217539128272147702022073030626465915}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{13} - \frac{8460507470719566348498133242964357767056076177672466991842134273313892920512258751470843292221280366746073}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{12} + \frac{12649204757851423849625734019474816257146638663473379520712939906727915788746882023912723872280485072304657}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{11} - \frac{13343195476544006165984763469257219804068129033752423858341101619374059878536564072097944926171904621318100}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{10} - \frac{1323856633383299035622738088976120483271914951963588432532266154400164830112171151686875483883262066943742}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{9} - \frac{11320156729506552407702374654488619026095973865187742922910490710422914944139449262290185620733201922444905}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{8} + \frac{11102544271362149869830465658688421683915963878751201206195236686381311207935071008822251078385142262944289}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{7} - \frac{1150373971207650769535250536353258807848321985960090333435877455677574685321821859078002185800404689352242}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{6} - \frac{7143515613402065188942341295596952391599499293887917113151060987636883318622970114523625355314943167910388}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{5} - \frac{14356389999376855750480205574846739814468063853910157127746075443256106720352891701971975032065285905273903}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{4} - \frac{127229638529111004196115806337140800391142171735837138420946731656368529753423855264831761285799499341681}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{3} + \frac{8616969019987116534210633556087332210966417396181541389995180725458103859243673339817393445954009454885436}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a^{2} - \frac{5673341627110617956982965554731498046471426970138688066227882195260259669596793014215826236603070349379707}{28984286034773570804410892996798705285512291929172742348302810749783673504341077389452764891783312031236177} a - \frac{95864333232608466759430671343806422561999822478933225998898284841040607844801352018909793862028138651642}{298807072523438874272277247389677374077446308548172601528894956183336840250938942159306854554467134342641}$

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

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $32$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 503657750536712100000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{33}\cdot(2\pi)^{0}\cdot 503657750536712100000000000 \cdot 1}{2\sqrt{23681050358190252966666038984115482490423545779778728084912954382239302601}}\approx 0.444523681073377$ (assuming GRH)

Galois group

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

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

Intermediate fields

\(\Q(\zeta_{7})^+\), 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 $33$ $33$ $33$ R $33$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{3}$ $33$ $33$ $33$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{33}$ $33$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{11}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{3}$ $33$ $33$ $33$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
67Data not computed