↑

Properties

Label 36.0.22878331820...2289.3
Degree $36$
Signature $[0, 18]$
Discriminant $3^{18}\cdot 7^{30}\cdot 13^{30}$
Root discriminant $74.32$
Ramified primes $3, 7, 13$
Class number Not computed
Class group Not computed
Galois group $C_6^2$ (as 36T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![244140625, 97656250, -19531250, -119140625, -156250000, -67890625, 45812500, -226762500, -21712500, 61986125, 105945575, 108765185, 9950131, -70160559, 41430373, -11693491, -1135357, 5140127, -4311109, 2341779, -705229, -325139, 415523, -284405, 208009, 40455, -23107, -6441, 2666, 1041, -449, -22, 53, 6, -7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 7*x^34 + 6*x^33 + 53*x^32 - 22*x^31 - 449*x^30 + 1041*x^29 + 2666*x^28 - 6441*x^27 - 23107*x^26 + 40455*x^25 + 208009*x^24 - 284405*x^23 + 415523*x^22 - 325139*x^21 - 705229*x^20 + 2341779*x^19 - 4311109*x^18 + 5140127*x^17 - 1135357*x^16 - 11693491*x^15 + 41430373*x^14 - 70160559*x^13 + 9950131*x^12 + 108765185*x^11 + 105945575*x^10 + 61986125*x^9 - 21712500*x^8 - 226762500*x^7 + 45812500*x^6 - 67890625*x^5 - 156250000*x^4 - 119140625*x^3 - 19531250*x^2 + 97656250*x + 244140625)
 
gp: K = bnfinit(x^36 - x^35 - 7*x^34 + 6*x^33 + 53*x^32 - 22*x^31 - 449*x^30 + 1041*x^29 + 2666*x^28 - 6441*x^27 - 23107*x^26 + 40455*x^25 + 208009*x^24 - 284405*x^23 + 415523*x^22 - 325139*x^21 - 705229*x^20 + 2341779*x^19 - 4311109*x^18 + 5140127*x^17 - 1135357*x^16 - 11693491*x^15 + 41430373*x^14 - 70160559*x^13 + 9950131*x^12 + 108765185*x^11 + 105945575*x^10 + 61986125*x^9 - 21712500*x^8 - 226762500*x^7 + 45812500*x^6 - 67890625*x^5 - 156250000*x^4 - 119140625*x^3 - 19531250*x^2 + 97656250*x + 244140625, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 7 x^{34} + 6 x^{33} + 53 x^{32} - 22 x^{31} - 449 x^{30} + 1041 x^{29} + 2666 x^{28} - 6441 x^{27} - 23107 x^{26} + 40455 x^{25} + 208009 x^{24} - 284405 x^{23} + 415523 x^{22} - 325139 x^{21} - 705229 x^{20} + 2341779 x^{19} - 4311109 x^{18} + 5140127 x^{17} - 1135357 x^{16} - 11693491 x^{15} + 41430373 x^{14} - 70160559 x^{13} + 9950131 x^{12} + 108765185 x^{11} + 105945575 x^{10} + 61986125 x^{9} - 21712500 x^{8} - 226762500 x^{7} + 45812500 x^{6} - 67890625 x^{5} - 156250000 x^{4} - 119140625 x^{3} - 19531250 x^{2} + 97656250 x + 244140625 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(22878331820822683097801634238807198405761132789439230168181892642289=3^{18}\cdot 7^{30}\cdot 13^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.32$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(273=3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{273}(256,·)$, $\chi_{273}(1,·)$, $\chi_{273}(134,·)$, $\chi_{273}(257,·)$, $\chi_{273}(139,·)$, $\chi_{273}(16,·)$, $\chi_{273}(17,·)$, $\chi_{273}(22,·)$, $\chi_{273}(23,·)$, $\chi_{273}(155,·)$, $\chi_{273}(157,·)$, $\chi_{273}(38,·)$, $\chi_{273}(40,·)$, $\chi_{273}(172,·)$, $\chi_{273}(173,·)$, $\chi_{273}(178,·)$, $\chi_{273}(179,·)$, $\chi_{273}(55,·)$, $\chi_{273}(61,·)$, $\chi_{273}(62,·)$, $\chi_{273}(194,·)$, $\chi_{273}(79,·)$, $\chi_{273}(211,·)$, $\chi_{273}(212,·)$, $\chi_{273}(218,·)$, $\chi_{273}(94,·)$, $\chi_{273}(95,·)$, $\chi_{273}(272,·)$, $\chi_{273}(100,·)$, $\chi_{273}(101,·)$, $\chi_{273}(233,·)$, $\chi_{273}(235,·)$, $\chi_{273}(116,·)$, $\chi_{273}(118,·)$, $\chi_{273}(250,·)$, $\chi_{273}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{4}$, $\frac{1}{20} a^{19} + \frac{1}{10} a^{18} + \frac{1}{20} a^{15} + \frac{1}{10} a^{14} + \frac{1}{10} a^{12} + \frac{1}{10} a^{10} + \frac{1}{10} a^{8} - \frac{2}{5} a^{6} + \frac{9}{20} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2} + \frac{9}{20} a - \frac{1}{2}$, $\frac{1}{20} a^{20} + \frac{1}{20} a^{18} + \frac{1}{20} a^{16} + \frac{1}{20} a^{14} + \frac{1}{10} a^{13} - \frac{1}{5} a^{12} + \frac{1}{10} a^{11} - \frac{1}{5} a^{10} + \frac{1}{10} a^{9} - \frac{1}{5} a^{8} + \frac{1}{10} a^{7} + \frac{1}{4} a^{6} - \frac{2}{5} a^{5} + \frac{1}{4} a^{4} - \frac{2}{5} a^{3} + \frac{1}{4} a^{2} - \frac{2}{5} a + \frac{1}{4}$, $\frac{1}{80} a^{21} + \frac{1}{10} a^{18} - \frac{1}{20} a^{17} - \frac{1}{16} a^{14} + \frac{1}{5} a^{13} + \frac{1}{5} a^{11} + \frac{1}{5} a^{9} + \frac{3}{16} a^{7} + \frac{1}{5} a^{5} - \frac{1}{10} a^{4} + \frac{1}{4} a^{3} + \frac{1}{5} a - \frac{7}{16}$, $\frac{1}{80} a^{22} + \frac{7}{80} a^{15} - \frac{1}{80} a^{8} + \frac{33}{80} a$, $\frac{1}{80} a^{23} + \frac{7}{80} a^{16} - \frac{1}{80} a^{9} + \frac{33}{80} a^{2}$, $\frac{1}{80} a^{24} + \frac{7}{80} a^{17} - \frac{1}{80} a^{10} + \frac{33}{80} a^{3}$, $\frac{1}{80} a^{25} + \frac{7}{80} a^{18} - \frac{1}{80} a^{11} + \frac{33}{80} a^{4}$, $\frac{1}{112400} a^{26} + \frac{119}{112400} a^{25} + \frac{3}{7025} a^{24} - \frac{99}{112400} a^{23} + \frac{11}{14050} a^{22} - \frac{27}{7025} a^{21} + \frac{187}{14050} a^{20} - \frac{2669}{112400} a^{19} - \frac{1599}{112400} a^{18} - \frac{27}{14050} a^{17} + \frac{10403}{112400} a^{16} + \frac{407}{5620} a^{15} - \frac{427}{14050} a^{14} - \frac{95}{562} a^{13} + \frac{16223}{112400} a^{12} - \frac{1679}{112400} a^{11} + \frac{1327}{14050} a^{10} + \frac{25339}{112400} a^{9} + \frac{591}{7025} a^{8} + \frac{299}{14050} a^{7} + \frac{3178}{7025} a^{6} - \frac{1451}{112400} a^{5} + \frac{17263}{112400} a^{4} - \frac{1019}{7025} a^{3} - \frac{24979}{112400} a^{2} - \frac{1601}{5620} a - \frac{68}{281}$, $\frac{1}{1124000} a^{27} - \frac{1}{1124000} a^{26} + \frac{6843}{1124000} a^{25} + \frac{5381}{1124000} a^{24} - \frac{6297}{1124000} a^{23} + \frac{1653}{1124000} a^{22} + \frac{1351}{1124000} a^{21} + \frac{8891}{1124000} a^{20} - \frac{1659}{1124000} a^{19} + \frac{7609}{1124000} a^{18} + \frac{41943}{1124000} a^{17} + \frac{23121}{224800} a^{16} - \frac{31841}{1124000} a^{15} - \frac{27191}{224800} a^{14} + \frac{273023}{1124000} a^{13} + \frac{18561}{1124000} a^{12} + \frac{22421}{1124000} a^{11} - \frac{192021}{1124000} a^{10} - \frac{146759}{1124000} a^{9} + \frac{260027}{1124000} a^{8} - \frac{240407}{1124000} a^{7} + \frac{109}{1124000} a^{6} + \frac{118323}{1124000} a^{5} + \frac{456591}{1124000} a^{4} - \frac{265919}{1124000} a^{3} - \frac{37753}{224800} a^{2} + \frac{20257}{44960} a + \frac{3351}{8992}$, $\frac{1}{5620000} a^{28} - \frac{1}{5620000} a^{27} - \frac{7}{5620000} a^{26} + \frac{5131}{5620000} a^{25} - \frac{11947}{5620000} a^{24} - \frac{8647}{5620000} a^{23} + \frac{30801}{5620000} a^{22} + \frac{3541}{5620000} a^{21} + \frac{91541}{5620000} a^{20} - \frac{30941}{5620000} a^{19} + \frac{401393}{5620000} a^{18} + \frac{108291}{1124000} a^{17} + \frac{545259}{5620000} a^{16} + \frac{35719}{1124000} a^{15} - \frac{170227}{5620000} a^{14} + \frac{571361}{5620000} a^{13} + \frac{845271}{5620000} a^{12} + \frac{1165029}{5620000} a^{11} - \frac{1141109}{5620000} a^{10} + \frac{697127}{5620000} a^{9} + \frac{220143}{5620000} a^{8} + \frac{854259}{5620000} a^{7} + \frac{2385123}{5620000} a^{6} - \frac{282059}{5620000} a^{5} - \frac{1031369}{5620000} a^{4} - \frac{150363}{1124000} a^{3} - \frac{5947}{224800} a^{2} - \frac{15063}{44960} a - \frac{1279}{4496}$, $\frac{1}{28100000} a^{29} - \frac{1}{28100000} a^{28} - \frac{7}{28100000} a^{27} - \frac{119}{28100000} a^{26} + \frac{65803}{28100000} a^{25} + \frac{160853}{28100000} a^{24} + \frac{58801}{28100000} a^{23} + \frac{173791}{28100000} a^{22} - \frac{28959}{28100000} a^{21} + \frac{264059}{28100000} a^{20} + \frac{363643}{28100000} a^{19} + \frac{297941}{5620000} a^{18} - \frac{2676241}{28100000} a^{17} - \frac{111081}{5620000} a^{16} + \frac{2265523}{28100000} a^{15} + \frac{3471861}{28100000} a^{14} - \frac{5622729}{28100000} a^{13} + \frac{1418279}{28100000} a^{12} + \frac{1913141}{28100000} a^{11} + \frac{4675627}{28100000} a^{10} + \frac{314143}{28100000} a^{9} + \frac{5092009}{28100000} a^{8} + \frac{6827623}{28100000} a^{7} - \frac{12929059}{28100000} a^{6} + \frac{7148381}{28100000} a^{5} - \frac{264413}{5620000} a^{4} + \frac{242593}{1124000} a^{3} + \frac{49993}{224800} a^{2} + \frac{427}{11240} a + \frac{1027}{2248}$, $\frac{1}{702500000} a^{30} - \frac{3}{351250000} a^{29} - \frac{1}{351250000} a^{28} + \frac{41}{702500000} a^{27} + \frac{23}{702500000} a^{26} + \frac{530963}{702500000} a^{25} + \frac{2718411}{702500000} a^{24} - \frac{3481089}{702500000} a^{23} - \frac{1252539}{702500000} a^{22} + \frac{1448979}{702500000} a^{21} - \frac{6225277}{702500000} a^{20} - \frac{2396927}{140500000} a^{19} - \frac{73947391}{702500000} a^{18} - \frac{461103}{28100000} a^{17} + \frac{54571923}{702500000} a^{16} + \frac{14050371}{702500000} a^{15} - \frac{43032659}{702500000} a^{14} + \frac{52289799}{702500000} a^{13} - \frac{26691879}{702500000} a^{12} - \frac{157101203}{702500000} a^{11} - \frac{84757867}{702500000} a^{10} + \frac{150530169}{702500000} a^{9} + \frac{25632203}{702500000} a^{8} - \frac{44859299}{702500000} a^{7} - \frac{118762699}{702500000} a^{6} - \frac{66718969}{140500000} a^{5} + \frac{1412911}{28100000} a^{4} - \frac{278309}{5620000} a^{3} - \frac{43801}{281000} a^{2} - \frac{110609}{224800} a + \frac{4981}{44960}$, $\frac{1}{368929686880865322287500000} a^{31} - \frac{128016170762712933}{184464843440432661143750000} a^{30} - \frac{90732901549917217}{368929686880865322287500000} a^{29} + \frac{267373954219165517}{46116210860108165285937500} a^{28} + \frac{222790663093711597}{92232421720216330571875000} a^{27} + \frac{812747994197363802529}{184464843440432661143750000} a^{26} + \frac{308358707813200807541}{64543332204490084375000} a^{25} + \frac{793316113666107310740413}{184464843440432661143750000} a^{24} + \frac{510782175696955905655813}{184464843440432661143750000} a^{23} - \frac{210813440689173139814303}{184464843440432661143750000} a^{22} - \frac{154233968015530547285421}{184464843440432661143750000} a^{21} - \frac{102943214329939462938327}{18446484344043266114375000} a^{20} + \frac{3854717994348104992160167}{184464843440432661143750000} a^{19} + \frac{697228656011797543454903}{18446484344043266114375000} a^{18} + \frac{11124895777692909041961887}{92232421720216330571875000} a^{17} - \frac{12648294936626366958883267}{184464843440432661143750000} a^{16} - \frac{3662667089919326157002043}{46116210860108165285937500} a^{15} + \frac{4906673766630780257258957}{184464843440432661143750000} a^{14} + \frac{1866304644720578649341839}{92232421720216330571875000} a^{13} + \frac{28746635805581368210645431}{184464843440432661143750000} a^{12} - \frac{13698195025813420566302353}{92232421720216330571875000} a^{11} + \frac{7402621483738786343655207}{184464843440432661143750000} a^{10} - \frac{25566032556447940268656381}{184464843440432661143750000} a^{9} - \frac{12066172749234495188633877}{184464843440432661143750000} a^{8} + \frac{31197523574595225970525533}{184464843440432661143750000} a^{7} - \frac{145418882618708385836429}{18446484344043266114375000} a^{6} - \frac{3392615387394466150653583}{7378593737617306445750000} a^{5} - \frac{178464762206649433753279}{737859373761730644575000} a^{4} + \frac{248941691187321240609259}{590287499009384515660000} a^{3} - \frac{1927808557634346515211}{7378593737617306445750} a^{2} + \frac{5910264527007999539289}{23611499960375380626400} a - \frac{632409702191110490781}{2361149996037538062640}$, $\frac{1}{1844648434404326611437500000} a^{32} - \frac{1}{1844648434404326611437500000} a^{31} - \frac{893133659970450607}{1844648434404326611437500000} a^{30} + \frac{32605261302892053}{922324217202163305718750000} a^{29} + \frac{7079858757157820753}{1844648434404326611437500000} a^{28} + \frac{218330000864980939}{922324217202163305718750000} a^{27} + \frac{4076713923483757145313}{922324217202163305718750000} a^{26} - \frac{2097636492305138438197317}{922324217202163305718750000} a^{25} - \frac{85055113877167504341287}{57645263575135206607421875} a^{24} - \frac{4543606163139696869450333}{922324217202163305718750000} a^{23} - \frac{1388756308565268996213083}{461162108601081652859375000} a^{22} - \frac{1034687466957273555923707}{184464843440432661143750000} a^{21} + \frac{9256791041049380364300917}{922324217202163305718750000} a^{20} - \frac{2308135134099740213838253}{184464843440432661143750000} a^{19} - \frac{24388832038688127265085219}{230581054300540826429687500} a^{18} - \frac{96064888710297309027082507}{922324217202163305718750000} a^{17} + \frac{1302025856784819151084503}{57645263575135206607421875} a^{16} + \frac{5422347523179049859895163}{230581054300540826429687500} a^{15} - \frac{15320793901710561666881771}{461162108601081652859375000} a^{14} + \frac{158498427342810009067005301}{922324217202163305718750000} a^{13} + \frac{173772407570788789133742959}{922324217202163305718750000} a^{12} + \frac{20808336913424480295997217}{922324217202163305718750000} a^{11} - \frac{34656356680282230470045819}{230581054300540826429687500} a^{10} + \frac{197231048286385526165513833}{922324217202163305718750000} a^{9} + \frac{27249648490685375356645239}{461162108601081652859375000} a^{8} + \frac{20368299154289656971143671}{184464843440432661143750000} a^{7} - \frac{1493122077419896822229253}{36892968688086532228750000} a^{6} + \frac{2682399940413644735382317}{7378593737617306445750000} a^{5} + \frac{5837558714083735531599}{2951437495046922578300000} a^{4} - \frac{194751437057741237528861}{590287499009384515660000} a^{3} + \frac{37199858670871166359039}{118057499801876903132000} a^{2} + \frac{3906253827339746157619}{11805749980187690313200} a - \frac{1048485595295014042653}{4722299992075076125280}$, $\frac{1}{18446484344043266114375000000} a^{33} + \frac{1}{4611621086010816528593750000} a^{32} + \frac{13}{18446484344043266114375000000} a^{31} + \frac{6098556620935529973}{9223242172021633057187500000} a^{30} + \frac{81403625519672200533}{18446484344043266114375000000} a^{29} - \frac{22372566431830085029}{2305810543005408264296875000} a^{28} - \frac{582022491879576463609}{18446484344043266114375000000} a^{27} + \frac{40116736211262791445121}{18446484344043266114375000000} a^{26} - \frac{40868330107080142767072729}{18446484344043266114375000000} a^{25} - \frac{33145221844056545608187711}{18446484344043266114375000000} a^{24} + \frac{9589819595105085640544463}{18446484344043266114375000000} a^{23} - \frac{4617218465139563424102471}{3689296868808653222875000000} a^{22} + \frac{29138122127174863538948859}{18446484344043266114375000000} a^{21} - \frac{28314529657856364229495847}{3689296868808653222875000000} a^{20} - \frac{296770806687874981269502527}{18446484344043266114375000000} a^{19} + \frac{1724721096079668321254876101}{18446484344043266114375000000} a^{18} + \frac{259212021350270170780993401}{18446484344043266114375000000} a^{17} + \frac{827538247511822763401623409}{18446484344043266114375000000} a^{16} - \frac{1841001433250183695213437939}{18446484344043266114375000000} a^{15} - \frac{1670581591285849616206415943}{18446484344043266114375000000} a^{14} - \frac{219389772502864673256577447}{18446484344043266114375000000} a^{13} + \frac{4148975870322640730251635399}{18446484344043266114375000000} a^{12} + \frac{1958706431754219327028243993}{18446484344043266114375000000} a^{11} + \frac{3196796681685561438381349031}{18446484344043266114375000000} a^{10} - \frac{1823694867019533127795455839}{18446484344043266114375000000} a^{9} + \frac{153370596190453619595548623}{3689296868808653222875000000} a^{8} + \frac{91539120394578707567978541}{737859373761730644575000000} a^{7} + \frac{49020026458351645519229239}{147571874752346128915000000} a^{6} - \frac{3079930954597015445666233}{14757187475234612891500000} a^{5} - \frac{2272013262734900399727021}{5902874990093845156600000} a^{4} + \frac{255319608897745605431997}{590287499009384515660000} a^{3} + \frac{90708809238558146161953}{236114999603753806264000} a^{2} + \frac{4854603639047029628103}{23611499960375380626400} a - \frac{745152454282713248677}{9444599984150152250560}$, $\frac{1}{92232421720216330571875000000} a^{34} - \frac{1}{92232421720216330571875000000} a^{33} - \frac{7}{92232421720216330571875000000} a^{32} - \frac{119}{92232421720216330571875000000} a^{31} + \frac{21889396973353998303}{92232421720216330571875000000} a^{30} - \frac{731758611613301831647}{92232421720216330571875000000} a^{29} + \frac{556643435826469846301}{92232421720216330571875000000} a^{28} - \frac{19024104518253618595917}{46116210860108165285937500000} a^{27} - \frac{185118510407965868839167}{46116210860108165285937500000} a^{26} + \frac{241380473098410232427614217}{46116210860108165285937500000} a^{25} - \frac{168314119985933358062844741}{46116210860108165285937500000} a^{24} + \frac{36144932951750580439821783}{9223242172021633057187500000} a^{23} + \frac{200988778511618064253035317}{46116210860108165285937500000} a^{22} + \frac{29548093892846085230634647}{9223242172021633057187500000} a^{21} + \frac{160081436368834068190314831}{11529052715027041321484375000} a^{20} + \frac{4344576890588682093745273}{2882263178756760330371093750} a^{19} + \frac{429940186354454673479125099}{23058105430054082642968750000} a^{18} - \frac{1439844048537238331051832637}{11529052715027041321484375000} a^{17} + \frac{2518071086537484679130536879}{23058105430054082642968750000} a^{16} + \frac{1513595491860130773452387813}{23058105430054082642968750000} a^{15} + \frac{9454322447267695415591399}{2882263178756760330371093750} a^{14} - \frac{9438413348090696634252331183}{46116210860108165285937500000} a^{13} + \frac{159619131062832785302023499}{46116210860108165285937500000} a^{12} + \frac{2854353703138572937053218283}{46116210860108165285937500000} a^{11} + \frac{945873751559239387609435753}{46116210860108165285937500000} a^{10} + \frac{1831874710321260111794672981}{9223242172021633057187500000} a^{9} + \frac{78662044635629606948926759}{1844648434404326611437500000} a^{8} - \frac{27215927890308360824167691}{368929686880865322287500000} a^{7} - \frac{14595431041910124838839741}{29514374950469225783000000} a^{6} + \frac{2677850829030012916441057}{5902874990093845156600000} a^{5} - \frac{142984723124973762391201}{1180574998018769031320000} a^{4} + \frac{446883210013151775283699}{1180574998018769031320000} a^{3} + \frac{24504193116498994754893}{236114999603753806264000} a^{2} - \frac{18364547851132251004793}{47222999920750761252800} a + \frac{146997679893319679187}{9444599984150152250560}$, $\frac{1}{461162108601081652859375000000} a^{35} - \frac{1}{461162108601081652859375000000} a^{34} - \frac{7}{461162108601081652859375000000} a^{33} - \frac{119}{461162108601081652859375000000} a^{32} - \frac{447}{461162108601081652859375000000} a^{31} - \frac{214559847190920032897}{461162108601081652859375000000} a^{30} - \frac{7898651273696746493699}{461162108601081652859375000000} a^{29} - \frac{8383374967400722313271}{115290527150270413214843750000} a^{28} + \frac{1541464823185861585944}{7205657946891900825927734375} a^{27} - \frac{452360849155797457951329}{115290527150270413214843750000} a^{26} + \frac{65280162512794504815526149}{14411315893783801651855468750} a^{25} + \frac{140167833335508830764287829}{23058105430054082642968750000} a^{24} - \frac{43764947320539502448642577}{57645263575135206607421875000} a^{23} - \frac{115488340574061226595307739}{23058105430054082642968750000} a^{22} - \frac{754969728531316093908363801}{230581054300540826429687500000} a^{21} - \frac{784366771413784544646872507}{230581054300540826429687500000} a^{20} - \frac{3370113383886815469413400427}{230581054300540826429687500000} a^{19} + \frac{23840297797401184225112203827}{230581054300540826429687500000} a^{18} + \frac{3924369618472918945777525633}{230581054300540826429687500000} a^{17} + \frac{3274241065061542927626920001}{230581054300540826429687500000} a^{16} - \frac{7943616212181130993205124491}{230581054300540826429687500000} a^{15} - \frac{1561802099451925394472079827}{57645263575135206607421875000} a^{14} - \frac{12416991440528766528354681469}{57645263575135206607421875000} a^{13} + \frac{10488878171807174960834942579}{115290527150270413214843750000} a^{12} - \frac{8783059033880895661803121843}{57645263575135206607421875000} a^{11} - \frac{3531516665749320291588892947}{23058105430054082642968750000} a^{10} - \frac{124143809943437891065336401}{576452635751352066074218750} a^{9} - \frac{55912981536374777224745443}{922324217202163305718750000} a^{8} + \frac{100466978638034269284921737}{737859373761730644575000000} a^{7} + \frac{13883500574518979097109559}{29514374950469225783000000} a^{6} + \frac{913823689909852172759129}{5902874990093845156600000} a^{5} + \frac{2593187486525221411614589}{5902874990093845156600000} a^{4} + \frac{364219174104026906696453}{1180574998018769031320000} a^{3} - \frac{98208804840968442846613}{236114999603753806264000} a^{2} + \frac{20050177491391731470057}{47222999920750761252800} a + \frac{486470017179084756067}{2361149996037538062640}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{8036833969378802283}{11529052715027041321484375000} a^{35} - \frac{2678944656459600761}{1441131589378380165185546875} a^{34} - \frac{18752612595217205327}{23058105430054082642968750000} a^{33} + \frac{224765211419218911641}{23058105430054082642968750000} a^{32} + \frac{222352406486146863163}{23058105430054082642968750000} a^{31} - \frac{85726229006707224352}{1441131589378380165185546875} a^{30} - \frac{3078107410272081274389}{23058105430054082642968750000} a^{29} + \frac{13643865135348746675773}{11529052715027041321484375000} a^{28} - \frac{15106568917775688691279}{23058105430054082642968750000} a^{27} - \frac{105231625050389577492841}{23058105430054082642968750000} a^{26} - \frac{794788929590751015697}{1441131589378380165185546875} a^{25} + \frac{33516276596966065120871}{922324217202163305718750000} a^{24} + \frac{712141179081999210695869}{23058105430054082642968750000} a^{23} - \frac{1486891973730115750777069}{4611621086010816528593750000} a^{22} + \frac{28229146249257931033771293}{23058105430054082642968750000} a^{21} - \frac{35405438900310154521919829}{23058105430054082642968750000} a^{20} + \frac{25203755111935661783157251}{23058105430054082642968750000} a^{19} + \frac{25752657110529350965594627}{11529052715027041321484375000} a^{18} - \frac{179262034428641126594843609}{23058105430054082642968750000} a^{17} + \frac{354950327217173924718904847}{23058105430054082642968750000} a^{16} - \frac{444681069485803524884455257}{23058105430054082642968750000} a^{15} + \frac{186711188308732386258903039}{23058105430054082642968750000} a^{14} + \frac{903161246270852973849182703}{23058105430054082642968750000} a^{13} - \frac{3020789672143603325503317199}{23058105430054082642968750000} a^{12} + \frac{305733449498684038199313771}{1441131589378380165185546875} a^{11} - \frac{128688376057176178589331887}{922324217202163305718750000} a^{10} - \frac{4355430901416675376834561}{184464843440432661143750000} a^{9} + \frac{1741606031666294591132949}{7378593737617306445750000} a^{8} + \frac{1627515136636993113923481}{7378593737617306445750000} a^{7} + \frac{69644524233980240983717}{1475718747523461289150000} a^{6} + \frac{17142566856684985269639}{73785937376173064457500} a^{5} + \frac{174139464759890353642421}{295143749504692257830000} a^{4} + \frac{608120437016329372747}{5902874990093845156600} a^{3} - \frac{233068185111985266207}{2361149996037538062640} a^{2} - \frac{77689395037328422069}{236114999603753806264} a - \frac{120552509540682034245}{472229999207507612528} \) (order $14$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{273}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-7}) \), 3.3.169.1, 3.3.8281.1, \(\Q(\zeta_{7})^+\), 3.3.8281.2, \(\Q(\sqrt{-7}, \sqrt{-39})\), 6.6.3438544473.1, 6.6.168488679177.2, 6.6.996974433.1, 6.6.168488679177.1, 6.0.10024911.1, 6.0.9796423.1, 6.0.24069811311.1, 6.0.480024727.1, 6.0.142424919.1, \(\Q(\zeta_{7})\), 6.0.24069811311.2, 6.0.480024727.2, 9.9.567869252041.1, 12.0.11823588092798847729.1, 12.0.28388435010810033397329.4, 12.0.993958020055671489.3, 12.0.28388435010810033397329.2, 18.18.4783129918873486243975221249718233.1, 18.0.13944985186220076513047292273231.1, 18.0.110609092182866440454328583.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{12}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$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}$