Properties

Label 42.0.447...375.1
Degree $42$
Signature $[0, 21]$
Discriminant $-4.473\times 10^{81}$
Root discriminant $87.91$
Ramified primes $5, 43$
Class number not computed
Class group not computed
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - x^41 + 44*x^40 - 44*x^39 + 904*x^38 - 904*x^37 + 11525*x^36 - 11525*x^35 + 102212*x^34 - 102212*x^33 + 670199*x^32 - 670199*x^31 + 3371975*x^30 - 3371975*x^29 + 13342815*x^28 - 13342815*x^27 + 42258251*x^26 - 42258251*x^25 + 108593663*x^24 - 108593663*x^23 + 229203503*x^22 - 229203503*x^21 + 402580148*x^20 - 402580148*x^19 + 598327973*x^18 - 598327973*x^17 + 769983758*x^16 - 769983758*x^15 + 884984678*x^14 - 884984678*x^13 + 942485138*x^12 - 942485138*x^11 + 963249193*x^10 - 963249193*x^9 + 968416718*x^8 - 968416718*x^7 + 969243522*x^6 - 969243522*x^5 + 969319675*x^4 - 969319675*x^3 + 969322986*x^2 - 969322986*x + 969323029)
 
gp: K = bnfinit(x^42 - x^41 + 44*x^40 - 44*x^39 + 904*x^38 - 904*x^37 + 11525*x^36 - 11525*x^35 + 102212*x^34 - 102212*x^33 + 670199*x^32 - 670199*x^31 + 3371975*x^30 - 3371975*x^29 + 13342815*x^28 - 13342815*x^27 + 42258251*x^26 - 42258251*x^25 + 108593663*x^24 - 108593663*x^23 + 229203503*x^22 - 229203503*x^21 + 402580148*x^20 - 402580148*x^19 + 598327973*x^18 - 598327973*x^17 + 769983758*x^16 - 769983758*x^15 + 884984678*x^14 - 884984678*x^13 + 942485138*x^12 - 942485138*x^11 + 963249193*x^10 - 963249193*x^9 + 968416718*x^8 - 968416718*x^7 + 969243522*x^6 - 969243522*x^5 + 969319675*x^4 - 969319675*x^3 + 969322986*x^2 - 969322986*x + 969323029, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![969323029, -969322986, 969322986, -969319675, 969319675, -969243522, 969243522, -968416718, 968416718, -963249193, 963249193, -942485138, 942485138, -884984678, 884984678, -769983758, 769983758, -598327973, 598327973, -402580148, 402580148, -229203503, 229203503, -108593663, 108593663, -42258251, 42258251, -13342815, 13342815, -3371975, 3371975, -670199, 670199, -102212, 102212, -11525, 11525, -904, 904, -44, 44, -1, 1]);
 

\( x^{42} - x^{41} + 44 x^{40} - 44 x^{39} + 904 x^{38} - 904 x^{37} + 11525 x^{36} - 11525 x^{35} + 102212 x^{34} - 102212 x^{33} + 670199 x^{32} - 670199 x^{31} + 3371975 x^{30} - 3371975 x^{29} + 13342815 x^{28} - 13342815 x^{27} + 42258251 x^{26} - 42258251 x^{25} + 108593663 x^{24} - 108593663 x^{23} + 229203503 x^{22} - 229203503 x^{21} + 402580148 x^{20} - 402580148 x^{19} + 598327973 x^{18} - 598327973 x^{17} + 769983758 x^{16} - 769983758 x^{15} + 884984678 x^{14} - 884984678 x^{13} + 942485138 x^{12} - 942485138 x^{11} + 963249193 x^{10} - 963249193 x^{9} + 968416718 x^{8} - 968416718 x^{7} + 969243522 x^{6} - 969243522 x^{5} + 969319675 x^{4} - 969319675 x^{3} + 969322986 x^{2} - 969322986 x + 969323029 \)

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-44\!\cdots\!375\)\(\medspace = -\,5^{21}\cdot 43^{41}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $87.91$
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:  $5, 43$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $42$
This field is Galois and abelian over $\Q$.
Conductor:  \(215=5\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{215}(1,·)$, $\chi_{215}(134,·)$, $\chi_{215}(11,·)$, $\chi_{215}(66,·)$, $\chi_{215}(16,·)$, $\chi_{215}(146,·)$, $\chi_{215}(19,·)$, $\chi_{215}(21,·)$, $\chi_{215}(81,·)$, $\chi_{215}(29,·)$, $\chi_{215}(31,·)$, $\chi_{215}(34,·)$, $\chi_{215}(36,·)$, $\chi_{215}(6,·)$, $\chi_{215}(39,·)$, $\chi_{215}(41,·)$, $\chi_{215}(174,·)$, $\chi_{215}(176,·)$, $\chi_{215}(179,·)$, $\chi_{215}(181,·)$, $\chi_{215}(56,·)$, $\chi_{215}(186,·)$, $\chi_{215}(159,·)$, $\chi_{215}(194,·)$, $\chi_{215}(196,·)$, $\chi_{215}(69,·)$, $\chi_{215}(199,·)$, $\chi_{215}(204,·)$, $\chi_{215}(184,·)$, $\chi_{215}(214,·)$, $\chi_{215}(89,·)$, $\chi_{215}(94,·)$, $\chi_{215}(96,·)$, $\chi_{215}(101,·)$, $\chi_{215}(209,·)$, $\chi_{215}(104,·)$, $\chi_{215}(111,·)$, $\chi_{215}(114,·)$, $\chi_{215}(119,·)$, $\chi_{215}(121,·)$, $\chi_{215}(126,·)$$\chi_{215}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{433494437} a^{22} + \frac{165580141}{433494437} a^{21} + \frac{22}{433494437} a^{20} + \frac{9227465}{433494437} a^{19} + \frac{209}{433494437} a^{18} + \frac{83047185}{433494437} a^{17} + \frac{1122}{433494437} a^{16} - \frac{159680836}{433494437} a^{15} + \frac{3740}{433494437} a^{14} - \frac{8638211}{433494437} a^{13} + \frac{8008}{433494437} a^{12} - \frac{81868677}{433494437} a^{11} + \frac{11011}{433494437} a^{10} + \frac{188934575}{433494437} a^{9} + \frac{9438}{433494437} a^{8} + \frac{156502726}{433494437} a^{7} + \frac{4719}{433494437} a^{6} + \frac{46530161}{433494437} a^{5} + \frac{1210}{433494437} a^{4} + \frac{24672046}{433494437} a^{3} + \frac{121}{433494437} a^{2} + \frac{9227465}{433494437} a + \frac{2}{433494437}$, $\frac{1}{433494437} a^{23} + \frac{23}{433494437} a^{21} - \frac{165580141}{433494437} a^{20} + \frac{230}{433494437} a^{19} + \frac{156352676}{433494437} a^{18} + \frac{1311}{433494437} a^{17} + \frac{28514435}{433494437} a^{16} + \frac{4692}{433494437} a^{15} + \frac{185184922}{433494437} a^{14} + \frac{10948}{433494437} a^{13} + \frac{11844978}{433494437} a^{12} + \frac{16744}{433494437} a^{11} - \frac{169890391}{433494437} a^{10} + \frac{16445}{433494437} a^{9} + \frac{158577353}{433494437} a^{8} + \frac{9867}{433494437} a^{7} - \frac{169179744}{433494437} a^{6} + \frac{3289}{433494437} a^{5} - \frac{52868670}{433494437} a^{4} + \frac{506}{433494437} a^{3} - \frac{85225494}{433494437} a^{2} + \frac{23}{433494437} a + \frac{102334155}{433494437}$, $\frac{1}{433494437} a^{24} - \frac{72473451}{433494437} a^{21} - \frac{276}{433494437} a^{20} - \frac{55879019}{433494437} a^{19} - \frac{3496}{433494437} a^{18} - \frac{147593072}{433494437} a^{17} - \frac{21114}{433494437} a^{16} - \frac{43605783}{433494437} a^{15} - \frac{75072}{433494437} a^{14} + \frac{210523831}{433494437} a^{13} - \frac{167440}{433494437} a^{12} - \frac{20888568}{433494437} a^{11} - \frac{236808}{433494437} a^{10} + \frac{148026498}{433494437} a^{9} - \frac{207207}{433494437} a^{8} + \frac{132707491}{433494437} a^{7} - \frac{105248}{433494437} a^{6} + \frac{177420938}{433494437} a^{5} - \frac{27324}{433494437} a^{4} + \frac{214306322}{433494437} a^{3} - \frac{2760}{433494437} a^{2} - \frac{109897540}{433494437} a - \frac{46}{433494437}$, $\frac{1}{433494437} a^{25} - \frac{300}{433494437} a^{21} - \frac{195440845}{433494437} a^{20} - \frac{4000}{433494437} a^{19} - \frac{172947108}{433494437} a^{18} - \frac{25650}{433494437} a^{17} + \frac{208146520}{433494437} a^{16} - \frac{97920}{433494437} a^{15} - \frac{106286991}{433494437} a^{14} - \frac{238000}{433494437} a^{13} - \frac{102544103}{433494437} a^{12} - \frac{374400}{433494437} a^{11} + \frac{89936942}{433494437} a^{10} - \frac{375375}{433494437} a^{9} + \frac{82916443}{433494437} a^{8} - \frac{228800}{433494437} a^{7} + \frac{152525414}{433494437} a^{6} - \frac{77220}{433494437} a^{5} - \frac{92188679}{433494437} a^{4} - \frac{12000}{433494437} a^{3} - \frac{10498709}{433494437} a^{2} - \frac{550}{433494437} a + \frac{144946902}{433494437}$, $\frac{1}{433494437} a^{26} + \frac{60235637}{433494437} a^{21} + \frac{2600}{433494437} a^{20} - \frac{5674230}{433494437} a^{19} + \frac{37050}{433494437} a^{18} - \frac{20375326}{433494437} a^{17} + \frac{238680}{433494437} a^{16} + \frac{107344716}{433494437} a^{15} + \frac{884000}{433494437} a^{14} - \frac{93040781}{433494437} a^{13} + \frac{2028000}{433494437} a^{12} - \frac{194977686}{433494437} a^{11} + \frac{2927925}{433494437} a^{10} - \frac{24482304}{433494437} a^{9} + \frac{2602600}{433494437} a^{8} - \frac{147550419}{433494437} a^{7} + \frac{1338480}{433494437} a^{6} - \frac{4962363}{433494437} a^{5} + \frac{351000}{433494437} a^{4} + \frac{21709662}{433494437} a^{3} + \frac{35750}{433494437} a^{2} - \frac{121274657}{433494437} a + \frac{600}{433494437}$, $\frac{1}{433494437} a^{27} + \frac{2925}{433494437} a^{21} - \frac{30374933}{433494437} a^{20} + \frac{43875}{433494437} a^{19} - \frac{38284786}{433494437} a^{18} + \frac{300105}{433494437} a^{17} + \frac{148092174}{433494437} a^{16} + \frac{1193400}{433494437} a^{15} + \frac{42784079}{433494437} a^{14} + \frac{2983500}{433494437} a^{13} - \frac{82650401}{433494437} a^{12} + \frac{4791150}{433494437} a^{11} - \frac{32592701}{433494437} a^{10} + \frac{4879875}{433494437} a^{9} + \frac{93208919}{433494437} a^{8} + \frac{3011580}{433494437} a^{7} + \frac{115417306}{433494437} a^{6} + \frac{1026675}{433494437} a^{5} - \frac{36345692}{433494437} a^{4} + \frac{160875}{433494437} a^{3} - \frac{40381305}{433494437} a^{2} + \frac{7425}{433494437} a - \frac{120471274}{433494437}$, $\frac{1}{433494437} a^{28} - \frac{139001229}{433494437} a^{21} - \frac{20475}{433494437} a^{20} - \frac{151964817}{433494437} a^{19} - \frac{311220}{433494437} a^{18} - \frac{8039231}{433494437} a^{17} - \frac{2088450}{433494437} a^{16} - \frac{197773707}{433494437} a^{15} - \frac{7956000}{433494437} a^{14} + \frac{41439428}{433494437} a^{13} - \frac{18632250}{433494437} a^{12} + \frac{144358300}{433494437} a^{11} - \frac{27327300}{433494437} a^{10} + \frac{164984219}{433494437} a^{9} - \frac{24594570}{433494437} a^{8} + \frac{115069228}{433494437} a^{7} - \frac{12776400}{433494437} a^{6} - \frac{19813399}{433494437} a^{5} - \frac{3378375}{433494437} a^{4} + \frac{187455124}{433494437} a^{3} - \frac{346500}{433494437} a^{2} + \frac{199343132}{433494437} a - \frac{5850}{433494437}$, $\frac{1}{433494437} a^{29} - \frac{23751}{433494437} a^{21} - \frac{128398838}{433494437} a^{20} - \frac{380016}{433494437} a^{19} - \frac{909649}{433494437} a^{18} - \frac{2707614}{433494437} a^{17} + \frac{137102348}{433494437} a^{16} - \frac{11074752}{433494437} a^{15} + \frac{146205925}{433494437} a^{14} - \frac{28263690}{433494437} a^{13} + \frac{52485916}{433494437} a^{12} - \frac{46108608}{433494437} a^{11} + \frac{38659691}{433494437} a^{10} - \frac{47549502}{433494437} a^{9} - \frac{178992269}{433494437} a^{8} - \frac{29641248}{433494437} a^{7} + \frac{49903071}{433494437} a^{6} - \frac{10189179}{433494437} a^{5} + \frac{183100658}{433494437} a^{4} - \frac{1607760}{433494437} a^{3} + \frac{112208798}{433494437} a^{2} - \frac{74646}{433494437} a - \frac{155491979}{433494437}$, $\frac{1}{433494437} a^{30} - \frac{96002411}{433494437} a^{21} + \frac{142506}{433494437} a^{20} - \frac{187573556}{433494437} a^{19} + \frac{2256345}{433494437} a^{18} + \frac{191104933}{433494437} a^{17} + \frac{15573870}{433494437} a^{16} + \frac{209499402}{433494437} a^{15} + \frac{60565050}{433494437} a^{14} - \frac{70794844}{433494437} a^{13} + \frac{144089400}{433494437} a^{12} - \frac{201737791}{433494437} a^{11} + \frac{213972759}{433494437} a^{10} + \frac{105181169}{433494437} a^{9} + \frac{194520690}{433494437} a^{8} - \frac{68648978}{433494437} a^{7} + \frac{101891790}{433494437} a^{6} - \frac{89859781}{433494437} a^{5} + \frac{27130950}{433494437} a^{4} + \frac{13494520}{433494437} a^{3} + \frac{2799225}{433494437} a^{2} + \frac{91338551}{433494437} a + \frac{47502}{433494437}$, $\frac{1}{433494437} a^{31} + \frac{169911}{433494437} a^{21} + \frac{190501738}{433494437} a^{20} + \frac{2831850}{433494437} a^{19} - \frac{118629707}{433494437} a^{18} + \frac{20753415}{433494437} a^{17} - \frac{15910269}{433494437} a^{16} + \frac{86654610}{433494437} a^{15} + \frac{44828460}{433494437} a^{14} - \frac{208834337}{433494437} a^{13} - \frac{67304}{433494437} a^{12} - \frac{62408813}{433494437} a^{11} - \frac{105203153}{433494437} a^{10} - \frac{46946912}{433494437} a^{9} - \frac{1267290}{433494437} a^{8} - \frac{190521707}{433494437} a^{7} - \frac{56168937}{433494437} a^{6} + \frac{84105945}{433494437} a^{5} - \frac{97286}{433494437} a^{4} + \frac{13350150}{433494437} a^{3} + \frac{3280483}{433494437} a^{2} + \frac{623007}{433494437} a + \frac{192004822}{433494437}$, $\frac{1}{433494437} a^{32} + \frac{92125587}{433494437} a^{21} - \frac{906192}{433494437} a^{20} - \frac{17056693}{433494437} a^{19} - \frac{14757984}{433494437} a^{18} + \frac{31257983}{433494437} a^{17} - \frac{103985532}{433494437} a^{16} + \frac{25531100}{433494437} a^{15} + \frac{22687397}{433494437} a^{14} - \frac{85161765}{433494437} a^{13} - \frac{122572790}{433494437} a^{12} - \frac{119414299}{433494437} a^{11} - \frac{183859185}{433494437} a^{10} - \frac{66802517}{433494437} a^{9} - \frac{60163977}{433494437} a^{8} - \frac{175091869}{433494437} a^{7} + \frac{149284810}{433494437} a^{6} + \frac{85259049}{433494437} a^{5} - \frac{192242160}{433494437} a^{4} - \frac{157521633}{433494437} a^{3} - \frac{19936224}{433494437} a^{2} - \frac{139916601}{433494437} a - \frac{339822}{433494437}$, $\frac{1}{433494437} a^{33} - \frac{1107568}{433494437} a^{21} + \frac{123652578}{433494437} a^{20} - \frac{18986880}{433494437} a^{19} - \frac{149234472}{433494437} a^{18} - \frac{142045596}{433494437} a^{17} - \frac{167701508}{433494437} a^{16} - \frac{169022555}{433494437} a^{15} - \frac{6779730}{433494437} a^{14} + \frac{152370644}{433494437} a^{13} - \frac{53583221}{433494437} a^{12} - \frac{37864482}{433494437} a^{11} - \frac{84658394}{433494437} a^{10} - \frac{170722298}{433494437} a^{9} - \frac{66541353}{433494437} a^{8} - \frac{20409964}{433494437} a^{7} + \frac{139534307}{433494437} a^{6} - \frac{177408427}{433494437} a^{5} + \frac{212082843}{433494437} a^{4} - \frac{97465984}{433494437} a^{3} - \frac{16257266}{433494437} a^{2} - \frac{4568718}{433494437} a - \frac{184251174}{433494437}$, $\frac{1}{433494437} a^{34} - \frac{166290932}{433494437} a^{21} + \frac{5379616}{433494437} a^{20} - \frac{169126064}{433494437} a^{19} + \frac{89436116}{433494437} a^{18} + \frac{86768601}{433494437} a^{17} + \frac{206679867}{433494437} a^{16} + \frac{102955119}{433494437} a^{15} - \frac{40269406}{433494437} a^{14} + \frac{196054958}{433494437} a^{13} + \frac{161651322}{433494437} a^{12} + \frac{120364671}{433494437} a^{11} - \frac{113135286}{433494437} a^{10} + \frac{87710296}{433494437} a^{9} + \frac{28950332}{433494437} a^{8} + \frac{31691418}{433494437} a^{7} - \frac{152728279}{433494437} a^{6} - \frac{23207017}{433494437} a^{5} - \frac{57792015}{433494437} a^{4} + \frac{197056130}{433494437} a^{3} + \frac{129447010}{433494437} a^{2} - \frac{204142766}{433494437} a + \frac{2215136}{433494437}$, $\frac{1}{433494437} a^{35} + \frac{6724520}{433494437} a^{21} + \frac{21318944}{433494437} a^{20} + \frac{117679100}{433494437} a^{19} + \frac{162018429}{433494437} a^{18} + \frac{27372286}{433494437} a^{17} - \frac{154721524}{433494437} a^{16} - \frac{60404109}{433494437} a^{15} + \frac{59623543}{433494437} a^{14} + \frac{214219149}{433494437} a^{13} + \frac{83237663}{433494437} a^{12} - \frac{205700520}{433494437} a^{11} + \frac{36660660}{433494437} a^{10} - \frac{84184954}{433494437} a^{9} - \frac{197848743}{433494437} a^{8} - \frac{165073479}{433494437} a^{7} + \frac{78770121}{433494437} a^{6} + \frac{137296779}{433494437} a^{5} - \frac{165829355}{433494437} a^{4} + \frac{213740613}{433494437} a^{3} - \frac{23684096}{433494437} a^{2} + \frac{30458120}{433494437} a - \frac{100912573}{433494437}$, $\frac{1}{433494437} a^{36} - \frac{14215270}{433494437} a^{21} - \frac{30260340}{433494437} a^{20} - \frac{150705628}{433494437} a^{19} - \frac{77569083}{433494437} a^{18} + \frac{65223022}{433494437} a^{17} + \frac{197584317}{433494437} a^{16} + \frac{43617901}{433494437} a^{15} + \frac{207191695}{433494437} a^{14} + \frac{84807820}{433494437} a^{13} + \frac{131147945}{433494437} a^{12} + \frac{61397188}{433494437} a^{11} - \frac{325947}{433494437} a^{10} + \frac{73215034}{433494437} a^{9} + \frac{92589000}{433494437} a^{8} + \frac{83288863}{433494437} a^{7} + \frac{49380800}{433494437} a^{6} + \frac{86088466}{433494437} a^{5} - \frac{120028721}{433494437} a^{4} + \frac{167465498}{433494437} a^{3} + \frac{83780074}{433494437} a^{2} + \frac{19857807}{433494437} a - \frac{13449040}{433494437}$, $\frac{1}{433494437} a^{37} - \frac{38608020}{433494437} a^{21} + \frac{162030312}{433494437} a^{20} + \frac{180624074}{433494437} a^{19} + \frac{1753393}{433494437} a^{18} - \frac{79643892}{433494437} a^{17} - \frac{46143328}{433494437} a^{16} + \frac{63100201}{433494437} a^{15} - \frac{69898131}{433494437} a^{14} - \frac{135342783}{433494437} a^{13} - \frac{111757583}{433494437} a^{12} - \frac{173187317}{433494437} a^{11} + \frac{106061247}{433494437} a^{10} + \frac{122144235}{433494437} a^{9} - \frac{136268347}{433494437} a^{8} - \frac{8526577}{433494437} a^{7} - \frac{23690139}{433494437} a^{6} - \frac{135077961}{433494437} a^{5} + \frac{28164718}{433494437} a^{4} - \frac{95615667}{433494437} a^{3} + \frac{5927729}{433494437} a^{2} - \frac{188750320}{433494437} a + \frac{28430540}{433494437}$, $\frac{1}{433494437} a^{38} - \frac{97344892}{433494437} a^{21} + \frac{163011640}{433494437} a^{20} + \frac{190301790}{433494437} a^{19} + \frac{186532422}{433494437} a^{18} - \frac{83185503}{433494437} a^{17} + \frac{31854941}{433494437} a^{16} + \frac{96736751}{433494437} a^{15} - \frac{94995504}{433494437} a^{14} - \frac{158142660}{433494437} a^{13} - \frac{81696738}{433494437} a^{12} - \frac{6064375}{433494437} a^{11} - \frac{22990242}{433494437} a^{10} + \frac{63851566}{433494437} a^{9} - \frac{194855334}{433494437} a^{8} - \frac{124374371}{433494437} a^{7} - \frac{11495121}{433494437} a^{6} + \frac{54195230}{433494437} a^{5} - \frac{197310663}{433494437} a^{4} - \frac{149805301}{433494437} a^{3} + \frac{147875730}{433494437} a^{2} - \frac{216515500}{433494437} a + \frac{77216040}{433494437}$, $\frac{1}{433494437} a^{39} + \frac{211915132}{433494437} a^{21} + \frac{164417229}{433494437} a^{20} - \frac{86977557}{433494437} a^{19} - \frac{112341614}{433494437} a^{18} + \frac{171231752}{433494437} a^{17} + \frac{77107451}{433494437} a^{16} + \frac{77224121}{433494437} a^{15} + \frac{209920777}{433494437} a^{14} + \frac{207911095}{433494437} a^{13} + \frac{108833035}{433494437} a^{12} - \frac{130171345}{433494437} a^{11} - \frac{103285323}{433494437} a^{10} + \frac{11307880}{433494437} a^{9} + \frac{42004322}{433494437} a^{8} - \frac{116586202}{433494437} a^{7} - \frac{79362642}{433494437} a^{6} + \frac{35350947}{433494437} a^{5} + \frac{160521592}{433494437} a^{4} - \frac{98035078}{433494437} a^{3} - \frac{142133367}{433494437} a^{2} - \frac{196293939}{433494437} a + \frac{194689784}{433494437}$, $\frac{1}{433494437} a^{40} - \frac{136245634}{433494437} a^{21} + \frac{19328346}{433494437} a^{20} + \frac{76154818}{433494437} a^{19} + \frac{97401738}{433494437} a^{18} + \frac{39597591}{433494437} a^{17} - \frac{136602507}{433494437} a^{16} + \frac{85203621}{433494437} a^{15} + \frac{73148251}{433494437} a^{14} + \frac{55172858}{433494437} a^{13} - \frac{15827546}{433494437} a^{12} + \frac{203897047}{433494437} a^{11} + \frac{114343799}{433494437} a^{10} + \frac{80287101}{433494437} a^{9} - \frac{28269700}{433494437} a^{8} + \frac{116853161}{433494437} a^{7} + \frac{79509198}{433494437} a^{6} + \frac{181407569}{433494437} a^{5} + \frac{113361906}{433494437} a^{4} - \frac{154320943}{433494437} a^{3} + \frac{171641309}{433494437} a^{2} - \frac{50308221}{433494437} a + \frac{9664173}{433494437}$, $\frac{1}{433494437} a^{41} + \frac{179383903}{433494437} a^{21} + \frac{39097707}{433494437} a^{20} - \frac{9387061}{433494437} a^{19} - \frac{95697745}{433494437} a^{18} + \frac{202781613}{433494437} a^{17} - \frac{70731292}{433494437} a^{16} - \frac{144003309}{433494437} a^{15} - \frac{175613894}{433494437} a^{14} + \frac{206045889}{433494437} a^{13} + \frac{153436190}{433494437} a^{12} + \frac{5128951}{433494437} a^{11} - \frac{43283382}{433494437} a^{10} + \frac{30909680}{433494437} a^{9} - \frac{174847726}{433494437} a^{8} - \frac{27408103}{433494437} a^{7} - \frac{181190093}{433494437} a^{6} - \frac{54862707}{433494437} a^{5} - \frac{24989863}{433494437} a^{4} - \frac{196673737}{433494437} a^{3} - \frac{37375113}{433494437} a^{2} - \frac{97124626}{433494437} a - \frac{161003169}{433494437}$

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $20$
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:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$C_{42}$ (as 42T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{-215}) \), 3.3.1849.1, 6.0.18376055375.1, 7.7.6321363049.1, 14.0.134239384709553967597109375.1, \(\Q(\zeta_{43})^+\)

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{6}$ $21^{2}$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{14}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{6}$ $42$ $42$ $42$ $42$ $42$ $21^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{14}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }^{3}$ $42$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{6}$

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
5Data not computed
43Data not computed