Properties

Label 42.0.563...875.1
Degree $42$
Signature $[0, 21]$
Discriminant $-5.635\times 10^{79}$
Root discriminant $79.22$
Ramified primes $5, 7$
Class number not computed
Class group not computed
Galois group $C_{42}$ (as 42T1)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 + 42*x^40 + 819*x^38 + 9842*x^36 - 29*x^35 + 81585*x^34 - 1015*x^33 + 494802*x^32 - 16240*x^31 + 2272424*x^30 - 157325*x^29 + 8070266*x^28 - 1030225*x^27 + 22451828*x^26 - 4821453*x^25 + 49380100*x^24 - 16625700*x^23 + 86841307*x^22 - 42945897*x^21 + 125155604*x^20 - 83971762*x^19 + 155582203*x^18 - 126598108*x^17 + 178243674*x^16 - 155985200*x^15 + 191428385*x^14 - 177440270*x^13 + 185472266*x^12 - 199822441*x^11 + 177861992*x^10 - 194249279*x^9 + 208189352*x^8 - 152082148*x^7 + 226081541*x^6 - 195873685*x^5 + 141819720*x^4 - 297986948*x^3 + 34945918*x^2 - 144775946*x + 599786069)
 
gp: K = bnfinit(x^42 + 42*x^40 + 819*x^38 + 9842*x^36 - 29*x^35 + 81585*x^34 - 1015*x^33 + 494802*x^32 - 16240*x^31 + 2272424*x^30 - 157325*x^29 + 8070266*x^28 - 1030225*x^27 + 22451828*x^26 - 4821453*x^25 + 49380100*x^24 - 16625700*x^23 + 86841307*x^22 - 42945897*x^21 + 125155604*x^20 - 83971762*x^19 + 155582203*x^18 - 126598108*x^17 + 178243674*x^16 - 155985200*x^15 + 191428385*x^14 - 177440270*x^13 + 185472266*x^12 - 199822441*x^11 + 177861992*x^10 - 194249279*x^9 + 208189352*x^8 - 152082148*x^7 + 226081541*x^6 - 195873685*x^5 + 141819720*x^4 - 297986948*x^3 + 34945918*x^2 - 144775946*x + 599786069, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![599786069, -144775946, 34945918, -297986948, 141819720, -195873685, 226081541, -152082148, 208189352, -194249279, 177861992, -199822441, 185472266, -177440270, 191428385, -155985200, 178243674, -126598108, 155582203, -83971762, 125155604, -42945897, 86841307, -16625700, 49380100, -4821453, 22451828, -1030225, 8070266, -157325, 2272424, -16240, 494802, -1015, 81585, -29, 9842, 0, 819, 0, 42, 0, 1]);
 

\( x^{42} + 42 x^{40} + 819 x^{38} + 9842 x^{36} - 29 x^{35} + 81585 x^{34} - 1015 x^{33} + 494802 x^{32} - 16240 x^{31} + 2272424 x^{30} - 157325 x^{29} + 8070266 x^{28} - 1030225 x^{27} + 22451828 x^{26} - 4821453 x^{25} + 49380100 x^{24} - 16625700 x^{23} + 86841307 x^{22} - 42945897 x^{21} + 125155604 x^{20} - 83971762 x^{19} + 155582203 x^{18} - 126598108 x^{17} + 178243674 x^{16} - 155985200 x^{15} + 191428385 x^{14} - 177440270 x^{13} + 185472266 x^{12} - 199822441 x^{11} + 177861992 x^{10} - 194249279 x^{9} + 208189352 x^{8} - 152082148 x^{7} + 226081541 x^{6} - 195873685 x^{5} + 141819720 x^{4} - 297986948 x^{3} + 34945918 x^{2} - 144775946 x + 599786069 \)

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:  \(-56\!\cdots\!875\)\(\medspace = -\,5^{21}\cdot 7^{77}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $79.22$
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:  $5, 7$
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:  \(245=5\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{245}(1,·)$, $\chi_{245}(129,·)$, $\chi_{245}(11,·)$, $\chi_{245}(141,·)$, $\chi_{245}(16,·)$, $\chi_{245}(19,·)$, $\chi_{245}(46,·)$, $\chi_{245}(151,·)$, $\chi_{245}(24,·)$, $\chi_{245}(156,·)$, $\chi_{245}(159,·)$, $\chi_{245}(34,·)$, $\chi_{245}(36,·)$, $\chi_{245}(71,·)$, $\chi_{245}(174,·)$, $\chi_{245}(176,·)$, $\chi_{245}(51,·)$, $\chi_{245}(54,·)$, $\chi_{245}(116,·)$, $\chi_{245}(186,·)$, $\chi_{245}(59,·)$, $\chi_{245}(191,·)$, $\chi_{245}(194,·)$, $\chi_{245}(139,·)$, $\chi_{245}(69,·)$, $\chi_{245}(199,·)$, $\chi_{245}(81,·)$, $\chi_{245}(211,·)$, $\chi_{245}(86,·)$, $\chi_{245}(164,·)$, $\chi_{245}(221,·)$, $\chi_{245}(94,·)$, $\chi_{245}(226,·)$, $\chi_{245}(229,·)$, $\chi_{245}(209,·)$, $\chi_{245}(104,·)$, $\chi_{245}(106,·)$, $\chi_{245}(244,·)$, $\chi_{245}(89,·)$, $\chi_{245}(121,·)$, $\chi_{245}(124,·)$$\chi_{245}(234,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{13} a^{21} - \frac{5}{13} a^{19} - \frac{6}{13} a^{17} + \frac{3}{13} a^{15} + \frac{5}{13} a^{14} + \frac{2}{13} a^{13} + \frac{5}{13} a^{12} - \frac{5}{13} a^{10} - \frac{3}{13} a^{8} - \frac{1}{13} a^{7} + \frac{1}{13} a^{6} + \frac{5}{13} a^{5} + \frac{5}{13} a^{4} - \frac{6}{13} a^{3} - \frac{2}{13} a^{2} + \frac{1}{13} a + \frac{5}{13}$, $\frac{1}{13} a^{22} - \frac{5}{13} a^{20} - \frac{6}{13} a^{18} + \frac{3}{13} a^{16} + \frac{5}{13} a^{15} + \frac{2}{13} a^{14} + \frac{5}{13} a^{13} - \frac{5}{13} a^{11} - \frac{3}{13} a^{9} - \frac{1}{13} a^{8} + \frac{1}{13} a^{7} + \frac{5}{13} a^{6} + \frac{5}{13} a^{5} - \frac{6}{13} a^{4} - \frac{2}{13} a^{3} + \frac{1}{13} a^{2} + \frac{5}{13} a$, $\frac{1}{13} a^{23} - \frac{5}{13} a^{19} - \frac{1}{13} a^{17} + \frac{5}{13} a^{16} + \frac{4}{13} a^{15} + \frac{4}{13} a^{14} - \frac{3}{13} a^{13} - \frac{6}{13} a^{12} - \frac{2}{13} a^{10} - \frac{1}{13} a^{9} - \frac{1}{13} a^{8} - \frac{3}{13} a^{6} + \frac{6}{13} a^{5} - \frac{3}{13} a^{4} - \frac{3}{13} a^{3} - \frac{5}{13} a^{2} + \frac{5}{13} a - \frac{1}{13}$, $\frac{1}{13} a^{24} - \frac{5}{13} a^{20} - \frac{1}{13} a^{18} + \frac{5}{13} a^{17} + \frac{4}{13} a^{16} + \frac{4}{13} a^{15} - \frac{3}{13} a^{14} - \frac{6}{13} a^{13} - \frac{2}{13} a^{11} - \frac{1}{13} a^{10} - \frac{1}{13} a^{9} - \frac{3}{13} a^{7} + \frac{6}{13} a^{6} - \frac{3}{13} a^{5} - \frac{3}{13} a^{4} - \frac{5}{13} a^{3} + \frac{5}{13} a^{2} - \frac{1}{13} a$, $\frac{1}{7778742049} a^{25} + \frac{20608792}{7778742049} a^{24} + \frac{25}{7778742049} a^{23} - \frac{103753765}{7778742049} a^{22} + \frac{275}{7778742049} a^{21} + \frac{406497400}{7778742049} a^{20} + \frac{1750}{7778742049} a^{19} + \frac{808760417}{7778742049} a^{18} - \frac{598357648}{7778742049} a^{17} + \frac{1941656407}{7778742049} a^{16} + \frac{2393478472}{7778742049} a^{15} - \frac{1267367788}{7778742049} a^{14} + \frac{3590224338}{7778742049} a^{13} - \frac{252935050}{598364773} a^{12} + \frac{3590232838}{7778742049} a^{11} + \frac{2773741759}{7778742049} a^{10} + \frac{1196765296}{7778742049} a^{9} - \frac{59844485}{7778742049} a^{8} + \frac{1375}{598364773} a^{7} + \frac{2279988080}{7778742049} a^{6} - \frac{2991818860}{7778742049} a^{5} + \frac{3651354103}{7778742049} a^{4} - \frac{1795093669}{7778742049} a^{3} + \frac{574206956}{7778742049} a^{2} + \frac{2991823890}{7778742049} a + \frac{1836311903}{7778742049}$, $\frac{1}{7778742049} a^{26} + \frac{2}{598364773} a^{24} - \frac{20608792}{7778742049} a^{23} + \frac{23}{598364773} a^{22} + \frac{124362557}{7778742049} a^{21} + \frac{154}{598364773} a^{20} - \frac{2944927841}{7778742049} a^{19} + \frac{665}{598364773} a^{18} - \frac{1886805846}{7778742049} a^{17} + \frac{598389967}{7778742049} a^{16} + \frac{2632100254}{7778742049} a^{15} - \frac{2991773477}{7778742049} a^{14} + \frac{2950288470}{7778742049} a^{13} + \frac{3590257590}{7778742049} a^{12} - \frac{216074084}{598364773} a^{11} - \frac{2991760659}{7778742049} a^{10} - \frac{1433852468}{7778742049} a^{9} - \frac{2393421912}{7778742049} a^{8} - \frac{2894751709}{7778742049} a^{7} + \frac{2991836878}{7778742049} a^{6} + \frac{1029632007}{7778742049} a^{5} + \frac{2393461458}{7778742049} a^{4} + \frac{26347474}{598364773} a^{3} + \frac{3590188807}{7778742049} a^{2} + \frac{55594410}{598364773} a - \frac{3590188636}{7778742049}$, $\frac{1}{7778742049} a^{27} + \frac{41927389}{7778742049} a^{24} - \frac{27}{598364773} a^{23} - \frac{169863418}{7778742049} a^{22} - \frac{396}{598364773} a^{21} - \frac{1546564781}{7778742049} a^{20} - \frac{2835}{598364773} a^{19} + \frac{2216743778}{7778742049} a^{18} + \frac{3590028582}{7778742049} a^{17} + \frac{18215512}{7778742049} a^{16} - \frac{34884}{598364773} a^{15} - \frac{2991859287}{7778742049} a^{14} + \frac{597505525}{7778742049} a^{13} - \frac{221775287}{598364773} a^{12} + \frac{2990737871}{7778742049} a^{11} + \frac{3637917515}{7778742049} a^{10} - \frac{1795986639}{7778742049} a^{9} + \frac{1653028766}{7778742049} a^{8} - \frac{1795546056}{7778742049} a^{7} + \frac{389689681}{7778742049} a^{6} + \frac{1196601782}{7778742049} a^{5} - \frac{649420155}{7778742049} a^{4} - \frac{1196746277}{7778742049} a^{3} + \frac{103910044}{598364773} a^{2} - \frac{3590189286}{7778742049} a - \frac{82435168}{598364773}$, $\frac{1}{7778742049} a^{28} - \frac{378}{7778742049} a^{24} - \frac{21318597}{7778742049} a^{23} - \frac{5796}{7778742049} a^{22} + \frac{6725250}{598364773} a^{21} - \frac{43659}{7778742049} a^{20} - \frac{2942602850}{7778742049} a^{19} - \frac{201096}{7778742049} a^{18} - \frac{3122233251}{7778742049} a^{17} + \frac{3589578168}{7778742049} a^{16} + \frac{624892265}{7778742049} a^{15} - \frac{1197985370}{7778742049} a^{14} - \frac{1983839212}{7778742049} a^{13} - \frac{600119071}{7778742049} a^{12} - \frac{46579055}{598364773} a^{11} - \frac{3591822270}{7778742049} a^{10} - \frac{2535925030}{7778742049} a^{9} + \frac{3589215666}{7778742049} a^{8} - \frac{2303151990}{7778742049} a^{7} - \frac{3590532618}{7778742049} a^{6} + \frac{2521857088}{7778742049} a^{5} - \frac{4851}{598364773} a^{4} - \frac{3164110904}{7778742049} a^{3} + \frac{2393454556}{7778742049} a^{2} + \frac{871981956}{7778742049} a + \frac{2991823811}{7778742049}$, $\frac{1}{7778742049} a^{29} - \frac{9937270}{7778742049} a^{24} + \frac{3654}{7778742049} a^{23} - \frac{237784675}{7778742049} a^{22} + \frac{60291}{7778742049} a^{21} + \frac{2917315419}{7778742049} a^{20} + \frac{460404}{7778742049} a^{19} - \frac{781735536}{7778742049} a^{18} + \frac{2993906645}{7778742049} a^{17} + \frac{3369261505}{7778742049} a^{16} + \frac{1202799362}{7778742049} a^{15} + \frac{1233143962}{7778742049} a^{14} + \frac{3003564167}{7778742049} a^{13} + \frac{82335725}{598364773} a^{12} + \frac{1159536}{598364773} a^{11} + \frac{3587025306}{7778742049} a^{10} - \frac{2979283337}{7778742049} a^{9} - \frac{1588141173}{7778742049} a^{8} - \frac{3583775868}{7778742049} a^{7} + \frac{916983889}{7778742049} a^{6} - \frac{2989995038}{7778742049} a^{5} + \frac{210397357}{7778742049} a^{4} + \frac{598605937}{7778742049} a^{3} + \frac{117433952}{7778742049} a^{2} - \frac{1795084923}{7778742049} a + \frac{1817856973}{7778742049}$, $\frac{1}{7778742049} a^{30} + \frac{4060}{7778742049} a^{24} + \frac{10647075}{7778742049} a^{23} + \frac{70035}{7778742049} a^{22} + \frac{20367824}{598364773} a^{21} + \frac{562716}{7778742049} a^{20} - \frac{3137550545}{7778742049} a^{19} + \frac{2699900}{7778742049} a^{18} - \frac{3016745462}{7778742049} a^{17} + \frac{138732663}{598364773} a^{16} - \frac{3642938528}{7778742049} a^{15} + \frac{17703630}{7778742049} a^{14} + \frac{1598957809}{7778742049} a^{13} - \frac{3565065358}{7778742049} a^{12} + \frac{1879022}{598364773} a^{11} - \frac{2369771428}{7778742049} a^{10} + \frac{3027504349}{7778742049} a^{9} + \frac{3006074465}{7778742049} a^{8} - \frac{2160476307}{7778742049} a^{7} + \frac{2996903940}{7778742049} a^{6} - \frac{316207225}{7778742049} a^{5} + \frac{599302633}{7778742049} a^{4} - \frac{2997176916}{7778742049} a^{3} - \frac{598296913}{7778742049} a^{2} + \frac{2664653496}{7778742049} a - \frac{1795093507}{7778742049}$, $\frac{1}{7778742049} a^{31} + \frac{110019775}{7778742049} a^{24} - \frac{31465}{7778742049} a^{23} + \frac{256267420}{7778742049} a^{22} - \frac{553784}{7778742049} a^{21} + \frac{2756697119}{7778742049} a^{20} - \frac{4405100}{7778742049} a^{19} - \frac{2758164258}{7778742049} a^{18} + \frac{2971326665}{7778742049} a^{17} - \frac{2115337127}{7778742049} a^{16} + \frac{537385603}{7778742049} a^{15} + \frac{576764516}{7778742049} a^{14} + \frac{2273640372}{7778742049} a^{13} + \frac{124231554}{598364773} a^{12} + \frac{442600437}{7778742049} a^{11} - \frac{729197969}{7778742049} a^{10} + \frac{3459294238}{7778742049} a^{9} + \frac{863858820}{7778742049} a^{8} - \frac{665857198}{7778742049} a^{7} - \frac{1561503261}{7778742049} a^{6} - \frac{3609571078}{7778742049} a^{5} + \frac{3574428482}{7778742049} a^{4} + \frac{2390887952}{7778742049} a^{3} + \frac{2613567744}{7778742049} a^{2} - \frac{2393559780}{7778742049} a - \frac{3391443238}{7778742049}$, $\frac{1}{7778742049} a^{32} - \frac{35960}{7778742049} a^{24} - \frac{100767863}{7778742049} a^{23} - \frac{661664}{7778742049} a^{22} + \frac{26038552}{7778742049} a^{21} - \frac{5537840}{7778742049} a^{20} - \frac{2619313602}{7778742049} a^{19} - \frac{27329600}{7778742049} a^{18} + \frac{2638536812}{7778742049} a^{17} - \frac{2480572192}{7778742049} a^{16} - \frac{3821342196}{7778742049} a^{15} + \frac{2207617812}{7778742049} a^{14} + \frac{2173777447}{7778742049} a^{13} - \frac{865389349}{7778742049} a^{12} - \frac{99315967}{598364773} a^{11} - \frac{2049403439}{7778742049} a^{10} + \frac{708555499}{7778742049} a^{9} - \frac{1949362719}{7778742049} a^{8} - \frac{3733431627}{7778742049} a^{7} - \frac{3047202265}{7778742049} a^{6} + \frac{1625951448}{7778742049} a^{5} - \frac{3600473198}{7778742049} a^{4} - \frac{3079160976}{7778742049} a^{3} - \frac{1197477514}{7778742049} a^{2} + \frac{440074890}{7778742049} a - \frac{2991832855}{7778742049}$, $\frac{1}{7778742049} a^{33} + \frac{215803483}{7778742049} a^{24} + \frac{237336}{7778742049} a^{23} - \frac{154991193}{7778742049} a^{22} + \frac{4351160}{7778742049} a^{21} - \frac{630754900}{7778742049} a^{20} + \frac{35600400}{7778742049} a^{19} - \frac{2643577717}{7778742049} a^{18} + \frac{3160925765}{7778742049} a^{17} + \frac{3334798976}{7778742049} a^{16} + \frac{1109428293}{7778742049} a^{15} + \frac{2679421285}{7778742049} a^{14} + \frac{3410206516}{7778742049} a^{13} + \frac{99200406}{598364773} a^{12} + \frac{1933487653}{7778742049} a^{11} - \frac{551990869}{7778742049} a^{10} - \frac{3057251811}{7778742049} a^{9} + \frac{1962259211}{7778742049} a^{8} - \frac{10958173}{7778742049} a^{7} - \frac{337347304}{7778742049} a^{6} - \frac{3420493398}{7778742049} a^{5} + \frac{2227337833}{7778742049} a^{4} + \frac{2416085124}{7778742049} a^{3} - \frac{47738807}{7778742049} a^{2} - \frac{2392569082}{7778742049} a + \frac{34777919}{7778742049}$, $\frac{1}{7778742049} a^{34} + \frac{278256}{7778742049} a^{24} - \frac{164795311}{7778742049} a^{23} + \frac{5333240}{7778742049} a^{22} - \frac{140235425}{7778742049} a^{21} + \frac{45912240}{7778742049} a^{20} + \frac{112080939}{598364773} a^{19} + \frac{231300300}{7778742049} a^{18} - \frac{1254468973}{7778742049} a^{17} - \frac{2242851465}{7778742049} a^{16} + \frac{1780954469}{7778742049} a^{15} - \frac{1374043481}{7778742049} a^{14} - \frac{743379766}{7778742049} a^{13} - \frac{2438943464}{7778742049} a^{12} - \frac{1729822622}{7778742049} a^{11} - \frac{3728846610}{7778742049} a^{10} - \frac{2483793161}{7778742049} a^{9} + \frac{779002427}{7778742049} a^{8} + \frac{3423274240}{7778742049} a^{7} - \frac{2494441265}{7778742049} a^{6} + \frac{384226364}{7778742049} a^{5} + \frac{1289574298}{7778742049} a^{4} + \frac{2689177990}{7778742049} a^{3} - \frac{2985041375}{7778742049} a^{2} + \frac{140005240}{598364773} a + \frac{81840}{7778742049}$, $\frac{1}{7778742049} a^{35} + \frac{43162369}{7778742049} a^{24} - \frac{1623160}{7778742049} a^{23} + \frac{63830711}{7778742049} a^{22} - \frac{30608160}{7778742049} a^{21} + \frac{9557997}{7778742049} a^{20} - \frac{255647700}{7778742049} a^{19} + \frac{3691522667}{7778742049} a^{18} - \frac{635236827}{7778742049} a^{17} + \frac{2748429302}{7778742049} a^{16} - \frac{782997031}{7778742049} a^{15} - \frac{913142091}{7778742049} a^{14} + \frac{1988071888}{7778742049} a^{13} - \frac{3888563700}{7778742049} a^{12} - \frac{3462101577}{7778742049} a^{11} - \frac{935955}{7778742049} a^{10} + \frac{1003551568}{7778742049} a^{9} - \frac{3563352693}{7778742049} a^{8} - \frac{287889989}{7778742049} a^{7} + \frac{2255258864}{7778742049} a^{6} - \frac{701461755}{7778742049} a^{5} + \frac{1124057843}{7778742049} a^{4} + \frac{1022645636}{7778742049} a^{3} - \frac{1144088475}{7778742049} a^{2} + \frac{2386584532}{7778742049} a - \frac{2575908505}{7778742049}$, $\frac{1}{7778742049} a^{36} - \frac{1947792}{7778742049} a^{24} + \frac{181501032}{7778742049} a^{23} - \frac{38399328}{7778742049} a^{22} + \frac{107201982}{7778742049} a^{21} - \frac{337454964}{7778742049} a^{20} + \frac{2354608769}{7778742049} a^{19} - \frac{1727042240}{7778742049} a^{18} - \frac{1412509912}{7778742049} a^{17} + \frac{321416386}{7778742049} a^{16} + \frac{69937974}{598364773} a^{15} - \frac{3378487701}{7778742049} a^{14} - \frac{1602387433}{7778742049} a^{13} + \frac{2264996938}{7778742049} a^{12} + \frac{2801074434}{7778742049} a^{11} - \frac{56089796}{598364773} a^{10} + \frac{3146717627}{7778742049} a^{9} - \frac{3563086884}{7778742049} a^{8} + \frac{1421740613}{7778742049} a^{7} + \frac{2682532919}{7778742049} a^{6} - \frac{90645495}{7778742049} a^{5} - \frac{132787649}{7778742049} a^{4} + \frac{3710434190}{7778742049} a^{3} - \frac{3045445433}{7778742049} a^{2} - \frac{64779092}{7778742049} a + \frac{2991174601}{7778742049}$, $\frac{1}{7778742049} a^{37} - \frac{77473182}{7778742049} a^{24} + \frac{10295472}{7778742049} a^{23} - \frac{124531424}{7778742049} a^{22} + \frac{198187836}{7778742049} a^{21} + \frac{519772779}{7778742049} a^{20} + \frac{1681593760}{7778742049} a^{19} + \frac{2260894080}{7778742049} a^{18} + \frac{2830503699}{7778742049} a^{17} + \frac{2513976653}{7778742049} a^{16} + \frac{3254753063}{7778742049} a^{15} - \frac{123906208}{598364773} a^{14} + \frac{1194128124}{7778742049} a^{13} + \frac{2354777882}{7778742049} a^{12} - \frac{3194747352}{7778742049} a^{11} + \frac{127471737}{7778742049} a^{10} - \frac{2143107006}{7778742049} a^{9} + \frac{2458947304}{7778742049} a^{8} + \frac{400698993}{7778742049} a^{7} - \frac{23847989}{7778742049} a^{6} + \frac{640439716}{7778742049} a^{5} - \frac{3883032744}{7778742049} a^{4} - \frac{582651087}{7778742049} a^{3} - \frac{3250273847}{7778742049} a^{2} - \frac{1148684010}{7778742049} a + \frac{2473875437}{7778742049}$, $\frac{1}{7778742049} a^{38} + \frac{12620256}{7778742049} a^{24} + \frac{17203807}{7778742049} a^{23} + \frac{253982652}{7778742049} a^{22} + \frac{283766001}{7778742049} a^{21} + \frac{2267439328}{7778742049} a^{20} - \frac{2178394302}{7778742049} a^{19} - \frac{3808025762}{7778742049} a^{18} - \frac{2567205260}{7778742049} a^{17} + \frac{3607022233}{7778742049} a^{16} - \frac{2076093774}{7778742049} a^{15} + \frac{1832128228}{7778742049} a^{14} + \frac{710300157}{7778742049} a^{13} - \frac{3693076765}{7778742049} a^{12} - \frac{184072218}{598364773} a^{11} - \frac{543773894}{7778742049} a^{10} + \frac{1097940041}{7778742049} a^{9} - \frac{195068635}{7778742049} a^{8} + \frac{1988289858}{7778742049} a^{7} + \frac{1306366798}{7778742049} a^{6} + \frac{916788446}{7778742049} a^{5} + \frac{414854392}{7778742049} a^{4} + \frac{434842159}{7778742049} a^{3} + \frac{1579543978}{7778742049} a^{2} + \frac{2017245895}{7778742049} a - \frac{1790444751}{7778742049}$, $\frac{1}{7778742049} a^{39} + \frac{10369100}{7778742049} a^{24} - \frac{4732596}{598364773} a^{23} + \frac{119733033}{7778742049} a^{22} - \frac{6401526}{7778742049} a^{21} + \frac{3878001583}{7778742049} a^{20} - \frac{762153296}{7778742049} a^{19} + \frac{1404794652}{7778742049} a^{18} - \frac{1942868774}{7778742049} a^{17} - \frac{1124275708}{7778742049} a^{16} + \frac{2581123878}{7778742049} a^{15} + \frac{111213106}{598364773} a^{14} + \frac{3512835380}{7778742049} a^{13} - \frac{2228375739}{7778742049} a^{12} + \frac{3505432753}{7778742049} a^{11} + \frac{732398598}{7778742049} a^{10} - \frac{153636624}{598364773} a^{9} + \frac{1086156510}{7778742049} a^{8} + \frac{1901174992}{7778742049} a^{7} - \frac{2954539505}{7778742049} a^{6} - \frac{2314684815}{7778742049} a^{5} - \frac{2318043458}{7778742049} a^{4} + \frac{556754854}{7778742049} a^{3} + \frac{1033048087}{7778742049} a^{2} + \frac{1484237487}{7778742049} a + \frac{3576519224}{7778742049}$, $\frac{1}{7778742049} a^{40} - \frac{5915745}{598364773} a^{24} - \frac{139494467}{7778742049} a^{23} + \frac{222820759}{7778742049} a^{22} - \frac{13094651}{598364773} a^{21} + \frac{148768764}{7778742049} a^{20} - \frac{2978740569}{7778742049} a^{19} + \frac{407792225}{7778742049} a^{18} + \frac{3381672055}{7778742049} a^{17} - \frac{679116528}{7778742049} a^{16} - \frac{850282986}{7778742049} a^{15} - \frac{2802495585}{7778742049} a^{14} - \frac{3812545571}{7778742049} a^{13} - \frac{2028807200}{7778742049} a^{12} - \frac{2824593922}{7778742049} a^{11} - \frac{55658327}{7778742049} a^{10} + \frac{3172085089}{7778742049} a^{9} - \frac{632345477}{7778742049} a^{8} - \frac{1014028056}{7778742049} a^{7} + \frac{1368103565}{7778742049} a^{6} - \frac{2756018480}{7778742049} a^{5} - \frac{3697436437}{7778742049} a^{4} - \frac{1518313502}{7778742049} a^{3} - \frac{1927346987}{7778742049} a^{2} + \frac{923832632}{7778742049} a - \frac{1825856193}{7778742049}$, $\frac{1}{7778742049} a^{41} - \frac{14718459}{598364773} a^{24} - \frac{248021208}{7778742049} a^{23} - \frac{217792917}{7778742049} a^{22} - \frac{243574689}{7778742049} a^{21} + \frac{1308771848}{7778742049} a^{20} - \frac{156503234}{598364773} a^{19} - \frac{3021350412}{7778742049} a^{18} + \frac{959726348}{7778742049} a^{17} + \frac{1689284698}{7778742049} a^{16} - \frac{1121255509}{7778742049} a^{15} - \frac{10704904}{7778742049} a^{14} - \frac{3624225543}{7778742049} a^{13} - \frac{2611971837}{7778742049} a^{12} + \frac{2812967125}{7778742049} a^{11} + \frac{818372964}{7778742049} a^{10} + \frac{420740884}{7778742049} a^{9} + \frac{1566318765}{7778742049} a^{8} - \frac{199629960}{7778742049} a^{7} + \frac{920558620}{7778742049} a^{6} + \frac{50039720}{598364773} a^{5} + \frac{3458077859}{7778742049} a^{4} - \frac{2800307442}{7778742049} a^{3} - \frac{558271638}{7778742049} a^{2} + \frac{96760932}{7778742049} a - \frac{702055773}{7778742049}$

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

\(\Q(\sqrt{-35}) \), \(\Q(\zeta_{7})^+\), 6.0.2100875.1, 7.7.13841287201.1, 14.0.104770985911247257875546875.1, \(\Q(\zeta_{49})^+\)

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 $42$ $21^{2}$ R R $21^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{6}$ $21^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{7}$ $42$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{7}$ $42$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{3}$ $21^{2}$ $42$ $42$

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
7Data not computed