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

\( x^{42} + 42 x^{40} + 819 x^{38} + 9842 x^{36} - 29 x^{35} + 81585 x^{34} - 1015 x^{33} + 494802 x^{32} - 16240 x^{31} + 2272424 x^{30} - 157325 x^{29} + 8070266 x^{28} + \cdots + 599786069 \) Copy content Toggle raw display

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-56353276529596271503862578540802938668269419115433656434196014026165008544921875\) \(\medspace = -\,5^{21}\cdot 7^{77}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(79.22\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{1/2}7^{11/6}\approx 79.21937447802398$
Ramified primes:   \(5\), \(7\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-35}) \)
$\card{ \Gal(K/\Q) }$:  $42$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
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.
Reflex fields:  unavailable$^{1048576}$

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

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

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $20$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

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

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display Deg $42$$2$$21$$21$
\(7\) Copy content Toggle raw display Deg $42$$42$$1$$77$