LMFDB - Number field 42.0.4472772095648327544266441640449519840379754819764800031247757188817501068115234375.1

Show commands: Magma / Oscar / PariGP / SageMath

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(y^42 - y^41 + 44*y^40 - 44*y^39 + 904*y^38 - 904*y^37 + 11525*y^36 - 11525*y^35 + 102212*y^34 - 102212*y^33 + 670199*y^32 - 670199*y^31 + 3371975*y^30 - 3371975*y^29 + 13342815*y^28 - 13342815*y^27 + 42258251*y^26 - 42258251*y^25 + 108593663*y^24 - 108593663*y^23 + 229203503*y^22 - 229203503*y^21 + 402580148*y^20 - 402580148*y^19 + 598327973*y^18 - 598327973*y^17 + 769983758*y^16 - 769983758*y^15 + 884984678*y^14 - 884984678*y^13 + 942485138*y^12 - 942485138*y^11 + 963249193*y^10 - 963249193*y^9 + 968416718*y^8 - 968416718*y^7 + 969243522*y^6 - 969243522*y^5 + 969319675*y^4 - 969319675*y^3 + 969322986*y^2 - 969322986*y + 969323029, 1)

magma: R<x> := PolynomialRing(Rationals()); 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);

oscar: Qx, x = PolynomialRing(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)

$ x^{42} - x^{41} + 44 x^{40} - 44 x^{39} + 904 x^{38} - 904 x^{37} + 11525 x^{36} - 11525 x^{35} + \cdots + 969323029 $ 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);

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:	$-447\!\cdots\!375$ -4472772095648327544266441640449519840379754819764800031247757188817501068115234375 $\medspace = -\,5^{21}\cdot 43^{41}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$87.91$	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:	$5^{1/2}43^{41/42}\approx 87.9146702848283$
Ramified primes:	$5$, $43$ 5, 43	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-215}) $
$\card{ \Gal(K/\Q) }$:	$42$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
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.
Reflex fields:	unavailable$^{1048576}$

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

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

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 $ -1 (order $2$)	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 - 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)

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 - 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);

[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 - 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);

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 - 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);

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 $C_{42}$

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.

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

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

# 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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$5$ 5		Deg $42$	$2$	$21$	$21$
$43$ 43		Deg $42$	$42$	$1$	$41$

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