↑

Properties

Label 42.42.6941495829...6073.1
Degree $42$
Signature $[42, 0]$
Discriminant $11^{21}\cdot 43^{41}$
Root discriminant $130.40$
Ramified primes $11, 43$
Class number Not computed
Class group Not computed
Galois group $C_{42}$ (as 42T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-937261883, -448857925846, 448857925846, 11095885225865, -11095885225865, -77413812270586, 77413812270586, 242906997716570, -242906997716570, -424428023090005, 424428023090005, 469396459323650, -469396459323650, -355672293673570, 355672293673570, 194373541657910, -194373541657910, -79301224450645, 79301224450645, 24727195415180, -24727195415180, -5985957116635, 5985957116635, 1135933325525, -1135933325525, -169746588871, 169746588871, 19967586725, -19967586725, -1838640355, 1838640355, 130954349, -130954349, -7066492, 7066492, 279155, -279155, -7612, 7612, 128, -128, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^42 - x^41 - 128*x^40 + 128*x^39 + 7612*x^38 - 7612*x^37 - 279155*x^36 + 279155*x^35 + 7066492*x^34 - 7066492*x^33 - 130954349*x^32 + 130954349*x^31 + 1838640355*x^30 - 1838640355*x^29 - 19967586725*x^28 + 19967586725*x^27 + 169746588871*x^26 - 169746588871*x^25 - 1135933325525*x^24 + 1135933325525*x^23 + 5985957116635*x^22 - 5985957116635*x^21 - 24727195415180*x^20 + 24727195415180*x^19 + 79301224450645*x^18 - 79301224450645*x^17 - 194373541657910*x^16 + 194373541657910*x^15 + 355672293673570*x^14 - 355672293673570*x^13 - 469396459323650*x^12 + 469396459323650*x^11 + 424428023090005*x^10 - 424428023090005*x^9 - 242906997716570*x^8 + 242906997716570*x^7 + 77413812270586*x^6 - 77413812270586*x^5 - 11095885225865*x^4 + 11095885225865*x^3 + 448857925846*x^2 - 448857925846*x - 937261883)
 
gp: K = bnfinit(x^42 - x^41 - 128*x^40 + 128*x^39 + 7612*x^38 - 7612*x^37 - 279155*x^36 + 279155*x^35 + 7066492*x^34 - 7066492*x^33 - 130954349*x^32 + 130954349*x^31 + 1838640355*x^30 - 1838640355*x^29 - 19967586725*x^28 + 19967586725*x^27 + 169746588871*x^26 - 169746588871*x^25 - 1135933325525*x^24 + 1135933325525*x^23 + 5985957116635*x^22 - 5985957116635*x^21 - 24727195415180*x^20 + 24727195415180*x^19 + 79301224450645*x^18 - 79301224450645*x^17 - 194373541657910*x^16 + 194373541657910*x^15 + 355672293673570*x^14 - 355672293673570*x^13 - 469396459323650*x^12 + 469396459323650*x^11 + 424428023090005*x^10 - 424428023090005*x^9 - 242906997716570*x^8 + 242906997716570*x^7 + 77413812270586*x^6 - 77413812270586*x^5 - 11095885225865*x^4 + 11095885225865*x^3 + 448857925846*x^2 - 448857925846*x - 937261883, 1)
 

Normalized defining polynomial

\( x^{42} - x^{41} - 128 x^{40} + 128 x^{39} + 7612 x^{38} - 7612 x^{37} - 279155 x^{36} + 279155 x^{35} + 7066492 x^{34} - 7066492 x^{33} - 130954349 x^{32} + 130954349 x^{31} + 1838640355 x^{30} - 1838640355 x^{29} - 19967586725 x^{28} + 19967586725 x^{27} + 169746588871 x^{26} - 169746588871 x^{25} - 1135933325525 x^{24} + 1135933325525 x^{23} + 5985957116635 x^{22} - 5985957116635 x^{21} - 24727195415180 x^{20} + 24727195415180 x^{19} + 79301224450645 x^{18} - 79301224450645 x^{17} - 194373541657910 x^{16} + 194373541657910 x^{15} + 355672293673570 x^{14} - 355672293673570 x^{13} - 469396459323650 x^{12} + 469396459323650 x^{11} + 424428023090005 x^{10} - 424428023090005 x^{9} - 242906997716570 x^{8} + 242906997716570 x^{7} + 77413812270586 x^{6} - 77413812270586 x^{5} - 11095885225865 x^{4} + 11095885225865 x^{3} + 448857925846 x^{2} - 448857925846 x - 937261883 \)

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

Invariants

Degree:  $42$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[42, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(69414958297778203866028207940161306079257307292058478902474460052228735844266189326476073=11^{21}\cdot 43^{41}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.40$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $11, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(473=11\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{473}(1,·)$, $\chi_{473}(131,·)$, $\chi_{473}(133,·)$, $\chi_{473}(263,·)$, $\chi_{473}(395,·)$, $\chi_{473}(397,·)$, $\chi_{473}(144,·)$, $\chi_{473}(406,·)$, $\chi_{473}(23,·)$, $\chi_{473}(408,·)$, $\chi_{473}(285,·)$, $\chi_{473}(32,·)$, $\chi_{473}(417,·)$, $\chi_{473}(298,·)$, $\chi_{473}(175,·)$, $\chi_{473}(56,·)$, $\chi_{473}(441,·)$, $\chi_{473}(188,·)$, $\chi_{473}(65,·)$, $\chi_{473}(450,·)$, $\chi_{473}(67,·)$, $\chi_{473}(329,·)$, $\chi_{473}(76,·)$, $\chi_{473}(78,·)$, $\chi_{473}(210,·)$, $\chi_{473}(340,·)$, $\chi_{473}(342,·)$, $\chi_{473}(472,·)$, $\chi_{473}(221,·)$, $\chi_{473}(351,·)$, $\chi_{473}(353,·)$, $\chi_{473}(98,·)$, $\chi_{473}(100,·)$, $\chi_{473}(232,·)$, $\chi_{473}(362,·)$, $\chi_{473}(111,·)$, $\chi_{473}(241,·)$, $\chi_{473}(373,·)$, $\chi_{473}(375,·)$, $\chi_{473}(120,·)$, $\chi_{473}(122,·)$, $\chi_{473}(252,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{10921827227} a^{22} + \frac{1983853805}{10921827227} a^{21} - \frac{66}{10921827227} a^{20} - \frac{4842690218}{10921827227} a^{19} + \frac{1881}{10921827227} a^{18} - \frac{309290838}{10921827227} a^{17} - \frac{30294}{10921827227} a^{16} + \frac{1033119143}{10921827227} a^{15} + \frac{302940}{10921827227} a^{14} + \frac{386591588}{10921827227} a^{13} - \frac{1945944}{10921827227} a^{12} + \frac{2107170101}{10921827227} a^{11} + \frac{8027019}{10921827227} a^{10} - \frac{445072219}{10921827227} a^{9} - \frac{20640906}{10921827227} a^{8} - \frac{4591385327}{10921827227} a^{7} + \frac{30961359}{10921827227} a^{6} - \frac{948449393}{10921827227} a^{5} - \frac{23816430}{10921827227} a^{4} + \frac{1740452688}{10921827227} a^{3} + \frac{7144929}{10921827227} a^{2} + \frac{3885350865}{10921827227} a - \frac{354294}{10921827227}$, $\frac{1}{10921827227} a^{23} - \frac{69}{10921827227} a^{21} - \frac{4970265812}{10921827227} a^{20} + \frac{2070}{10921827227} a^{19} + \frac{3326613591}{10921827227} a^{18} - \frac{35397}{10921827227} a^{17} - \frac{2914942368}{10921827227} a^{16} + \frac{380052}{10921827227} a^{15} - \frac{3820102210}{10921827227} a^{14} - \frac{2660364}{10921827227} a^{13} - \frac{2223077307}{10921827227} a^{12} + \frac{12206376}{10921827227} a^{11} - \frac{4426456115}{10921827227} a^{10} - \frac{35965215}{10921827227} a^{9} - \frac{5275368250}{10921827227} a^{8} + \frac{64737387}{10921827227} a^{7} - \frac{5383388622}{10921827227} a^{6} - \frac{64737387}{10921827227} a^{5} + \frac{951241623}{10921827227} a^{4} + \frac{29878794}{10921827227} a^{3} - \frac{1338799429}{10921827227} a^{2} - \frac{4074381}{10921827227} a + \frac{4230622312}{10921827227}$, $\frac{1}{10921827227} a^{24} + \frac{853720009}{10921827227} a^{21} - \frac{2484}{10921827227} a^{20} - \frac{3164194641}{10921827227} a^{19} + \frac{94392}{10921827227} a^{18} - \frac{2412355736}{10921827227} a^{17} - \frac{1710234}{10921827227} a^{16} + \frac{1934155295}{10921827227} a^{15} + \frac{18242496}{10921827227} a^{14} + \frac{2608087811}{10921827227} a^{13} - \frac{122063760}{10921827227} a^{12} - \frac{1015473097}{10921827227} a^{11} + \frac{517899096}{10921827227} a^{10} - \frac{3219869680}{10921827227} a^{9} - \frac{1359485127}{10921827227} a^{8} - \frac{5455986602}{10921827227} a^{7} + \frac{2071596384}{10921827227} a^{6} + \frac{1039196868}{10921827227} a^{5} - \frac{1613454876}{10921827227} a^{4} - \frac{1387663454}{10921827227} a^{3} + \frac{488925720}{10921827227} a^{2} - \frac{725848678}{10921827227} a - \frac{24446286}{10921827227}$, $\frac{1}{10921827227} a^{25} - \frac{2700}{10921827227} a^{21} - \frac{1427810182}{10921827227} a^{20} + \frac{108000}{10921827227} a^{19} - \frac{2751090296}{10921827227} a^{18} - \frac{2077650}{10921827227} a^{17} + \frac{1641234405}{10921827227} a^{16} + \frac{23794560}{10921827227} a^{15} - \frac{5384530516}{10921827227} a^{14} - \frac{173502000}{10921827227} a^{13} + \frac{3939703110}{10921827227} a^{12} + \frac{818812800}{10921827227} a^{11} + \frac{5009824937}{10921827227} a^{10} - \frac{2462835375}{10921827227} a^{9} - \frac{93620923}{10921827227} a^{8} + \frac{4503470400}{10921827227} a^{7} - \frac{1543438037}{10921827227} a^{6} - \frac{4559763780}{10921827227} a^{5} - \frac{3601723999}{10921827227} a^{4} + \frac{2125764000}{10921827227} a^{3} + \frac{5399283099}{10921827227} a^{2} - \frac{292292550}{10921827227} a - \frac{1206355892}{10921827227}$, $\frac{1}{10921827227} a^{26} + \frac{3282122088}{10921827227} a^{21} - \frac{70200}{10921827227} a^{20} - \frac{4587488177}{10921827227} a^{19} + \frac{3001050}{10921827227} a^{18} - \frac{3385158943}{10921827227} a^{17} - \frac{57999240}{10921827227} a^{16} - \frac{1028787301}{10921827227} a^{15} + \frac{644436000}{10921827227} a^{14} - \frac{758423082}{10921827227} a^{13} - \frac{4435236000}{10921827227} a^{12} + \frac{4097112370}{10921827227} a^{11} - \frac{2633538529}{10921827227} a^{10} - \frac{387617253}{10921827227} a^{9} + \frac{3382160335}{10921827227} a^{8} - \frac{2009918292}{10921827227} a^{7} + \frac{2583114931}{10921827227} a^{6} + \frac{2214313246}{10921827227} a^{5} + \frac{3352366362}{10921827227} a^{4} - \frac{2685994138}{10921827227} a^{3} - \frac{2844638704}{10921827227} a^{2} + \frac{4286841688}{10921827227} a - \frac{956593800}{10921827227}$, $\frac{1}{10921827227} a^{27} - \frac{78975}{10921827227} a^{21} + \frac{4517852318}{10921827227} a^{20} + \frac{3553875}{10921827227} a^{19} + \frac{4697404011}{10921827227} a^{18} - \frac{72925515}{10921827227} a^{17} - \frac{4737328037}{10921827227} a^{16} + \frac{869988600}{10921827227} a^{15} + \frac{3561502597}{10921827227} a^{14} + \frac{4396912727}{10921827227} a^{13} - \frac{3478799937}{10921827227} a^{12} - \frac{1330746531}{10921827227} a^{11} - \frac{3267689071}{10921827227} a^{10} + \frac{2245865418}{10921827227} a^{9} - \frac{3965585526}{10921827227} a^{8} + \frac{3081551788}{10921827227} a^{7} + \frac{5147814097}{10921827227} a^{6} + \frac{3798666634}{10921827227} a^{5} + \frac{4299012861}{10921827227} a^{4} - \frac{1879046941}{10921827227} a^{3} + \frac{3283522311}{10921827227} a^{2} - \frac{916021048}{10921827227} a + \frac{140014409}{10921827227}$, $\frac{1}{10921827227} a^{28} - \frac{5161296349}{10921827227} a^{21} - \frac{1658475}{10921827227} a^{20} + \frac{2861445320}{10921827227} a^{19} + \frac{75626460}{10921827227} a^{18} + \frac{1146247712}{10921827227} a^{17} - \frac{1522480050}{10921827227} a^{16} - \frac{2825391895}{10921827227} a^{15} - \frac{4443882454}{10921827227} a^{14} + \frac{1084762898}{10921827227} a^{13} - \frac{2106092753}{10921827227} a^{12} - \frac{5390420395}{10921827227} a^{11} + \frac{2713711777}{10921827227} a^{10} + \frac{3817762662}{10921827227} a^{9} + \frac{318257261}{10921827227} a^{8} + \frac{5155550672}{10921827227} a^{7} + \frac{2482694811}{10921827227} a^{6} + \frac{2399323452}{10921827227} a^{5} - \frac{4227323147}{10921827227} a^{4} + \frac{4338905316}{10921827227} a^{3} - \frac{4580269077}{10921827227} a^{2} - \frac{3011364781}{10921827227} a + \frac{4785113031}{10921827227}$, $\frac{1}{10921827227} a^{29} - \frac{1923831}{10921827227} a^{21} + \frac{792530323}{10921827227} a^{20} + \frac{92343888}{10921827227} a^{19} + \frac{40275378}{10921827227} a^{18} - \frac{1973850606}{10921827227} a^{17} - \frac{2258406769}{10921827227} a^{16} + \frac{2376828170}{10921827227} a^{15} - \frac{4585088362}{10921827227} a^{14} + \frac{232992769}{10921827227} a^{13} + \frac{4046698079}{10921827227} a^{12} + \frac{1044071423}{10921827227} a^{11} - \frac{5075326169}{10921827227} a^{10} - \frac{840946259}{10921827227} a^{9} + \frac{1499055829}{10921827227} a^{8} - \frac{2540736731}{10921827227} a^{7} + \frac{2423152103}{10921827227} a^{6} + \frac{2278827653}{10921827227} a^{5} - \frac{1560408307}{10921827227} a^{4} - \frac{3340651865}{10921827227} a^{3} + \frac{2034977474}{10921827227} a^{2} + \frac{3390794517}{10921827227} a + \frac{3360289550}{10921827227}$, $\frac{1}{10921827227} a^{30} + \frac{483063809}{10921827227} a^{21} - \frac{34628958}{10921827227} a^{20} - \frac{2306988694}{10921827227} a^{19} + \frac{1644875505}{10921827227} a^{18} - \frac{4413240187}{10921827227} a^{17} - \frac{1294572009}{10921827227} a^{16} - \frac{1148033762}{10921827227} a^{15} + \frac{4181512878}{10921827227} a^{14} - \frac{2740645312}{10921827227} a^{13} + \frac{3563418820}{10921827227} a^{12} + \frac{1319059626}{10921827227} a^{11} - \frac{1676655448}{10921827227} a^{10} - \frac{3743980041}{10921827227} a^{9} - \frac{391770245}{10921827227} a^{8} + \frac{2611615070}{10921827227} a^{7} - \frac{944622076}{10921827227} a^{6} - \frac{2838914135}{10921827227} a^{5} - \frac{5061777930}{10921827227} a^{4} + \frac{1531722131}{10921827227} a^{3} - \frac{1553781277}{10921827227} a^{2} + \frac{5229848516}{10921827227} a - \frac{4448492240}{10921827227}$, $\frac{1}{10921827227} a^{31} - \frac{41288373}{10921827227} a^{21} - \frac{3190258981}{10921827227} a^{20} + \frac{2064418650}{10921827227} a^{19} + \frac{4377222152}{10921827227} a^{18} - \frac{1700409697}{10921827227} a^{17} - \frac{2461488096}{10921827227} a^{16} + \frac{605880406}{10921827227} a^{15} - \frac{527929199}{10921827227} a^{14} + \frac{1355278635}{10921827227} a^{13} - \frac{3385973114}{10921827227} a^{12} + \frac{3049594214}{10921827227} a^{11} - \frac{718460829}{10921827227} a^{10} + \frac{4122302115}{10921827227} a^{9} + \frac{2633013687}{10921827227} a^{8} - \frac{3092700891}{10921827227} a^{7} + \frac{2589901553}{10921827227} a^{6} - \frac{4769684356}{10921827227} a^{5} + \frac{716272287}{10921827227} a^{4} + \frac{4351626308}{10921827227} a^{3} - \frac{1078612867}{10921827227} a^{2} - \frac{5414831663}{10921827227} a + \frac{1576498756}{10921827227}$, $\frac{1}{10921827227} a^{32} + \frac{3610362421}{10921827227} a^{21} - \frac{660613968}{10921827227} a^{20} + \frac{1889335679}{10921827227} a^{19} - \frac{489770673}{10921827227} a^{18} - \frac{819516687}{10921827227} a^{17} - \frac{5095787378}{10921827227} a^{16} + \frac{144321928}{10921827227} a^{15} + \frac{3762820340}{10921827227} a^{14} - \frac{2868347621}{10921827227} a^{13} - \frac{851033086}{10921827227} a^{12} - \frac{2108295879}{10921827227} a^{11} + \frac{3829648887}{10921827227} a^{10} - \frac{3192751690}{10921827227} a^{9} - \frac{2340164019}{10921827227} a^{8} + \frac{4954325656}{10921827227} a^{7} + \frac{5023337563}{10921827227} a^{6} - \frac{4038634704}{10921827227} a^{5} - \frac{1501186364}{10921827227} a^{4} + \frac{1563002128}{10921827227} a^{3} - \frac{1474622416}{10921827227} a^{2} - \frac{2827339053}{10921827227} a - \frac{3896166709}{10921827227}$, $\frac{1}{10921827227} a^{33} - \frac{807417072}{10921827227} a^{21} - \frac{106943529}{10921827227} a^{20} - \frac{2163002348}{10921827227} a^{19} + \frac{1465304606}{10921827227} a^{18} - \frac{3605841061}{10921827227} a^{17} + \frac{1285652524}{10921827227} a^{16} - \frac{1762414986}{10921827227} a^{15} - \frac{3359826354}{10921827227} a^{14} + \frac{226733981}{10921827227} a^{13} - \frac{2677538248}{10921827227} a^{12} - \frac{4113562539}{10921827227} a^{11} - \frac{4940312128}{10921827227} a^{10} + \frac{4941290089}{10921827227} a^{9} + \frac{86486302}{10921827227} a^{8} + \frac{2570978730}{10921827227} a^{7} + \frac{3263124327}{10921827227} a^{6} - \frac{2100656679}{10921827227} a^{5} + \frac{5187810527}{10921827227} a^{4} + \frac{1285489365}{10921827227} a^{3} - \frac{2822598850}{10921827227} a^{2} + \frac{1013802911}{10921827227} a - \frac{1895758785}{10921827227}$, $\frac{1}{10921827227} a^{34} - \frac{606013331}{10921827227} a^{21} - \frac{843392965}{10921827227} a^{20} + \frac{141220451}{10921827227} a^{19} - \frac{2988313182}{10921827227} a^{18} + \frac{4270443028}{10921827227} a^{17} + \frac{3237794326}{10921827227} a^{16} + \frac{2505199756}{10921827227} a^{15} + \frac{4833776996}{10921827227} a^{14} + \frac{163697686}{10921827227} a^{13} - \frac{299096741}{10921827227} a^{12} - \frac{2994436056}{10921827227} a^{11} + \frac{2858902706}{10921827227} a^{10} - \frac{3407783553}{10921827227} a^{9} - \frac{4556399116}{10921827227} a^{8} + \frac{3386369605}{10921827227} a^{7} - \frac{2331417137}{10921827227} a^{6} - \frac{422419481}{10921827227} a^{5} - \frac{2737705370}{10921827227} a^{4} + \frac{2721268316}{10921827227} a^{3} - \frac{3240152282}{10921827227} a^{2} + \frac{572900458}{10921827227} a + \frac{1474622416}{10921827227}$, $\frac{1}{10921827227} a^{35} - \frac{3784698013}{10921827227} a^{21} + \frac{3831649513}{10921827227} a^{20} - \frac{3357158017}{10921827227} a^{19} - \frac{2610340196}{10921827227} a^{18} + \frac{2274777644}{10921827227} a^{17} + \frac{3528919029}{10921827227} a^{16} - \frac{3671161733}{10921827227} a^{15} + \frac{848332183}{10921827227} a^{14} - \frac{4059505497}{10921827227} a^{13} + \frac{2373192578}{10921827227} a^{12} + \frac{5198004920}{10921827227} a^{11} + \frac{4485773206}{10921827227} a^{10} - \frac{2211280007}{10921827227} a^{9} - \frac{1308737451}{10921827227} a^{8} - \frac{367629229}{10921827227} a^{7} - \frac{4301586124}{10921827227} a^{6} - \frac{3436628839}{10921827227} a^{5} + \frac{3184820989}{10921827227} a^{4} - \frac{5278934183}{10921827227} a^{3} - \frac{2842713512}{10921827227} a^{2} - \frac{1567596234}{10921827227} a + \frac{5314362279}{10921827227}$, $\frac{1}{10921827227} a^{36} - \frac{5258550452}{10921827227} a^{21} - \frac{1945200654}{10921827227} a^{20} + \frac{2137241795}{10921827227} a^{19} + \frac{260388093}{10921827227} a^{18} - \frac{4515695231}{10921827227} a^{17} + \frac{29461491}{10921827227} a^{16} + \frac{4698365332}{10921827227} a^{15} + \frac{2621571171}{10921827227} a^{14} + \frac{1075400559}{10921827227} a^{13} + \frac{1343506288}{10921827227} a^{12} - \frac{3758792962}{10921827227} a^{11} + \frac{2766699623}{10921827227} a^{10} - \frac{203534107}{10921827227} a^{9} - \frac{3806582083}{10921827227} a^{8} - \frac{3617022450}{10921827227} a^{7} + \frac{1721800442}{10921827227} a^{6} - \frac{2265327289}{10921827227} a^{5} + \frac{3355151632}{10921827227} a^{4} + \frac{1398683001}{10921827227} a^{3} + \frac{1303091635}{10921827227} a^{2} - \frac{5110039589}{10921827227} a - \frac{1225504578}{10921827227}$, $\frac{1}{10921827227} a^{37} - \frac{2105192999}{10921827227} a^{21} + \frac{4571383227}{10921827227} a^{20} - \frac{581921399}{10921827227} a^{19} + \frac{2564064546}{10921827227} a^{18} + \frac{928163321}{10921827227} a^{17} - \frac{2978921761}{10921827227} a^{16} - \frac{2464057528}{10921827227} a^{15} + \frac{1395480900}{10921827227} a^{14} - \frac{3900164236}{10921827227} a^{13} - \frac{2859520491}{10921827227} a^{12} - \frac{1507674938}{10921827227} a^{11} + \frac{2913108967}{10921827227} a^{10} - \frac{2696311662}{10921827227} a^{9} + \frac{850840762}{10921827227} a^{8} - \frac{502431767}{10921827227} a^{7} + \frac{1954705120}{10921827227} a^{6} + \frac{1639156582}{10921827227} a^{5} - \frac{3527776887}{10921827227} a^{4} - \frac{5024795933}{10921827227} a^{3} + \frac{5105360159}{10921827227} a^{2} - \frac{1381504128}{10921827227} a + \frac{5180022453}{10921827227}$, $\frac{1}{10921827227} a^{38} + \frac{2600872681}{10921827227} a^{21} + \frac{2459094618}{10921827227} a^{20} - \frac{4355697780}{10921827227} a^{19} - \frac{3827088961}{10921827227} a^{18} - \frac{1816401818}{10921827227} a^{17} - \frac{4631590781}{10921827227} a^{16} - \frac{4036123542}{10921827227} a^{15} - \frac{4068486160}{10921827227} a^{14} - \frac{3934136876}{10921827227} a^{13} + \frac{2528863847}{10921827227} a^{12} + \frac{1008244399}{10921827227} a^{11} - \frac{4329521713}{10921827227} a^{10} - \frac{3086483049}{10921827227} a^{9} - \frac{983531876}{10921827227} a^{8} - \frac{2072094479}{10921827227} a^{7} - \frac{3100106868}{10921827227} a^{6} + \frac{2401021399}{10921827227} a^{5} + \frac{1141213004}{10921827227} a^{4} + \frac{3025119692}{10921827227} a^{3} + \frac{45241359}{10921827227} a^{2} + \frac{754537194}{10921827227} a + \frac{5254771351}{10921827227}$, $\frac{1}{10921827227} a^{39} + \frac{1012457558}{10921827227} a^{21} + \frac{3474490761}{10921827227} a^{20} - \frac{63571997}{10921827227} a^{19} - \frac{1079317083}{10921827227} a^{18} + \frac{2971723461}{10921827227} a^{17} - \frac{3260740906}{10921827227} a^{16} + \frac{2390623372}{10921827227} a^{15} - \frac{766136009}{10921827227} a^{14} - \frac{3346210519}{10921827227} a^{13} + \frac{703262917}{10921827227} a^{12} + \frac{4919460706}{10921827227} a^{11} + \frac{902022371}{10921827227} a^{10} + \frac{5150232420}{10921827227} a^{9} + \frac{3274359051}{10921827227} a^{8} - \frac{3691546966}{10921827227} a^{7} + \frac{5303168331}{10921827227} a^{6} - \frac{4443288739}{10921827227} a^{5} + \frac{1632839531}{10921827227} a^{4} - \frac{4133475824}{10921827227} a^{3} + \frac{2264403965}{10921827227} a^{2} - \frac{217377547}{10921827227} a - \frac{977499776}{10921827227}$, $\frac{1}{10921827227} a^{40} + \frac{4520353933}{10921827227} a^{21} + \frac{1227663469}{10921827227} a^{20} - \frac{1437733557}{10921827227} a^{19} - \frac{1063005639}{10921827227} a^{18} - \frac{4505323335}{10921827227} a^{17} + \frac{5289032008}{10921827227} a^{16} - \frac{5016303450}{10921827227} a^{15} + \frac{435184802}{10921827227} a^{14} + \frac{1210958489}{10921827227} a^{13} + \frac{2216226928}{10921827227} a^{12} + \frac{4485099611}{10921827227} a^{11} - \frac{2812125893}{10921827227} a^{10} + \frac{1058236398}{10921827227} a^{9} - \frac{5060565758}{10921827227} a^{8} + \frac{157434072}{10921827227} a^{7} - \frac{2389760551}{10921827227} a^{6} + \frac{3876426858}{10921827227} a^{5} - \frac{2352550668}{10921827227} a^{4} - \frac{1409106156}{10921827227} a^{3} + \frac{3617075797}{10921827227} a^{2} - \frac{1766430676}{10921827227} a + \frac{2066437691}{10921827227}$, $\frac{1}{10921827227} a^{41} - \frac{2604120145}{10921827227} a^{21} + \frac{2016290892}{10921827227} a^{20} + \frac{59163049}{10921827227} a^{19} + \frac{812338525}{10921827227} a^{18} - \frac{3165868114}{10921827227} a^{17} - \frac{3284029274}{10921827227} a^{16} - \frac{436734028}{10921827227} a^{15} - \frac{5189956044}{10921827227} a^{14} - \frac{1912449636}{10921827227} a^{13} - \frac{3096937848}{10921827227} a^{12} - \frac{2687387893}{10921827227} a^{11} - \frac{1796569168}{10921827227} a^{10} - \frac{1877329349}{10921827227} a^{9} + \frac{2817579573}{10921827227} a^{8} + \frac{5305159139}{10921827227} a^{7} + \frac{5347122220}{10921827227} a^{6} + \frac{506796230}{10921827227} a^{5} + \frac{1660292499}{10921827227} a^{4} + \frac{815013384}{10921827227} a^{3} + \frac{2773663341}{10921827227} a^{2} - \frac{1645195219}{10921827227} a + \frac{1219079930}{10921827227}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $41$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{473}) \), 3.3.1849.1, 6.6.195668237633.1, 7.7.6321363049.1, 14.14.33484106813054252803754617553.1, \(\Q(\zeta_{43})^+\)

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{6}$ $42$ $42$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{14}$ R $42$ $42$ $21^{2}$ $21^{2}$ $21^{2}$ $21^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{6}$ $21^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{6}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$11$11.14.7.2$x^{14} - 1771561 x^{2} + 77948684$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
11.14.7.2$x^{14} - 1771561 x^{2} + 77948684$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
11.14.7.2$x^{14} - 1771561 x^{2} + 77948684$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
43Data not computed