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Properties

Label 40.0.86004774278...3761.1
Degree $40$
Signature $[0, 20]$
Discriminant $3^{20}\cdot 7^{20}\cdot 11^{36}$
Root discriminant $39.66$
Ramified primes $3, 7, 11$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1048576, -524288, 524288, -655360, 327680, -327680, 278528, -139264, 139264, -92160, 46080, -11776, 5632, 5760, -10048, 8000, -9360, 7752, -5704, 5130, -3589, 2565, -1426, 969, -585, 250, -157, 45, 22, -23, 45, -45, 34, -17, 17, -10, 5, -5, 2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + 2*x^38 - 5*x^37 + 5*x^36 - 10*x^35 + 17*x^34 - 17*x^33 + 34*x^32 - 45*x^31 + 45*x^30 - 23*x^29 + 22*x^28 + 45*x^27 - 157*x^26 + 250*x^25 - 585*x^24 + 969*x^23 - 1426*x^22 + 2565*x^21 - 3589*x^20 + 5130*x^19 - 5704*x^18 + 7752*x^17 - 9360*x^16 + 8000*x^15 - 10048*x^14 + 5760*x^13 + 5632*x^12 - 11776*x^11 + 46080*x^10 - 92160*x^9 + 139264*x^8 - 139264*x^7 + 278528*x^6 - 327680*x^5 + 327680*x^4 - 655360*x^3 + 524288*x^2 - 524288*x + 1048576)
 
gp: K = bnfinit(x^40 - x^39 + 2*x^38 - 5*x^37 + 5*x^36 - 10*x^35 + 17*x^34 - 17*x^33 + 34*x^32 - 45*x^31 + 45*x^30 - 23*x^29 + 22*x^28 + 45*x^27 - 157*x^26 + 250*x^25 - 585*x^24 + 969*x^23 - 1426*x^22 + 2565*x^21 - 3589*x^20 + 5130*x^19 - 5704*x^18 + 7752*x^17 - 9360*x^16 + 8000*x^15 - 10048*x^14 + 5760*x^13 + 5632*x^12 - 11776*x^11 + 46080*x^10 - 92160*x^9 + 139264*x^8 - 139264*x^7 + 278528*x^6 - 327680*x^5 + 327680*x^4 - 655360*x^3 + 524288*x^2 - 524288*x + 1048576, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} + 2 x^{38} - 5 x^{37} + 5 x^{36} - 10 x^{35} + 17 x^{34} - 17 x^{33} + 34 x^{32} - 45 x^{31} + 45 x^{30} - 23 x^{29} + 22 x^{28} + 45 x^{27} - 157 x^{26} + 250 x^{25} - 585 x^{24} + 969 x^{23} - 1426 x^{22} + 2565 x^{21} - 3589 x^{20} + 5130 x^{19} - 5704 x^{18} + 7752 x^{17} - 9360 x^{16} + 8000 x^{15} - 10048 x^{14} + 5760 x^{13} + 5632 x^{12} - 11776 x^{11} + 46080 x^{10} - 92160 x^{9} + 139264 x^{8} - 139264 x^{7} + 278528 x^{6} - 327680 x^{5} + 327680 x^{4} - 655360 x^{3} + 524288 x^{2} - 524288 x + 1048576 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8600477427850631837563790410386763343300628153464218251684553761=3^{20}\cdot 7^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.66$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(231=3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{231}(1,·)$, $\chi_{231}(134,·)$, $\chi_{231}(8,·)$, $\chi_{231}(139,·)$, $\chi_{231}(13,·)$, $\chi_{231}(146,·)$, $\chi_{231}(20,·)$, $\chi_{231}(155,·)$, $\chi_{231}(29,·)$, $\chi_{231}(160,·)$, $\chi_{231}(34,·)$, $\chi_{231}(167,·)$, $\chi_{231}(41,·)$, $\chi_{231}(43,·)$, $\chi_{231}(50,·)$, $\chi_{231}(181,·)$, $\chi_{231}(188,·)$, $\chi_{231}(62,·)$, $\chi_{231}(64,·)$, $\chi_{231}(197,·)$, $\chi_{231}(71,·)$, $\chi_{231}(202,·)$, $\chi_{231}(76,·)$, $\chi_{231}(83,·)$, $\chi_{231}(85,·)$, $\chi_{231}(218,·)$, $\chi_{231}(92,·)$, $\chi_{231}(223,·)$, $\chi_{231}(97,·)$, $\chi_{231}(230,·)$, $\chi_{231}(104,·)$, $\chi_{231}(106,·)$, $\chi_{231}(113,·)$, $\chi_{231}(211,·)$, $\chi_{231}(190,·)$, $\chi_{231}(118,·)$, $\chi_{231}(169,·)$, $\chi_{231}(148,·)$, $\chi_{231}(125,·)$, $\chi_{231}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{20} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{92} a^{22} - \frac{1}{4} a^{21} + \frac{1}{4} a^{19} + \frac{1}{4} a^{18} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} + \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{92} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{6}{23}$, $\frac{1}{184} a^{23} - \frac{1}{184} a^{22} - \frac{1}{4} a^{21} - \frac{3}{8} a^{20} + \frac{3}{8} a^{19} + \frac{1}{4} a^{18} - \frac{1}{8} a^{17} + \frac{1}{8} a^{16} - \frac{1}{4} a^{15} - \frac{3}{8} a^{14} + \frac{3}{8} a^{13} - \frac{47}{184} a^{12} + \frac{35}{92} a^{11} + \frac{3}{8} a^{10} - \frac{3}{8} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} + \frac{3}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{3}{23} a - \frac{3}{23}$, $\frac{1}{368} a^{24} - \frac{1}{368} a^{23} - \frac{1}{184} a^{22} + \frac{1}{16} a^{21} + \frac{3}{16} a^{20} + \frac{3}{8} a^{19} + \frac{3}{16} a^{18} - \frac{7}{16} a^{17} + \frac{1}{8} a^{16} - \frac{7}{16} a^{15} - \frac{5}{16} a^{14} + \frac{45}{368} a^{13} - \frac{11}{184} a^{12} - \frac{159}{368} a^{11} + \frac{5}{16} a^{10} - \frac{3}{8} a^{9} + \frac{5}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{8} a^{6} - \frac{1}{16} a^{5} - \frac{3}{16} a^{4} - \frac{3}{8} a^{3} + \frac{29}{92} a^{2} - \frac{3}{46} a - \frac{3}{23}$, $\frac{1}{736} a^{25} - \frac{1}{736} a^{24} - \frac{1}{368} a^{23} - \frac{1}{736} a^{22} - \frac{5}{32} a^{21} - \frac{5}{16} a^{20} + \frac{11}{32} a^{19} - \frac{15}{32} a^{18} + \frac{1}{16} a^{17} + \frac{1}{32} a^{16} - \frac{13}{32} a^{15} + \frac{45}{736} a^{14} + \frac{81}{368} a^{13} + \frac{25}{736} a^{12} - \frac{229}{736} a^{11} - \frac{3}{16} a^{10} + \frac{13}{32} a^{9} - \frac{9}{32} a^{8} + \frac{7}{16} a^{7} + \frac{7}{32} a^{6} + \frac{5}{32} a^{5} + \frac{5}{16} a^{4} + \frac{75}{184} a^{3} - \frac{13}{46} a^{2} + \frac{10}{23} a + \frac{5}{23}$, $\frac{1}{1472} a^{26} - \frac{1}{1472} a^{25} - \frac{1}{736} a^{24} - \frac{1}{1472} a^{23} - \frac{3}{1472} a^{22} + \frac{3}{32} a^{21} + \frac{11}{64} a^{20} + \frac{1}{64} a^{19} + \frac{9}{32} a^{18} + \frac{1}{64} a^{17} - \frac{29}{64} a^{16} + \frac{413}{1472} a^{15} + \frac{81}{736} a^{14} + \frac{393}{1472} a^{13} + \frac{139}{1472} a^{12} - \frac{125}{736} a^{11} + \frac{13}{64} a^{10} + \frac{7}{64} a^{9} - \frac{1}{32} a^{8} + \frac{7}{64} a^{7} - \frac{11}{64} a^{6} - \frac{3}{32} a^{5} - \frac{109}{368} a^{4} + \frac{5}{46} a^{3} - \frac{3}{92} a^{2} + \frac{5}{46} a - \frac{4}{23}$, $\frac{1}{2944} a^{27} - \frac{1}{2944} a^{26} - \frac{1}{1472} a^{25} - \frac{1}{2944} a^{24} - \frac{3}{2944} a^{23} + \frac{5}{1472} a^{22} + \frac{11}{128} a^{21} + \frac{1}{128} a^{20} - \frac{23}{64} a^{19} + \frac{1}{128} a^{18} + \frac{35}{128} a^{17} + \frac{413}{2944} a^{16} - \frac{655}{1472} a^{15} + \frac{393}{2944} a^{14} + \frac{139}{2944} a^{13} + \frac{611}{1472} a^{12} + \frac{427}{2944} a^{11} - \frac{57}{128} a^{10} + \frac{31}{64} a^{9} - \frac{57}{128} a^{8} + \frac{53}{128} a^{7} - \frac{3}{64} a^{6} - \frac{109}{736} a^{5} + \frac{5}{92} a^{4} + \frac{89}{184} a^{3} + \frac{5}{92} a^{2} - \frac{2}{23} a - \frac{1}{23}$, $\frac{1}{5888} a^{28} - \frac{1}{5888} a^{27} - \frac{1}{2944} a^{26} - \frac{1}{5888} a^{25} - \frac{3}{5888} a^{24} + \frac{5}{2944} a^{23} - \frac{3}{5888} a^{22} + \frac{1}{256} a^{21} - \frac{23}{128} a^{20} + \frac{1}{256} a^{19} - \frac{93}{256} a^{18} + \frac{413}{5888} a^{17} + \frac{817}{2944} a^{16} + \frac{393}{5888} a^{15} - \frac{2805}{5888} a^{14} - \frac{861}{2944} a^{13} - \frac{2517}{5888} a^{12} - \frac{1055}{5888} a^{11} + \frac{31}{128} a^{10} - \frac{57}{256} a^{9} - \frac{75}{256} a^{8} + \frac{61}{128} a^{7} + \frac{627}{1472} a^{6} - \frac{87}{184} a^{5} + \frac{89}{368} a^{4} - \frac{87}{184} a^{3} - \frac{1}{23} a^{2} - \frac{1}{46} a - \frac{1}{23}$, $\frac{1}{11776} a^{29} - \frac{1}{11776} a^{28} - \frac{1}{5888} a^{27} - \frac{1}{11776} a^{26} - \frac{3}{11776} a^{25} + \frac{5}{5888} a^{24} - \frac{3}{11776} a^{23} + \frac{1}{512} a^{22} - \frac{23}{256} a^{21} + \frac{1}{512} a^{20} - \frac{93}{512} a^{19} - \frac{5475}{11776} a^{18} - \frac{2127}{5888} a^{17} - \frac{5495}{11776} a^{16} - \frac{2805}{11776} a^{15} + \frac{2083}{5888} a^{14} - \frac{2517}{11776} a^{13} - \frac{1055}{11776} a^{12} + \frac{31}{256} a^{11} - \frac{57}{512} a^{10} + \frac{181}{512} a^{9} - \frac{67}{256} a^{8} + \frac{627}{2944} a^{7} + \frac{97}{368} a^{6} - \frac{279}{736} a^{5} + \frac{97}{368} a^{4} - \frac{1}{46} a^{3} - \frac{1}{92} a^{2} - \frac{1}{46} a$, $\frac{1}{23552} a^{30} - \frac{1}{23552} a^{29} - \frac{1}{11776} a^{28} - \frac{1}{23552} a^{27} - \frac{3}{23552} a^{26} + \frac{5}{11776} a^{25} - \frac{3}{23552} a^{24} + \frac{1}{1024} a^{23} - \frac{17}{11776} a^{22} + \frac{1}{1024} a^{21} - \frac{93}{1024} a^{20} + \frac{6301}{23552} a^{19} - \frac{2127}{11776} a^{18} + \frac{6281}{23552} a^{17} + \frac{8971}{23552} a^{16} + \frac{2083}{11776} a^{15} + \frac{9259}{23552} a^{14} - \frac{1055}{23552} a^{13} - \frac{225}{512} a^{12} - \frac{2335}{23552} a^{11} + \frac{181}{1024} a^{10} - \frac{67}{512} a^{9} + \frac{627}{5888} a^{8} - \frac{271}{736} a^{7} + \frac{457}{1472} a^{6} - \frac{271}{736} a^{5} - \frac{1}{92} a^{4} + \frac{91}{184} a^{3} - \frac{1}{92} a^{2} + \frac{1}{23}$, $\frac{1}{114980864} a^{31} - \frac{111}{114980864} a^{30} - \frac{149}{3593152} a^{29} + \frac{4655}{114980864} a^{28} - \frac{9205}{114980864} a^{27} + \frac{745}{3593152} a^{26} - \frac{23283}{114980864} a^{25} + \frac{46913}{114980864} a^{24} - \frac{2533}{3593152} a^{23} + \frac{79175}{114980864} a^{22} - \frac{441707}{4999168} a^{21} - \frac{7023005}{114980864} a^{20} + \frac{6035717}{14372608} a^{19} + \frac{14880041}{114980864} a^{18} - \frac{26892115}{114980864} a^{17} - \frac{834603}{3593152} a^{16} + \frac{6873883}{114980864} a^{15} - \frac{39317113}{114980864} a^{14} - \frac{712115}{3593152} a^{13} - \frac{38428879}{114980864} a^{12} - \frac{48200971}{114980864} a^{11} - \frac{9983}{156224} a^{10} + \frac{1794351}{28745216} a^{9} - \frac{1829293}{14372608} a^{8} + \frac{68093}{449144} a^{7} - \frac{532219}{3593152} a^{6} + \frac{547217}{1796576} a^{5} - \frac{13857}{56143} a^{4} + \frac{108343}{449144} a^{3} + \frac{111223}{224572} a^{2} + \frac{1192}{56143} a - \frac{1187}{56143}$, $\frac{1}{229961728} a^{32} - \frac{1}{229961728} a^{31} + \frac{1275}{114980864} a^{30} + \frac{7431}{229961728} a^{29} - \frac{4883}{229961728} a^{28} + \frac{3609}{114980864} a^{27} - \frac{37163}{229961728} a^{26} + \frac{24423}{229961728} a^{25} - \frac{28245}{114980864} a^{24} + \frac{126367}{229961728} a^{23} - \frac{83051}{229961728} a^{22} - \frac{12012027}{229961728} a^{21} + \frac{5951101}{114980864} a^{20} - \frac{8640959}{229961728} a^{19} - \frac{16965285}{229961728} a^{18} + \frac{16284015}{114980864} a^{17} - \frac{79860205}{229961728} a^{16} - \frac{49039087}{229961728} a^{15} - \frac{14048019}{114980864} a^{14} + \frac{8505241}{229961728} a^{13} - \frac{79005741}{229961728} a^{12} + \frac{54948839}{114980864} a^{11} + \frac{2863805}{57490432} a^{10} - \frac{470547}{14372608} a^{9} + \frac{1340493}{14372608} a^{8} - \frac{849377}{7186304} a^{7} + \frac{139567}{1796576} a^{6} - \frac{451761}{1796576} a^{5} + \frac{172885}{898288} a^{4} - \frac{28411}{224572} a^{3} - \frac{109783}{224572} a^{2} - \frac{1881}{112286} a + \frac{622}{56143}$, $\frac{1}{459923456} a^{33} - \frac{1}{459923456} a^{32} - \frac{1}{229961728} a^{31} - \frac{2217}{459923456} a^{30} - \frac{2891}{459923456} a^{29} + \frac{341}{229961728} a^{28} + \frac{869}{459923456} a^{27} + \frac{14463}{459923456} a^{26} - \frac{1697}{229961728} a^{25} + \frac{13391}{459923456} a^{24} - \frac{49187}{459923456} a^{23} - \frac{96821}{19996672} a^{22} + \frac{16079749}{229961728} a^{21} - \frac{85377999}{459923456} a^{20} - \frac{156652301}{459923456} a^{19} + \frac{111254067}{229961728} a^{18} + \frac{122881027}{459923456} a^{17} - \frac{165403207}{459923456} a^{16} + \frac{4976857}{229961728} a^{15} + \frac{68618857}{459923456} a^{14} - \frac{219382837}{459923456} a^{13} + \frac{4930091}{229961728} a^{12} - \frac{884209}{4999168} a^{11} + \frac{28187831}{57490432} a^{10} - \frac{14307429}{28745216} a^{9} - \frac{14917}{898288} a^{8} - \frac{3427811}{7186304} a^{7} + \frac{1777289}{3593152} a^{6} + \frac{5897}{112286} a^{5} + \frac{415479}{898288} a^{4} - \frac{220665}{449144} a^{3} - \frac{7284}{56143} a^{2} - \frac{27885}{56143} a - \frac{11609}{56143}$, $\frac{1}{919846912} a^{34} - \frac{1}{919846912} a^{33} - \frac{1}{459923456} a^{32} - \frac{1}{919846912} a^{31} - \frac{14531}{919846912} a^{30} - \frac{10043}{459923456} a^{29} + \frac{5565}{919846912} a^{28} + \frac{3415}{919846912} a^{27} + \frac{50223}{459923456} a^{26} - \frac{27817}{919846912} a^{25} + \frac{99173}{919846912} a^{24} + \frac{2419229}{919846912} a^{23} - \frac{57961}{19996672} a^{22} - \frac{186014775}{919846912} a^{21} - \frac{421786741}{919846912} a^{20} - \frac{141821533}{459923456} a^{19} + \frac{239707755}{919846912} a^{18} + \frac{144434849}{919846912} a^{17} - \frac{204917367}{459923456} a^{16} + \frac{289579425}{919846912} a^{15} - \frac{107421053}{919846912} a^{14} - \frac{140458181}{459923456} a^{13} + \frac{43463555}{229961728} a^{12} + \frac{599793}{1249792} a^{11} - \frac{23493951}{57490432} a^{10} + \frac{11047319}{28745216} a^{9} + \frac{1378033}{7186304} a^{8} - \frac{324389}{7186304} a^{7} + \frac{369717}{3593152} a^{6} - \frac{61825}{898288} a^{5} + \frac{444623}{898288} a^{4} + \frac{46147}{112286} a^{3} - \frac{42655}{224572} a^{2} + \frac{4335}{112286} a - \frac{1705}{56143}$, $\frac{1}{1839693824} a^{35} - \frac{1}{1839693824} a^{34} - \frac{1}{919846912} a^{33} - \frac{1}{1839693824} a^{32} - \frac{3}{1839693824} a^{31} + \frac{3829}{919846912} a^{30} + \frac{21405}{1839693824} a^{29} - \frac{13737}{1839693824} a^{28} + \frac{9919}{919846912} a^{27} - \frac{107017}{1839693824} a^{26} + \frac{68709}{1839693824} a^{25} - \frac{2398851}{1839693824} a^{24} + \frac{1301121}{919846912} a^{23} + \frac{4243145}{1839693824} a^{22} + \frac{282835371}{1839693824} a^{21} + \frac{342819475}{919846912} a^{20} + \frac{296072331}{1839693824} a^{19} - \frac{42789215}{1839693824} a^{18} - \frac{162032007}{919846912} a^{17} + \frac{625961985}{1839693824} a^{16} - \frac{209043069}{1839693824} a^{15} - \frac{92701941}{919846912} a^{14} - \frac{143551413}{459923456} a^{13} + \frac{4460431}{57490432} a^{12} - \frac{50609553}{114980864} a^{11} + \frac{11294785}{28745216} a^{10} - \frac{7517255}{28745216} a^{9} + \frac{5870803}{14372608} a^{8} + \frac{296323}{3593152} a^{7} + \frac{996451}{3593152} a^{6} + \frac{61549}{1796576} a^{5} + \frac{99763}{224572} a^{4} - \frac{132269}{449144} a^{3} - \frac{82109}{224572} a^{2} + \frac{5123}{112286} a + \frac{6008}{56143}$, $\frac{1}{3679387648} a^{36} - \frac{1}{3679387648} a^{35} - \frac{1}{1839693824} a^{34} - \frac{1}{3679387648} a^{33} - \frac{3}{3679387648} a^{32} + \frac{5}{1839693824} a^{31} - \frac{67011}{3679387648} a^{30} - \frac{4543}{159973376} a^{29} + \frac{18735}{1839693824} a^{28} - \frac{7977}{3679387648} a^{27} + \frac{522469}{3679387648} a^{26} - \frac{2425827}{3679387648} a^{25} + \frac{1407185}{1839693824} a^{24} + \frac{117407}{159973376} a^{23} + \frac{2875787}{3679387648} a^{22} - \frac{357990525}{1839693824} a^{21} - \frac{97668245}{3679387648} a^{20} + \frac{260453921}{3679387648} a^{19} + \frac{22958431}{79986688} a^{18} - \frac{1415760863}{3679387648} a^{17} - \frac{1337773821}{3679387648} a^{16} + \frac{223704635}{1839693824} a^{15} - \frac{23380893}{919846912} a^{14} + \frac{47523395}{114980864} a^{13} + \frac{4190791}{9998336} a^{12} - \frac{50751637}{114980864} a^{11} + \frac{19042823}{57490432} a^{10} - \frac{336289}{7186304} a^{9} + \frac{3383549}{14372608} a^{8} + \frac{71499}{312448} a^{7} - \frac{12619}{56143} a^{6} + \frac{305329}{1796576} a^{5} + \frac{185001}{898288} a^{4} - \frac{104445}{224572} a^{3} + \frac{51695}{224572} a^{2} + \frac{3713}{56143} a + \frac{15183}{56143}$, $\frac{1}{7358775296} a^{37} - \frac{1}{7358775296} a^{36} - \frac{1}{3679387648} a^{35} - \frac{1}{7358775296} a^{34} - \frac{3}{7358775296} a^{33} + \frac{5}{3679387648} a^{32} - \frac{3}{7358775296} a^{31} - \frac{43625}{7358775296} a^{30} - \frac{67409}{3679387648} a^{29} + \frac{91159}{7358775296} a^{28} - \frac{138715}{7358775296} a^{27} - \frac{1564387}{7358775296} a^{26} + \frac{891313}{3679387648} a^{25} + \frac{5519369}{7358775296} a^{24} - \frac{53109}{7358775296} a^{23} + \frac{179669}{159973376} a^{22} + \frac{613406379}{7358775296} a^{21} + \frac{1932461153}{7358775296} a^{20} - \frac{288842743}{3679387648} a^{19} + \frac{1998275297}{7358775296} a^{18} - \frac{1535733309}{7358775296} a^{17} - \frac{1520232517}{3679387648} a^{16} + \frac{725853043}{1839693824} a^{15} - \frac{109585515}{229961728} a^{14} - \frac{66419983}{459923456} a^{13} - \frac{107742383}{229961728} a^{12} - \frac{752425}{2499584} a^{11} - \frac{1709671}{28745216} a^{10} - \frac{2146177}{28745216} a^{9} + \frac{3728945}{14372608} a^{8} - \frac{96247}{3593152} a^{7} - \frac{1173289}{3593152} a^{6} + \frac{142829}{1796576} a^{5} + \frac{28769}{224572} a^{4} - \frac{129529}{449144} a^{3} + \frac{110083}{224572} a^{2} - \frac{19807}{56143} a - \frac{5202}{56143}$, $\frac{1}{14717550592} a^{38} - \frac{1}{14717550592} a^{37} - \frac{1}{7358775296} a^{36} - \frac{1}{14717550592} a^{35} - \frac{3}{14717550592} a^{34} + \frac{5}{7358775296} a^{33} - \frac{3}{14717550592} a^{32} + \frac{1}{639893504} a^{31} + \frac{9711}{7358775296} a^{30} - \frac{557033}{14717550592} a^{29} + \frac{576421}{14717550592} a^{28} + \frac{1588381}{14717550592} a^{27} + \frac{12209}{7358775296} a^{26} - \frac{8403831}{14717550592} a^{25} + \frac{2945803}{14717550592} a^{24} - \frac{8875997}{7358775296} a^{23} + \frac{37407787}{14717550592} a^{22} + \frac{2212191201}{14717550592} a^{21} - \frac{116315873}{319946752} a^{20} - \frac{157302047}{14717550592} a^{19} - \frac{6845227965}{14717550592} a^{18} - \frac{3088664581}{7358775296} a^{17} - \frac{521830285}{3679387648} a^{16} - \frac{140246703}{459923456} a^{15} + \frac{138923529}{919846912} a^{14} - \frac{226931663}{459923456} a^{13} - \frac{10255617}{57490432} a^{12} - \frac{45686725}{114980864} a^{11} + \frac{25841235}{57490432} a^{10} - \frac{186853}{624896} a^{9} + \frac{5611077}{14372608} a^{8} - \frac{316869}{3593152} a^{7} + \frac{1001291}{3593152} a^{6} + \frac{220461}{898288} a^{5} - \frac{316185}{898288} a^{4} - \frac{177527}{449144} a^{3} + \frac{24577}{56143} a^{2} - \frac{6217}{56143} a - \frac{14207}{56143}$, $\frac{1}{29435101184} a^{39} - \frac{1}{29435101184} a^{38} - \frac{1}{14717550592} a^{37} - \frac{1}{29435101184} a^{36} - \frac{3}{29435101184} a^{35} + \frac{5}{14717550592} a^{34} - \frac{3}{29435101184} a^{33} + \frac{1}{1279787008} a^{32} - \frac{17}{14717550592} a^{31} + \frac{352791}{29435101184} a^{30} - \frac{391771}{29435101184} a^{29} - \frac{1493859}{29435101184} a^{28} + \frac{198321}{14717550592} a^{27} + \frac{6435465}{29435101184} a^{26} - \frac{1484021}{29435101184} a^{25} + \frac{6550819}{14717550592} a^{24} - \frac{29043157}{29435101184} a^{23} + \frac{19370977}{29435101184} a^{22} - \frac{29314529}{639893504} a^{21} + \frac{14171389153}{29435101184} a^{20} + \frac{12706638403}{29435101184} a^{19} - \frac{6239020037}{14717550592} a^{18} + \frac{3169759475}{7358775296} a^{17} + \frac{327640417}{919846912} a^{16} - \frac{411174007}{1839693824} a^{15} + \frac{380695649}{919846912} a^{14} - \frac{8362811}{114980864} a^{13} - \frac{112947381}{229961728} a^{12} + \frac{55730469}{114980864} a^{11} - \frac{1105771}{2499584} a^{10} - \frac{26327}{112286} a^{9} + \frac{5487403}{14372608} a^{8} + \frac{2033907}{7186304} a^{7} - \frac{508761}{1796576} a^{6} - \frac{466261}{1796576} a^{5} + \frac{327649}{898288} a^{4} - \frac{42671}{224572} a^{3} + \frac{46385}{224572} a^{2} - \frac{9875}{112286} a + \frac{12100}{56143}$

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:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{18157}{7358775296} a^{39} - \frac{271}{3014656} a^{28} - \frac{7917}{3014656} a^{17} - \frac{8672}{56143} a^{6} \) (order $66$)
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

$C_2^2\times C_{10}$ (as 40T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{77}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-231}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}, \sqrt{33})\), \(\Q(\sqrt{-7}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{77})\), \(\Q(\sqrt{-11}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{-7}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), 8.0.2847396321.1, 10.10.39630026842637.1, 10.10.875463320250981.1, \(\Q(\zeta_{33})^+\), \(\Q(\zeta_{11})\), 10.0.3602729712967.1, 10.0.9630096522760791.1, 10.0.52089208083.1, 20.20.92738759037689478010716606945681.1, 20.0.1570539027548129147161113769.2, 20.0.92738759037689478010716606945681.1, 20.0.92738759037689478010716606945681.4, 20.0.766436025104871719096831462361.1, \(\Q(\zeta_{33})\), 20.0.92738759037689478010716606945681.3

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
3Data not computed
7Data not computed
11Data not computed