Properties

Label 40.0.860...761.1
Degree $40$
Signature $[0, 20]$
Discriminant $8.600\times 10^{63}$
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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Normalized defining polynomial

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

\( x^{40} - x^{39} + 2 x^{38} - 5 x^{37} + 5 x^{36} - 10 x^{35} + 17 x^{34} - 17 x^{33} + 34 x^{32} - 45 x^{31} + 45 x^{30} - 23 x^{29} + 22 x^{28} + 45 x^{27} - 157 x^{26} + 250 x^{25} - 585 x^{24} + \cdots + 1048576 \) Copy content Toggle raw display

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

Invariants

Degree:  $40$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 20]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(8600477427850631837563790410386763343300628153464218251684553761\) \(\medspace = 3^{20}\cdot 7^{20}\cdot 11^{36}\) 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:  \(39.66\)
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:  $3^{1/2}7^{1/2}11^{9/10}\approx 39.66094555677728$
Ramified primes:   \(3\), \(7\), \(11\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $40$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
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.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{92}a^{22}-\frac{1}{4}a^{21}+\frac{1}{4}a^{19}+\frac{1}{4}a^{18}-\frac{1}{4}a^{16}-\frac{1}{4}a^{15}+\frac{1}{4}a^{13}+\frac{1}{4}a^{12}-\frac{1}{92}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{6}{23}$, $\frac{1}{184}a^{23}-\frac{1}{184}a^{22}-\frac{1}{4}a^{21}-\frac{3}{8}a^{20}+\frac{3}{8}a^{19}+\frac{1}{4}a^{18}-\frac{1}{8}a^{17}+\frac{1}{8}a^{16}-\frac{1}{4}a^{15}-\frac{3}{8}a^{14}+\frac{3}{8}a^{13}-\frac{47}{184}a^{12}+\frac{35}{92}a^{11}+\frac{3}{8}a^{10}-\frac{3}{8}a^{9}-\frac{1}{4}a^{8}+\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{4}a^{5}+\frac{3}{8}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}+\frac{3}{23}a-\frac{3}{23}$, $\frac{1}{368}a^{24}-\frac{1}{368}a^{23}-\frac{1}{184}a^{22}+\frac{1}{16}a^{21}+\frac{3}{16}a^{20}+\frac{3}{8}a^{19}+\frac{3}{16}a^{18}-\frac{7}{16}a^{17}+\frac{1}{8}a^{16}-\frac{7}{16}a^{15}-\frac{5}{16}a^{14}+\frac{45}{368}a^{13}-\frac{11}{184}a^{12}-\frac{159}{368}a^{11}+\frac{5}{16}a^{10}-\frac{3}{8}a^{9}+\frac{5}{16}a^{8}-\frac{1}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{16}a^{5}-\frac{3}{16}a^{4}-\frac{3}{8}a^{3}+\frac{29}{92}a^{2}-\frac{3}{46}a-\frac{3}{23}$, $\frac{1}{736}a^{25}-\frac{1}{736}a^{24}-\frac{1}{368}a^{23}-\frac{1}{736}a^{22}-\frac{5}{32}a^{21}-\frac{5}{16}a^{20}+\frac{11}{32}a^{19}-\frac{15}{32}a^{18}+\frac{1}{16}a^{17}+\frac{1}{32}a^{16}-\frac{13}{32}a^{15}+\frac{45}{736}a^{14}+\frac{81}{368}a^{13}+\frac{25}{736}a^{12}-\frac{229}{736}a^{11}-\frac{3}{16}a^{10}+\frac{13}{32}a^{9}-\frac{9}{32}a^{8}+\frac{7}{16}a^{7}+\frac{7}{32}a^{6}+\frac{5}{32}a^{5}+\frac{5}{16}a^{4}+\frac{75}{184}a^{3}-\frac{13}{46}a^{2}+\frac{10}{23}a+\frac{5}{23}$, $\frac{1}{1472}a^{26}-\frac{1}{1472}a^{25}-\frac{1}{736}a^{24}-\frac{1}{1472}a^{23}-\frac{3}{1472}a^{22}+\frac{3}{32}a^{21}+\frac{11}{64}a^{20}+\frac{1}{64}a^{19}+\frac{9}{32}a^{18}+\frac{1}{64}a^{17}-\frac{29}{64}a^{16}+\frac{413}{1472}a^{15}+\frac{81}{736}a^{14}+\frac{393}{1472}a^{13}+\frac{139}{1472}a^{12}-\frac{125}{736}a^{11}+\frac{13}{64}a^{10}+\frac{7}{64}a^{9}-\frac{1}{32}a^{8}+\frac{7}{64}a^{7}-\frac{11}{64}a^{6}-\frac{3}{32}a^{5}-\frac{109}{368}a^{4}+\frac{5}{46}a^{3}-\frac{3}{92}a^{2}+\frac{5}{46}a-\frac{4}{23}$, $\frac{1}{2944}a^{27}-\frac{1}{2944}a^{26}-\frac{1}{1472}a^{25}-\frac{1}{2944}a^{24}-\frac{3}{2944}a^{23}+\frac{5}{1472}a^{22}+\frac{11}{128}a^{21}+\frac{1}{128}a^{20}-\frac{23}{64}a^{19}+\frac{1}{128}a^{18}+\frac{35}{128}a^{17}+\frac{413}{2944}a^{16}-\frac{655}{1472}a^{15}+\frac{393}{2944}a^{14}+\frac{139}{2944}a^{13}+\frac{611}{1472}a^{12}+\frac{427}{2944}a^{11}-\frac{57}{128}a^{10}+\frac{31}{64}a^{9}-\frac{57}{128}a^{8}+\frac{53}{128}a^{7}-\frac{3}{64}a^{6}-\frac{109}{736}a^{5}+\frac{5}{92}a^{4}+\frac{89}{184}a^{3}+\frac{5}{92}a^{2}-\frac{2}{23}a-\frac{1}{23}$, $\frac{1}{5888}a^{28}-\frac{1}{5888}a^{27}-\frac{1}{2944}a^{26}-\frac{1}{5888}a^{25}-\frac{3}{5888}a^{24}+\frac{5}{2944}a^{23}-\frac{3}{5888}a^{22}+\frac{1}{256}a^{21}-\frac{23}{128}a^{20}+\frac{1}{256}a^{19}-\frac{93}{256}a^{18}+\frac{413}{5888}a^{17}+\frac{817}{2944}a^{16}+\frac{393}{5888}a^{15}-\frac{2805}{5888}a^{14}-\frac{861}{2944}a^{13}-\frac{2517}{5888}a^{12}-\frac{1055}{5888}a^{11}+\frac{31}{128}a^{10}-\frac{57}{256}a^{9}-\frac{75}{256}a^{8}+\frac{61}{128}a^{7}+\frac{627}{1472}a^{6}-\frac{87}{184}a^{5}+\frac{89}{368}a^{4}-\frac{87}{184}a^{3}-\frac{1}{23}a^{2}-\frac{1}{46}a-\frac{1}{23}$, $\frac{1}{11776}a^{29}-\frac{1}{11776}a^{28}-\frac{1}{5888}a^{27}-\frac{1}{11776}a^{26}-\frac{3}{11776}a^{25}+\frac{5}{5888}a^{24}-\frac{3}{11776}a^{23}+\frac{1}{512}a^{22}-\frac{23}{256}a^{21}+\frac{1}{512}a^{20}-\frac{93}{512}a^{19}-\frac{5475}{11776}a^{18}-\frac{2127}{5888}a^{17}-\frac{5495}{11776}a^{16}-\frac{2805}{11776}a^{15}+\frac{2083}{5888}a^{14}-\frac{2517}{11776}a^{13}-\frac{1055}{11776}a^{12}+\frac{31}{256}a^{11}-\frac{57}{512}a^{10}+\frac{181}{512}a^{9}-\frac{67}{256}a^{8}+\frac{627}{2944}a^{7}+\frac{97}{368}a^{6}-\frac{279}{736}a^{5}+\frac{97}{368}a^{4}-\frac{1}{46}a^{3}-\frac{1}{92}a^{2}-\frac{1}{46}a$, $\frac{1}{23552}a^{30}-\frac{1}{23552}a^{29}-\frac{1}{11776}a^{28}-\frac{1}{23552}a^{27}-\frac{3}{23552}a^{26}+\frac{5}{11776}a^{25}-\frac{3}{23552}a^{24}+\frac{1}{1024}a^{23}-\frac{17}{11776}a^{22}+\frac{1}{1024}a^{21}-\frac{93}{1024}a^{20}+\frac{6301}{23552}a^{19}-\frac{2127}{11776}a^{18}+\frac{6281}{23552}a^{17}+\frac{8971}{23552}a^{16}+\frac{2083}{11776}a^{15}+\frac{9259}{23552}a^{14}-\frac{1055}{23552}a^{13}-\frac{225}{512}a^{12}-\frac{2335}{23552}a^{11}+\frac{181}{1024}a^{10}-\frac{67}{512}a^{9}+\frac{627}{5888}a^{8}-\frac{271}{736}a^{7}+\frac{457}{1472}a^{6}-\frac{271}{736}a^{5}-\frac{1}{92}a^{4}+\frac{91}{184}a^{3}-\frac{1}{92}a^{2}+\frac{1}{23}$, $\frac{1}{114980864}a^{31}-\frac{111}{114980864}a^{30}-\frac{149}{3593152}a^{29}+\frac{4655}{114980864}a^{28}-\frac{9205}{114980864}a^{27}+\frac{745}{3593152}a^{26}-\frac{23283}{114980864}a^{25}+\frac{46913}{114980864}a^{24}-\frac{2533}{3593152}a^{23}+\frac{79175}{114980864}a^{22}-\frac{441707}{4999168}a^{21}-\frac{7023005}{114980864}a^{20}+\frac{6035717}{14372608}a^{19}+\frac{14880041}{114980864}a^{18}-\frac{26892115}{114980864}a^{17}-\frac{834603}{3593152}a^{16}+\frac{6873883}{114980864}a^{15}-\frac{39317113}{114980864}a^{14}-\frac{712115}{3593152}a^{13}-\frac{38428879}{114980864}a^{12}-\frac{48200971}{114980864}a^{11}-\frac{9983}{156224}a^{10}+\frac{1794351}{28745216}a^{9}-\frac{1829293}{14372608}a^{8}+\frac{68093}{449144}a^{7}-\frac{532219}{3593152}a^{6}+\frac{547217}{1796576}a^{5}-\frac{13857}{56143}a^{4}+\frac{108343}{449144}a^{3}+\frac{111223}{224572}a^{2}+\frac{1192}{56143}a-\frac{1187}{56143}$, $\frac{1}{229961728}a^{32}-\frac{1}{229961728}a^{31}+\frac{1275}{114980864}a^{30}+\frac{7431}{229961728}a^{29}-\frac{4883}{229961728}a^{28}+\frac{3609}{114980864}a^{27}-\frac{37163}{229961728}a^{26}+\frac{24423}{229961728}a^{25}-\frac{28245}{114980864}a^{24}+\frac{126367}{229961728}a^{23}-\frac{83051}{229961728}a^{22}-\frac{12012027}{229961728}a^{21}+\frac{5951101}{114980864}a^{20}-\frac{8640959}{229961728}a^{19}-\frac{16965285}{229961728}a^{18}+\frac{16284015}{114980864}a^{17}-\frac{79860205}{229961728}a^{16}-\frac{49039087}{229961728}a^{15}-\frac{14048019}{114980864}a^{14}+\frac{8505241}{229961728}a^{13}-\frac{79005741}{229961728}a^{12}+\frac{54948839}{114980864}a^{11}+\frac{2863805}{57490432}a^{10}-\frac{470547}{14372608}a^{9}+\frac{1340493}{14372608}a^{8}-\frac{849377}{7186304}a^{7}+\frac{139567}{1796576}a^{6}-\frac{451761}{1796576}a^{5}+\frac{172885}{898288}a^{4}-\frac{28411}{224572}a^{3}-\frac{109783}{224572}a^{2}-\frac{1881}{112286}a+\frac{622}{56143}$, $\frac{1}{459923456}a^{33}-\frac{1}{459923456}a^{32}-\frac{1}{229961728}a^{31}-\frac{2217}{459923456}a^{30}-\frac{2891}{459923456}a^{29}+\frac{341}{229961728}a^{28}+\frac{869}{459923456}a^{27}+\frac{14463}{459923456}a^{26}-\frac{1697}{229961728}a^{25}+\frac{13391}{459923456}a^{24}-\frac{49187}{459923456}a^{23}-\frac{96821}{19996672}a^{22}+\frac{16079749}{229961728}a^{21}-\frac{85377999}{459923456}a^{20}-\frac{156652301}{459923456}a^{19}+\frac{111254067}{229961728}a^{18}+\frac{122881027}{459923456}a^{17}-\frac{165403207}{459923456}a^{16}+\frac{4976857}{229961728}a^{15}+\frac{68618857}{459923456}a^{14}-\frac{219382837}{459923456}a^{13}+\frac{4930091}{229961728}a^{12}-\frac{884209}{4999168}a^{11}+\frac{28187831}{57490432}a^{10}-\frac{14307429}{28745216}a^{9}-\frac{14917}{898288}a^{8}-\frac{3427811}{7186304}a^{7}+\frac{1777289}{3593152}a^{6}+\frac{5897}{112286}a^{5}+\frac{415479}{898288}a^{4}-\frac{220665}{449144}a^{3}-\frac{7284}{56143}a^{2}-\frac{27885}{56143}a-\frac{11609}{56143}$, $\frac{1}{919846912}a^{34}-\frac{1}{919846912}a^{33}-\frac{1}{459923456}a^{32}-\frac{1}{919846912}a^{31}-\frac{14531}{919846912}a^{30}-\frac{10043}{459923456}a^{29}+\frac{5565}{919846912}a^{28}+\frac{3415}{919846912}a^{27}+\frac{50223}{459923456}a^{26}-\frac{27817}{919846912}a^{25}+\frac{99173}{919846912}a^{24}+\frac{2419229}{919846912}a^{23}-\frac{57961}{19996672}a^{22}-\frac{186014775}{919846912}a^{21}-\frac{421786741}{919846912}a^{20}-\frac{141821533}{459923456}a^{19}+\frac{239707755}{919846912}a^{18}+\frac{144434849}{919846912}a^{17}-\frac{204917367}{459923456}a^{16}+\frac{289579425}{919846912}a^{15}-\frac{107421053}{919846912}a^{14}-\frac{140458181}{459923456}a^{13}+\frac{43463555}{229961728}a^{12}+\frac{599793}{1249792}a^{11}-\frac{23493951}{57490432}a^{10}+\frac{11047319}{28745216}a^{9}+\frac{1378033}{7186304}a^{8}-\frac{324389}{7186304}a^{7}+\frac{369717}{3593152}a^{6}-\frac{61825}{898288}a^{5}+\frac{444623}{898288}a^{4}+\frac{46147}{112286}a^{3}-\frac{42655}{224572}a^{2}+\frac{4335}{112286}a-\frac{1705}{56143}$, $\frac{1}{1839693824}a^{35}-\frac{1}{1839693824}a^{34}-\frac{1}{919846912}a^{33}-\frac{1}{1839693824}a^{32}-\frac{3}{1839693824}a^{31}+\frac{3829}{919846912}a^{30}+\frac{21405}{1839693824}a^{29}-\frac{13737}{1839693824}a^{28}+\frac{9919}{919846912}a^{27}-\frac{107017}{1839693824}a^{26}+\frac{68709}{1839693824}a^{25}-\frac{2398851}{1839693824}a^{24}+\frac{1301121}{919846912}a^{23}+\frac{4243145}{1839693824}a^{22}+\frac{282835371}{1839693824}a^{21}+\frac{342819475}{919846912}a^{20}+\frac{296072331}{1839693824}a^{19}-\frac{42789215}{1839693824}a^{18}-\frac{162032007}{919846912}a^{17}+\frac{625961985}{1839693824}a^{16}-\frac{209043069}{1839693824}a^{15}-\frac{92701941}{919846912}a^{14}-\frac{143551413}{459923456}a^{13}+\frac{4460431}{57490432}a^{12}-\frac{50609553}{114980864}a^{11}+\frac{11294785}{28745216}a^{10}-\frac{7517255}{28745216}a^{9}+\frac{5870803}{14372608}a^{8}+\frac{296323}{3593152}a^{7}+\frac{996451}{3593152}a^{6}+\frac{61549}{1796576}a^{5}+\frac{99763}{224572}a^{4}-\frac{132269}{449144}a^{3}-\frac{82109}{224572}a^{2}+\frac{5123}{112286}a+\frac{6008}{56143}$, $\frac{1}{3679387648}a^{36}-\frac{1}{3679387648}a^{35}-\frac{1}{1839693824}a^{34}-\frac{1}{3679387648}a^{33}-\frac{3}{3679387648}a^{32}+\frac{5}{1839693824}a^{31}-\frac{67011}{3679387648}a^{30}-\frac{4543}{159973376}a^{29}+\frac{18735}{1839693824}a^{28}-\frac{7977}{3679387648}a^{27}+\frac{522469}{3679387648}a^{26}-\frac{2425827}{3679387648}a^{25}+\frac{1407185}{1839693824}a^{24}+\frac{117407}{159973376}a^{23}+\frac{2875787}{3679387648}a^{22}-\frac{357990525}{1839693824}a^{21}-\frac{97668245}{3679387648}a^{20}+\frac{260453921}{3679387648}a^{19}+\frac{22958431}{79986688}a^{18}-\frac{1415760863}{3679387648}a^{17}-\frac{1337773821}{3679387648}a^{16}+\frac{223704635}{1839693824}a^{15}-\frac{23380893}{919846912}a^{14}+\frac{47523395}{114980864}a^{13}+\frac{4190791}{9998336}a^{12}-\frac{50751637}{114980864}a^{11}+\frac{19042823}{57490432}a^{10}-\frac{336289}{7186304}a^{9}+\frac{3383549}{14372608}a^{8}+\frac{71499}{312448}a^{7}-\frac{12619}{56143}a^{6}+\frac{305329}{1796576}a^{5}+\frac{185001}{898288}a^{4}-\frac{104445}{224572}a^{3}+\frac{51695}{224572}a^{2}+\frac{3713}{56143}a+\frac{15183}{56143}$, $\frac{1}{7358775296}a^{37}-\frac{1}{7358775296}a^{36}-\frac{1}{3679387648}a^{35}-\frac{1}{7358775296}a^{34}-\frac{3}{7358775296}a^{33}+\frac{5}{3679387648}a^{32}-\frac{3}{7358775296}a^{31}-\frac{43625}{7358775296}a^{30}-\frac{67409}{3679387648}a^{29}+\frac{91159}{7358775296}a^{28}-\frac{138715}{7358775296}a^{27}-\frac{1564387}{7358775296}a^{26}+\frac{891313}{3679387648}a^{25}+\frac{5519369}{7358775296}a^{24}-\frac{53109}{7358775296}a^{23}+\frac{179669}{159973376}a^{22}+\frac{613406379}{7358775296}a^{21}+\frac{1932461153}{7358775296}a^{20}-\frac{288842743}{3679387648}a^{19}+\frac{1998275297}{7358775296}a^{18}-\frac{1535733309}{7358775296}a^{17}-\frac{1520232517}{3679387648}a^{16}+\frac{725853043}{1839693824}a^{15}-\frac{109585515}{229961728}a^{14}-\frac{66419983}{459923456}a^{13}-\frac{107742383}{229961728}a^{12}-\frac{752425}{2499584}a^{11}-\frac{1709671}{28745216}a^{10}-\frac{2146177}{28745216}a^{9}+\frac{3728945}{14372608}a^{8}-\frac{96247}{3593152}a^{7}-\frac{1173289}{3593152}a^{6}+\frac{142829}{1796576}a^{5}+\frac{28769}{224572}a^{4}-\frac{129529}{449144}a^{3}+\frac{110083}{224572}a^{2}-\frac{19807}{56143}a-\frac{5202}{56143}$, $\frac{1}{14717550592}a^{38}-\frac{1}{14717550592}a^{37}-\frac{1}{7358775296}a^{36}-\frac{1}{14717550592}a^{35}-\frac{3}{14717550592}a^{34}+\frac{5}{7358775296}a^{33}-\frac{3}{14717550592}a^{32}+\frac{1}{639893504}a^{31}+\frac{9711}{7358775296}a^{30}-\frac{557033}{14717550592}a^{29}+\frac{576421}{14717550592}a^{28}+\frac{1588381}{14717550592}a^{27}+\frac{12209}{7358775296}a^{26}-\frac{8403831}{14717550592}a^{25}+\frac{2945803}{14717550592}a^{24}-\frac{8875997}{7358775296}a^{23}+\frac{37407787}{14717550592}a^{22}+\frac{2212191201}{14717550592}a^{21}-\frac{116315873}{319946752}a^{20}-\frac{157302047}{14717550592}a^{19}-\frac{6845227965}{14717550592}a^{18}-\frac{3088664581}{7358775296}a^{17}-\frac{521830285}{3679387648}a^{16}-\frac{140246703}{459923456}a^{15}+\frac{138923529}{919846912}a^{14}-\frac{226931663}{459923456}a^{13}-\frac{10255617}{57490432}a^{12}-\frac{45686725}{114980864}a^{11}+\frac{25841235}{57490432}a^{10}-\frac{186853}{624896}a^{9}+\frac{5611077}{14372608}a^{8}-\frac{316869}{3593152}a^{7}+\frac{1001291}{3593152}a^{6}+\frac{220461}{898288}a^{5}-\frac{316185}{898288}a^{4}-\frac{177527}{449144}a^{3}+\frac{24577}{56143}a^{2}-\frac{6217}{56143}a-\frac{14207}{56143}$, $\frac{1}{29435101184}a^{39}-\frac{1}{29435101184}a^{38}-\frac{1}{14717550592}a^{37}-\frac{1}{29435101184}a^{36}-\frac{3}{29435101184}a^{35}+\frac{5}{14717550592}a^{34}-\frac{3}{29435101184}a^{33}+\frac{1}{1279787008}a^{32}-\frac{17}{14717550592}a^{31}+\frac{352791}{29435101184}a^{30}-\frac{391771}{29435101184}a^{29}-\frac{1493859}{29435101184}a^{28}+\frac{198321}{14717550592}a^{27}+\frac{6435465}{29435101184}a^{26}-\frac{1484021}{29435101184}a^{25}+\frac{6550819}{14717550592}a^{24}-\frac{29043157}{29435101184}a^{23}+\frac{19370977}{29435101184}a^{22}-\frac{29314529}{639893504}a^{21}+\frac{14171389153}{29435101184}a^{20}+\frac{12706638403}{29435101184}a^{19}-\frac{6239020037}{14717550592}a^{18}+\frac{3169759475}{7358775296}a^{17}+\frac{327640417}{919846912}a^{16}-\frac{411174007}{1839693824}a^{15}+\frac{380695649}{919846912}a^{14}-\frac{8362811}{114980864}a^{13}-\frac{112947381}{229961728}a^{12}+\frac{55730469}{114980864}a^{11}-\frac{1105771}{2499584}a^{10}-\frac{26327}{112286}a^{9}+\frac{5487403}{14372608}a^{8}+\frac{2033907}{7186304}a^{7}-\frac{508761}{1796576}a^{6}-\frac{466261}{1796576}a^{5}+\frac{327649}{898288}a^{4}-\frac{42671}{224572}a^{3}+\frac{46385}{224572}a^{2}-\frac{9875}{112286}a+\frac{12100}{56143}$ 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:  $19$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{18157}{7358775296} a^{39} - \frac{271}{3014656} a^{28} - \frac{7917}{3014656} a^{17} - \frac{8672}{56143} a^{6} \)  (order $66$) 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^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)
 
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^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);
 
[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^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);
 
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^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);
 
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

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

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)
 
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.

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

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

# 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
\(3\) Copy content Toggle raw display 3.20.10.1$x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$$2$$10$$10$20T3$[\ ]_{2}^{10}$
3.20.10.1$x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$$2$$10$$10$20T3$[\ ]_{2}^{10}$
\(7\) Copy content Toggle raw display 7.20.10.1$x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$$2$$10$$10$20T3$[\ ]_{2}^{10}$
7.20.10.1$x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$$2$$10$$10$20T3$[\ ]_{2}^{10}$
\(11\) Copy content Toggle raw display 11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$
11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$