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Properties

Label 40.0.32177243693...0761.1
Degree $40$
Signature $[0, 20]$
Discriminant $17^{20}\cdot 41^{39}$
Root discriminant $154.06$
Ramified primes $17, 41$
Class number Not computed
Class group Not computed
Galois group $C_{40}$ (as 40T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![56057779182769621, -56012699206030805, 56012699206030805, -55223799613101525, 55223799613101525, -51101799240046037, 51101799240046037, -40944012606445013, 40944012606445013, -26553814875510229, 26553814875510229, -13471816938296789, 13471816938296789, -5295568227538389, 5295568227538389, -1616256307697109, 1616256307697109, -385310040397269, 385310040397269, -72174586435029, 72174586435029, -10665836549589, 10665836549589, -1245029995989, 1245029995989, -114533209557, 114533209557, -8247186901, 8247186901, -458986965, 458986965, -19330517, 19330517, -595157, 595157, -12629, 12629, -165, 165, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + 165*x^38 - 165*x^37 + 12629*x^36 - 12629*x^35 + 595157*x^34 - 595157*x^33 + 19330517*x^32 - 19330517*x^31 + 458986965*x^30 - 458986965*x^29 + 8247186901*x^28 - 8247186901*x^27 + 114533209557*x^26 - 114533209557*x^25 + 1245029995989*x^24 - 1245029995989*x^23 + 10665836549589*x^22 - 10665836549589*x^21 + 72174586435029*x^20 - 72174586435029*x^19 + 385310040397269*x^18 - 385310040397269*x^17 + 1616256307697109*x^16 - 1616256307697109*x^15 + 5295568227538389*x^14 - 5295568227538389*x^13 + 13471816938296789*x^12 - 13471816938296789*x^11 + 26553814875510229*x^10 - 26553814875510229*x^9 + 40944012606445013*x^8 - 40944012606445013*x^7 + 51101799240046037*x^6 - 51101799240046037*x^5 + 55223799613101525*x^4 - 55223799613101525*x^3 + 56012699206030805*x^2 - 56012699206030805*x + 56057779182769621)
 
gp: K = bnfinit(x^40 - x^39 + 165*x^38 - 165*x^37 + 12629*x^36 - 12629*x^35 + 595157*x^34 - 595157*x^33 + 19330517*x^32 - 19330517*x^31 + 458986965*x^30 - 458986965*x^29 + 8247186901*x^28 - 8247186901*x^27 + 114533209557*x^26 - 114533209557*x^25 + 1245029995989*x^24 - 1245029995989*x^23 + 10665836549589*x^22 - 10665836549589*x^21 + 72174586435029*x^20 - 72174586435029*x^19 + 385310040397269*x^18 - 385310040397269*x^17 + 1616256307697109*x^16 - 1616256307697109*x^15 + 5295568227538389*x^14 - 5295568227538389*x^13 + 13471816938296789*x^12 - 13471816938296789*x^11 + 26553814875510229*x^10 - 26553814875510229*x^9 + 40944012606445013*x^8 - 40944012606445013*x^7 + 51101799240046037*x^6 - 51101799240046037*x^5 + 55223799613101525*x^4 - 55223799613101525*x^3 + 56012699206030805*x^2 - 56012699206030805*x + 56057779182769621, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} + 165 x^{38} - 165 x^{37} + 12629 x^{36} - 12629 x^{35} + 595157 x^{34} - 595157 x^{33} + 19330517 x^{32} - 19330517 x^{31} + 458986965 x^{30} - 458986965 x^{29} + 8247186901 x^{28} - 8247186901 x^{27} + 114533209557 x^{26} - 114533209557 x^{25} + 1245029995989 x^{24} - 1245029995989 x^{23} + 10665836549589 x^{22} - 10665836549589 x^{21} + 72174586435029 x^{20} - 72174586435029 x^{19} + 385310040397269 x^{18} - 385310040397269 x^{17} + 1616256307697109 x^{16} - 1616256307697109 x^{15} + 5295568227538389 x^{14} - 5295568227538389 x^{13} + 13471816938296789 x^{12} - 13471816938296789 x^{11} + 26553814875510229 x^{10} - 26553814875510229 x^{9} + 40944012606445013 x^{8} - 40944012606445013 x^{7} + 51101799240046037 x^{6} - 51101799240046037 x^{5} + 55223799613101525 x^{4} - 55223799613101525 x^{3} + 56012699206030805 x^{2} - 56012699206030805 x + 56057779182769621 \)

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:  \(3217724369317177596907031731223092219803204403094256869508157994902742845244016168900761=17^{20}\cdot 41^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $154.06$
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:  $17, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(697=17\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{697}(256,·)$, $\chi_{697}(1,·)$, $\chi_{697}(645,·)$, $\chi_{697}(135,·)$, $\chi_{697}(392,·)$, $\chi_{697}(526,·)$, $\chi_{697}(528,·)$, $\chi_{697}(664,·)$, $\chi_{697}(18,·)$, $\chi_{697}(67,·)$, $\chi_{697}(662,·)$, $\chi_{697}(407,·)$, $\chi_{697}(152,·)$, $\chi_{697}(409,·)$, $\chi_{697}(154,·)$, $\chi_{697}(424,·)$, $\chi_{697}(681,·)$, $\chi_{697}(426,·)$, $\chi_{697}(647,·)$, $\chi_{697}(560,·)$, $\chi_{697}(307,·)$, $\chi_{697}(186,·)$, $\chi_{697}(443,·)$, $\chi_{697}(577,·)$, $\chi_{697}(322,·)$, $\chi_{697}(579,·)$, $\chi_{697}(324,·)$, $\chi_{697}(458,·)$, $\chi_{697}(460,·)$, $\chi_{697}(339,·)$, $\chi_{697}(86,·)$, $\chi_{697}(475,·)$, $\chi_{697}(220,·)$, $\chi_{697}(101,·)$, $\chi_{697}(356,·)$, $\chi_{697}(613,·)$, $\chi_{697}(103,·)$, $\chi_{697}(494,·)$, $\chi_{697}(628,·)$, $\chi_{697}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{13596008554085389} a^{21} + \frac{339835402508973}{13596008554085389} a^{20} + \frac{84}{13596008554085389} a^{19} - \frac{5184907452938}{13596008554085389} a^{18} + \frac{3024}{13596008554085389} a^{17} - \frac{176286853399892}{13596008554085389} a^{16} + \frac{60928}{13596008554085389} a^{15} - \frac{3318340769880320}{13596008554085389} a^{14} + \frac{752640}{13596008554085389} a^{13} + \frac{3041899404867527}{13596008554085389} a^{12} + \frac{5870592}{13596008554085389} a^{11} - \frac{1970162902759479}{13596008554085389} a^{10} + \frac{28700672}{13596008554085389} a^{9} + \frac{3210166320246852}{13596008554085389} a^{8} + \frac{84344832}{13596008554085389} a^{7} - \frac{2556520253304202}{13596008554085389} a^{6} + \frac{136249344}{13596008554085389} a^{5} + \frac{3602353960412442}{13596008554085389} a^{4} + \frac{100925440}{13596008554085389} a^{3} + \frac{510595081949785}{13596008554085389} a^{2} + \frac{22020096}{13596008554085389} a - \frac{2678354104261095}{13596008554085389}$, $\frac{1}{13596008554085389} a^{22} + \frac{88}{13596008554085389} a^{20} - \frac{1359341610035892}{13596008554085389} a^{19} + \frac{3344}{13596008554085389} a^{18} + \frac{5458106069955320}{13596008554085389} a^{17} + \frac{71808}{13596008554085389} a^{16} - \frac{2088716964539817}{13596008554085389} a^{15} + \frac{957440}{13596008554085389} a^{14} - \frac{2562525494233325}{13596008554085389} a^{13} + \frac{8200192}{13596008554085389} a^{12} + \frac{541751968144198}{13596008554085389} a^{11} + \frac{45101056}{13596008554085389} a^{10} + \frac{3405298085477816}{13596008554085389} a^{9} + \frac{154632192}{13596008554085389} a^{8} + \frac{1069093385230453}{13596008554085389} a^{7} + \frac{309264384}{13596008554085389} a^{6} + \frac{2138186770460906}{13596008554085389} a^{5} + \frac{317194240}{13596008554085389} a^{4} + \frac{1943806154964460}{13596008554085389} a^{3} + \frac{126877696}{13596008554085389} a^{2} + \frac{5050351159352319}{13596008554085389} a + \frac{8388608}{13596008554085389}$, $\frac{1}{13596008554085389} a^{23} - \frac{4072839922654738}{13596008554085389} a^{20} - \frac{4048}{13596008554085389} a^{19} + \frac{5914377925813864}{13596008554085389} a^{18} - \frac{194304}{13596008554085389} a^{17} - \frac{171482419434710}{13596008554085389} a^{16} - \frac{4404224}{13596008554085389} a^{15} + \frac{3935282619441666}{13596008554085389} a^{14} - \frac{58032128}{13596008554085389} a^{13} + \frac{4774775421509602}{13596008554085389} a^{12} - \frac{471511040}{13596008554085389} a^{11} + \frac{31522325201911}{13596008554085389} a^{10} - \frac{2371026944}{13596008554085389} a^{9} + \frac{4090636839300646}{13596008554085389} a^{8} - \frac{7113080832}{13596008554085389} a^{7} - \frac{4020176358220931}{13596008554085389} a^{6} - \frac{11672748032}{13596008554085389} a^{5} - \frac{2355145617366489}{13596008554085389} a^{4} - \frac{8754561024}{13596008554085389} a^{3} + \frac{906009610027406}{13596008554085389} a^{2} - \frac{1929379840}{13596008554085389} a + \frac{4563015755524747}{13596008554085389}$, $\frac{1}{13596008554085389} a^{24} - \frac{4416}{13596008554085389} a^{20} - \frac{5463290977408258}{13596008554085389} a^{19} - \frac{223744}{13596008554085389} a^{18} - \frac{1887306312869432}{13596008554085389} a^{17} - \frac{5405184}{13596008554085389} a^{16} - \frac{422039039201498}{13596008554085389} a^{15} - \frac{76873728}{13596008554085389} a^{14} + \frac{3733541083543204}{13596008554085389} a^{13} - \frac{685834240}{13596008554085389} a^{12} + \frac{67499621515240}{13596008554085389} a^{11} - \frac{3879862272}{13596008554085389} a^{10} - \frac{265477007862653}{13596008554085389} a^{9} - \frac{13579517952}{13596008554085389} a^{8} + \frac{2323582748725534}{13596008554085389} a^{7} - \frac{27590131712}{13596008554085389} a^{6} - \frac{3611849682679067}{13596008554085389} a^{5} - \frac{28651290624}{13596008554085389} a^{4} + \frac{5010676433531196}{13596008554085389} a^{3} - \frac{11576279040}{13596008554085389} a^{2} + \frac{514576315530887}{13596008554085389} a - \frac{771751936}{13596008554085389}$, $\frac{1}{13596008554085389} a^{25} - \frac{311094447176280}{13596008554085389} a^{20} + \frac{147200}{13596008554085389} a^{19} + \frac{2408159483127138}{13596008554085389} a^{18} + \frac{7948800}{13596008554085389} a^{17} - \frac{3932296070257397}{13596008554085389} a^{16} + \frac{192184320}{13596008554085389} a^{15} + \frac{6437922596099426}{13596008554085389} a^{14} + \frac{2637824000}{13596008554085389} a^{13} + \frac{238820080150140}{13596008554085389} a^{12} + \frac{22044672000}{13596008554085389} a^{11} + \frac{940619020927043}{13596008554085389} a^{10} + \frac{113162649600}{13596008554085389} a^{9} - \frac{2218868952236761}{13596008554085389} a^{8} + \frac{344876646400}{13596008554085389} a^{7} + \frac{5077820170923160}{13596008554085389} a^{6} + \frac{573025812480}{13596008554085389} a^{5} + \frac{5675757334969938}{13596008554085389} a^{4} + \frac{434110464000}{13596008554085389} a^{3} - \frac{1634961772393127}{13596008554085389} a^{2} + \frac{96468992000}{13596008554085389} a + \frac{915717637292910}{13596008554085389}$, $\frac{1}{13596008554085389} a^{26} + \frac{166400}{13596008554085389} a^{20} + \frac{1348075937763880}{13596008554085389} a^{19} + \frac{9484800}{13596008554085389} a^{18} - \frac{1307278041078518}{13596008554085389} a^{17} + \frac{244408320}{13596008554085389} a^{16} - \frac{5631532796630389}{13596008554085389} a^{15} + \frac{3620864000}{13596008554085389} a^{14} + \frac{5500232931045371}{13596008554085389} a^{13} + \frac{33226752000}{13596008554085389} a^{12} - \frac{1527588115163400}{13596008554085389} a^{11} + \frac{191884492800}{13596008554085389} a^{10} - \frac{3710986075501402}{13596008554085389} a^{9} + \frac{682255974400}{13596008554085389} a^{8} + \frac{5132335907969240}{13596008554085389} a^{7} + \frac{1403498004480}{13596008554085389} a^{6} - \frac{6786340524342138}{13596008554085389} a^{5} + \frac{1472200704000}{13596008554085389} a^{4} - \frac{1802139669159961}{13596008554085389} a^{3} + \frac{599785472000}{13596008554085389} a^{2} - \frac{4806361117346471}{13596008554085389} a + \frac{40265318400}{13596008554085389}$, $\frac{1}{13596008554085389} a^{27} - \frac{1463325114210469}{13596008554085389} a^{20} - \frac{4492800}{13596008554085389} a^{19} + \frac{4912783220425175}{13596008554085389} a^{18} - \frac{258785280}{13596008554085389} a^{17} + \frac{1910421783214338}{13596008554085389} a^{16} - \frac{6517555200}{13596008554085389} a^{15} + \frac{2708933946389914}{13596008554085389} a^{14} - \frac{92012544000}{13596008554085389} a^{13} + \frac{5809910527376270}{13596008554085389} a^{12} - \frac{784982016000}{13596008554085389} a^{11} + \frac{4437776994904630}{13596008554085389} a^{10} - \frac{4093535846400}{13596008554085389} a^{9} - \frac{6559280261440528}{13596008554085389} a^{8} - \frac{12631482040320}{13596008554085389} a^{7} + \frac{6268169071219630}{13596008554085389} a^{6} - \frac{21199690137600}{13596008554085389} a^{5} + \frac{919988771206860}{13596008554085389} a^{4} - \frac{16194207744000}{13596008554085389} a^{3} - \frac{6370543081974610}{13596008554085389} a^{2} - \frac{3623878656000}{13596008554085389} a + \frac{962546127156580}{13596008554085389}$, $\frac{1}{13596008554085389} a^{28} - \frac{5241600}{13596008554085389} a^{20} + \frac{5468015827336070}{13596008554085389} a^{19} - \frac{318689280}{13596008554085389} a^{18} - \frac{5293221476164220}{13596008554085389} a^{17} - \frac{8554291200}{13596008554085389} a^{16} - \frac{2442605130135916}{13596008554085389} a^{15} - \frac{130351104000}{13596008554085389} a^{14} + \frac{4554937653743096}{13596008554085389} a^{13} - \frac{1221083136000}{13596008554085389} a^{12} + \frac{5525795423834184}{13596008554085389} a^{11} - \frac{7163687731200}{13596008554085389} a^{10} - \frac{2750746670461085}{13596008554085389} a^{9} - \frac{25789275832320}{13596008554085389} a^{8} + \frac{4142055326422510}{13596008554085389} a^{7} - \frac{53588105625600}{13596008554085389} a^{6} - \frac{2514502662696180}{13596008554085389} a^{5} - \frac{56679727104000}{13596008554085389} a^{4} + \frac{149394891931916}{13596008554085389} a^{3} - \frac{23253221376000}{13596008554085389} a^{2} + \frac{6587480692726215}{13596008554085389} a - \frac{1570347417600}{13596008554085389}$, $\frac{1}{13596008554085389} a^{29} + \frac{5653093362973035}{13596008554085389} a^{20} + \frac{121605120}{13596008554085389} a^{19} - \frac{4083027179292409}{13596008554085389} a^{18} + \frac{7296307200}{13596008554085389} a^{17} - \frac{2084024698750509}{13596008554085389} a^{16} + \frac{189009100800}{13596008554085389} a^{15} + \frac{4106800068834963}{13596008554085389} a^{14} + \frac{2723954688000}{13596008554085389} a^{13} + \frac{5526733603286192}{13596008554085389} a^{12} + \frac{23607607296000}{13596008554085389} a^{11} - \frac{1112620860628702}{13596008554085389} a^{10} + \frac{124648166522880}{13596008554085389} a^{9} + \frac{5335733702524699}{13596008554085389} a^{8} + \frac{388513765785600}{13596008554085389} a^{7} - \frac{5851298300330202}{13596008554085389} a^{6} + \frac{657484834406400}{13596008554085389} a^{5} + \frac{2776404621932083}{13596008554085389} a^{4} + \frac{505757564928000}{13596008554085389} a^{3} - \frac{5322825914871657}{13596008554085389} a^{2} + \frac{113850187776000}{13596008554085389} a - \frac{3128185897261492}{13596008554085389}$, $\frac{1}{13596008554085389} a^{30} + \frac{145926144}{13596008554085389} a^{20} - \frac{3082570276038734}{13596008554085389} a^{19} + \frac{9241989120}{13596008554085389} a^{18} + \frac{6740406710211013}{13596008554085389} a^{17} + \frac{255162286080}{13596008554085389} a^{16} + \frac{119081492918020}{13596008554085389} a^{15} + \frac{3969191116800}{13596008554085389} a^{14} - \frac{3745058940142548}{13596008554085389} a^{13} + \frac{37772171673600}{13596008554085389} a^{12} - \frac{5452559928945929}{13596008554085389} a^{11} + \frac{224366699741184}{13596008554085389} a^{10} + \frac{735597601300398}{13596008554085389} a^{9} + \frac{815878908149760}{13596008554085389} a^{8} + \frac{3382438492704544}{13596008554085389} a^{7} + \frac{1709460569456640}{13596008554085389} a^{6} - \frac{5316188880848601}{13596008554085389} a^{5} + \frac{1820727233740800}{13596008554085389} a^{4} + \frac{4796448295317927}{13596008554085389} a^{3} + \frac{751411239321600}{13596008554085389} a^{2} - \frac{6358747673036102}{13596008554085389} a + \frac{51004884123648}{13596008554085389}$, $\frac{1}{13596008554085389} a^{31} - \frac{2796725446537684}{13596008554085389} a^{20} - \frac{3015806976}{13596008554085389} a^{19} + \frac{2415975962124235}{13596008554085389} a^{18} - \frac{186118373376}{13596008554085389} a^{17} + \frac{2644530152906847}{13596008554085389} a^{16} - \frac{4921796984832}{13596008554085389} a^{15} + \frac{2205007317738888}{13596008554085389} a^{14} - \frac{72057681346560}{13596008554085389} a^{13} + \frac{1047405261071155}{13596008554085389} a^{12} - \frac{632306153816064}{13596008554085389} a^{11} + \frac{2444284823784009}{13596008554085389} a^{10} - \frac{3372299487019008}{13596008554085389} a^{9} - \frac{4832911758463282}{13596008554085389} a^{8} + \frac{2997353023454221}{13596008554085389} a^{7} + \frac{1653427609062913}{13596008554085389} a^{6} - \frac{4465605604623347}{13596008554085389} a^{5} + \frac{6710464630664625}{13596008554085389} a^{4} - \frac{380240497296371}{13596008554085389} a^{3} - \frac{5430739752868006}{13596008554085389} a^{2} - \frac{3162302815666176}{13596008554085389} a - \frac{1557280392876799}{13596008554085389}$, $\frac{1}{13596008554085389} a^{32} - \frac{3711762432}{13596008554085389} a^{20} + \frac{6208768051838078}{13596008554085389} a^{19} - \frac{241794809856}{13596008554085389} a^{18} + \frac{3224959841751305}{13596008554085389} a^{17} - \frac{6814795825152}{13596008554085389} a^{16} + \frac{2317805613569303}{13596008554085389} a^{15} - \frac{107690600693760}{13596008554085389} a^{14} - \frac{5556856116368065}{13596008554085389} a^{13} - \frac{1037630611390464}{13596008554085389} a^{12} + \frac{5315080296099649}{13596008554085389} a^{11} - \frac{6225783668342784}{13596008554085389} a^{10} - \frac{2883252608551610}{13596008554085389} a^{9} + \frac{4364143657580570}{13596008554085389} a^{8} + \frac{5163281673442457}{13596008554085389} a^{7} + \frac{6219729793118260}{13596008554085389} a^{6} - \frac{169932853724163}{13596008554085389} a^{5} + \frac{2779422334316596}{13596008554085389} a^{4} + \frac{3068678072040503}{13596008554085389} a^{3} + \frac{5785659586738202}{13596008554085389} a^{2} - \frac{6377266157629588}{13596008554085389} a - \frac{1459524376461312}{13596008554085389}$, $\frac{1}{13596008554085389} a^{33} - \frac{6639801377146738}{13596008554085389} a^{20} + \frac{69993234432}{13596008554085389} a^{19} - \frac{4959952920629800}{13596008554085389} a^{18} + \frac{4409573769216}{13596008554085389} a^{17} - \frac{4049057225680487}{13596008554085389} a^{16} + \frac{118459660763136}{13596008554085389} a^{15} + \frac{4158277836684889}{13596008554085389} a^{14} + \frac{1755990265430016}{13596008554085389} a^{13} - \frac{3140345836170153}{13596008554085389} a^{12} + \frac{1968450616771571}{13596008554085389} a^{11} + \frac{1763707795552020}{13596008554085389} a^{10} + \frac{2126151327651762}{13596008554085389} a^{9} + \frac{5655956899478887}{13596008554085389} a^{8} + \frac{6579511800105737}{13596008554085389} a^{7} - \frac{2094734271346502}{13596008554085389} a^{6} + \frac{5452302277001811}{13596008554085389} a^{5} - \frac{2736325838508768}{13596008554085389} a^{4} - \frac{291323302582610}{13596008554085389} a^{3} + \frac{5228195564315875}{13596008554085389} a^{2} - \frac{1302210619140174}{13596008554085389} a - \frac{3966126141715103}{13596008554085389}$, $\frac{1}{13596008554085389} a^{34} + \frac{88139628544}{13596008554085389} a^{20} - \frac{4652987957804757}{13596008554085389} a^{19} + \frac{5861285298176}{13596008554085389} a^{18} - \frac{6594327118064328}{13596008554085389} a^{17} + \frac{167817852747776}{13596008554085389} a^{16} + \frac{4742057822388058}{13596008554085389} a^{15} + \frac{2685085643964416}{13596008554085389} a^{14} + \frac{871993150966549}{13596008554085389} a^{13} - \frac{1059098006485018}{13596008554085389} a^{12} + \frac{429531365413362}{13596008554085389} a^{11} - \frac{5047942083825820}{13596008554085389} a^{10} - \frac{866683425643441}{13596008554085389} a^{9} - \frac{859159584937519}{13596008554085389} a^{8} - \frac{453092746065095}{13596008554085389} a^{7} + \frac{1783581925910369}{13596008554085389} a^{6} - \frac{84443374592649}{13596008554085389} a^{5} + \frac{1920780535595782}{13596008554085389} a^{4} - \frac{1871452757729111}{13596008554085389} a^{3} - \frac{1465676202515989}{13596008554085389} a^{2} + \frac{3878465589024987}{13596008554085389} a - \frac{2732278957487143}{13596008554085389}$, $\frac{1}{13596008554085389} a^{35} + \frac{3927077319146993}{13596008554085389} a^{20} - \frac{1542443499520}{13596008554085389} a^{19} + \frac{10467333806222}{13596008554085389} a^{18} - \frac{98716383969280}{13596008554085389} a^{17} - \frac{1827076233464855}{13596008554085389} a^{16} - \frac{2685085643964416}{13596008554085389} a^{15} - \frac{418217972437430}{13596008554085389} a^{14} + \frac{583534736585767}{13596008554085389} a^{13} - \frac{5749114817003531}{13596008554085389} a^{12} - \frac{5831415241959086}{13596008554085389} a^{11} + \frac{2949757218358786}{13596008554085389} a^{10} - \frac{1668137568236733}{13596008554085389} a^{9} - \frac{5462074856761162}{13596008554085389} a^{8} + \frac{4678098924533544}{13596008554085389} a^{7} - \frac{5093317810260066}{13596008554085389} a^{6} - \frac{1770235730680867}{13596008554085389} a^{5} - \frac{4733543663792035}{13596008554085389} a^{4} - \frac{5206874070430943}{13596008554085389} a^{3} - \frac{4101218997713589}{13596008554085389} a^{2} + \frac{653862333503260}{13596008554085389} a - \frac{4445257162082995}{13596008554085389}$, $\frac{1}{13596008554085389} a^{36} - \frac{1983141642240}{13596008554085389} a^{20} - \frac{3559822176491854}{13596008554085389} a^{19} - \frac{133972235386880}{13596008554085389} a^{18} + \frac{5602586936658299}{13596008554085389} a^{17} - \frac{3883784592162816}{13596008554085389} a^{16} + \frac{6769424388334077}{13596008554085389} a^{15} + \frac{5211806937492545}{13596008554085389} a^{14} - \frac{4134999324844174}{13596008554085389} a^{13} - \frac{4169851034464695}{13596008554085389} a^{12} - \frac{3494553918517774}{13596008554085389} a^{11} - \frac{303692048581116}{13596008554085389} a^{10} + \frac{5331097308996866}{13596008554085389} a^{9} - \frac{3070285973884915}{13596008554085389} a^{8} - \frac{2231442254254442}{13596008554085389} a^{7} - \frac{2017940559592689}{13596008554085389} a^{6} - \frac{1156360731327980}{13596008554085389} a^{5} + \frac{1281499170471974}{13596008554085389} a^{4} - \frac{6487012586290359}{13596008554085389} a^{3} + \frac{4623740786915910}{13596008554085389} a^{2} + \frac{1214209943175432}{13596008554085389} a + \frac{317590276273012}{13596008554085389}$, $\frac{1}{13596008554085389} a^{37} - \frac{6267171150710783}{13596008554085389} a^{20} + \frac{32611662561280}{13596008554085389} a^{19} + \frac{6550805864510203}{13596008554085389} a^{18} + \frac{2113235733970944}{13596008554085389} a^{17} - \frac{1779157726957354}{13596008554085389} a^{16} + \frac{3676583929122764}{13596008554085389} a^{15} + \frac{4500738120617270}{13596008554085389} a^{14} + \frac{6456942185741504}{13596008554085389} a^{13} + \frac{6362903922051123}{13596008554085389} a^{12} + \frac{3728445455331980}{13596008554085389} a^{11} - \frac{5760770133583602}{13596008554085389} a^{10} + \frac{1535710096262011}{13596008554085389} a^{9} - \frac{5095417473318741}{13596008554085389} a^{8} - \frac{5962534535229876}{13596008554085389} a^{7} - \frac{4022758638536904}{13596008554085389} a^{6} - \frac{4044690439858452}{13596008554085389} a^{5} + \frac{3525998583783875}{13596008554085389} a^{4} - \frac{6371367063595348}{13596008554085389} a^{3} + \frac{4009001941683335}{13596008554085389} a^{2} - \frac{1092541723541416}{13596008554085389} a - \frac{1951778008773030}{13596008554085389}$, $\frac{1}{13596008554085389} a^{38} + \frac{42732523356160}{13596008554085389} a^{20} + \frac{2748848914885804}{13596008554085389} a^{19} + \frac{2922904597561344}{13596008554085389} a^{18} - \frac{2689522372581828}{13596008554085389} a^{17} + \frac{4013308183025586}{13596008554085389} a^{16} - \frac{6791641415031560}{13596008554085389} a^{15} - \frac{5599318725732667}{13596008554085389} a^{14} - \frac{1169936728664472}{13596008554085389} a^{13} - \frac{2549940079614846}{13596008554085389} a^{12} + \frac{3463115561261702}{13596008554085389} a^{11} + \frac{3018464671963263}{13596008554085389} a^{10} + \frac{6764447235938461}{13596008554085389} a^{9} + \frac{392127932061204}{13596008554085389} a^{8} - \frac{5517791341641983}{13596008554085389} a^{7} - \frac{3044299732464674}{13596008554085389} a^{6} + \frac{5294333003101944}{13596008554085389} a^{5} + \frac{1677125597689569}{13596008554085389} a^{4} + \frac{5822435161699882}{13596008554085389} a^{3} - \frac{1310244843550362}{13596008554085389} a^{2} - \frac{6747323117783841}{13596008554085389} a + \frac{1614104776889531}{13596008554085389}$, $\frac{1}{13596008554085389} a^{39} - \frac{6216332774275368}{13596008554085389} a^{20} - \frac{666627364356096}{13596008554085389} a^{19} + \frac{684395861050029}{13596008554085389} a^{18} - \frac{2845765459233753}{13596008554085389} a^{17} - \frac{846489904231534}{13596008554085389} a^{16} + \frac{1227140614545541}{13596008554085389} a^{15} - \frac{3627627303852843}{13596008554085389} a^{14} + \frac{3399920106153128}{13596008554085389} a^{13} - \frac{2399600831117261}{13596008554085389} a^{12} - \frac{2237458384571018}{13596008554085389} a^{11} + \frac{1928226758539404}{13596008554085389} a^{10} + \frac{3399188825407207}{13596008554085389} a^{9} + \frac{4832140450271519}{13596008554085389} a^{8} + \frac{4127959804623328}{13596008554085389} a^{7} - \frac{1650162984237671}{13596008554085389} a^{6} - \frac{3470465678196445}{13596008554085389} a^{5} - \frac{6257186960995984}{13596008554085389} a^{4} + \frac{3437174412069317}{13596008554085389} a^{3} - \frac{656102084288616}{13596008554085389} a^{2} - \frac{6496500412814528}{13596008554085389} a + \frac{487297682679618}{13596008554085389}$

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:  \( -1 \) (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

$C_{40}$ (as 40T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 40
The 40 conjugacy class representatives for $C_{40}$
Character table for $C_{40}$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 5.5.2825761.1, 8.0.16266071708815001.1, 10.10.327381934393961.1, \(\Q(\zeta_{41})^+\)

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 $20^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{5}$ $20^{2}$ $40$ $40$ $40$ R $40$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{8}$ $40$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ R $20^{2}$ $40$ $40$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

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
17Data not computed
41Data not computed