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Properties

Label 46.0.20927324417...1875.1
Degree $46$
Signature $[0, 23]$
Discriminant $-\,5^{23}\cdot 47^{45}$
Root discriminant $96.66$
Ramified primes $5, 47$
Class number Not computed
Class group Not computed
Galois group $C_{46}$ (as 46T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6643838879, -6643838832, 6643838832, -6643834508, 6643834508, -6643715598, 6643715598, -6642169768, 6642169768, -6630576043, 6630576043, -6574504573, 6574504573, -6386880808, 6386880808, -5931223093, 5931223093, -5100317848, 5100317848, -3934135048, 3934135048, -2651333968, 2651333968, -1530784408, 1530784408, -746399716, 746399716, -303926300, 303926300, -102306640, 102306640, -28162636, 28162636, -6256453, 6256453, -1102057, 1102057, -150119, 150119, -15229, 15229, -1082, 1082, -48, 48, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^46 - x^45 + 48*x^44 - 48*x^43 + 1082*x^42 - 1082*x^41 + 15229*x^40 - 15229*x^39 + 150119*x^38 - 150119*x^37 + 1102057*x^36 - 1102057*x^35 + 6256453*x^34 - 6256453*x^33 + 28162636*x^32 - 28162636*x^31 + 102306640*x^30 - 102306640*x^29 + 303926300*x^28 - 303926300*x^27 + 746399716*x^26 - 746399716*x^25 + 1530784408*x^24 - 1530784408*x^23 + 2651333968*x^22 - 2651333968*x^21 + 3934135048*x^20 - 3934135048*x^19 + 5100317848*x^18 - 5100317848*x^17 + 5931223093*x^16 - 5931223093*x^15 + 6386880808*x^14 - 6386880808*x^13 + 6574504573*x^12 - 6574504573*x^11 + 6630576043*x^10 - 6630576043*x^9 + 6642169768*x^8 - 6642169768*x^7 + 6643715598*x^6 - 6643715598*x^5 + 6643834508*x^4 - 6643834508*x^3 + 6643838832*x^2 - 6643838832*x + 6643838879)
 
gp: K = bnfinit(x^46 - x^45 + 48*x^44 - 48*x^43 + 1082*x^42 - 1082*x^41 + 15229*x^40 - 15229*x^39 + 150119*x^38 - 150119*x^37 + 1102057*x^36 - 1102057*x^35 + 6256453*x^34 - 6256453*x^33 + 28162636*x^32 - 28162636*x^31 + 102306640*x^30 - 102306640*x^29 + 303926300*x^28 - 303926300*x^27 + 746399716*x^26 - 746399716*x^25 + 1530784408*x^24 - 1530784408*x^23 + 2651333968*x^22 - 2651333968*x^21 + 3934135048*x^20 - 3934135048*x^19 + 5100317848*x^18 - 5100317848*x^17 + 5931223093*x^16 - 5931223093*x^15 + 6386880808*x^14 - 6386880808*x^13 + 6574504573*x^12 - 6574504573*x^11 + 6630576043*x^10 - 6630576043*x^9 + 6642169768*x^8 - 6642169768*x^7 + 6643715598*x^6 - 6643715598*x^5 + 6643834508*x^4 - 6643834508*x^3 + 6643838832*x^2 - 6643838832*x + 6643838879, 1)
 

Normalized defining polynomial

\( x^{46} - x^{45} + 48 x^{44} - 48 x^{43} + 1082 x^{42} - 1082 x^{41} + 15229 x^{40} - 15229 x^{39} + 150119 x^{38} - 150119 x^{37} + 1102057 x^{36} - 1102057 x^{35} + 6256453 x^{34} - 6256453 x^{33} + 28162636 x^{32} - 28162636 x^{31} + 102306640 x^{30} - 102306640 x^{29} + 303926300 x^{28} - 303926300 x^{27} + 746399716 x^{26} - 746399716 x^{25} + 1530784408 x^{24} - 1530784408 x^{23} + 2651333968 x^{22} - 2651333968 x^{21} + 3934135048 x^{20} - 3934135048 x^{19} + 5100317848 x^{18} - 5100317848 x^{17} + 5931223093 x^{16} - 5931223093 x^{15} + 6386880808 x^{14} - 6386880808 x^{13} + 6574504573 x^{12} - 6574504573 x^{11} + 6630576043 x^{10} - 6630576043 x^{9} + 6642169768 x^{8} - 6642169768 x^{7} + 6643715598 x^{6} - 6643715598 x^{5} + 6643834508 x^{4} - 6643834508 x^{3} + 6643838832 x^{2} - 6643838832 x + 6643838879 \)

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

Invariants

Degree:  $46$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 23]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-20927324417353262576794873573459132405730192352302940670222843367282253010356426239013671875=-\,5^{23}\cdot 47^{45}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.66$
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:  $5, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(235=5\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{235}(1,·)$, $\chi_{235}(131,·)$, $\chi_{235}(179,·)$, $\chi_{235}(134,·)$, $\chi_{235}(129,·)$, $\chi_{235}(136,·)$, $\chi_{235}(139,·)$, $\chi_{235}(16,·)$, $\chi_{235}(19,·)$, $\chi_{235}(21,·)$, $\chi_{235}(154,·)$, $\chi_{235}(29,·)$, $\chi_{235}(36,·)$, $\chi_{235}(6,·)$, $\chi_{235}(166,·)$, $\chi_{235}(39,·)$, $\chi_{235}(71,·)$, $\chi_{235}(44,·)$, $\chi_{235}(174,·)$, $\chi_{235}(51,·)$, $\chi_{235}(184,·)$, $\chi_{235}(61,·)$, $\chi_{235}(191,·)$, $\chi_{235}(196,·)$, $\chi_{235}(69,·)$, $\chi_{235}(199,·)$, $\chi_{235}(206,·)$, $\chi_{235}(56,·)$, $\chi_{235}(214,·)$, $\chi_{235}(216,·)$, $\chi_{235}(164,·)$, $\chi_{235}(219,·)$, $\chi_{235}(101,·)$, $\chi_{235}(96,·)$, $\chi_{235}(99,·)$, $\chi_{235}(229,·)$, $\chi_{235}(81,·)$, $\chi_{235}(104,·)$, $\chi_{235}(106,·)$, $\chi_{235}(109,·)$, $\chi_{235}(111,·)$, $\chi_{235}(114,·)$, $\chi_{235}(121,·)$, $\chi_{235}(124,·)$, $\chi_{235}(234,·)$, $\chi_{235}(126,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2971215073} a^{24} + \frac{1134903170}{2971215073} a^{23} + \frac{24}{2971215073} a^{22} - \frac{638162747}{2971215073} a^{21} + \frac{252}{2971215073} a^{20} - \frac{439197324}{2971215073} a^{19} + \frac{1520}{2971215073} a^{18} - \frac{720695703}{2971215073} a^{17} + \frac{5814}{2971215073} a^{16} + \frac{548262824}{2971215073} a^{15} + \frac{14688}{2971215073} a^{14} - \frac{701530126}{2971215073} a^{13} + \frac{24752}{2971215073} a^{12} - \frac{1072928428}{2971215073} a^{11} + \frac{27456}{2971215073} a^{10} + \frac{1280757137}{2971215073} a^{9} + \frac{19305}{2971215073} a^{8} - \frac{420031747}{2971215073} a^{7} + \frac{8008}{2971215073} a^{6} + \frac{850394442}{2971215073} a^{5} + \frac{1716}{2971215073} a^{4} + \frac{816494931}{2971215073} a^{3} + \frac{144}{2971215073} a^{2} - \frac{638162747}{2971215073} a + \frac{2}{2971215073}$, $\frac{1}{2971215073} a^{25} + \frac{25}{2971215073} a^{23} - \frac{1134903170}{2971215073} a^{22} + \frac{275}{2971215073} a^{21} - \frac{1198149156}{2971215073} a^{20} + \frac{1750}{2971215073} a^{19} + \frac{502443310}{2971215073} a^{18} + \frac{7125}{2971215073} a^{17} + \frac{1289909577}{2971215073} a^{16} + \frac{19380}{2971215073} a^{15} + \frac{1328483517}{2971215073} a^{14} + \frac{35700}{2971215073} a^{13} + \frac{642322947}{2971215073} a^{12} + \frac{44200}{2971215073} a^{11} + \frac{511792168}{2971215073} a^{10} + \frac{35750}{2971215073} a^{9} + \frac{14219705}{2971215073} a^{8} + \frac{17875}{2971215073} a^{7} - \frac{1478497684}{2971215073} a^{6} + \frac{5005}{2971215073} a^{5} - \frac{531471974}{2971215073} a^{4} + \frac{650}{2971215073} a^{3} - \frac{647390212}{2971215073} a^{2} + \frac{25}{2971215073} a + \frac{701408733}{2971215073}$, $\frac{1}{2971215073} a^{26} + \frac{204668310}{2971215073} a^{23} - \frac{325}{2971215073} a^{22} - \frac{100155846}{2971215073} a^{21} - \frac{4550}{2971215073} a^{20} - \frac{402483882}{2971215073} a^{19} - \frac{30875}{2971215073} a^{18} + \frac{1480011714}{2971215073} a^{17} - \frac{125970}{2971215073} a^{16} - \frac{493226791}{2971215073} a^{15} - \frac{331500}{2971215073} a^{14} + \frac{353285659}{2971215073} a^{13} - \frac{574600}{2971215073} a^{12} + \frac{594067211}{2971215073} a^{11} - \frac{650650}{2971215073} a^{10} + \frac{678657083}{2971215073} a^{9} - \frac{464750}{2971215073} a^{8} + \frac{108650772}{2971215073} a^{7} - \frac{195195}{2971215073} a^{6} - \frac{992827513}{2971215073} a^{5} - \frac{42250}{2971215073} a^{4} - \frac{261257976}{2971215073} a^{3} - \frac{3575}{2971215073} a^{2} - \frac{1171813030}{2971215073} a - \frac{50}{2971215073}$, $\frac{1}{2971215073} a^{27} - \frac{351}{2971215073} a^{23} + \frac{930234860}{2971215073} a^{22} - \frac{5148}{2971215073} a^{21} - \frac{1468241761}{2971215073} a^{20} - \frac{36855}{2971215073} a^{19} - \frac{609451894}{2971215073} a^{18} - \frac{160056}{2971215073} a^{17} + \frac{1022463142}{2971215073} a^{16} - \frac{453492}{2971215073} a^{15} + \frac{1054802255}{2971215073} a^{14} - \frac{859248}{2971215073} a^{13} + \frac{565757556}{2971215073} a^{12} - \frac{1085994}{2971215073} a^{11} - \frac{126759234}{2971215073} a^{10} - \frac{892320}{2971215073} a^{9} + \frac{702973312}{2971215073} a^{8} - \frac{451737}{2971215073} a^{7} + \frac{134066303}{2971215073} a^{6} - \frac{127764}{2971215073} a^{5} - \frac{868699322}{2971215073} a^{4} - \frac{16731}{2971215073} a^{3} - \frac{931898940}{2971215073} a^{2} - \frac{648}{2971215073} a - \frac{409336620}{2971215073}$, $\frac{1}{2971215073} a^{28} + \frac{1138427748}{2971215073} a^{23} + \frac{3276}{2971215073} a^{22} + \frac{348979590}{2971215073} a^{21} + \frac{51597}{2971215073} a^{20} - \frac{264528822}{2971215073} a^{19} + \frac{373464}{2971215073} a^{18} + \frac{611552594}{2971215073} a^{17} + \frac{1587222}{2971215073} a^{16} + \frac{366073734}{2971215073} a^{15} + \frac{4296240}{2971215073} a^{14} + \frac{939534389}{2971215073} a^{13} + \frac{7601958}{2971215073} a^{12} + \frac{619676809}{2971215073} a^{11} + \frac{8744736}{2971215073} a^{10} - \frac{1375962697}{2971215073} a^{9} + \frac{6324318}{2971215073} a^{8} + \frac{1263676756}{2971215073} a^{7} + \frac{2683044}{2971215073} a^{6} + \frac{498242520}{2971215073} a^{5} + \frac{585585}{2971215073} a^{4} + \frac{421174833}{2971215073} a^{3} + \frac{49896}{2971215073} a^{2} + \frac{1407884731}{2971215073} a + \frac{702}{2971215073}$, $\frac{1}{2971215073} a^{29} + \frac{3654}{2971215073} a^{23} - \frac{232350705}{2971215073} a^{22} + \frac{60291}{2971215073} a^{21} + \frac{1059540763}{2971215073} a^{20} + \frac{460404}{2971215073} a^{19} - \frac{551451880}{2971215073} a^{18} + \frac{2082780}{2971215073} a^{17} + \frac{1414329506}{2971215073} a^{16} + \frac{6069816}{2971215073} a^{15} - \frac{1260012464}{2971215073} a^{14} + \frac{11740302}{2971215073} a^{13} + \frac{1259810645}{2971215073} a^{12} + \frac{15073968}{2971215073} a^{11} - \frac{865643825}{2971215073} a^{10} + \frac{12540528}{2971215073} a^{9} - \frac{977318476}{2971215073} a^{8} + \frac{6412770}{2971215073} a^{7} - \frac{343319500}{2971215073} a^{6} + \frac{1828827}{2971215073} a^{5} - \frac{1032537774}{2971215073} a^{4} + \frac{241164}{2971215073} a^{3} + \frac{891118034}{2971215073} a^{2} + \frac{9396}{2971215073} a + \frac{694359577}{2971215073}$, $\frac{1}{2971215073} a^{30} + \frac{647708023}{2971215073} a^{23} - \frac{27405}{2971215073} a^{22} + \frac{502385996}{2971215073} a^{21} - \frac{460404}{2971215073} a^{20} - \frac{180569404}{2971215073} a^{19} - \frac{3471300}{2971215073} a^{18} - \frac{631341483}{2971215073} a^{17} - \frac{15174540}{2971215073} a^{16} + \frac{957802915}{2971215073} a^{15} - \frac{41929650}{2971215073} a^{14} + \frac{492283050}{2971215073} a^{13} - \frac{75369840}{2971215073} a^{12} + \frac{582150800}{2971215073} a^{11} - \frac{87783696}{2971215073} a^{10} - \frac{1200157099}{2971215073} a^{9} - \frac{64127700}{2971215073} a^{8} + \frac{1305706370}{2971215073} a^{7} - \frac{27432405}{2971215073} a^{6} - \frac{482862484}{2971215073} a^{5} - \frac{6029100}{2971215073} a^{4} + \frac{518573452}{2971215073} a^{3} - \frac{516780}{2971215073} a^{2} + \frac{137204810}{2971215073} a - \frac{7308}{2971215073}$, $\frac{1}{2971215073} a^{31} - \frac{31465}{2971215073} a^{23} - \frac{186531191}{2971215073} a^{22} - \frac{553784}{2971215073} a^{21} + \frac{13837815}{2971215073} a^{20} - \frac{4405100}{2971215073} a^{19} + \frac{1295867793}{2971215073} a^{18} - \frac{20497200}{2971215073} a^{17} - \frac{287145316}{2971215073} a^{16} - \frac{60979170}{2971215073} a^{15} + \frac{787504972}{2971215073} a^{14} - \frac{119818720}{2971215073} a^{13} + \frac{1189699412}{2971215073} a^{12} - \frac{155764336}{2971215073} a^{11} + \frac{1021790391}{2971215073} a^{10} - \frac{130894400}{2971215073} a^{9} + \frac{175349539}{2971215073} a^{8} - \frac{67492425}{2971215073} a^{7} + \frac{412806790}{2971215073} a^{6} - \frac{19382440}{2971215073} a^{5} + \frac{286043286}{2971215073} a^{4} - \frac{2571140}{2971215073} a^{3} - \frac{1025083239}{2971215073} a^{2} - \frac{100688}{2971215073} a - \frac{1295416046}{2971215073}$, $\frac{1}{2971215073} a^{32} + \frac{1478965545}{2971215073} a^{23} + \frac{201376}{2971215073} a^{22} - \frac{305533206}{2971215073} a^{21} + \frac{3524080}{2971215073} a^{20} + \frac{1073372656}{2971215073} a^{19} + \frac{27329600}{2971215073} a^{18} - \frac{664003075}{2971215073} a^{17} + \frac{121958340}{2971215073} a^{16} + \frac{1002548294}{2971215073} a^{15} + \frac{342339200}{2971215073} a^{14} + \frac{701062139}{2971215073} a^{13} + \frac{623057344}{2971215073} a^{12} + \frac{274462797}{2971215073} a^{11} + \frac{733008640}{2971215073} a^{10} + \frac{608630145}{2971215073} a^{9} + \frac{539939400}{2971215073} a^{8} + \frac{78532139}{2971215073} a^{7} + \frac{232589280}{2971215073} a^{6} - \frac{815786622}{2971215073} a^{5} + \frac{51422800}{2971215073} a^{4} + \frac{862399518}{2971215073} a^{3} + \frac{4430272}{2971215073} a^{2} + \frac{1356428006}{2971215073} a + \frac{62930}{2971215073}$, $\frac{1}{2971215073} a^{33} + \frac{237336}{2971215073} a^{23} - \frac{146125410}{2971215073} a^{22} + \frac{4351160}{2971215073} a^{21} - \frac{224060559}{2971215073} a^{20} + \frac{35600400}{2971215073} a^{19} + \frac{518178786}{2971215073} a^{18} + \frac{169101900}{2971215073} a^{17} + \frac{993290926}{2971215073} a^{16} + \frac{511063520}{2971215073} a^{15} + \frac{208535882}{2971215073} a^{14} + \frac{1016747424}{2971215073} a^{13} + \frac{1260207390}{2971215073} a^{12} + \frac{1335122880}{2971215073} a^{11} - \frac{1244185757}{2971215073} a^{10} + \frac{1131301600}{2971215073} a^{9} - \frac{945677629}{2971215073} a^{8} + \frac{587406600}{2971215073} a^{7} - \frac{1108590004}{2971215073} a^{6} + \frac{169695240}{2971215073} a^{5} + \frac{375196640}{2971215073} a^{4} + \frac{22626032}{2971215073} a^{3} - \frac{658340291}{2971215073} a^{2} + \frac{890010}{2971215073} a + \frac{13283983}{2971215073}$, $\frac{1}{2971215073} a^{34} - \frac{993652788}{2971215073} a^{23} - \frac{1344904}{2971215073} a^{22} + \frac{1081315258}{2971215073} a^{21} - \frac{24208272}{2971215073} a^{20} - \frac{1284138409}{2971215073} a^{19} - \frac{191648820}{2971215073} a^{18} + \frac{1119335670}{2971215073} a^{17} - \frac{868807984}{2971215073} a^{16} - \frac{904154020}{2971215073} a^{15} + \frac{501971329}{2971215073} a^{14} - \frac{1336069048}{2971215073} a^{13} + \frac{1403012354}{2971215073} a^{12} + \frac{1251800732}{2971215073} a^{11} + \frac{557434530}{2971215073} a^{10} - \frac{563501396}{2971215073} a^{9} - \frac{1023149807}{2971215073} a^{8} + \frac{309201765}{2971215073} a^{7} + \frac{1240323625}{2971215073} a^{6} - \frac{142611128}{2971215073} a^{5} - \frac{384642544}{2971215073} a^{4} + \frac{1318992026}{2971215073} a^{3} - \frac{33286374}{2971215073} a^{2} + \frac{1318659800}{2971215073} a - \frac{474672}{2971215073}$, $\frac{1}{2971215073} a^{35} - \frac{1623160}{2971215073} a^{23} + \frac{1159261586}{2971215073} a^{22} - \frac{30608160}{2971215073} a^{21} - \frac{465701965}{2971215073} a^{20} - \frac{255647700}{2971215073} a^{19} - \frac{876898727}{2971215073} a^{18} - \frac{1233601600}{2971215073} a^{17} + \frac{151053500}{2971215073} a^{16} - \frac{803605823}{2971215073} a^{15} - \frac{1172357480}{2971215073} a^{14} + \frac{1327880739}{2971215073} a^{13} + \frac{427235014}{2971215073} a^{12} - \frac{1130468861}{2971215073} a^{11} - \frac{529354354}{2971215073} a^{10} + \frac{343360419}{2971215073} a^{9} + \frac{611762817}{2971215073} a^{8} + \frac{1465986746}{2971215073} a^{7} + \frac{114949682}{2971215073} a^{6} - \frac{1299826528}{2971215073} a^{5} + \frac{949724332}{2971215073} a^{4} - \frac{174083910}{2971215073} a^{3} - \frac{1184877305}{2971215073} a^{2} - \frac{6874560}{2971215073} a - \frac{983909497}{2971215073}$, $\frac{1}{2971215073} a^{36} + \frac{1013139370}{2971215073} a^{23} + \frac{8347680}{2971215073} a^{22} - \frac{855297860}{2971215073} a^{21} + \frac{153388620}{2971215073} a^{20} + \frac{1169572469}{2971215073} a^{19} + \frac{1233601600}{2971215073} a^{18} - \frac{1257407004}{2971215073} a^{17} - \frac{280198802}{2971215073} a^{16} - \frac{427113089}{2971215073} a^{15} + \frac{1399134235}{2971215073} a^{14} - \frac{805076480}{2971215073} a^{13} + \frac{420191510}{2971215073} a^{12} + \frac{1081481094}{2971215073} a^{11} + \frac{340615284}{2971215073} a^{10} + \frac{1344914754}{2971215073} a^{9} + \frac{117724743}{2971215073} a^{8} - \frac{633645185}{2971215073} a^{7} - \frac{186421540}{2971215073} a^{6} - \frac{1280617339}{2971215073} a^{5} - \frac{359956423}{2971215073} a^{4} - \frac{842341776}{2971215073} a^{3} + \frac{226860480}{2971215073} a^{2} - \frac{1373505392}{2971215073} a + \frac{3246320}{2971215073}$, $\frac{1}{2971215073} a^{37} + \frac{10295472}{2971215073} a^{23} - \frac{1400922156}{2971215073} a^{22} + \frac{198187836}{2971215073} a^{21} + \frac{1382947507}{2971215073} a^{20} - \frac{1289621313}{2971215073} a^{19} + \frac{831373483}{2971215073} a^{18} - \frac{697858563}{2971215073} a^{17} + \frac{1100079490}{2971215073} a^{16} - \frac{1346685993}{2971215073} a^{15} + \frac{1020157617}{2971215073} a^{14} + \frac{946112815}{2971215073} a^{13} + \frac{911010974}{2971215073} a^{12} + \frac{271078729}{2971215073} a^{11} + \frac{1105885460}{2971215073} a^{10} - \frac{534201620}{2971215073} a^{9} + \frac{219642524}{2971215073} a^{8} + \frac{1205151686}{2971215073} a^{7} - \frac{112327936}{2971215073} a^{6} + \frac{103901319}{2971215073} a^{5} - \frac{1228682991}{2971215073} a^{4} + \frac{1212443232}{2971215073} a^{3} + \frac{1295178978}{2971215073} a^{2} + \frac{48045536}{2971215073} a + \frac{944936333}{2971215073}$, $\frac{1}{2971215073} a^{38} + \frac{459720637}{2971215073} a^{23} - \frac{48903492}{2971215073} a^{22} - \frac{390494349}{2971215073} a^{21} - \frac{912865184}{2971215073} a^{20} + \frac{924245361}{2971215073} a^{19} + \frac{1480314435}{2971215073} a^{18} + \frac{1035206180}{2971215073} a^{17} + \frac{1190956332}{2971215073} a^{16} - \frac{1244957174}{2971215073} a^{15} + \frac{1258188802}{2971215073} a^{14} + \frac{577768250}{2971215073} a^{13} + \frac{962052063}{2971215073} a^{12} - \frac{452444026}{2971215073} a^{11} - \frac{941248917}{2971215073} a^{10} + \frac{917551655}{2971215073} a^{9} - \frac{1448740456}{2971215073} a^{8} - \frac{287825472}{2971215073} a^{7} + \frac{851783587}{2971215073} a^{6} + \frac{693627879}{2971215073} a^{5} + \frac{1372703718}{2971215073} a^{4} + \frac{905254095}{2971215073} a^{3} - \frac{1434502432}{2971215073} a^{2} - \frac{828505523}{2971215073} a - \frac{20590944}{2971215073}$, $\frac{1}{2971215073} a^{39} - \frac{61523748}{2971215073} a^{23} + \frac{461070655}{2971215073} a^{22} - \frac{1203131072}{2971215073} a^{21} + \frac{952032684}{2971215073} a^{20} - \frac{1422344445}{2971215073} a^{19} + \frac{495380095}{2971215073} a^{18} - \frac{498123919}{2971215073} a^{17} + \frac{32825008}{2971215073} a^{16} + \frac{1468249110}{2971215073} a^{15} - \frac{1198302150}{2971215073} a^{14} - \frac{528617267}{2971215073} a^{13} + \frac{296078540}{2971215073} a^{12} - \frac{1298545198}{2971215073} a^{11} + \frac{549372287}{2971215073} a^{10} - \frac{1003795266}{2971215073} a^{9} - \frac{175299706}{2971215073} a^{8} + \frac{1118750701}{2971215073} a^{7} + \frac{586242230}{2971215073} a^{6} + \frac{1461692756}{2971215073} a^{5} - \frac{603364652}{2971215073} a^{4} + \frac{1093293251}{2971215073} a^{3} + \frac{1309669428}{2971215073} a^{2} - \frac{310856832}{2971215073} a - \frac{919441274}{2971215073}$, $\frac{1}{2971215073} a^{40} - \frac{665861798}{2971215073} a^{23} + \frac{273438880}{2971215073} a^{22} - \frac{1197693151}{2971215073} a^{21} - \frac{774435314}{2971215073} a^{20} - \frac{203372744}{2971215073} a^{19} + \frac{910305778}{2971215073} a^{18} - \frac{681729594}{2971215073} a^{17} - \frac{349703851}{2971215073} a^{16} - \frac{2541686}{2971215073} a^{15} - \frac{117188835}{2971215073} a^{14} - \frac{875433808}{2971215073} a^{13} + \frac{275147922}{2971215073} a^{12} + \frac{1379307878}{2971215073} a^{11} + \frac{542068358}{2971215073} a^{10} - \frac{1111078990}{2971215073} a^{9} + \frac{348676641}{2971215073} a^{8} - \frac{569468355}{2971215073} a^{7} + \frac{922164622}{2971215073} a^{6} + \frac{992208535}{2971215073} a^{5} - \frac{295697809}{2971215073} a^{4} - \frac{1416072847}{2971215073} a^{3} - \frac{365082339}{2971215073} a^{2} - \frac{97952036}{2971215073} a + \frac{123047496}{2971215073}$, $\frac{1}{2971215073} a^{41} + \frac{350343565}{2971215073} a^{23} - \frac{73085364}{2971215073} a^{22} + \frac{994372441}{2971215073} a^{21} + \frac{1205756264}{2971215073} a^{20} + \frac{771092890}{2971215073} a^{19} + \frac{1215078546}{2971215073} a^{18} - \frac{548974298}{2971215073} a^{17} - \frac{175288233}{2971215073} a^{16} + \frac{1202569352}{2971215073} a^{15} + \frac{1033849973}{2971215073} a^{14} + \frac{1373773643}{2971215073} a^{13} + \frac{1460522043}{2971215073} a^{12} - \frac{1273227684}{2971215073} a^{11} - \frac{1095897271}{2971215073} a^{10} - \frac{691399632}{2971215073} a^{9} + \frac{416136237}{2971215073} a^{8} - \frac{890367171}{2971215073} a^{7} - \frac{117569116}{2971215073} a^{6} + \frac{90529951}{2971215073} a^{5} + \frac{256184489}{2971215073} a^{4} - \frac{77107678}{2971215073} a^{3} + \frac{707264540}{2971215073} a^{2} - \frac{1079359822}{2971215073} a + \frac{1331723596}{2971215073}$, $\frac{1}{2971215073} a^{42} - \frac{341065032}{2971215073} a^{23} - \frac{1471442973}{2971215073} a^{22} + \frac{984654046}{2971215073} a^{21} - \frac{1350248373}{2971215073} a^{20} - \frac{995943634}{2971215073} a^{19} - \frac{1223695031}{2971215073} a^{18} - \frac{893254367}{2971215073} a^{17} - \frac{412592553}{2971215073} a^{16} + \frac{497223596}{2971215073} a^{15} - \frac{1299217714}{2971215073} a^{14} + \frac{208393830}{2971215073} a^{13} - \frac{350477}{2971215073} a^{12} + \frac{1358517040}{2971215073} a^{11} + \frac{1070086102}{2971215073} a^{10} - \frac{197550764}{2971215073} a^{9} + \frac{1183831725}{2971215073} a^{8} + \frac{108461698}{2971215073} a^{7} - \frac{633709657}{2971215073} a^{6} + \frac{1321933118}{2971215073} a^{5} - \frac{1081220472}{2971215073} a^{4} - \frac{1185769183}{2971215073} a^{3} - \frac{1018176941}{2971215073} a^{2} + \frac{1110621378}{2971215073} a - \frac{700687130}{2971215073}$, $\frac{1}{2971215073} a^{43} + \frac{1053880290}{2971215073} a^{23} + \frac{256569595}{2971215073} a^{22} + \frac{279100289}{2971215073} a^{21} - \frac{1212792687}{2971215073} a^{20} + \frac{213716224}{2971215073} a^{19} + \frac{534171571}{2971215073} a^{18} + \frac{352613918}{2971215073} a^{17} - \frac{1322349120}{2971215073} a^{16} + \frac{1283036231}{2971215073} a^{15} + \frac{302970768}{2971215073} a^{14} + \frac{82478582}{2971215073} a^{13} - \frac{793048362}{2971215073} a^{12} - \frac{1250117162}{2971215073} a^{11} - \frac{1185942268}{2971215073} a^{10} + \frac{1451816439}{2971215073} a^{9} + \frac{156302690}{2971215073} a^{8} + \frac{121324186}{2971215073} a^{7} - \frac{947157786}{2971215073} a^{6} + \frac{160862500}{2971215073} a^{5} - \frac{1247543652}{2971215073} a^{4} - \frac{827087253}{2971215073} a^{3} - \frac{286670255}{2971215073} a^{2} + \frac{928661532}{2971215073} a + \frac{682130064}{2971215073}$, $\frac{1}{2971215073} a^{44} + \frac{1128906218}{2971215073} a^{23} - \frac{1244306087}{2971215073} a^{22} + \frac{1023659204}{2971215073} a^{21} - \frac{925975359}{2971215073} a^{20} - \frac{871030103}{2971215073} a^{19} - \frac{60502535}{2971215073} a^{18} + \frac{1364375601}{2971215073} a^{17} + \frac{668510697}{2971215073} a^{16} - \frac{1027964710}{2971215073} a^{15} + \frac{719309392}{2971215073} a^{14} + \frac{63055620}{2971215073} a^{13} + \frac{373285698}{2971215073} a^{12} - \frac{1274825624}{2971215073} a^{11} - \frac{193044927}{2971215073} a^{10} + \frac{705695481}{2971215073} a^{9} - \frac{1128069433}{2971215073} a^{8} - \frac{1211765126}{2971215073} a^{7} - \frac{1061692500}{2971215073} a^{6} + \frac{645063950}{2971215073} a^{5} + \frac{184314564}{2971215073} a^{4} + \frac{1375830179}{2971215073} a^{3} + \frac{701868495}{2971215073} a^{2} - \frac{52633118}{2971215073} a + \frac{863454493}{2971215073}$, $\frac{1}{2971215073} a^{45} + \frac{1236950691}{2971215073} a^{23} + \frac{670845629}{2971215073} a^{22} - \frac{310155852}{2971215073} a^{21} - \frac{118750031}{2971215073} a^{20} + \frac{155262389}{2971215073} a^{19} - \frac{181978638}{2971215073} a^{18} + \frac{115641720}{2971215073} a^{17} - \frac{1074619905}{2971215073} a^{16} - \frac{1415759319}{2971215073} a^{15} + \frac{1039848049}{2971215073} a^{14} - \frac{646921197}{2971215073} a^{13} + \frac{316228005}{2971215073} a^{12} - \frac{529919854}{2971215073} a^{11} + \frac{1172215609}{2971215073} a^{10} - \frac{556952732}{2971215073} a^{9} - \frac{883743161}{2971215073} a^{8} - \frac{1313265494}{2971215073} a^{7} - \frac{1199677728}{2971215073} a^{6} - \frac{889948125}{2971215073} a^{5} + \frac{1404987687}{2971215073} a^{4} - \frac{606177716}{2971215073} a^{3} + \frac{801700505}{2971215073} a^{2} + \frac{1479274000}{2971215073} a + \frac{713402637}{2971215073}$

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:  $22$
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

$C_{46}$ (as 46T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 46
The 46 conjugacy class representatives for $C_{46}$
Character table for $C_{46}$ is not computed

Intermediate fields

\(\Q(\sqrt{-235}) \), \(\Q(\zeta_{47})^+\)

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 $46$ $46$ R $46$ $46$ $23^{2}$ $46$ $46$ $23^{2}$ $46$ $46$ $46$ $46$ $23^{2}$ R $46$ $23^{2}$

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
5Data not computed
47Data not computed