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Properties

Label 46.0.45232166103...2227.1
Degree $46$
Signature $[0, 23]$
Discriminant $-\,3^{23}\cdot 7^{23}\cdot 47^{45}$
Root discriminant $198.09$
Ramified primes $3, 7, 47$
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![897051474401885688211, -896491190740997016336, 896491190740997016336, -886181971380645453836, 886181971380645453836, -829481264898711860086, 829481264898711860086, -682059428045684516336, 682059428045684516336, -460926672766143500711, 460926672766143500711, -247030989477569281961, 247030989477569281961, -103885416815215766336, 103885416815215766336, -34357567236358344461, 34357567236358344461, -9000351507598578836, 9000351507598578836, -1882536566192328836, 1882536566192328836, -316617279082953836, 316617279082953836, -43045609161078836, 43045609161078836, -4745575372016336, 4745575372016336, -424545918891336, 424545918891336, -30757520453836, 30757520453836, -1795018891336, 1795018891336, -83598344461, 83598344461, -3060906961, 3060906961, -86100711, 86100711, -1794461, 1794461, -26086, 26086, -236, 236, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^46 - x^45 + 236*x^44 - 236*x^43 + 26086*x^42 - 26086*x^41 + 1794461*x^40 - 1794461*x^39 + 86100711*x^38 - 86100711*x^37 + 3060906961*x^36 - 3060906961*x^35 + 83598344461*x^34 - 83598344461*x^33 + 1795018891336*x^32 - 1795018891336*x^31 + 30757520453836*x^30 - 30757520453836*x^29 + 424545918891336*x^28 - 424545918891336*x^27 + 4745575372016336*x^26 - 4745575372016336*x^25 + 43045609161078836*x^24 - 43045609161078836*x^23 + 316617279082953836*x^22 - 316617279082953836*x^21 + 1882536566192328836*x^20 - 1882536566192328836*x^19 + 9000351507598578836*x^18 - 9000351507598578836*x^17 + 34357567236358344461*x^16 - 34357567236358344461*x^15 + 103885416815215766336*x^14 - 103885416815215766336*x^13 + 247030989477569281961*x^12 - 247030989477569281961*x^11 + 460926672766143500711*x^10 - 460926672766143500711*x^9 + 682059428045684516336*x^8 - 682059428045684516336*x^7 + 829481264898711860086*x^6 - 829481264898711860086*x^5 + 886181971380645453836*x^4 - 886181971380645453836*x^3 + 896491190740997016336*x^2 - 896491190740997016336*x + 897051474401885688211)
 
gp: K = bnfinit(x^46 - x^45 + 236*x^44 - 236*x^43 + 26086*x^42 - 26086*x^41 + 1794461*x^40 - 1794461*x^39 + 86100711*x^38 - 86100711*x^37 + 3060906961*x^36 - 3060906961*x^35 + 83598344461*x^34 - 83598344461*x^33 + 1795018891336*x^32 - 1795018891336*x^31 + 30757520453836*x^30 - 30757520453836*x^29 + 424545918891336*x^28 - 424545918891336*x^27 + 4745575372016336*x^26 - 4745575372016336*x^25 + 43045609161078836*x^24 - 43045609161078836*x^23 + 316617279082953836*x^22 - 316617279082953836*x^21 + 1882536566192328836*x^20 - 1882536566192328836*x^19 + 9000351507598578836*x^18 - 9000351507598578836*x^17 + 34357567236358344461*x^16 - 34357567236358344461*x^15 + 103885416815215766336*x^14 - 103885416815215766336*x^13 + 247030989477569281961*x^12 - 247030989477569281961*x^11 + 460926672766143500711*x^10 - 460926672766143500711*x^9 + 682059428045684516336*x^8 - 682059428045684516336*x^7 + 829481264898711860086*x^6 - 829481264898711860086*x^5 + 886181971380645453836*x^4 - 886181971380645453836*x^3 + 896491190740997016336*x^2 - 896491190740997016336*x + 897051474401885688211, 1)
 

Normalized defining polynomial

\( x^{46} - x^{45} + 236 x^{44} - 236 x^{43} + 26086 x^{42} - 26086 x^{41} + 1794461 x^{40} - 1794461 x^{39} + 86100711 x^{38} - 86100711 x^{37} + 3060906961 x^{36} - 3060906961 x^{35} + 83598344461 x^{34} - 83598344461 x^{33} + 1795018891336 x^{32} - 1795018891336 x^{31} + 30757520453836 x^{30} - 30757520453836 x^{29} + 424545918891336 x^{28} - 424545918891336 x^{27} + 4745575372016336 x^{26} - 4745575372016336 x^{25} + 43045609161078836 x^{24} - 43045609161078836 x^{23} + 316617279082953836 x^{22} - 316617279082953836 x^{21} + 1882536566192328836 x^{20} - 1882536566192328836 x^{19} + 9000351507598578836 x^{18} - 9000351507598578836 x^{17} + 34357567236358344461 x^{16} - 34357567236358344461 x^{15} + 103885416815215766336 x^{14} - 103885416815215766336 x^{13} + 247030989477569281961 x^{12} - 247030989477569281961 x^{11} + 460926672766143500711 x^{10} - 460926672766143500711 x^{9} + 682059428045684516336 x^{8} - 682059428045684516336 x^{7} + 829481264898711860086 x^{6} - 829481264898711860086 x^{5} + 886181971380645453836 x^{4} - 886181971380645453836 x^{3} + 896491190740997016336 x^{2} - 896491190740997016336 x + 897051474401885688211 \)

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:  \(-4523216610394124899776106583774876615554155445428856762643591351453121704871884692071299440370187945622227=-\,3^{23}\cdot 7^{23}\cdot 47^{45}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $198.09$
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:  $3, 7, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(987=3\cdot 7\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{987}(1,·)$, $\chi_{987}(650,·)$, $\chi_{987}(526,·)$, $\chi_{987}(400,·)$, $\chi_{987}(146,·)$, $\chi_{987}(20,·)$, $\chi_{987}(923,·)$, $\chi_{987}(925,·)$, $\chi_{987}(671,·)$, $\chi_{987}(419,·)$, $\chi_{987}(293,·)$, $\chi_{987}(167,·)$, $\chi_{987}(41,·)$, $\chi_{987}(839,·)$, $\chi_{987}(818,·)$, $\chi_{987}(797,·)$, $\chi_{987}(946,·)$, $\chi_{987}(820,·)$, $\chi_{987}(862,·)$, $\chi_{987}(694,·)$, $\chi_{987}(568,·)$, $\chi_{987}(316,·)$, $\chi_{987}(62,·)$, $\chi_{987}(64,·)$, $\chi_{987}(967,·)$, $\chi_{987}(841,·)$, $\chi_{987}(587,·)$, $\chi_{987}(589,·)$, $\chi_{987}(461,·)$, $\chi_{987}(337,·)$, $\chi_{987}(398,·)$, $\chi_{987}(169,·)$, $\chi_{987}(986,·)$, $\chi_{987}(734,·)$, $\chi_{987}(608,·)$, $\chi_{987}(484,·)$, $\chi_{987}(104,·)$, $\chi_{987}(106,·)$, $\chi_{987}(125,·)$, $\chi_{987}(881,·)$, $\chi_{987}(883,·)$, $\chi_{987}(190,·)$, $\chi_{987}(503,·)$, $\chi_{987}(148,·)$, $\chi_{987}(379,·)$, $\chi_{987}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{195752680522760891501} a^{24} + \frac{29446513144251866572}{195752680522760891501} a^{23} + \frac{120}{195752680522760891501} a^{22} + \frac{58553442702029500263}{195752680522760891501} a^{21} + \frac{6300}{195752680522760891501} a^{20} - \frac{8618072739938359365}{195752680522760891501} a^{19} + \frac{190000}{195752680522760891501} a^{18} + \frac{48013947695898095349}{195752680522760891501} a^{17} + \frac{3633750}{195752680522760891501} a^{16} - \frac{88658125867823663128}{195752680522760891501} a^{15} + \frac{45900000}{195752680522760891501} a^{14} + \frac{9669494330115351512}{195752680522760891501} a^{13} + \frac{386750000}{195752680522760891501} a^{12} - \frac{29690580105285430997}{195752680522760891501} a^{11} + \frac{2145000000}{195752680522760891501} a^{10} - \frac{79385867839661367601}{195752680522760891501} a^{9} + \frac{7541015625}{195752680522760891501} a^{8} - \frac{42404922996223211302}{195752680522760891501} a^{7} + \frac{15640625000}{195752680522760891501} a^{6} - \frac{5423978152785055003}{195752680522760891501} a^{5} + \frac{16757812500}{195752680522760891501} a^{4} - \frac{49345986392010248041}{195752680522760891501} a^{3} + \frac{7031250000}{195752680522760891501} a^{2} - \frac{37908544251287905214}{195752680522760891501} a + \frac{488281250}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{25} + \frac{125}{195752680522760891501} a^{23} + \frac{48520114801501558641}{195752680522760891501} a^{22} + \frac{6875}{195752680522760891501} a^{21} + \frac{51890254050627379983}{195752680522760891501} a^{20} + \frac{218750}{195752680522760891501} a^{19} + \frac{17878560870289405430}{195752680522760891501} a^{18} + \frac{4453125}{195752680522760891501} a^{17} - \frac{4332102143271849013}{195752680522760891501} a^{16} + \frac{60562500}{195752680522760891501} a^{15} - \frac{6950808880277186383}{195752680522760891501} a^{14} + \frac{557812500}{195752680522760891501} a^{13} - \frac{28355087889376726806}{195752680522760891501} a^{12} + \frac{3453125000}{195752680522760891501} a^{11} + \frac{49749621413986117585}{195752680522760891501} a^{10} + \frac{13964843750}{195752680522760891501} a^{9} - \frac{10504685966071943494}{195752680522760891501} a^{8} + \frac{34912109375}{195752680522760891501} a^{7} - \frac{26261714915179858735}{195752680522760891501} a^{6} + \frac{48876953125}{195752680522760891501} a^{5} - \frac{43707464277038838968}{195752680522760891501} a^{4} + \frac{31738281250}{195752680522760891501} a^{3} - \frac{21853732138519419484}{195752680522760891501} a^{2} + \frac{6103515625}{195752680522760891501} a - \frac{26072954786996756755}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{26} + \frac{87006901702475175660}{195752680522760891501} a^{23} - \frac{8125}{195752680522760891501} a^{22} - \frac{24440904360907167355}{195752680522760891501} a^{21} - \frac{568750}{195752680522760891501} a^{20} - \frac{79378429773981022951}{195752680522760891501} a^{19} - \frac{19296875}{195752680522760891501} a^{18} + \frac{62257532075053868893}{195752680522760891501} a^{17} - \frac{393656250}{195752680522760891501} a^{16} - \frac{82587865199690110940}{195752680522760891501} a^{15} - \frac{5179687500}{195752680522760891501} a^{14} - \frac{62525796017230316800}{195752680522760891501} a^{13} - \frac{44890625000}{195752680522760891501} a^{12} + \frac{41771204642208053691}{195752680522760891501} a^{11} - \frac{254160156250}{195752680522760891501} a^{10} - \frac{70657912669206459920}{195752680522760891501} a^{9} - \frac{907714843750}{195752680522760891501} a^{8} - \frac{10968714501822516512}{195752680522760891501} a^{7} - \frac{1906201171875}{195752680522760891501} a^{6} + \frac{47031763252810361904}{195752680522760891501} a^{5} - \frac{2062988281250}{195752680522760891501} a^{4} + \frac{78061470657173949110}{195752680522760891501} a^{3} - \frac{872802734375}{195752680522760891501} a^{2} + \frac{14430744077729998971}{195752680522760891501} a - \frac{61035156250}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{27} - \frac{8775}{195752680522760891501} a^{23} - \frac{90377040951600997002}{195752680522760891501} a^{22} - \frac{643500}{195752680522760891501} a^{21} + \frac{80398988885669413350}{195752680522760891501} a^{20} - \frac{23034375}{195752680522760891501} a^{19} + \frac{64804208948965728343}{195752680522760891501} a^{18} - \frac{500175000}{195752680522760891501} a^{17} - \frac{82820840116816021834}{195752680522760891501} a^{16} - \frac{7085812500}{195752680522760891501} a^{15} - \frac{55165864416570196961}{195752680522760891501} a^{14} - \frac{67128750000}{195752680522760891501} a^{13} + \frac{63909897096837400362}{195752680522760891501} a^{12} - \frac{424216406250}{195752680522760891501} a^{11} - \frac{23540990685942865014}{195752680522760891501} a^{10} - \frac{1742812500000}{195752680522760891501} a^{9} - \frac{47193237443508610031}{195752680522760891501} a^{8} - \frac{4411494140625}{195752680522760891501} a^{7} + \frac{15400015045482366050}{195752680522760891501} a^{6} - \frac{6238476562500}{195752680522760891501} a^{5} - \frac{67688362720826555877}{195752680522760891501} a^{4} - \frac{4084716796875}{195752680522760891501} a^{3} + \frac{80584655168196660582}{195752680522760891501} a^{2} - \frac{791015625000}{195752680522760891501} a + \frac{61688553422643222633}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{28} - \frac{90762490185848609022}{195752680522760891501} a^{23} + \frac{409500}{195752680522760891501} a^{22} + \frac{36072326947194031050}{195752680522760891501} a^{21} + \frac{32248125}{195752680522760891501} a^{20} + \frac{1750597775566419854}{195752680522760891501} a^{19} + \frac{1167075000}{195752680522760891501} a^{18} - \frac{20198293592467844511}{195752680522760891501} a^{17} + \frac{24800343750}{195752680522760891501} a^{16} + \frac{86684723405329571314}{195752680522760891501} a^{15} + \frac{335643750000}{195752680522760891501} a^{14} - \frac{42940703019179993272}{195752680522760891501} a^{13} + \frac{2969514843750}{195752680522760891501} a^{12} - \frac{11563638770853275858}{195752680522760891501} a^{11} + \frac{17079562500000}{195752680522760891501} a^{10} + \frac{25606450034003543253}{195752680522760891501} a^{9} + \frac{61760917968750}{195752680522760891501} a^{8} + \frac{38046396955257934401}{195752680522760891501} a^{7} + \frac{131008007812500}{195752680522760891501} a^{6} - \frac{95195286378787572459}{195752680522760891501} a^{5} + \frac{142965087890625}{195752680522760891501} a^{4} + \frac{74483381625362101019}{195752680522760891501} a^{3} + \frac{60908203125000}{195752680522760891501} a^{2} - \frac{1983043457970370018}{195752680522760891501} a + \frac{4284667968750}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{29} + \frac{456750}{195752680522760891501} a^{23} - \frac{34578960025582810366}{195752680522760891501} a^{22} + \frac{37681875}{195752680522760891501} a^{21} + \frac{11858961637239184033}{195752680522760891501} a^{20} + \frac{1438762500}{195752680522760891501} a^{19} + \frac{20546365022509554894}{195752680522760891501} a^{18} + \frac{32543437500}{195752680522760891501} a^{17} + \frac{58446512260383657493}{195752680522760891501} a^{16} + \frac{474204375000}{195752680522760891501} a^{15} + \frac{84596418548332774281}{195752680522760891501} a^{14} + \frac{4586055468750}{195752680522760891501} a^{13} + \frac{80666248109704273028}{195752680522760891501} a^{12} + \frac{29441343750000}{195752680522760891501} a^{11} + \frac{33769518111340933719}{195752680522760891501} a^{10} + \frac{122466093750000}{195752680522760891501} a^{9} + \frac{128677319690269781}{696628756308757621} a^{8} + \frac{313123535156250}{195752680522760891501} a^{7} - \frac{84611085112225989853}{195752680522760891501} a^{6} + \frac{446490966796875}{195752680522760891501} a^{5} - \frac{81964414608545538904}{195752680522760891501} a^{4} + \frac{294389648437500}{195752680522760891501} a^{3} + \frac{7664017064528865207}{195752680522760891501} a^{2} + \frac{57348632812500}{195752680522760891501} a + \frac{63202041036611065009}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{30} + \frac{45715760789693679342}{195752680522760891501} a^{23} - \frac{17128125}{195752680522760891501} a^{22} + \frac{45375870824273599906}{195752680522760891501} a^{21} - \frac{1438762500}{195752680522760891501} a^{20} - \frac{65382300330617674965}{195752680522760891501} a^{19} - \frac{54239062500}{195752680522760891501} a^{18} + \frac{56388056230768750274}{195752680522760891501} a^{17} - \frac{1185510937500}{195752680522760891501} a^{16} - \frac{86175154970875649086}{195752680522760891501} a^{15} - \frac{16378769531250}{195752680522760891501} a^{14} - \frac{84643758068625678911}{195752680522760891501} a^{13} - \frac{147206718750000}{195752680522760891501} a^{12} + \frac{47784031925668298692}{195752680522760891501} a^{11} - \frac{857262656250000}{195752680522760891501} a^{10} + \frac{66528180639923943480}{195752680522760891501} a^{9} - \frac{3131235351562500}{195752680522760891501} a^{8} + \frac{6498476308648415204}{195752680522760891501} a^{7} - \frac{6697364501953125}{195752680522760891501} a^{6} + \frac{69884854426245136191}{195752680522760891501} a^{5} - \frac{7359741210937500}{195752680522760891501} a^{4} + \frac{19065857579035058318}{195752680522760891501} a^{3} - \frac{3154174804687500}{195752680522760891501} a^{2} + \frac{74691217540942513057}{195752680522760891501} a - \frac{223022460937500}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{31} - \frac{19665625}{195752680522760891501} a^{23} + \frac{40559630698337040894}{195752680522760891501} a^{22} - \frac{1730575000}{195752680522760891501} a^{21} + \frac{73270454103234759907}{195752680522760891501} a^{20} - \frac{68829687500}{195752680522760891501} a^{19} - 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\frac{73638215691168739096}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{32} - \frac{67028315458825455850}{195752680522760891501} a^{23} + \frac{629300000}{195752680522760891501} a^{22} + \frac{33075518963483798910}{195752680522760891501} a^{21} + \frac{55063750000}{195752680522760891501} a^{20} - \frac{52441781381967682194}{195752680522760891501} a^{19} + \frac{2135125000000}{195752680522760891501} a^{18} - \frac{14446660006617273898}{195752680522760891501} a^{17} + \frac{47639976562500}{195752680522760891501} a^{16} - \frac{77826375573175377264}{195752680522760891501} a^{15} + \frac{668631250000000}{195752680522760891501} a^{14} - \frac{67944553959819743259}{195752680522760891501} a^{13} + \frac{6084544375000000}{195752680522760891501} a^{12} + \frac{79627958223884378909}{195752680522760891501} a^{11} + \frac{35791437500000000}{195752680522760891501} a^{10} - \frac{86805853803257232988}{195752680522760891501} a^{9} + \frac{131821142578125000}{195752680522760891501} a^{8} - \frac{96614518035011118517}{195752680522760891501} a^{7} + \frac{283922460937500000}{195752680522760891501} a^{6} + \frac{88537741165649778463}{195752680522760891501} a^{5} + \frac{313859863281250000}{195752680522760891501} a^{4} + \frac{48084212174155854931}{195752680522760891501} a^{3} + \frac{135201171875000000}{195752680522760891501} a^{2} + \frac{18946948607287700007}{195752680522760891501} a + \frac{9602355957031250}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{33} + \frac{741675000}{195752680522760891501} a^{23} + \frac{50613472589341949369}{195752680522760891501} a^{22} + \frac{67986875000}{195752680522760891501} a^{21} - \frac{12586278376838794851}{195752680522760891501} a^{20} + \frac{2781281250000}{195752680522760891501} a^{19} + \frac{87601067051914954044}{195752680522760891501} a^{18} + \frac{66055429687500}{195752680522760891501} a^{17} - \frac{34752230675665210008}{195752680522760891501} a^{16} + \frac{998170937500000}{195752680522760891501} a^{15} - \frac{49291497710260023528}{195752680522760891501} a^{14} + \frac{9929174062500000}{195752680522760891501} a^{13} - \frac{78582090145199418924}{195752680522760891501} a^{12} + \frac{65191546875000000}{195752680522760891501} a^{11} - \frac{36504342112149365452}{195752680522760891501} a^{10} + \frac{276196679687500000}{195752680522760891501} a^{9} + \frac{31288563873349755414}{195752680522760891501} a^{8} + \frac{717049072265625000}{195752680522760891501} a^{7} + \frac{63814236201122862911}{195752680522760891501} a^{6} + \frac{1035737548828125000}{195752680522760891501} a^{5} + \frac{49559411215414287054}{195752680522760891501} a^{4} + \frac{690491699218750000}{195752680522760891501} a^{3} + \frac{26410412558961199202}{195752680522760891501} a^{2} + \frac{135804748535156250}{195752680522760891501} a - \frac{94639256924142551480}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{34} - \frac{86484283542548705593}{195752680522760891501} a^{23} - \frac{21014125000}{195752680522760891501} a^{22} + \frac{77138888799368096100}{195752680522760891501} a^{21} - \frac{1891271250000}{195752680522760891501} a^{20} + \frac{71547773838758997071}{195752680522760891501} a^{19} - 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\frac{362145996093750000}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{35} - \frac{25361875000}{195752680522760891501} a^{23} + \frac{80360846198885517707}{195752680522760891501} a^{22} - \frac{2391262500000}{195752680522760891501} a^{21} - \frac{52928483470717705813}{195752680522760891501} a^{20} - \frac{99862382812500}{195752680522760891501} a^{19} - \frac{56277423807160353429}{195752680522760891501} a^{18} - \frac{2409378125000000}{195752680522760891501} a^{17} - \frac{81610361053912235871}{195752680522760891501} a^{16} - \frac{36863485312500000}{195752680522760891501} a^{15} + \frac{67212811318124322919}{195752680522760891501} a^{14} - \frac{370398656250000000}{195752680522760891501} a^{13} - \frac{35085112986675709351}{195752680522760891501} a^{12} - \frac{2452176289062500000}{195752680522760891501} a^{11} + \frac{90207580577745635090}{195752680522760891501} a^{10} - \frac{10461773437500000000}{195752680522760891501} a^{9} - \frac{23776228109962976374}{195752680522760891501} a^{8} - 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\frac{73834810173318350827}{195752680522760891501} a^{13} + \frac{7356528867187500000}{195752680522760891501} a^{12} + \frac{58728228681272524069}{195752680522760891501} a^{11} + \frac{43939448437500000000}{195752680522760891501} a^{10} - \frac{311631864973738}{696628756308757621} a^{9} - \frac{31820427104792141501}{195752680522760891501} a^{8} - \frac{41897539897093548021}{195752680522760891501} a^{7} - \frac{34497342490834283002}{195752680522760891501} a^{6} - \frac{20666168909713303425}{195752680522760891501} a^{5} + \frac{6941088234263373248}{195752680522760891501} a^{4} + \frac{27060643940136009315}{195752680522760891501} a^{3} - \frac{22671869975885891501}{195752680522760891501} a^{2} + \frac{77812134705355933260}{195752680522760891501} a + \frac{12383728027343750000}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{37} + \frac{804333750000}{195752680522760891501} a^{23} + \frac{39909649056535716732}{195752680522760891501} a^{22} + \frac{77417123437500}{195752680522760891501} a^{21} + \frac{95025298858377215937}{195752680522760891501} a^{20} + \frac{3284362812500000}{195752680522760891501} a^{19} - 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\frac{12473505718073391501}{195752680522760891501} a + \frac{2647344622768600634}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{38} - \frac{90958974514573224385}{195752680522760891501} a^{23} - \frac{19102926562500}{195752680522760891501} a^{22} - \frac{77662692089323258442}{195752680522760891501} a^{21} - \frac{1782939812500000}{195752680522760891501} a^{20} - \frac{48383355226988608028}{195752680522760891501} a^{19} - \frac{72591120937500000}{195752680522760891501} a^{18} - \frac{35206915031894595770}{195752680522760891501} a^{17} - \frac{1682794167187500000}{195752680522760891501} a^{16} - \frac{12588466148482167482}{195752680522760891501} a^{15} - \frac{24356231367187500000}{195752680522760891501} a^{14} + \frac{74622386091687316079}{195752680522760891501} a^{13} - \frac{31572145570989108499}{195752680522760891501} a^{12} - \frac{25825620392961876565}{195752680522760891501} a^{11} + \frac{4409514440576240507}{195752680522760891501} a^{10} + \frac{17530531602597975903}{195752680522760891501} a^{9} - 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\frac{38812833210937500000}{195752680522760891501} a^{15} - \frac{80401757658192341941}{195752680522760891501} a^{14} - \frac{4479819891978216998}{195752680522760891501} a^{13} - \frac{27029702207801199875}{195752680522760891501} a^{12} + \frac{84922623412402481014}{195752680522760891501} a^{11} + \frac{61846826547233557852}{195752680522760891501} a^{10} + \frac{93814447717892598559}{195752680522760891501} a^{9} - \frac{30914798544942773686}{195752680522760891501} a^{8} - \frac{59265908906443802596}{195752680522760891501} a^{7} + \frac{72119496193610677658}{195752680522760891501} a^{6} + \frac{18144811815836479226}{195752680522760891501} a^{5} + \frac{88323062245769568799}{195752680522760891501} a^{4} - \frac{77867829095031991848}{195752680522760891501} a^{3} + \frac{82331615512765962438}{195752680522760891501} a^{2} - \frac{56543119473423254970}{195752680522760891501} a - \frac{82155861640424971848}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{40} - \frac{36421256507448606987}{195752680522760891501} a^{23} + \frac{534060312500000}{195752680522760891501} a^{22} + \frac{2100166664858906758}{195752680522760891501} a^{21} + \frac{50468699531250000}{195752680522760891501} a^{20} - \frac{24687185367597518400}{195752680522760891501} a^{19} + \frac{2075552578125000000}{195752680522760891501} a^{18} - \frac{3267856060034587508}{195752680522760891501} a^{17} + \frac{48516041513671875000}{195752680522760891501} a^{16} - \frac{64894171665891680950}{195752680522760891501} a^{15} - \frac{75894327559793566004}{195752680522760891501} a^{14} - \frac{41860546153721046481}{195752680522760891501} a^{13} - \frac{16553878008245311034}{195752680522760891501} a^{12} + \frac{52068262531661085189}{195752680522760891501} a^{11} - \frac{34721546228482757705}{195752680522760891501} a^{10} + \frac{64384394261320915050}{195752680522760891501} a^{9} - \frac{95175816513302770022}{195752680522760891501} a^{8} - \frac{33488286433120039136}{195752680522760891501} a^{7} + \frac{59666591903987297306}{195752680522760891501} a^{6} - \frac{22439015291194662024}{195752680522760891501} a^{5} - \frac{5415538925904559405}{195752680522760891501} a^{4} + \frac{57361611414412213036}{195752680522760891501} a^{3} - \frac{11085658662947620333}{195752680522760891501} a^{2} - \frac{87830862571869005803}{195752680522760891501} a - \frac{10437168035575365060}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{41} + \frac{684264775390625}{195752680522760891501} a^{23} + \frac{66091976057952132176}{195752680522760891501} a^{22} + \frac{67742212763671875}{195752680522760891501} a^{21} + \frac{7087238882861660528}{195752680522760891501} a^{20} + \frac{2939228239746093750}{195752680522760891501} a^{19} - \frac{17540600944982509359}{195752680522760891501} a^{18} + \frac{73130797869873046875}{195752680522760891501} a^{17} + \frac{29175858389655985214}{195752680522760891501} a^{16} - \frac{26925101178557536506}{195752680522760891501} a^{15} - \frac{89940254016871946521}{195752680522760891501} a^{14} + \frac{32746615045479322440}{195752680522760891501} a^{13} - \frac{70293922597196507054}{195752680522760891501} a^{12} - \frac{83767727543656324406}{195752680522760891501} a^{11} + \frac{78273775059945490189}{195752680522760891501} a^{10} + \frac{65964512030353867118}{195752680522760891501} a^{9} + \frac{67338373700902844642}{195752680522760891501} a^{8} - \frac{44170464033325465777}{195752680522760891501} a^{7} - \frac{59820429004846099782}{195752680522760891501} a^{6} + \frac{17275381701335060291}{195752680522760891501} a^{5} - \frac{66583909217111852348}{195752680522760891501} a^{4} - \frac{83978495344806785875}{195752680522760891501} a^{3} - \frac{60439784023416643677}{195752680522760891501} a^{2} - \frac{62595118285590792047}{195752680522760891501} a + \frac{7339732691496994356}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{42} - \frac{17825245934158202347}{195752680522760891501} a^{23} - \frac{14369560283203125}{195752680522760891501} a^{22} - \frac{51511944594543126068}{195752680522760891501} a^{21} - \frac{1371639845214843750}{195752680522760891501} a^{20} - \frac{11209566973808466644}{195752680522760891501} a^{19} - \frac{56879509454345703125}{195752680522760891501} a^{18} - \frac{84018694216025134660}{195752680522760891501} a^{17} + \frac{31412618041650459257}{195752680522760891501} a^{16} + \frac{6844182917715574113}{195752680522760891501} a^{15} - \frac{54577691742465537400}{195752680522760891501} a^{14} + \frac{14843545518266479004}{195752680522760891501} a^{13} - \frac{65545543095149765054}{195752680522760891501} a^{12} - \frac{7626266025143266783}{195752680522760891501} a^{11} + \frac{71619858800893341616}{195752680522760891501} a^{10} + \frac{50995510169489545478}{195752680522760891501} a^{9} - \frac{54874741874829015042}{195752680522760891501} a^{8} + \frac{86903660923674894227}{195752680522760891501} a^{7} + \frac{74825008613415469464}{195752680522760891501} a^{6} + \frac{47454453648582543333}{195752680522760891501} a^{5} - \frac{64265183765312252797}{195752680522760891501} a^{4} - \frac{78123255946676819346}{195752680522760891501} a^{3} - \frac{89915195200430730469}{195752680522760891501} a^{2} + \frac{65605705602543601727}{195752680522760891501} a + \frac{36165793649228510957}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{43} - \frac{18723972490234375}{195752680522760891501} a^{23} - \frac{65761918245928650939}{195752680522760891501} a^{22} - \frac{1872397249023437500}{195752680522760891501} a^{21} - \frac{74198801841885402118}{195752680522760891501} a^{20} - \frac{81917379644775390625}{195752680522760891501} a^{19} - \frac{4416928443763063461}{195752680522760891501} a^{18} - \frac{94908640817312959990}{195752680522760891501} a^{17} + \frac{85551934456866110974}{195752680522760891501} a^{16} + \frac{95785425640612676666}{195752680522760891501} a^{15} - \frac{62441633690733724652}{195752680522760891501} a^{14} - \frac{73083298330751957793}{195752680522760891501} a^{13} + \frac{95184863764471043749}{195752680522760891501} a^{12} - \frac{66625843804260264065}{195752680522760891501} a^{11} + \frac{53690223453463809203}{195752680522760891501} a^{10} - \frac{38138756024611589213}{195752680522760891501} a^{9} - \frac{4557713581901669545}{195752680522760891501} a^{8} - \frac{70719714358693144550}{195752680522760891501} a^{7} - \frac{55242020649164007527}{195752680522760891501} a^{6} + \frac{19414745485774876348}{195752680522760891501} a^{5} - \frac{20367180817610216838}{195752680522760891501} a^{4} - \frac{70719714358693144550}{195752680522760891501} a^{3} + \frac{44665328333263089356}{195752680522760891501} a^{2} - \frac{75143635559280746312}{195752680522760891501} a - \frac{58548724685060851158}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{44} + \frac{23851848554331362270}{195752680522760891501} a^{23} + \frac{374479449804687500}{195752680522760891501} a^{22} + \frac{52244993004449758865}{195752680522760891501} a^{21} + \frac{36043647043701171875}{195752680522760891501} a^{20} + \frac{53687037144646457657}{195752680522760891501} a^{19} - \frac{60902117082477757008}{195752680522760891501} a^{18} - \frac{20259170978310560672}{195752680522760891501} a^{17} + \frac{12087640108982590568}{195752680522760891501} a^{16} + \frac{11926057461227428877}{195752680522760891501} a^{15} + \frac{2986508506746852817}{195752680522760891501} a^{14} - \frac{81206368917373586418}{195752680522760891501} a^{13} - \frac{49175824153388310558}{195752680522760891501} a^{12} - \frac{51578644321925934303}{195752680522760891501} a^{11} - \frac{86115419187867632385}{195752680522760891501} a^{10} + \frac{86728496645387895759}{195752680522760891501} a^{9} - \frac{80338617867271946982}{195752680522760891501} a^{8} + \frac{22863929509943796884}{195752680522760891501} a^{7} + \frac{28464824212094529304}{195752680522760891501} a^{6} + \frac{44558509273315826680}{195752680522760891501} a^{5} - \frac{5093864145418975954}{195752680522760891501} a^{4} - \frac{40912919651012878977}{195752680522760891501} a^{3} - \frac{21599216379358319359}{195752680522760891501} a^{2} + \frac{2082315532369520744}{195752680522760891501} a - \frac{64251318294019585455}{195752680522760891501}$, $\frac{1}{195752680522760891501} a^{45} + \frac{495634565917968750}{195752680522760891501} a^{23} - \frac{69439306196661232521}{195752680522760891501} a^{22} + \frac{49976485396728515625}{195752680522760891501} a^{21} - \frac{70652894185332062076}{195752680522760891501} a^{20} + \frac{48481759000606755989}{195752680522760891501} a^{19} - \frac{1177711499742721021}{195752680522760891501} a^{18} + \frac{95361560374097392717}{195752680522760891501} a^{17} + \frac{9822694008660690638}{195752680522760891501} a^{16} - \frac{750638107005596498}{195752680522760891501} a^{15} - \frac{15050216491645647147}{195752680522760891501} a^{14} - \frac{92775867980500219233}{195752680522760891501} a^{13} + \frac{81717387762207177969}{195752680522760891501} a^{12} + \frac{64096750484568209691}{195752680522760891501} a^{11} - \frac{68851982588679926737}{195752680522760891501} a^{10} - \frac{27823952198366199898}{195752680522760891501} a^{9} - \frac{34716991490423662826}{195752680522760891501} a^{8} - \frac{9743859303949120945}{195752680522760891501} a^{7} - \frac{45868189020370624348}{195752680522760891501} a^{6} + \frac{61726783676348867965}{195752680522760891501} a^{5} - \frac{81869330530602393221}{195752680522760891501} a^{4} - \frac{60546109065426543191}{195752680522760891501} a^{3} + \frac{20489203929687089446}{195752680522760891501} a^{2} + \frac{15875606333128950720}{195752680522760891501} a + \frac{28466128433530510535}{195752680522760891501}$

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

\(\Q(\sqrt{-987}) \), \(\Q(\zeta_{47})^+\)

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 $46$ R $46$ R $23^{2}$ $23^{2}$ $23^{2}$ $23^{2}$ $23^{2}$ $23^{2}$ $23^{2}$ $23^{2}$ $46$ $46$ 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
3Data not computed
7Data not computed
47Data not computed