↑

Properties

Label 40.40.2297193591...5625.1
Degree $40$
Signature $[40, 0]$
Discriminant $5^{30}\cdot 7^{20}\cdot 11^{36}$
Root discriminant $76.56$
Ramified primes $5, 7, 11$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{20}$ (as 40T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-440429, 11969793, -55346148, -331925242, 1401644442, 2893122177, -11039037933, -13143856660, 42561792020, 36333104654, -96911211847, -65437791794, 144757030391, 80642756914, -151353912514, -70671998879, 115530235482, 45356621385, -66197093248, -21780235070, 28998395977, 7941388686, -9822625214, -2217966289, 2587528870, 476024791, -530401956, -78304622, 84201027, 9779770, -10236731, -910316, 934252, 61132, -61891, -2797, 2808, 78, -78, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - 78*x^38 + 78*x^37 + 2808*x^36 - 2797*x^35 - 61891*x^34 + 61132*x^33 + 934252*x^32 - 910316*x^31 - 10236731*x^30 + 9779770*x^29 + 84201027*x^28 - 78304622*x^27 - 530401956*x^26 + 476024791*x^25 + 2587528870*x^24 - 2217966289*x^23 - 9822625214*x^22 + 7941388686*x^21 + 28998395977*x^20 - 21780235070*x^19 - 66197093248*x^18 + 45356621385*x^17 + 115530235482*x^16 - 70671998879*x^15 - 151353912514*x^14 + 80642756914*x^13 + 144757030391*x^12 - 65437791794*x^11 - 96911211847*x^10 + 36333104654*x^9 + 42561792020*x^8 - 13143856660*x^7 - 11039037933*x^6 + 2893122177*x^5 + 1401644442*x^4 - 331925242*x^3 - 55346148*x^2 + 11969793*x - 440429)
 
gp: K = bnfinit(x^40 - x^39 - 78*x^38 + 78*x^37 + 2808*x^36 - 2797*x^35 - 61891*x^34 + 61132*x^33 + 934252*x^32 - 910316*x^31 - 10236731*x^30 + 9779770*x^29 + 84201027*x^28 - 78304622*x^27 - 530401956*x^26 + 476024791*x^25 + 2587528870*x^24 - 2217966289*x^23 - 9822625214*x^22 + 7941388686*x^21 + 28998395977*x^20 - 21780235070*x^19 - 66197093248*x^18 + 45356621385*x^17 + 115530235482*x^16 - 70671998879*x^15 - 151353912514*x^14 + 80642756914*x^13 + 144757030391*x^12 - 65437791794*x^11 - 96911211847*x^10 + 36333104654*x^9 + 42561792020*x^8 - 13143856660*x^7 - 11039037933*x^6 + 2893122177*x^5 + 1401644442*x^4 - 331925242*x^3 - 55346148*x^2 + 11969793*x - 440429, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} - 78 x^{38} + 78 x^{37} + 2808 x^{36} - 2797 x^{35} - 61891 x^{34} + 61132 x^{33} + 934252 x^{32} - 910316 x^{31} - 10236731 x^{30} + 9779770 x^{29} + 84201027 x^{28} - 78304622 x^{27} - 530401956 x^{26} + 476024791 x^{25} + 2587528870 x^{24} - 2217966289 x^{23} - 9822625214 x^{22} + 7941388686 x^{21} + 28998395977 x^{20} - 21780235070 x^{19} - 66197093248 x^{18} + 45356621385 x^{17} + 115530235482 x^{16} - 70671998879 x^{15} - 151353912514 x^{14} + 80642756914 x^{13} + 144757030391 x^{12} - 65437791794 x^{11} - 96911211847 x^{10} + 36333104654 x^{9} + 42561792020 x^{8} - 13143856660 x^{7} - 11039037933 x^{6} + 2893122177 x^{5} + 1401644442 x^{4} - 331925242 x^{3} - 55346148 x^{2} + 11969793 x - 440429 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[40, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2297193591531201418681029625209696069890454162435033694840967655181884765625=5^{30}\cdot 7^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.56$
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:  $5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(385=5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{385}(384,·)$, $\chi_{385}(1,·)$, $\chi_{385}(258,·)$, $\chi_{385}(6,·)$, $\chi_{385}(8,·)$, $\chi_{385}(139,·)$, $\chi_{385}(141,·)$, $\chi_{385}(216,·)$, $\chi_{385}(27,·)$, $\chi_{385}(288,·)$, $\chi_{385}(162,·)$, $\chi_{385}(36,·)$, $\chi_{385}(41,·)$, $\chi_{385}(43,·)$, $\chi_{385}(48,·)$, $\chi_{385}(309,·)$, $\chi_{385}(183,·)$, $\chi_{385}(57,·)$, $\chi_{385}(314,·)$, $\chi_{385}(188,·)$, $\chi_{385}(64,·)$, $\chi_{385}(321,·)$, $\chi_{385}(197,·)$, $\chi_{385}(71,·)$, $\chi_{385}(328,·)$, $\chi_{385}(202,·)$, $\chi_{385}(76,·)$, $\chi_{385}(337,·)$, $\chi_{385}(342,·)$, $\chi_{385}(344,·)$, $\chi_{385}(349,·)$, $\chi_{385}(223,·)$, $\chi_{385}(97,·)$, $\chi_{385}(358,·)$, $\chi_{385}(244,·)$, $\chi_{385}(246,·)$, $\chi_{385}(169,·)$, $\chi_{385}(377,·)$, $\chi_{385}(379,·)$, $\chi_{385}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{23} a^{22} + \frac{2}{23} a^{20} + \frac{8}{23} a^{18} - \frac{6}{23} a^{16} - \frac{6}{23} a^{14} + \frac{10}{23} a^{12} - \frac{1}{23} a^{11} + \frac{7}{23} a^{10} - \frac{1}{23} a^{9} + \frac{11}{23} a^{8} + \frac{8}{23} a^{7} - \frac{11}{23} a^{6} - \frac{5}{23} a^{5} + \frac{8}{23} a^{4} - \frac{6}{23} a^{3} + \frac{3}{23} a^{2} + \frac{7}{23} a - \frac{1}{23}$, $\frac{1}{23} a^{23} + \frac{2}{23} a^{21} + \frac{8}{23} a^{19} - \frac{6}{23} a^{17} - \frac{6}{23} a^{15} + \frac{10}{23} a^{13} - \frac{1}{23} a^{12} + \frac{7}{23} a^{11} - \frac{1}{23} a^{10} + \frac{11}{23} a^{9} + \frac{8}{23} a^{8} - \frac{11}{23} a^{7} - \frac{5}{23} a^{6} + \frac{8}{23} a^{5} - \frac{6}{23} a^{4} + \frac{3}{23} a^{3} + \frac{7}{23} a^{2} - \frac{1}{23} a$, $\frac{1}{23} a^{24} + \frac{4}{23} a^{20} + \frac{1}{23} a^{18} + \frac{6}{23} a^{16} - \frac{1}{23} a^{14} - \frac{1}{23} a^{13} + \frac{10}{23} a^{12} + \frac{1}{23} a^{11} - \frac{3}{23} a^{10} + \frac{10}{23} a^{9} - \frac{10}{23} a^{8} + \frac{2}{23} a^{7} + \frac{7}{23} a^{6} + \frac{4}{23} a^{5} + \frac{10}{23} a^{4} - \frac{4}{23} a^{3} - \frac{7}{23} a^{2} + \frac{9}{23} a + \frac{2}{23}$, $\frac{1}{23} a^{25} + \frac{4}{23} a^{21} + \frac{1}{23} a^{19} + \frac{6}{23} a^{17} - \frac{1}{23} a^{15} - \frac{1}{23} a^{14} + \frac{10}{23} a^{13} + \frac{1}{23} a^{12} - \frac{3}{23} a^{11} + \frac{10}{23} a^{10} - \frac{10}{23} a^{9} + \frac{2}{23} a^{8} + \frac{7}{23} a^{7} + \frac{4}{23} a^{6} + \frac{10}{23} a^{5} - \frac{4}{23} a^{4} - \frac{7}{23} a^{3} + \frac{9}{23} a^{2} + \frac{2}{23} a$, $\frac{1}{23} a^{26} - \frac{7}{23} a^{20} - \frac{3}{23} a^{18} - \frac{1}{23} a^{15} + \frac{11}{23} a^{14} + \frac{1}{23} a^{13} + \frac{3}{23} a^{12} - \frac{9}{23} a^{11} + \frac{8}{23} a^{10} + \frac{6}{23} a^{9} + \frac{9}{23} a^{8} - \frac{5}{23} a^{7} + \frac{8}{23} a^{6} - \frac{7}{23} a^{5} + \frac{7}{23} a^{4} + \frac{10}{23} a^{3} - \frac{10}{23} a^{2} - \frac{5}{23} a + \frac{4}{23}$, $\frac{1}{74404793} a^{27} - \frac{768712}{74404793} a^{26} - \frac{556574}{74404793} a^{25} - \frac{825057}{74404793} a^{24} + \frac{90359}{74404793} a^{23} + \frac{1168843}{74404793} a^{22} + \frac{17351258}{74404793} a^{21} - \frac{22379180}{74404793} a^{20} - \frac{12878684}{74404793} a^{19} - \frac{9211044}{74404793} a^{18} - \frac{15774323}{74404793} a^{17} + \frac{25741281}{74404793} a^{16} - \frac{1216293}{74404793} a^{15} + \frac{26834020}{74404793} a^{14} + \frac{27556110}{74404793} a^{13} + \frac{35213977}{74404793} a^{12} - \frac{11984298}{74404793} a^{11} - \frac{6518688}{74404793} a^{10} + \frac{490173}{3234991} a^{9} + \frac{22167060}{74404793} a^{8} + \frac{26370820}{74404793} a^{7} - \frac{7169107}{74404793} a^{6} + \frac{28554481}{74404793} a^{5} + \frac{23080657}{74404793} a^{4} - \frac{6918754}{74404793} a^{3} + \frac{15754476}{74404793} a^{2} + \frac{10170842}{74404793} a - \frac{19847882}{74404793}$, $\frac{1}{74404793} a^{28} + \frac{935497}{74404793} a^{26} + \frac{1031951}{74404793} a^{25} - \frac{1435702}{74404793} a^{24} - \frac{22187}{3234991} a^{23} - \frac{993767}{74404793} a^{22} - \frac{21462}{169487} a^{21} - \frac{8730359}{74404793} a^{20} - \frac{23418590}{74404793} a^{19} + \frac{11596374}{74404793} a^{18} + \frac{9049110}{74404793} a^{17} - \frac{23460408}{74404793} a^{16} + \frac{31378206}{74404793} a^{15} - \frac{26453798}{74404793} a^{14} - \frac{9008604}{74404793} a^{13} + \frac{25212446}{74404793} a^{12} + \frac{10707260}{74404793} a^{11} + \frac{15408132}{74404793} a^{10} - \frac{20782045}{74404793} a^{9} - \frac{839151}{3234991} a^{8} - \frac{4593180}{74404793} a^{7} + \frac{9187374}{74404793} a^{6} - \frac{20933639}{74404793} a^{5} - \frac{14694254}{74404793} a^{4} - \frac{10341867}{74404793} a^{3} + \frac{29688496}{74404793} a^{2} + \frac{31029020}{74404793} a - \frac{3789999}{74404793}$, $\frac{1}{74404793} a^{29} + \frac{7488}{74404793} a^{26} + \frac{70126}{74404793} a^{25} + \frac{1335338}{74404793} a^{24} - \frac{1252360}{74404793} a^{23} + \frac{766121}{74404793} a^{22} + \frac{6995529}{74404793} a^{21} - \frac{26001478}{74404793} a^{20} - \frac{23969176}{74404793} a^{19} - \frac{19224037}{74404793} a^{18} - \frac{182707}{3234991} a^{17} - \frac{848299}{74404793} a^{16} - \frac{31385607}{74404793} a^{15} - \frac{31665410}{74404793} a^{14} + \frac{1025327}{3234991} a^{13} + \frac{7686873}{74404793} a^{12} - \frac{1330065}{74404793} a^{11} - \frac{24548389}{74404793} a^{10} - \frac{770043}{74404793} a^{9} - \frac{11674556}{74404793} a^{8} + \frac{20516410}{74404793} a^{7} - \frac{22663948}{74404793} a^{6} - \frac{8677974}{74404793} a^{5} + \frac{9651221}{74404793} a^{4} - \frac{1993746}{74404793} a^{3} - \frac{9678454}{74404793} a^{2} + \frac{14555574}{74404793} a - \frac{1597214}{3234991}$, $\frac{1}{74404793} a^{30} + \frac{1136593}{74404793} a^{26} - \frac{941949}{74404793} a^{25} + \frac{1176637}{74404793} a^{24} + \frac{271048}{74404793} a^{23} - \frac{1120182}{74404793} a^{22} + \frac{32951989}{74404793} a^{21} + \frac{27321729}{74404793} a^{20} + \frac{36274892}{74404793} a^{19} + \frac{11027055}{74404793} a^{18} - \frac{21354004}{74404793} a^{17} - \frac{8984045}{74404793} a^{16} - \frac{11153145}{74404793} a^{15} - \frac{10148157}{74404793} a^{14} - \frac{1503836}{74404793} a^{13} + \frac{10231542}{74404793} a^{12} + \frac{7574605}{74404793} a^{11} + \frac{27501421}{74404793} a^{10} + \frac{2035792}{74404793} a^{9} - \frac{27565525}{74404793} a^{8} + \frac{31481379}{74404793} a^{7} + \frac{18034516}{74404793} a^{6} - \frac{34862236}{74404793} a^{5} - \frac{32909097}{74404793} a^{4} + \frac{8980579}{74404793} a^{3} - \frac{3953863}{74404793} a^{2} + \frac{10682169}{74404793} a - \frac{33366016}{74404793}$, $\frac{1}{74404793} a^{31} - \frac{1102995}{74404793} a^{26} + \frac{33960}{74404793} a^{25} + \frac{1560751}{74404793} a^{24} - \frac{1267792}{74404793} a^{23} - \frac{590805}{74404793} a^{22} + \frac{3411539}{74404793} a^{21} - \frac{8981231}{74404793} a^{20} - \frac{22385665}{74404793} a^{19} + \frac{2150185}{74404793} a^{18} - \frac{20117625}{74404793} a^{17} + \frac{26759835}{74404793} a^{16} - \frac{21092348}{74404793} a^{15} + \frac{17890628}{74404793} a^{14} - \frac{486155}{3234991} a^{13} - \frac{35355448}{74404793} a^{12} - \frac{1676339}{74404793} a^{11} + \frac{13473431}{74404793} a^{10} - \frac{34133405}{74404793} a^{9} - \frac{13909343}{74404793} a^{8} - \frac{19828580}{74404793} a^{7} - \frac{12180477}{74404793} a^{6} - \frac{35374200}{74404793} a^{5} - \frac{1959137}{74404793} a^{4} - \frac{35925920}{74404793} a^{3} + \frac{15714795}{74404793} a^{2} + \frac{8571430}{74404793} a + \frac{16041797}{74404793}$, $\frac{1}{74404793} a^{32} - \frac{787362}{74404793} a^{26} + \frac{993709}{74404793} a^{25} + \frac{304703}{74404793} a^{24} + \frac{1331672}{74404793} a^{23} + \frac{138067}{74404793} a^{22} + \frac{28325776}{74404793} a^{21} + \frac{16071950}{74404793} a^{20} + \frac{14721795}{74404793} a^{19} + \frac{93276}{3234991} a^{18} + \frac{35022039}{74404793} a^{17} - \frac{30015543}{74404793} a^{16} + \frac{10030775}{74404793} a^{15} + \frac{355942}{3234991} a^{14} - \frac{12346858}{74404793} a^{13} - \frac{22326886}{74404793} a^{12} + \frac{25295643}{74404793} a^{11} + \frac{21882509}{74404793} a^{10} + \frac{23723204}{74404793} a^{9} - \frac{3101664}{74404793} a^{8} - \frac{4437598}{74404793} a^{7} - \frac{6364004}{74404793} a^{6} + \frac{34804180}{74404793} a^{5} + \frac{12847403}{74404793} a^{4} - \frac{20699354}{74404793} a^{3} - \frac{35233343}{74404793} a^{2} - \frac{1607862}{3234991} a + \frac{27047746}{74404793}$, $\frac{1}{74404793} a^{33} + \frac{252101}{74404793} a^{26} - \frac{92261}{74404793} a^{25} + \frac{1344748}{74404793} a^{24} + \frac{1458953}{74404793} a^{23} - \frac{1406621}{74404793} a^{22} + \frac{20901318}{74404793} a^{21} + \frac{19231958}{74404793} a^{20} + \frac{24442880}{74404793} a^{19} - \frac{10255584}{74404793} a^{18} - \frac{2460250}{74404793} a^{17} - \frac{16457072}{74404793} a^{16} - \frac{13256634}{74404793} a^{15} - \frac{9996619}{74404793} a^{14} - \frac{20183326}{74404793} a^{13} + \frac{32055626}{74404793} a^{12} - \frac{1340945}{74404793} a^{11} - \frac{1150953}{3234991} a^{10} + \frac{7177392}{74404793} a^{9} - \frac{8059853}{74404793} a^{8} + \frac{25438112}{74404793} a^{7} - \frac{28285330}{74404793} a^{6} + \frac{17445193}{74404793} a^{5} + \frac{32870583}{74404793} a^{4} - \frac{16715895}{74404793} a^{3} + \frac{4866599}{74404793} a^{2} + \frac{18965834}{74404793} a + \frac{26170795}{74404793}$, $\frac{1}{74404793} a^{34} + \frac{835796}{74404793} a^{26} - \frac{292912}{74404793} a^{25} - \frac{1062617}{74404793} a^{24} - \frac{8446}{3234991} a^{23} + \frac{627446}{74404793} a^{22} + \frac{10754361}{74404793} a^{21} + \frac{17031051}{74404793} a^{20} + \frac{33367035}{74404793} a^{19} - \frac{23826444}{74404793} a^{18} - \frac{6825920}{74404793} a^{17} - \frac{7817060}{74404793} a^{16} + \frac{22412949}{74404793} a^{15} - \frac{29384759}{74404793} a^{14} + \frac{19421664}{74404793} a^{13} + \frac{17730479}{74404793} a^{12} + \frac{36593369}{74404793} a^{11} - \frac{10192093}{74404793} a^{10} - \frac{1579862}{3234991} a^{9} + \frac{34112831}{74404793} a^{8} + \frac{36631103}{74404793} a^{7} + \frac{2602201}{74404793} a^{6} - \frac{7200996}{74404793} a^{5} + \frac{36950637}{74404793} a^{4} - \frac{6308645}{74404793} a^{3} - \frac{30567740}{74404793} a^{2} - \frac{18811593}{74404793} a - \frac{2904303}{74404793}$, $\frac{1}{74404793} a^{35} + \frac{734285}{74404793} a^{26} - \frac{740540}{74404793} a^{25} - \frac{1240419}{74404793} a^{24} - \frac{198423}{74404793} a^{23} - \frac{967487}{74404793} a^{22} - \frac{18546237}{74404793} a^{21} + \frac{13036568}{74404793} a^{20} + \frac{7322098}{74404793} a^{19} - \frac{566930}{74404793} a^{18} + \frac{258620}{3234991} a^{17} - \frac{3471578}{74404793} a^{16} + \frac{28392494}{74404793} a^{15} - \frac{13923978}{74404793} a^{14} + \frac{29662031}{74404793} a^{13} - \frac{26968459}{74404793} a^{12} + \frac{15790536}{74404793} a^{11} - \frac{4833684}{74404793} a^{10} + \frac{7595797}{74404793} a^{9} - \frac{35017422}{74404793} a^{8} + \frac{33585636}{74404793} a^{7} + \frac{4898111}{74404793} a^{6} + \frac{18857494}{74404793} a^{5} - \frac{9934913}{74404793} a^{4} + \frac{25238124}{74404793} a^{3} + \frac{18192307}{74404793} a^{2} + \frac{15045661}{74404793} a - \frac{1174546}{3234991}$, $\frac{1}{74404793} a^{36} - \frac{1219264}{74404793} a^{26} - \frac{183841}{74404793} a^{25} + \frac{311279}{74404793} a^{24} - \frac{560392}{74404793} a^{23} - \frac{496300}{74404793} a^{22} + \frac{1045249}{74404793} a^{21} - \frac{25279455}{74404793} a^{20} + \frac{14113044}{74404793} a^{19} - \frac{13571468}{74404793} a^{18} - \frac{17338086}{74404793} a^{17} - \frac{35061701}{74404793} a^{16} - \frac{24533753}{74404793} a^{15} + \frac{36808735}{74404793} a^{14} - \frac{18182523}{74404793} a^{13} - \frac{16227297}{74404793} a^{12} + \frac{30555136}{74404793} a^{11} - \frac{7319453}{74404793} a^{10} + \frac{15926500}{74404793} a^{9} + \frac{2909892}{74404793} a^{8} + \frac{15608048}{74404793} a^{7} - \frac{30153545}{74404793} a^{6} - \frac{11923193}{74404793} a^{5} + \frac{13482851}{74404793} a^{4} - \frac{13497816}{74404793} a^{3} + \frac{15692145}{74404793} a^{2} + \frac{25014937}{74404793} a - \frac{33559514}{74404793}$, $\frac{1}{74404793} a^{37} + \frac{1185648}{74404793} a^{26} + \frac{201795}{74404793} a^{25} - \frac{15309}{3234991} a^{24} + \frac{125980}{74404793} a^{23} - \frac{758375}{74404793} a^{22} - \frac{21698349}{74404793} a^{21} + \frac{2712395}{74404793} a^{20} + \frac{3749271}{74404793} a^{19} + \frac{2760565}{74404793} a^{18} - \frac{32822880}{74404793} a^{17} - \frac{20904874}{74404793} a^{16} + \frac{3529702}{74404793} a^{15} + \frac{4667066}{74404793} a^{14} - \frac{481535}{74404793} a^{13} - \frac{1097919}{74404793} a^{12} + \frac{29241973}{74404793} a^{11} + \frac{5478930}{74404793} a^{10} - \frac{18541257}{74404793} a^{9} + \frac{8379539}{74404793} a^{8} + \frac{457173}{3234991} a^{7} - \frac{29148639}{74404793} a^{6} - \frac{20065743}{74404793} a^{5} - \frac{35124675}{74404793} a^{4} - \frac{21823833}{74404793} a^{3} - \frac{13808713}{74404793} a^{2} - \frac{19213869}{74404793} a + \frac{1545374}{74404793}$, $\frac{1}{74404793} a^{38} - \frac{1082178}{74404793} a^{26} + \frac{1153737}{74404793} a^{25} + \frac{1614417}{74404793} a^{24} - \frac{1529060}{74404793} a^{23} - \frac{859177}{74404793} a^{22} - \frac{35146957}{74404793} a^{21} - \frac{25027802}{74404793} a^{20} + \frac{7383976}{74404793} a^{19} + \frac{781777}{74404793} a^{18} - \frac{22765916}{74404793} a^{17} - \frac{34819599}{74404793} a^{16} + \frac{12211932}{74404793} a^{15} + \frac{23014648}{74404793} a^{14} - \frac{27087780}{74404793} a^{13} + \frac{17514275}{74404793} a^{12} - \frac{19169897}{74404793} a^{11} - \frac{24664092}{74404793} a^{10} - \frac{31221709}{74404793} a^{9} + \frac{34570517}{74404793} a^{8} + \frac{17663975}{74404793} a^{7} + \frac{818309}{74404793} a^{6} + \frac{12267659}{74404793} a^{5} + \frac{15102253}{74404793} a^{4} + \frac{35490710}{74404793} a^{3} - \frac{26826478}{74404793} a^{2} - \frac{35384335}{74404793} a + \frac{25122019}{74404793}$, $\frac{1}{74404793} a^{39} - \frac{890358}{74404793} a^{26} - \frac{489429}{74404793} a^{25} + \frac{687785}{74404793} a^{24} - \frac{410232}{74404793} a^{23} - \frac{1037957}{74404793} a^{22} + \frac{14061524}{74404793} a^{21} + \frac{16297813}{74404793} a^{20} - \frac{20075829}{74404793} a^{19} - \frac{24466475}{74404793} a^{18} - \frac{827013}{74404793} a^{17} + \frac{3891364}{74404793} a^{16} - \frac{17898291}{74404793} a^{15} - \frac{832519}{3234991} a^{14} - \frac{11704759}{74404793} a^{13} - \frac{1399161}{74404793} a^{12} + \frac{20603558}{74404793} a^{11} - \frac{6195086}{74404793} a^{10} - \frac{23140441}{74404793} a^{9} + \frac{28289093}{74404793} a^{8} - \frac{705980}{74404793} a^{7} - \frac{11021385}{74404793} a^{6} + \frac{7835996}{74404793} a^{5} + \frac{23330728}{74404793} a^{4} + \frac{460734}{3234991} a^{3} - \frac{78983}{169487} a^{2} - \frac{4747622}{74404793} a - \frac{8228081}{74404793}$

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:  $39$
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

$C_2\times C_{20}$ (as 40T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{385}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{77}) \), \(\Q(\sqrt{5}, \sqrt{77})\), 4.4.15125.1, 4.4.6125.1, \(\Q(\zeta_{11})^+\), 8.8.549266265625.1, 10.10.123843833883240625.1, 10.10.669871503125.1, 10.10.39630026842637.1, 20.20.15337295190899698702745251650390625.1, \(\Q(\zeta_{55})^+\), 20.20.396107830343483954099825714111328125.1

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 $20^{2}$ $20^{2}$ R R R $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
7Data not computed
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$