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Properties

Label 36.0.44387950739...5625.2
Degree $36$
Signature $[0, 18]$
Discriminant $3^{18}\cdot 5^{18}\cdot 19^{34}$
Root discriminant $62.48$
Ramified primes $3, 5, 19$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{18}$ (as 36T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![87403801, -87226170, -177270, 82074510, -81902655, 11551791, 70408263, -58512762, -12157587, 19867883, -6904183, 3200721, 2074269, 942095, -600913, -3183278, 1113874, 2247035, -1062783, -1174903, 679648, 495255, -326116, -169139, 122532, 46607, -36329, -10278, 8512, 1766, -1538, -228, 208, 20, -19, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 19*x^34 + 20*x^33 + 208*x^32 - 228*x^31 - 1538*x^30 + 1766*x^29 + 8512*x^28 - 10278*x^27 - 36329*x^26 + 46607*x^25 + 122532*x^24 - 169139*x^23 - 326116*x^22 + 495255*x^21 + 679648*x^20 - 1174903*x^19 - 1062783*x^18 + 2247035*x^17 + 1113874*x^16 - 3183278*x^15 - 600913*x^14 + 942095*x^13 + 2074269*x^12 + 3200721*x^11 - 6904183*x^10 + 19867883*x^9 - 12157587*x^8 - 58512762*x^7 + 70408263*x^6 + 11551791*x^5 - 81902655*x^4 + 82074510*x^3 - 177270*x^2 - 87226170*x + 87403801)
 
gp: K = bnfinit(x^36 - x^35 - 19*x^34 + 20*x^33 + 208*x^32 - 228*x^31 - 1538*x^30 + 1766*x^29 + 8512*x^28 - 10278*x^27 - 36329*x^26 + 46607*x^25 + 122532*x^24 - 169139*x^23 - 326116*x^22 + 495255*x^21 + 679648*x^20 - 1174903*x^19 - 1062783*x^18 + 2247035*x^17 + 1113874*x^16 - 3183278*x^15 - 600913*x^14 + 942095*x^13 + 2074269*x^12 + 3200721*x^11 - 6904183*x^10 + 19867883*x^9 - 12157587*x^8 - 58512762*x^7 + 70408263*x^6 + 11551791*x^5 - 81902655*x^4 + 82074510*x^3 - 177270*x^2 - 87226170*x + 87403801, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 19 x^{34} + 20 x^{33} + 208 x^{32} - 228 x^{31} - 1538 x^{30} + 1766 x^{29} + 8512 x^{28} - 10278 x^{27} - 36329 x^{26} + 46607 x^{25} + 122532 x^{24} - 169139 x^{23} - 326116 x^{22} + 495255 x^{21} + 679648 x^{20} - 1174903 x^{19} - 1062783 x^{18} + 2247035 x^{17} + 1113874 x^{16} - 3183278 x^{15} - 600913 x^{14} + 942095 x^{13} + 2074269 x^{12} + 3200721 x^{11} - 6904183 x^{10} + 19867883 x^{9} - 12157587 x^{8} - 58512762 x^{7} + 70408263 x^{6} + 11551791 x^{5} - 81902655 x^{4} + 82074510 x^{3} - 177270 x^{2} - 87226170 x + 87403801 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(44387950739803436916061690678602018428037077267652774810791015625=3^{18}\cdot 5^{18}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.48$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(285=3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{285}(256,·)$, $\chi_{285}(1,·)$, $\chi_{285}(131,·)$, $\chi_{285}(11,·)$, $\chi_{285}(269,·)$, $\chi_{285}(14,·)$, $\chi_{285}(271,·)$, $\chi_{285}(16,·)$, $\chi_{285}(274,·)$, $\chi_{285}(259,·)$, $\chi_{285}(154,·)$, $\chi_{285}(284,·)$, $\chi_{285}(29,·)$, $\chi_{285}(161,·)$, $\chi_{285}(34,·)$, $\chi_{285}(164,·)$, $\chi_{285}(176,·)$, $\chi_{285}(179,·)$, $\chi_{285}(184,·)$, $\chi_{285}(59,·)$, $\chi_{285}(61,·)$, $\chi_{285}(191,·)$, $\chi_{285}(196,·)$, $\chi_{285}(206,·)$, $\chi_{285}(79,·)$, $\chi_{285}(89,·)$, $\chi_{285}(26,·)$, $\chi_{285}(94,·)$, $\chi_{285}(224,·)$, $\chi_{285}(226,·)$, $\chi_{285}(101,·)$, $\chi_{285}(106,·)$, $\chi_{285}(109,·)$, $\chi_{285}(121,·)$, $\chi_{285}(251,·)$, $\chi_{285}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{37} a^{18} - \frac{12}{37} a^{17} + \frac{16}{37} a^{16} + \frac{8}{37} a^{15} + \frac{14}{37} a^{14} - \frac{2}{37} a^{13} + \frac{9}{37} a^{12} + \frac{12}{37} a^{11} + \frac{2}{37} a^{10} - \frac{7}{37} a^{9} + \frac{7}{37} a^{8} + \frac{6}{37} a^{7} + \frac{8}{37} a^{6} - \frac{8}{37} a^{5} + \frac{10}{37} a^{4} + \frac{16}{37} a^{3} - \frac{14}{37} a^{2} + \frac{4}{37} a + \frac{1}{37}$, $\frac{1}{37} a^{19} - \frac{17}{37} a^{17} + \frac{15}{37} a^{16} - \frac{1}{37} a^{15} + \frac{18}{37} a^{14} - \frac{15}{37} a^{13} + \frac{9}{37} a^{12} - \frac{2}{37} a^{11} + \frac{17}{37} a^{10} - \frac{3}{37} a^{9} + \frac{16}{37} a^{8} + \frac{6}{37} a^{7} + \frac{14}{37} a^{6} - \frac{12}{37} a^{5} - \frac{12}{37} a^{4} - \frac{7}{37} a^{3} - \frac{16}{37} a^{2} + \frac{12}{37} a + \frac{12}{37}$, $\frac{1}{4181} a^{20} + \frac{15}{4181} a^{19} - \frac{11}{4181} a^{18} + \frac{1575}{4181} a^{17} + \frac{1430}{4181} a^{16} + \frac{1383}{4181} a^{15} + \frac{487}{4181} a^{14} + \frac{1215}{4181} a^{13} - \frac{812}{4181} a^{12} + \frac{873}{4181} a^{11} + \frac{671}{4181} a^{10} + \frac{1890}{4181} a^{9} + \frac{2027}{4181} a^{8} + \frac{917}{4181} a^{7} + \frac{1171}{4181} a^{6} - \frac{536}{4181} a^{5} - \frac{1126}{4181} a^{4} - \frac{1024}{4181} a^{3} + \frac{95}{4181} a^{2} + \frac{1400}{4181} a + \frac{1925}{4181}$, $\frac{1}{4181} a^{21} - \frac{10}{4181} a^{19} + \frac{45}{4181} a^{18} - \frac{1516}{4181} a^{17} - \frac{1987}{4181} a^{16} - \frac{596}{4181} a^{15} - \frac{18}{113} a^{14} + \frac{1868}{4181} a^{13} - \frac{168}{4181} a^{12} + \frac{232}{4181} a^{11} + \frac{639}{4181} a^{10} + \frac{1588}{4181} a^{9} - \frac{108}{4181} a^{8} - \frac{493}{4181} a^{7} + \frac{770}{4181} a^{6} + \frac{1038}{4181} a^{5} + \frac{385}{4181} a^{4} - \frac{704}{4181} a^{3} - \frac{816}{4181} a^{2} + \frac{1943}{4181} a + \frac{1409}{4181}$, $\frac{1}{4181} a^{22} - \frac{31}{4181} a^{19} - \frac{44}{4181} a^{18} - \frac{1379}{4181} a^{17} - \frac{2003}{4181} a^{16} + \frac{960}{4181} a^{15} - \frac{268}{4181} a^{14} - \frac{335}{4181} a^{13} + \frac{135}{4181} a^{12} - \frac{462}{4181} a^{11} - \frac{742}{4181} a^{10} + \frac{34}{4181} a^{9} - \frac{2032}{4181} a^{8} + \frac{1352}{4181} a^{7} + \frac{1335}{4181} a^{6} + \frac{1805}{4181} a^{5} - \frac{1794}{4181} a^{4} - \frac{886}{4181} a^{3} + \frac{1085}{4181} a^{2} - \frac{1880}{4181} a + \frac{1396}{4181}$, $\frac{1}{4181} a^{23} - \frac{31}{4181} a^{19} - \frac{25}{4181} a^{18} + \frac{718}{4181} a^{17} - \frac{1266}{4181} a^{16} - \frac{1917}{4181} a^{15} + \frac{1089}{4181} a^{14} - \frac{620}{4181} a^{13} - \frac{1904}{4181} a^{12} + \frac{1574}{4181} a^{11} - \frac{183}{4181} a^{10} + \frac{58}{4181} a^{9} + \frac{1926}{4181} a^{8} - \frac{409}{4181} a^{7} - \frac{653}{4181} a^{6} - \frac{1460}{4181} a^{5} - \frac{875}{4181} a^{4} - \frac{375}{4181} a^{3} + \frac{1291}{4181} a^{2} + \frac{161}{4181} a + \frac{1593}{4181}$, $\frac{1}{4181} a^{24} - \frac{12}{4181} a^{19} + \frac{38}{4181} a^{18} + \frac{21}{113} a^{17} + \frac{942}{4181} a^{16} - \frac{108}{4181} a^{15} + \frac{1595}{4181} a^{14} + \frac{1409}{4181} a^{13} - \frac{1450}{4181} a^{12} - \frac{1370}{4181} a^{11} - \frac{46}{4181} a^{10} + \frac{1530}{4181} a^{9} - \frac{1530}{4181} a^{8} - \frac{2058}{4181} a^{7} + \frac{715}{4181} a^{6} - \frac{993}{4181} a^{5} + \frac{201}{4181} a^{4} + \frac{735}{4181} a^{3} - \frac{1640}{4181} a^{2} + \frac{584}{4181} a - \frac{441}{4181}$, $\frac{1}{4181} a^{25} - \frac{8}{4181} a^{19} - \frac{33}{4181} a^{18} - \frac{44}{113} a^{17} - \frac{1367}{4181} a^{16} + \frac{450}{4181} a^{15} + \frac{2055}{4181} a^{14} + \frac{1152}{4181} a^{13} + \frac{1655}{4181} a^{12} - \frac{1435}{4181} a^{11} + \frac{203}{4181} a^{10} + \frac{1488}{4181} a^{9} + \frac{1361}{4181} a^{8} - \frac{2067}{4181} a^{7} + \frac{290}{4181} a^{6} + \frac{1905}{4181} a^{5} - \frac{121}{4181} a^{4} + \frac{1892}{4181} a^{3} - \frac{1892}{4181} a^{2} - \frac{1608}{4181} a - \frac{1195}{4181}$, $\frac{1}{4181} a^{26} - \frac{26}{4181} a^{19} - \frac{21}{4181} a^{18} + \frac{1176}{4181} a^{17} - \frac{314}{4181} a^{16} + \frac{1706}{4181} a^{15} + \frac{1658}{4181} a^{14} + \frac{1318}{4181} a^{13} - \frac{2055}{4181} a^{12} - \frac{1514}{4181} a^{11} - \frac{1}{113} a^{10} + \frac{774}{4181} a^{9} - \frac{880}{4181} a^{8} + \frac{394}{4181} a^{7} - \frac{1835}{4181} a^{6} + \frac{3}{113} a^{5} - \frac{1353}{4181} a^{4} + \frac{1103}{4181} a^{3} - \frac{1865}{4181} a^{2} - \frac{35}{113} a - \frac{985}{4181}$, $\frac{1}{4181} a^{27} + \frac{30}{4181} a^{19} - \frac{14}{4181} a^{18} - \frac{1287}{4181} a^{17} - \frac{1568}{4181} a^{16} + \frac{1456}{4181} a^{15} - \frac{597}{4181} a^{14} - \frac{1201}{4181} a^{13} - \frac{365}{4181} a^{12} - \frac{52}{4181} a^{11} - \frac{1894}{4181} a^{10} + \frac{1252}{4181} a^{9} - \frac{466}{4181} a^{8} + \frac{2006}{4181} a^{7} + \frac{1855}{4181} a^{6} + \frac{192}{4181} a^{5} + \frac{303}{4181} a^{4} + \frac{1230}{4181} a^{3} - \frac{1650}{4181} a^{2} - \frac{1536}{4181} a - \frac{913}{4181}$, $\frac{1}{4181} a^{28} - \frac{12}{4181} a^{19} - \frac{53}{4181} a^{18} - \frac{454}{4181} a^{17} + \frac{705}{4181} a^{16} - \frac{1859}{4181} a^{15} + \frac{800}{4181} a^{14} + \frac{588}{4181} a^{13} - \frac{1117}{4181} a^{12} - \frac{1416}{4181} a^{11} - \frac{1024}{4181} a^{10} + \frac{2046}{4181} a^{9} + \frac{747}{4181} a^{8} - \frac{795}{4181} a^{7} - \frac{473}{4181} a^{6} - \frac{454}{4181} a^{5} + \frac{997}{4181} a^{4} - \frac{1440}{4181} a^{3} + \frac{812}{4181} a^{2} - \frac{425}{4181} a - \frac{1250}{4181}$, $\frac{1}{4181} a^{29} + \frac{14}{4181} a^{19} - \frac{21}{4181} a^{18} - \frac{1978}{4181} a^{17} + \frac{1741}{4181} a^{16} + \frac{1124}{4181} a^{15} - \frac{235}{4181} a^{14} + \frac{1485}{4181} a^{13} + \frac{1270}{4181} a^{12} - \frac{266}{4181} a^{11} + \frac{945}{4181} a^{10} - \frac{1094}{4181} a^{9} + \frac{590}{4181} a^{8} + \frac{700}{4181} a^{7} - \frac{188}{4181} a^{6} - \frac{237}{4181} a^{5} + \frac{416}{4181} a^{4} - \frac{1645}{4181} a^{3} - \frac{1206}{4181} a^{2} - \frac{270}{4181} a + \frac{1404}{4181}$, $\frac{1}{4181} a^{30} - \frac{5}{4181} a^{19} - \frac{16}{4181} a^{18} + \frac{144}{4181} a^{17} + \frac{879}{4181} a^{16} - \frac{1178}{4181} a^{15} - \frac{1039}{4181} a^{14} - \frac{1841}{4181} a^{13} + \frac{141}{4181} a^{12} + \frac{1605}{4181} a^{11} + \frac{1151}{4181} a^{10} - \frac{1575}{4181} a^{9} + \frac{1137}{4181} a^{8} - \frac{822}{4181} a^{7} + \frac{997}{4181} a^{6} - \frac{894}{4181} a^{5} + \frac{220}{4181} a^{4} - \frac{1334}{4181} a^{3} - \frac{1261}{4181} a^{2} + \frac{110}{4181} a - \frac{1525}{4181}$, $\frac{1}{4181} a^{31} - \frac{54}{4181} a^{19} - \frac{24}{4181} a^{18} - \frac{512}{4181} a^{17} - \frac{1712}{4181} a^{16} + \frac{8}{37} a^{15} + \frac{1159}{4181} a^{14} - \frac{225}{4181} a^{13} - \frac{308}{4181} a^{12} + \frac{205}{4181} a^{11} - \frac{367}{4181} a^{10} - \frac{826}{4181} a^{9} - \frac{1648}{4181} a^{8} + \frac{45}{4181} a^{7} - \frac{1706}{4181} a^{6} - \frac{200}{4181} a^{5} + \frac{1624}{4181} a^{4} + \frac{964}{4181} a^{3} - \frac{206}{4181} a^{2} - \frac{514}{4181} a - \frac{206}{4181}$, $\frac{1}{154697} a^{32} - \frac{18}{154697} a^{30} + \frac{6}{154697} a^{29} - \frac{5}{154697} a^{28} - \frac{3}{154697} a^{27} - \frac{13}{154697} a^{26} + \frac{6}{154697} a^{25} - \frac{11}{154697} a^{24} - \frac{1}{154697} a^{23} + \frac{3}{154697} a^{22} - \frac{3}{154697} a^{21} + \frac{12}{154697} a^{20} - \frac{308}{154697} a^{19} + \frac{793}{154697} a^{18} - \frac{77183}{154697} a^{17} + \frac{22340}{154697} a^{16} - \frac{22106}{154697} a^{15} - \frac{53227}{154697} a^{14} - \frac{32626}{154697} a^{13} - \frac{66786}{154697} a^{12} - \frac{10296}{154697} a^{11} + \frac{40284}{154697} a^{10} + \frac{27526}{154697} a^{9} - \frac{37837}{154697} a^{8} - \frac{53293}{154697} a^{7} + \frac{16582}{154697} a^{6} + \frac{1788}{4181} a^{5} + \frac{50885}{154697} a^{4} + \frac{65066}{154697} a^{3} - \frac{2211}{154697} a^{2} + \frac{4456}{154697} a + \frac{65055}{154697}$, $\frac{1}{154697} a^{33} - \frac{18}{154697} a^{31} + \frac{6}{154697} a^{30} - \frac{5}{154697} a^{29} - \frac{3}{154697} a^{28} - \frac{13}{154697} a^{27} + \frac{6}{154697} a^{26} - \frac{11}{154697} a^{25} - \frac{1}{154697} a^{24} + \frac{3}{154697} a^{23} - \frac{3}{154697} a^{22} + \frac{12}{154697} a^{21} - \frac{12}{154697} a^{20} + \frac{1052}{154697} a^{19} - \frac{1000}{154697} a^{18} + \frac{70440}{154697} a^{17} + \frac{62513}{154697} a^{16} + \frac{67652}{154697} a^{15} + \frac{65535}{154697} a^{14} + \frac{41994}{154697} a^{13} - \frac{37417}{154697} a^{12} + \frac{22746}{154697} a^{11} + \frac{4549}{154697} a^{10} - \frac{21927}{154697} a^{9} - \frac{47003}{154697} a^{8} - \frac{33923}{154697} a^{7} + \frac{1664}{4181} a^{6} - \frac{74323}{154697} a^{5} - \frac{42456}{154697} a^{4} + \frac{66794}{154697} a^{3} + \frac{70205}{154697} a^{2} - \frac{26446}{154697} a - \frac{533}{4181}$, $\frac{1}{5723789} a^{34} + \frac{18}{5723789} a^{33} - \frac{14}{5723789} a^{32} + \frac{533}{5723789} a^{31} - \frac{6}{5723789} a^{30} - \frac{624}{5723789} a^{29} - \frac{420}{5723789} a^{28} + \frac{93}{5723789} a^{27} - \frac{140}{5723789} a^{26} - \frac{471}{5723789} a^{25} + \frac{533}{5723789} a^{24} + \frac{528}{5723789} a^{23} - \frac{511}{5723789} a^{22} + \frac{525}{5723789} a^{21} - \frac{189}{5723789} a^{20} - \frac{63105}{5723789} a^{19} + \frac{7068}{5723789} a^{18} - \frac{1995834}{5723789} a^{17} - \frac{1513955}{5723789} a^{16} + \frac{246722}{5723789} a^{15} + \frac{2443243}{5723789} a^{14} - \frac{1269688}{5723789} a^{13} + \frac{550700}{5723789} a^{12} + \frac{1032281}{5723789} a^{11} - \frac{171368}{5723789} a^{10} + \frac{2235697}{5723789} a^{9} - \frac{458269}{5723789} a^{8} - \frac{1381376}{5723789} a^{7} - \frac{21184}{50653} a^{6} - \frac{390890}{5723789} a^{5} + \frac{824658}{5723789} a^{4} - \frac{896955}{5723789} a^{3} + \frac{31145}{154697} a^{2} - \frac{2715352}{5723789} a + \frac{2466678}{5723789}$, $\frac{1}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{35} - \frac{3062043228483641355878159701646612967936427915544524198654463106145440165591961893}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{34} + \frac{261315667888294879211939333999891462239214992648006037889973910436504388205266445418}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{33} - \frac{398770080952527870977034630222300668415777358540561959422296574325183743146683704829}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{32} - \frac{2473623606906983010409956458015398075882263085389856430106732976594758040048765289405}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{31} + \frac{13574732485534963857357981491008408798841326960761982938405933884434420061985040427909}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{30} - \frac{4176803631487605975624892573680362901014898867128953916309740187011569054272580815645}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{29} - \frac{3206673295321289995196239176378298625978165888391174513171923453259688965734134206821}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{28} + \frac{2594242464904176851204400092821215133717950868871284155361587937511545924042710725304}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{27} - \frac{11673710549470159019309342033230922498526529913509362974541159431135561725863915695979}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{26} - \frac{17516675135068031341026627430836682816801356037564149047700353216951214173724071202600}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{25} + \frac{16736164510430030567241532889359659806411459441909197180181128606108511279133870830979}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{24} + \frac{15557769077140301350442640281563245578311760115329778428990919665362866549569108335382}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{23} + \frac{13054611009217673699227341497964537759716750353138339480983578628797592298047982778160}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{22} - \frac{18405141813953184698731760602407952303594941606262303522313719799459755280330279580016}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{21} - \frac{17640570551298040311367371163174915571219870159485648589204505727981837467896950966427}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{20} - \frac{2085831989772320739328094455370909844011469351325038336341417463763047462910125047191073}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{19} - \frac{1877937757767669353758924235338346514019537231278307590709286067650097374452601434025605}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{18} - \frac{28557455780397355054567992343873918209838557235301778608597348012489006366515752627908981}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{17} + \frac{32638583537340390755856980074962964232606305762906707282283553717646746010661249372981519}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{16} - \frac{76512961978145562918220795001427486387218190750179967641765823143748559888961036821185769}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{15} + \frac{63925743317962163030808490908933978251094099957223491886023989632325372063352156386443430}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{14} + \frac{9141272204067746190897330022767965852032758412836478681801569261547509795655345154330784}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{13} - \frac{14482271738012341350030129237888101565206476434239650551552999585571711258153347787640367}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{12} + \frac{176736815376131583376841339172643442156696615217318169771408715504200457269254882844166}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{11} - \frac{24568693172730987866571980772082073101867723829144956258410621647337799431178715886146942}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{10} + \frac{51731541978652024565298305177056512134084848696357929845245966557100419141621055632599158}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{9} + \frac{12859907643401812582883340009790694624779075736958685830762612455616702153940211009263960}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{8} + \frac{976681125788532607991825006124118749541170516477303609790261369734381048524383119064521}{4237156826774291086423158933481855546317390463028886156893130058720817798487054396909011} a^{7} + \frac{18220522924290242051420462675610910468312505673285016128677064034635427995582154602122480}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{6} + \frac{75074801409354307513576946507552677250829181301867188141752934641654241321388752030560960}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{5} + \frac{11674418889706446257027487857776110499768714930956551688301440725119805010558270535179993}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{4} + \frac{39020206714927107759720432064197357131182812804940148741935961949939520501140327204053151}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{3} + \frac{34083125507698616092968484816755298732068167559046319807203415288123613869409004969366891}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a^{2} - \frac{43861683571641655653400178979114598632332838073909837529411783712559520835604312440788988}{156774802590648770197656880538828655213743447132068787805045812172670258544021012685633407} a + \frac{3463181112967437199648404143557349513159959203193437823094856476449984029902371697065}{16769152058043509487395109695029271067894261111570091753668393643456012252007809678643}$

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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{95791106472146210571782233856602475761970734586696802223606514720160656851736487}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{35} - \frac{197487018764293965402285709299031116755654892729062126928833481784474538351072385}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{34} - \frac{1512298809810659472763137828867018687186167685592505276027233752713439385825390427}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{33} + \frac{3378188993540423288861015700179496080693560937166460521894722065975156460799669979}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{32} + \frac{14765799617460620719908517603709692105533882162547611644779009180421455341083982010}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{31} - \frac{35098231375209425063256138813618639685521356716048044233890716129745757646460843657}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{30} - \frac{94569514292769583897951925157949174427456715055023816728574888117970123221377149506}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{29} + \frac{244700844520741531928074021464488372704123666858648573614434004383090607296056557962}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{28} + \frac{454982766382856151744504303914482911877888163162636307128370942421537350836471058028}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{27} - \frac{1296713845263022571876411421071415968977097642177696762407132208744651378467090257292}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{26} - \frac{1612613151710812928525536132664547383311384904262359044906301456908807495154698926543}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{25} + \frac{5286613709411040259451360744816947761816312148003569005008253710766300716660833389231}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{24} + \frac{4348684999506104879137283704555672931338295559008520886725854867717200696865845641487}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{23} - \frac{17279649403527370930237468778042892293244525848503904748015051902169081571215635936433}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{22} - \frac{7975009646368186878229491844332679755366872668658357500618697622645346048780015707373}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{21} + \frac{44721075699442143139282629241526678647754584454254173514859274801939296212805873890261}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{20} + \frac{8115982088558921906069470462472322433922924672506003489266132827393040775775633611732}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{19} - \frac{93709815620696760427375460209224254402175146722600764287392275429395893690171473202741}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{18} + \frac{8607125699244672264210801457733954932513013169958475337506326086097845104317532072644}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{17} + \frac{153086198155489299176061082815564524081153653926901102857432986961844482939127391714590}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{16} - \frac{49578718557738153049392895223887749219427337654304478860877215677667394310714765915273}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{15} - \frac{178107058818976599783856077487342380403179236882722701366744453121409641271956134181486}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{14} + \frac{74948606734200872106079120541110574872109733883793583086027115878886278496272049636621}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{13} - \frac{38079060483730403333613931995868921134522293506611966317981719358959413121455715383695}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{12} + \frac{362097646281947382765083055795371408633972686835393839022139800515222643071887583014073}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{11} - \frac{299552495076498803083908279934449738485802175603564136011266856558535647364606784346866}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{10} - \frac{231118755005050492422910164086479102690656467427908417914808063581390133837089325032299}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{9} + \frac{2100999340271337860714906707305262130876607879113312863238716722096905766884673041464903}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{8} - \frac{3230613094581858221578517776084965115989601684990939920437816008640281080418885630932579}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{7} - \frac{202044026070252676620243995666882302459747100124053315509357703530043503099775224451925}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{6} + \frac{4035226722401281360608681864005666387410234637441984576651831835557809107492354388612818}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{5} - \frac{3521231930944645544536531070507309847401749606461842825024507678222294147088186214738005}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{4} - \frac{752778692493440259184118832891163372217631178925069490766199196310727421443836475794842}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{3} + \frac{5254803761491860995004968666032456175529836373694462718589407802112080309401682733920735}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a^{2} - \frac{5159357329001948327431964763322655640075147054923956786197247848485892799438030629275648}{1387387633545564338032361774679899603661446434797068918628724001528055385345318696333039} a + \frac{302363825828074885492494338455847020606229735078173230210191111471897368353097898007}{148399575734898314047744333584329832459241248774956564191755695959787719044316899811} \) (order $6$)
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

$C_2\times C_{18}$ (as 36T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_2\times C_{18}$
Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{285}) \), 3.3.361.1, \(\Q(\sqrt{-3}, \sqrt{-95})\), 6.0.3518667.1, 6.0.309512375.1, 6.6.8356834125.1, \(\Q(\zeta_{19})^+\), 12.0.69836676592764515625.2, 18.0.5677392343251487443465123.1, 18.0.10703880581610941769412109375.1, 18.18.210684481487848166847338548828125.1

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 $18^{2}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{4}$ $18^{2}$ R $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{36}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$

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
5Data not computed
19Data not computed