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Properties

Label 36.0.33775576763...5009.1
Degree $36$
Signature $[0, 18]$
Discriminant $13^{18}\cdot 19^{34}$
Root discriminant $58.17$
Ramified primes $13, 19$
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![387420489, 129140163, 172186884, 100442349, 90876411, 63772920, 51549777, 38440899, 29996892, 22812597, 17603163, 13471920, 10358361, 7943427, 6100596, 4681341, 3593979, 2758440, 2117473, -919480, 399331, -173383, 75316, -32689, 14209, -6160, 2683, -1159, 508, -217, 97, -40, 19, -7, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 4*x^34 - 7*x^33 + 19*x^32 - 40*x^31 + 97*x^30 - 217*x^29 + 508*x^28 - 1159*x^27 + 2683*x^26 - 6160*x^25 + 14209*x^24 - 32689*x^23 + 75316*x^22 - 173383*x^21 + 399331*x^20 - 919480*x^19 + 2117473*x^18 + 2758440*x^17 + 3593979*x^16 + 4681341*x^15 + 6100596*x^14 + 7943427*x^13 + 10358361*x^12 + 13471920*x^11 + 17603163*x^10 + 22812597*x^9 + 29996892*x^8 + 38440899*x^7 + 51549777*x^6 + 63772920*x^5 + 90876411*x^4 + 100442349*x^3 + 172186884*x^2 + 129140163*x + 387420489)
 
gp: K = bnfinit(x^36 - x^35 + 4*x^34 - 7*x^33 + 19*x^32 - 40*x^31 + 97*x^30 - 217*x^29 + 508*x^28 - 1159*x^27 + 2683*x^26 - 6160*x^25 + 14209*x^24 - 32689*x^23 + 75316*x^22 - 173383*x^21 + 399331*x^20 - 919480*x^19 + 2117473*x^18 + 2758440*x^17 + 3593979*x^16 + 4681341*x^15 + 6100596*x^14 + 7943427*x^13 + 10358361*x^12 + 13471920*x^11 + 17603163*x^10 + 22812597*x^9 + 29996892*x^8 + 38440899*x^7 + 51549777*x^6 + 63772920*x^5 + 90876411*x^4 + 100442349*x^3 + 172186884*x^2 + 129140163*x + 387420489, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} + 4 x^{34} - 7 x^{33} + 19 x^{32} - 40 x^{31} + 97 x^{30} - 217 x^{29} + 508 x^{28} - 1159 x^{27} + 2683 x^{26} - 6160 x^{25} + 14209 x^{24} - 32689 x^{23} + 75316 x^{22} - 173383 x^{21} + 399331 x^{20} - 919480 x^{19} + 2117473 x^{18} + 2758440 x^{17} + 3593979 x^{16} + 4681341 x^{15} + 6100596 x^{14} + 7943427 x^{13} + 10358361 x^{12} + 13471920 x^{11} + 17603163 x^{10} + 22812597 x^{9} + 29996892 x^{8} + 38440899 x^{7} + 51549777 x^{6} + 63772920 x^{5} + 90876411 x^{4} + 100442349 x^{3} + 172186884 x^{2} + 129140163 x + 387420489 \)

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:  \(3377557676335881126900903346326957671873387511586368229584565009=13^{18}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.17$
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:  $13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(247=13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{247}(1,·)$, $\chi_{247}(131,·)$, $\chi_{247}(129,·)$, $\chi_{247}(12,·)$, $\chi_{247}(66,·)$, $\chi_{247}(142,·)$, $\chi_{247}(144,·)$, $\chi_{247}(25,·)$, $\chi_{247}(155,·)$, $\chi_{247}(157,·)$, $\chi_{247}(27,·)$, $\chi_{247}(40,·)$, $\chi_{247}(92,·)$, $\chi_{247}(170,·)$, $\chi_{247}(51,·)$, $\chi_{247}(53,·)$, $\chi_{247}(183,·)$, $\chi_{247}(181,·)$, $\chi_{247}(64,·)$, $\chi_{247}(194,·)$, $\chi_{247}(196,·)$, $\chi_{247}(118,·)$, $\chi_{247}(77,·)$, $\chi_{247}(79,·)$, $\chi_{247}(14,·)$, $\chi_{247}(90,·)$, $\chi_{247}(207,·)$, $\chi_{247}(220,·)$, $\chi_{247}(222,·)$, $\chi_{247}(103,·)$, $\chi_{247}(105,·)$, $\chi_{247}(235,·)$, $\chi_{247}(168,·)$, $\chi_{247}(116,·)$, $\chi_{247}(246,·)$, $\chi_{247}(233,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{6352419} a^{19} - \frac{1}{3} a^{18} + \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{919480}{2117473}$, $\frac{1}{19057257} a^{20} - \frac{1}{19057257} a^{19} - \frac{2}{9} a^{18} - \frac{1}{9} a^{17} + \frac{4}{9} a^{16} + \frac{2}{9} a^{15} + \frac{1}{9} a^{14} - \frac{4}{9} a^{13} - \frac{2}{9} a^{12} - \frac{1}{9} a^{11} + \frac{4}{9} a^{10} + \frac{2}{9} a^{9} + \frac{1}{9} a^{8} - \frac{4}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{9} a^{5} + \frac{4}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{919480}{6352419} a + \frac{399331}{2117473}$, $\frac{1}{57171771} a^{21} - \frac{1}{57171771} a^{20} + \frac{4}{57171771} a^{19} - \frac{10}{27} a^{18} + \frac{4}{27} a^{17} - \frac{7}{27} a^{16} - \frac{8}{27} a^{15} - \frac{13}{27} a^{14} - \frac{11}{27} a^{13} - \frac{1}{27} a^{12} - \frac{5}{27} a^{11} + \frac{2}{27} a^{10} + \frac{10}{27} a^{9} - \frac{4}{27} a^{8} + \frac{7}{27} a^{7} + \frac{8}{27} a^{6} + \frac{13}{27} a^{5} + \frac{11}{27} a^{4} + \frac{1}{27} a^{3} + \frac{919480}{19057257} a^{2} + \frac{399331}{6352419} a + \frac{173383}{2117473}$, $\frac{1}{171515313} a^{22} - \frac{1}{171515313} a^{21} + \frac{4}{171515313} a^{20} - \frac{7}{171515313} a^{19} - \frac{23}{81} a^{18} - \frac{7}{81} a^{17} + \frac{19}{81} a^{16} - \frac{40}{81} a^{15} + \frac{16}{81} a^{14} + \frac{26}{81} a^{13} + \frac{22}{81} a^{12} - \frac{25}{81} a^{11} + \frac{10}{81} a^{10} - \frac{4}{81} a^{9} + \frac{34}{81} a^{8} + \frac{35}{81} a^{7} - \frac{14}{81} a^{6} + \frac{38}{81} a^{5} + \frac{1}{81} a^{4} + \frac{919480}{57171771} a^{3} + \frac{399331}{19057257} a^{2} + \frac{173383}{6352419} a + \frac{75316}{2117473}$, $\frac{1}{514545939} a^{23} - \frac{1}{514545939} a^{22} + \frac{4}{514545939} a^{21} - \frac{7}{514545939} a^{20} + \frac{19}{514545939} a^{19} - \frac{7}{243} a^{18} - \frac{62}{243} a^{17} + \frac{41}{243} a^{16} + \frac{16}{243} a^{15} + \frac{107}{243} a^{14} - \frac{59}{243} a^{13} - \frac{106}{243} a^{12} - \frac{71}{243} a^{11} - \frac{4}{243} a^{10} + \frac{34}{243} a^{9} - \frac{46}{243} a^{8} - \frac{95}{243} a^{7} - \frac{43}{243} a^{6} + \frac{1}{243} a^{5} + \frac{919480}{171515313} a^{4} + \frac{399331}{57171771} a^{3} + \frac{173383}{19057257} a^{2} + \frac{75316}{6352419} a + \frac{32689}{2117473}$, $\frac{1}{1543637817} a^{24} - \frac{1}{1543637817} a^{23} + \frac{4}{1543637817} a^{22} - \frac{7}{1543637817} a^{21} + \frac{19}{1543637817} a^{20} - \frac{40}{1543637817} a^{19} - \frac{62}{729} a^{18} + \frac{41}{729} a^{17} - \frac{227}{729} a^{16} + \frac{350}{729} a^{15} - \frac{302}{729} a^{14} - \frac{106}{729} a^{13} - \frac{71}{729} a^{12} - \frac{247}{729} a^{11} + \frac{34}{729} a^{10} - \frac{46}{729} a^{9} + \frac{148}{729} a^{8} - \frac{286}{729} a^{7} + \frac{1}{729} a^{6} + \frac{919480}{514545939} a^{5} + \frac{399331}{171515313} a^{4} + \frac{173383}{57171771} a^{3} + \frac{75316}{19057257} a^{2} + \frac{32689}{6352419} a + \frac{14209}{2117473}$, $\frac{1}{4630913451} a^{25} - \frac{1}{4630913451} a^{24} + \frac{4}{4630913451} a^{23} - \frac{7}{4630913451} a^{22} + \frac{19}{4630913451} a^{21} - \frac{40}{4630913451} a^{20} + \frac{97}{4630913451} a^{19} + \frac{770}{2187} a^{18} - \frac{956}{2187} a^{17} + \frac{1079}{2187} a^{16} + \frac{427}{2187} a^{15} + \frac{623}{2187} a^{14} + \frac{658}{2187} a^{13} - \frac{976}{2187} a^{12} + \frac{763}{2187} a^{11} + \frac{683}{2187} a^{10} - \frac{581}{2187} a^{9} + \frac{443}{2187} a^{8} + \frac{1}{2187} a^{7} + \frac{919480}{1543637817} a^{6} + \frac{399331}{514545939} a^{5} + \frac{173383}{171515313} a^{4} + \frac{75316}{57171771} a^{3} + \frac{32689}{19057257} a^{2} + \frac{14209}{6352419} a + \frac{6160}{2117473}$, $\frac{1}{13892740353} a^{26} - \frac{1}{13892740353} a^{25} + \frac{4}{13892740353} a^{24} - \frac{7}{13892740353} a^{23} + \frac{19}{13892740353} a^{22} - \frac{40}{13892740353} a^{21} + \frac{97}{13892740353} a^{20} - \frac{217}{13892740353} a^{19} - \frac{956}{6561} a^{18} + \frac{3266}{6561} a^{17} + \frac{427}{6561} a^{16} + \frac{2810}{6561} a^{15} - \frac{1529}{6561} a^{14} - \frac{3163}{6561} a^{13} - \frac{1424}{6561} a^{12} - \frac{1504}{6561} a^{11} - \frac{2768}{6561} a^{10} - \frac{1744}{6561} a^{9} + \frac{1}{6561} a^{8} + \frac{919480}{4630913451} a^{7} + \frac{399331}{1543637817} a^{6} + \frac{173383}{514545939} a^{5} + \frac{75316}{171515313} a^{4} + \frac{32689}{57171771} a^{3} + \frac{14209}{19057257} a^{2} + \frac{6160}{6352419} a + \frac{2683}{2117473}$, $\frac{1}{41678221059} a^{27} - \frac{1}{41678221059} a^{26} + \frac{4}{41678221059} a^{25} - \frac{7}{41678221059} a^{24} + \frac{19}{41678221059} a^{23} - \frac{40}{41678221059} a^{22} + \frac{97}{41678221059} a^{21} - \frac{217}{41678221059} a^{20} + \frac{508}{41678221059} a^{19} - \frac{3295}{19683} a^{18} + \frac{427}{19683} a^{17} + \frac{9371}{19683} a^{16} - \frac{8090}{19683} a^{15} - \frac{3163}{19683} a^{14} - \frac{1424}{19683} a^{13} - \frac{8065}{19683} a^{12} + \frac{3793}{19683} a^{11} - \frac{8305}{19683} a^{10} + \frac{1}{19683} a^{9} + \frac{919480}{13892740353} a^{8} + \frac{399331}{4630913451} a^{7} + \frac{173383}{1543637817} a^{6} + \frac{75316}{514545939} a^{5} + \frac{32689}{171515313} a^{4} + \frac{14209}{57171771} a^{3} + \frac{6160}{19057257} a^{2} + \frac{2683}{6352419} a + \frac{1159}{2117473}$, $\frac{1}{125034663177} a^{28} - \frac{1}{125034663177} a^{27} + \frac{4}{125034663177} a^{26} - \frac{7}{125034663177} a^{25} + \frac{19}{125034663177} a^{24} - \frac{40}{125034663177} a^{23} + \frac{97}{125034663177} a^{22} - \frac{217}{125034663177} a^{21} + \frac{508}{125034663177} a^{20} - \frac{1159}{125034663177} a^{19} + \frac{427}{59049} a^{18} - \frac{10312}{59049} a^{17} + \frac{11593}{59049} a^{16} + \frac{16520}{59049} a^{15} + \frac{18259}{59049} a^{14} - \frac{27748}{59049} a^{13} + \frac{23476}{59049} a^{12} + \frac{11378}{59049} a^{11} + \frac{1}{59049} a^{10} + \frac{919480}{41678221059} a^{9} + \frac{399331}{13892740353} a^{8} + \frac{173383}{4630913451} a^{7} + \frac{75316}{1543637817} a^{6} + \frac{32689}{514545939} a^{5} + \frac{14209}{171515313} a^{4} + \frac{6160}{57171771} a^{3} + \frac{2683}{19057257} a^{2} + \frac{1159}{6352419} a + \frac{508}{2117473}$, $\frac{1}{375103989531} a^{29} - \frac{1}{375103989531} a^{28} + \frac{4}{375103989531} a^{27} - \frac{7}{375103989531} a^{26} + \frac{19}{375103989531} a^{25} - \frac{40}{375103989531} a^{24} + \frac{97}{375103989531} a^{23} - \frac{217}{375103989531} a^{22} + \frac{508}{375103989531} a^{21} - \frac{1159}{375103989531} a^{20} + \frac{2683}{375103989531} a^{19} + \frac{48737}{177147} a^{18} - \frac{47456}{177147} a^{17} + \frac{16520}{177147} a^{16} + \frac{18259}{177147} a^{15} + \frac{31301}{177147} a^{14} + \frac{23476}{177147} a^{13} + \frac{70427}{177147} a^{12} + \frac{1}{177147} a^{11} + \frac{919480}{125034663177} a^{10} + \frac{399331}{41678221059} a^{9} + \frac{173383}{13892740353} a^{8} + \frac{75316}{4630913451} a^{7} + \frac{32689}{1543637817} a^{6} + \frac{14209}{514545939} a^{5} + \frac{6160}{171515313} a^{4} + \frac{2683}{57171771} a^{3} + \frac{1159}{19057257} a^{2} + \frac{508}{6352419} a + \frac{217}{2117473}$, $\frac{1}{1125311968593} a^{30} - \frac{1}{1125311968593} a^{29} + \frac{4}{1125311968593} a^{28} - \frac{7}{1125311968593} a^{27} + \frac{19}{1125311968593} a^{26} - \frac{40}{1125311968593} a^{25} + \frac{97}{1125311968593} a^{24} - \frac{217}{1125311968593} a^{23} + \frac{508}{1125311968593} a^{22} - \frac{1159}{1125311968593} a^{21} + \frac{2683}{1125311968593} a^{20} - \frac{6160}{1125311968593} a^{19} - \frac{224603}{531441} a^{18} - \frac{160627}{531441} a^{17} + \frac{18259}{531441} a^{16} + \frac{31301}{531441} a^{15} + \frac{23476}{531441} a^{14} + \frac{70427}{531441} a^{13} + \frac{1}{531441} a^{12} + \frac{919480}{375103989531} a^{11} + \frac{399331}{125034663177} a^{10} + \frac{173383}{41678221059} a^{9} + \frac{75316}{13892740353} a^{8} + \frac{32689}{4630913451} a^{7} + \frac{14209}{1543637817} a^{6} + \frac{6160}{514545939} a^{5} + \frac{2683}{171515313} a^{4} + \frac{1159}{57171771} a^{3} + \frac{508}{19057257} a^{2} + \frac{217}{6352419} a + \frac{97}{2117473}$, $\frac{1}{3375935905779} a^{31} - \frac{1}{3375935905779} a^{30} + \frac{4}{3375935905779} a^{29} - \frac{7}{3375935905779} a^{28} + \frac{19}{3375935905779} a^{27} - \frac{40}{3375935905779} a^{26} + \frac{97}{3375935905779} a^{25} - \frac{217}{3375935905779} a^{24} + \frac{508}{3375935905779} a^{23} - \frac{1159}{3375935905779} a^{22} + \frac{2683}{3375935905779} a^{21} - \frac{6160}{3375935905779} a^{20} + \frac{14209}{3375935905779} a^{19} - \frac{160627}{1594323} a^{18} - \frac{513182}{1594323} a^{17} + \frac{31301}{1594323} a^{16} + \frac{23476}{1594323} a^{15} + \frac{70427}{1594323} a^{14} + \frac{1}{1594323} a^{13} + \frac{919480}{1125311968593} a^{12} + \frac{399331}{375103989531} a^{11} + \frac{173383}{125034663177} a^{10} + \frac{75316}{41678221059} a^{9} + \frac{32689}{13892740353} a^{8} + \frac{14209}{4630913451} a^{7} + \frac{6160}{1543637817} a^{6} + \frac{2683}{514545939} a^{5} + \frac{1159}{171515313} a^{4} + \frac{508}{57171771} a^{3} + \frac{217}{19057257} a^{2} + \frac{97}{6352419} a + \frac{40}{2117473}$, $\frac{1}{10127807717337} a^{32} - \frac{1}{10127807717337} a^{31} + \frac{4}{10127807717337} a^{30} - \frac{7}{10127807717337} a^{29} + \frac{19}{10127807717337} a^{28} - \frac{40}{10127807717337} a^{27} + \frac{97}{10127807717337} a^{26} - \frac{217}{10127807717337} a^{25} + \frac{508}{10127807717337} a^{24} - \frac{1159}{10127807717337} a^{23} + \frac{2683}{10127807717337} a^{22} - \frac{6160}{10127807717337} a^{21} + \frac{14209}{10127807717337} a^{20} - \frac{32689}{10127807717337} a^{19} + \frac{1081141}{4782969} a^{18} - \frac{1563022}{4782969} a^{17} + \frac{23476}{4782969} a^{16} + \frac{70427}{4782969} a^{15} + \frac{1}{4782969} a^{14} + \frac{919480}{3375935905779} a^{13} + \frac{399331}{1125311968593} a^{12} + \frac{173383}{375103989531} a^{11} + \frac{75316}{125034663177} a^{10} + \frac{32689}{41678221059} a^{9} + \frac{14209}{13892740353} a^{8} + \frac{6160}{4630913451} a^{7} + \frac{2683}{1543637817} a^{6} + \frac{1159}{514545939} a^{5} + \frac{508}{171515313} a^{4} + \frac{217}{57171771} a^{3} + \frac{97}{19057257} a^{2} + \frac{40}{6352419} a + \frac{19}{2117473}$, $\frac{1}{30383423152011} a^{33} - \frac{1}{30383423152011} a^{32} + \frac{4}{30383423152011} a^{31} - \frac{7}{30383423152011} a^{30} + \frac{19}{30383423152011} a^{29} - \frac{40}{30383423152011} a^{28} + \frac{97}{30383423152011} a^{27} - \frac{217}{30383423152011} a^{26} + \frac{508}{30383423152011} a^{25} - \frac{1159}{30383423152011} a^{24} + \frac{2683}{30383423152011} a^{23} - \frac{6160}{30383423152011} a^{22} + \frac{14209}{30383423152011} a^{21} - \frac{32689}{30383423152011} a^{20} + \frac{75316}{30383423152011} a^{19} - \frac{6345991}{14348907} a^{18} - \frac{4759493}{14348907} a^{17} + \frac{70427}{14348907} a^{16} + \frac{1}{14348907} a^{15} + \frac{919480}{10127807717337} a^{14} + \frac{399331}{3375935905779} a^{13} + \frac{173383}{1125311968593} a^{12} + \frac{75316}{375103989531} a^{11} + \frac{32689}{125034663177} a^{10} + \frac{14209}{41678221059} a^{9} + \frac{6160}{13892740353} a^{8} + \frac{2683}{4630913451} a^{7} + \frac{1159}{1543637817} a^{6} + \frac{508}{514545939} a^{5} + \frac{217}{171515313} a^{4} + \frac{97}{57171771} a^{3} + \frac{40}{19057257} a^{2} + \frac{19}{6352419} a + \frac{7}{2117473}$, $\frac{1}{91150269456033} a^{34} - \frac{1}{91150269456033} a^{33} + \frac{4}{91150269456033} a^{32} - \frac{7}{91150269456033} a^{31} + \frac{19}{91150269456033} a^{30} - \frac{40}{91150269456033} a^{29} + \frac{97}{91150269456033} a^{28} - \frac{217}{91150269456033} a^{27} + \frac{508}{91150269456033} a^{26} - \frac{1159}{91150269456033} a^{25} + \frac{2683}{91150269456033} a^{24} - \frac{6160}{91150269456033} a^{23} + \frac{14209}{91150269456033} a^{22} - \frac{32689}{91150269456033} a^{21} + \frac{75316}{91150269456033} a^{20} - \frac{173383}{91150269456033} a^{19} - \frac{4759493}{43046721} a^{18} - \frac{14278480}{43046721} a^{17} + \frac{1}{43046721} a^{16} + \frac{919480}{30383423152011} a^{15} + \frac{399331}{10127807717337} a^{14} + \frac{173383}{3375935905779} a^{13} + \frac{75316}{1125311968593} a^{12} + \frac{32689}{375103989531} a^{11} + \frac{14209}{125034663177} a^{10} + \frac{6160}{41678221059} a^{9} + \frac{2683}{13892740353} a^{8} + \frac{1159}{4630913451} a^{7} + \frac{508}{1543637817} a^{6} + \frac{217}{514545939} a^{5} + \frac{97}{171515313} a^{4} + \frac{40}{57171771} a^{3} + \frac{19}{19057257} a^{2} + \frac{7}{6352419} a + \frac{4}{2117473}$, $\frac{1}{273450808368099} a^{35} - \frac{1}{273450808368099} a^{34} + \frac{4}{273450808368099} a^{33} - \frac{7}{273450808368099} a^{32} + \frac{19}{273450808368099} a^{31} - \frac{40}{273450808368099} a^{30} + \frac{97}{273450808368099} a^{29} - \frac{217}{273450808368099} a^{28} + \frac{508}{273450808368099} a^{27} - \frac{1159}{273450808368099} a^{26} + \frac{2683}{273450808368099} a^{25} - \frac{6160}{273450808368099} a^{24} + \frac{14209}{273450808368099} a^{23} - \frac{32689}{273450808368099} a^{22} + \frac{75316}{273450808368099} a^{21} - \frac{173383}{273450808368099} a^{20} + \frac{399331}{273450808368099} a^{19} - \frac{14278480}{129140163} a^{18} + \frac{1}{129140163} a^{17} + \frac{919480}{91150269456033} a^{16} + \frac{399331}{30383423152011} a^{15} + \frac{173383}{10127807717337} a^{14} + \frac{75316}{3375935905779} a^{13} + \frac{32689}{1125311968593} a^{12} + \frac{14209}{375103989531} a^{11} + \frac{6160}{125034663177} a^{10} + \frac{2683}{41678221059} a^{9} + \frac{1159}{13892740353} a^{8} + \frac{508}{4630913451} a^{7} + \frac{217}{1543637817} a^{6} + \frac{97}{514545939} a^{5} + \frac{40}{171515313} a^{4} + \frac{19}{57171771} a^{3} + \frac{7}{19057257} a^{2} + \frac{4}{6352419} a + \frac{1}{2117473}$

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:  \( -\frac{75316}{30383423152011} a^{34} - \frac{574888488199}{30383423152011} a^{15} \) (order $38$)
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

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-247}) \), 3.3.361.1, \(\Q(\sqrt{13}, \sqrt{-19})\), 6.0.2476099.1, 6.6.286315237.1, 6.0.5439989503.1, \(\Q(\zeta_{19})^+\), 12.0.29593485792750187009.1, \(\Q(\zeta_{19})\), 18.18.3058776789325072365774692364013.1, 18.0.58116758997176374949719154916247.1

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 $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{4}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$ $18^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{4}$ $18^{2}$ $18^{2}$ $18^{2}$

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
13Data not computed
19Data not computed