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Properties

Label 36.0.14555723661...0377.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{90}\cdot 17^{27}$
Root discriminant $130.51$
Ramified primes $3, 17$
Class number Not computed
Class group Not computed
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![88739165414377, 1197785699568, 16930660662, 3992608699248, 112871077641, 3593254105584, 289702503252, 1197349695360, 372475050696, 132056987296, 269011171638, -1517431968, 114129248142, -1478523456, 28224185886, -985682304, 3769723674, -456602832, 215269598, -146860560, 3996135, -32169456, 1937520, -4577472, 712530, -381456, 197316, -14128, 40455, 0, 5952, 0, 594, 0, 36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 36*x^34 + 594*x^32 + 5952*x^30 + 40455*x^28 - 14128*x^27 + 197316*x^26 - 381456*x^25 + 712530*x^24 - 4577472*x^23 + 1937520*x^22 - 32169456*x^21 + 3996135*x^20 - 146860560*x^19 + 215269598*x^18 - 456602832*x^17 + 3769723674*x^16 - 985682304*x^15 + 28224185886*x^14 - 1478523456*x^13 + 114129248142*x^12 - 1517431968*x^11 + 269011171638*x^10 + 132056987296*x^9 + 372475050696*x^8 + 1197349695360*x^7 + 289702503252*x^6 + 3593254105584*x^5 + 112871077641*x^4 + 3992608699248*x^3 + 16930660662*x^2 + 1197785699568*x + 88739165414377)
 
gp: K = bnfinit(x^36 + 36*x^34 + 594*x^32 + 5952*x^30 + 40455*x^28 - 14128*x^27 + 197316*x^26 - 381456*x^25 + 712530*x^24 - 4577472*x^23 + 1937520*x^22 - 32169456*x^21 + 3996135*x^20 - 146860560*x^19 + 215269598*x^18 - 456602832*x^17 + 3769723674*x^16 - 985682304*x^15 + 28224185886*x^14 - 1478523456*x^13 + 114129248142*x^12 - 1517431968*x^11 + 269011171638*x^10 + 132056987296*x^9 + 372475050696*x^8 + 1197349695360*x^7 + 289702503252*x^6 + 3593254105584*x^5 + 112871077641*x^4 + 3992608699248*x^3 + 16930660662*x^2 + 1197785699568*x + 88739165414377, 1)
 

Normalized defining polynomial

\( x^{36} + 36 x^{34} + 594 x^{32} + 5952 x^{30} + 40455 x^{28} - 14128 x^{27} + 197316 x^{26} - 381456 x^{25} + 712530 x^{24} - 4577472 x^{23} + 1937520 x^{22} - 32169456 x^{21} + 3996135 x^{20} - 146860560 x^{19} + 215269598 x^{18} - 456602832 x^{17} + 3769723674 x^{16} - 985682304 x^{15} + 28224185886 x^{14} - 1478523456 x^{13} + 114129248142 x^{12} - 1517431968 x^{11} + 269011171638 x^{10} + 132056987296 x^{9} + 372475050696 x^{8} + 1197349695360 x^{7} + 289702503252 x^{6} + 3593254105584 x^{5} + 112871077641 x^{4} + 3992608699248 x^{3} + 16930660662 x^{2} + 1197785699568 x + 88739165414377 \)

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:  \(14555723661975701129520414713591505717264293810334661879947459060497463940377=3^{90}\cdot 17^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.51$
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:  $3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(459=3^{3}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{459}(256,·)$, $\chi_{459}(1,·)$, $\chi_{459}(395,·)$, $\chi_{459}(140,·)$, $\chi_{459}(271,·)$, $\chi_{459}(16,·)$, $\chi_{459}(404,·)$, $\chi_{459}(149,·)$, $\chi_{459}(409,·)$, $\chi_{459}(154,·)$, $\chi_{459}(293,·)$, $\chi_{459}(38,·)$, $\chi_{459}(424,·)$, $\chi_{459}(169,·)$, $\chi_{459}(302,·)$, $\chi_{459}(47,·)$, $\chi_{459}(307,·)$, $\chi_{459}(52,·)$, $\chi_{459}(446,·)$, $\chi_{459}(191,·)$, $\chi_{459}(322,·)$, $\chi_{459}(67,·)$, $\chi_{459}(455,·)$, $\chi_{459}(200,·)$, $\chi_{459}(205,·)$, $\chi_{459}(344,·)$, $\chi_{459}(89,·)$, $\chi_{459}(220,·)$, $\chi_{459}(353,·)$, $\chi_{459}(98,·)$, $\chi_{459}(358,·)$, $\chi_{459}(103,·)$, $\chi_{459}(242,·)$, $\chi_{459}(373,·)$, $\chi_{459}(118,·)$, $\chi_{459}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{106} a^{14} - \frac{23}{106} a^{13} + \frac{7}{53} a^{12} + \frac{19}{106} a^{11} + \frac{12}{53} a^{10} - \frac{11}{106} a^{9} + \frac{51}{106} a^{8} + \frac{8}{53} a^{7} + \frac{29}{106} a^{6} - \frac{26}{53} a^{5} - \frac{8}{53} a^{4} + \frac{27}{106} a^{3} - \frac{2}{53} a^{2} - \frac{17}{53} a + \frac{1}{53}$, $\frac{1}{106} a^{15} + \frac{15}{106} a^{13} + \frac{23}{106} a^{12} - \frac{8}{53} a^{11} + \frac{11}{106} a^{10} + \frac{5}{53} a^{9} + \frac{23}{106} a^{8} - \frac{27}{106} a^{7} - \frac{21}{106} a^{6} + \frac{7}{106} a^{5} - \frac{23}{106} a^{4} + \frac{17}{53} a^{3} + \frac{33}{106} a^{2} + \frac{15}{106} a - \frac{7}{106}$, $\frac{1}{106} a^{16} - \frac{3}{106} a^{13} - \frac{7}{53} a^{12} - \frac{9}{106} a^{11} + \frac{21}{106} a^{10} - \frac{12}{53} a^{9} + \frac{3}{106} a^{8} + \frac{2}{53} a^{7} + \frac{49}{106} a^{6} + \frac{15}{106} a^{5} + \frac{9}{106} a^{4} + \frac{26}{53} a^{3} + \frac{11}{53} a^{2} - \frac{27}{106} a + \frac{23}{106}$, $\frac{1}{106} a^{17} + \frac{23}{106} a^{13} - \frac{10}{53} a^{12} + \frac{25}{106} a^{11} - \frac{5}{106} a^{10} + \frac{23}{106} a^{9} - \frac{1}{53} a^{8} + \frac{22}{53} a^{7} - \frac{2}{53} a^{6} - \frac{41}{106} a^{5} - \frac{49}{106} a^{4} - \frac{3}{106} a^{3} + \frac{7}{53} a^{2} + \frac{27}{106} a + \frac{3}{53}$, $\frac{1}{198008} a^{18} + \frac{9}{99004} a^{16} + \frac{135}{198008} a^{14} + \frac{273}{99004} a^{12} + \frac{1287}{198008} a^{10} - \frac{11573}{49502} a^{9} + \frac{891}{99004} a^{8} - \frac{5153}{49502} a^{7} + \frac{693}{99004} a^{6} - \frac{15459}{49502} a^{5} + \frac{135}{49502} a^{4} - \frac{338}{24751} a^{3} + \frac{81}{198008} a^{2} - \frac{5153}{49502} a - \frac{91317}{198008}$, $\frac{1}{198008} a^{19} + \frac{9}{99004} a^{17} + \frac{135}{198008} a^{15} + \frac{273}{99004} a^{13} + \frac{1287}{198008} a^{11} - \frac{11573}{49502} a^{10} + \frac{891}{99004} a^{9} - \frac{5153}{49502} a^{8} + \frac{693}{99004} a^{7} - \frac{15459}{49502} a^{6} + \frac{135}{49502} a^{5} - \frac{338}{24751} a^{4} + \frac{81}{198008} a^{3} - \frac{5153}{49502} a^{2} - \frac{91317}{198008} a$, $\frac{1}{198008} a^{20} - \frac{189}{198008} a^{16} - \frac{2}{24751} a^{14} - \frac{23}{106} a^{13} + \frac{17611}{198008} a^{12} - \frac{1350}{24751} a^{11} + \frac{2931}{24751} a^{10} + \frac{8}{24751} a^{9} + \frac{32289}{99004} a^{8} - \frac{14237}{49502} a^{7} + \frac{7441}{49502} a^{6} + \frac{2896}{24751} a^{5} - \frac{39527}{198008} a^{4} + \frac{9812}{24751} a^{3} + \frac{97761}{198008} a^{2} - \frac{11064}{24751} a + \frac{31689}{99004}$, $\frac{1}{198008} a^{21} - \frac{189}{198008} a^{17} - \frac{2}{24751} a^{15} + \frac{19479}{198008} a^{13} - \frac{416}{24751} a^{12} + \frac{11933}{49502} a^{11} + \frac{5145}{24751} a^{10} - \frac{6005}{99004} a^{9} - \frac{5484}{24751} a^{8} - \frac{18711}{49502} a^{7} + \frac{20269}{49502} a^{6} - \frac{95567}{198008} a^{5} - \frac{1863}{24751} a^{4} + \frac{69741}{198008} a^{3} - \frac{7795}{24751} a^{2} - \frac{107}{1868} a + \frac{23}{53}$, $\frac{1}{198008} a^{22} - \frac{175}{99004} a^{16} + \frac{81}{99004} a^{14} + \frac{6122}{24751} a^{13} - \frac{14201}{99004} a^{12} + \frac{1876}{24751} a^{11} - \frac{32155}{198008} a^{10} + \frac{886}{24751} a^{9} + \frac{21699}{99004} a^{8} - \frac{22917}{49502} a^{7} - \frac{29753}{198008} a^{6} - \frac{2663}{24751} a^{5} + \frac{63457}{198008} a^{4} + \frac{241}{24751} a^{3} + \frac{2099}{198008} a^{2} - \frac{1623}{49502} a + \frac{89203}{198008}$, $\frac{1}{198008} a^{23} - \frac{175}{99004} a^{17} + \frac{81}{99004} a^{15} + \frac{51}{24751} a^{14} - \frac{191}{99004} a^{13} + \frac{7021}{49502} a^{12} + \frac{35093}{198008} a^{11} + \frac{3688}{24751} a^{10} - \frac{8189}{99004} a^{9} + \frac{1367}{49502} a^{8} + \frac{84195}{198008} a^{7} + \frac{13821}{49502} a^{6} - \frac{84115}{198008} a^{5} - \frac{1627}{24751} a^{4} + \frac{76819}{198008} a^{3} - \frac{2557}{49502} a^{2} + \frac{57447}{198008} a - \frac{26}{53}$, $\frac{1}{198008} a^{24} + \frac{429}{99004} a^{16} + \frac{51}{24751} a^{15} + \frac{21}{24751} a^{14} + \frac{3744}{24751} a^{13} + \frac{46865}{198008} a^{12} - \frac{1916}{24751} a^{11} - \frac{3095}{49502} a^{10} - \frac{1227}{49502} a^{9} + \frac{91455}{198008} a^{8} - \frac{2053}{49502} a^{7} - \frac{39863}{198008} a^{6} + \frac{23377}{49502} a^{5} + \frac{71547}{198008} a^{4} - \frac{8459}{49502} a^{3} + \frac{50305}{198008} a^{2} + \frac{8875}{24751} a - \frac{3471}{99004}$, $\frac{1}{198008} a^{25} + \frac{429}{99004} a^{17} + \frac{51}{24751} a^{16} + \frac{21}{24751} a^{15} + \frac{8}{24751} a^{14} + \frac{41261}{198008} a^{13} - \frac{4718}{24751} a^{12} + \frac{3443}{49502} a^{11} - \frac{3649}{24751} a^{10} + \frac{24207}{198008} a^{9} - \frac{5930}{24751} a^{8} - \frac{23051}{198008} a^{7} - \frac{10027}{24751} a^{6} - \frac{57345}{198008} a^{5} - \frac{6331}{24751} a^{4} - \frac{63643}{198008} a^{3} - \frac{932}{24751} a^{2} - \frac{39897}{99004} a - \frac{16}{53}$, $\frac{1}{198008} a^{26} + \frac{51}{24751} a^{17} - \frac{83}{49502} a^{16} + \frac{8}{24751} a^{15} + \frac{151}{198008} a^{14} - \frac{2383}{24751} a^{13} - \frac{1731}{24751} a^{12} - \frac{7765}{49502} a^{11} + \frac{37025}{198008} a^{10} - \frac{5509}{49502} a^{9} - \frac{72551}{198008} a^{8} - \frac{12397}{49502} a^{7} + \frac{68539}{198008} a^{6} + \frac{4921}{24751} a^{5} - \frac{3923}{198008} a^{4} + \frac{14475}{49502} a^{3} - \frac{2550}{24751} a^{2} + \frac{3595}{24751} a + \frac{17977}{99004}$, $\frac{1}{16831531193226256} a^{27} + \frac{27}{16831531193226256} a^{25} + \frac{81}{4207882798306564} a^{23} + \frac{2277}{16831531193226256} a^{21} + \frac{10395}{16831531193226256} a^{19} + \frac{2059}{4267864432} a^{18} + \frac{32319}{16831531193226256} a^{17} + \frac{18531}{2133932216} a^{16} + \frac{8721}{2103941399153282} a^{15} + \frac{277965}{4267864432} a^{14} + \frac{26163}{4207882798306564} a^{13} + \frac{43239}{164148632} a^{12} + \frac{4131}{647366584354856} a^{11} + \frac{203841}{328297264} a^{10} + \frac{112018068637735}{647366584354856} a^{9} + \frac{1834569}{2133932216} a^{8} - \frac{143285275497077}{323683292177428} a^{7} + \frac{1426887}{2133932216} a^{6} - \frac{424690137261773}{1294733168709712} a^{5} + \frac{277965}{1066966108} a^{4} + \frac{247418274547303}{1294733168709712} a^{3} + \frac{166779}{4267864432} a^{2} - \frac{7450834325878865}{16831531193226256} a + \frac{7260028847717371}{16831531193226256}$, $\frac{1}{16831531193226256} a^{28} + \frac{27}{16831531193226256} a^{26} + \frac{81}{4207882798306564} a^{24} + \frac{2277}{16831531193226256} a^{22} + \frac{10395}{16831531193226256} a^{20} + \frac{2059}{4267864432} a^{19} + \frac{32319}{16831531193226256} a^{18} + \frac{18531}{2133932216} a^{17} + \frac{8721}{2103941399153282} a^{16} + \frac{277965}{4267864432} a^{15} + \frac{26163}{4207882798306564} a^{14} + \frac{43239}{164148632} a^{13} + \frac{4131}{647366584354856} a^{12} + \frac{203841}{328297264} a^{11} + \frac{112018068637735}{647366584354856} a^{10} + \frac{1834569}{2133932216} a^{9} - \frac{143285275497077}{323683292177428} a^{8} + \frac{1426887}{2133932216} a^{7} - \frac{424690137261773}{1294733168709712} a^{6} + \frac{277965}{1066966108} a^{5} + \frac{247418274547303}{1294733168709712} a^{4} + \frac{166779}{4267864432} a^{3} - \frac{7450834325878865}{16831531193226256} a^{2} + \frac{7260028847717371}{16831531193226256} a$, $\frac{1}{16831531193226256} a^{29} - \frac{405}{16831531193226256} a^{25} - \frac{6471}{16831531193226256} a^{23} - \frac{12771}{4207882798306564} a^{21} + \frac{2059}{4267864432} a^{20} - \frac{124173}{8415765596613128} a^{19} + \frac{3023}{4267864432} a^{18} - \frac{802845}{16831531193226256} a^{17} - \frac{25749}{328297264} a^{16} - \frac{444771}{4207882798306564} a^{15} - \frac{3471051}{4267864432} a^{14} - \frac{1359099}{8415765596613128} a^{13} - \frac{1225797}{328297264} a^{12} + \frac{56009034263099}{323683292177428} a^{11} - \frac{40139055}{4267864432} a^{10} + \frac{98110925153465}{647366584354856} a^{9} - \frac{14450931}{1066966108} a^{8} + \frac{25910591145887}{1294733168709712} a^{7} - \frac{23033097}{2133932216} a^{6} + \frac{152243645618859}{647366584354856} a^{5} - \frac{18214281}{4267864432} a^{4} + \frac{3232343421734395}{8415765596613128} a^{3} + \frac{278813661958649}{647366584354856} a^{2} + \frac{5857809470132827}{16831531193226256} a + \frac{6611023475083289}{16831531193226256}$, $\frac{1}{16831531193226256} a^{30} - \frac{405}{16831531193226256} a^{26} - \frac{6471}{16831531193226256} a^{24} - \frac{12771}{4207882798306564} a^{22} + \frac{2059}{4267864432} a^{21} - \frac{124173}{8415765596613128} a^{20} + \frac{3023}{4267864432} a^{19} - \frac{802845}{16831531193226256} a^{18} - \frac{25749}{328297264} a^{17} - \frac{444771}{4207882798306564} a^{16} - \frac{3471051}{4267864432} a^{15} - \frac{1359099}{8415765596613128} a^{14} - \frac{1225797}{328297264} a^{13} + \frac{56009034263099}{323683292177428} a^{12} - \frac{40139055}{4267864432} a^{11} + \frac{98110925153465}{647366584354856} a^{10} - \frac{14450931}{1066966108} a^{9} + \frac{25910591145887}{1294733168709712} a^{8} - \frac{23033097}{2133932216} a^{7} + \frac{152243645618859}{647366584354856} a^{6} - \frac{18214281}{4267864432} a^{5} + \frac{3232343421734395}{8415765596613128} a^{4} + \frac{278813661958649}{647366584354856} a^{3} + \frac{5857809470132827}{16831531193226256} a^{2} + \frac{6611023475083289}{16831531193226256} a$, $\frac{1}{16831531193226256} a^{31} + \frac{279}{1051970699576641} a^{25} + \frac{189}{39697007531194} a^{23} + \frac{2059}{4267864432} a^{22} + \frac{673839}{16831531193226256} a^{21} + \frac{3023}{4267864432} a^{20} + \frac{1703565}{8415765596613128} a^{19} + \frac{427}{533483054} a^{18} + \frac{11310111}{16831531193226256} a^{17} + \frac{2615703}{4267864432} a^{16} + \frac{12768921}{8415765596613128} a^{15} - \frac{5273789}{2133932216} a^{14} - \frac{231396384959759}{2103941399153282} a^{13} - \frac{419187725}{4267864432} a^{12} - \frac{4481619950781}{161841646088714} a^{11} - \frac{588909741}{4267864432} a^{10} + \frac{104507743462477}{1294733168709712} a^{9} - \frac{374221005}{1066966108} a^{8} + \frac{127282441772117}{647366584354856} a^{7} - \frac{717157511}{4267864432} a^{6} + \frac{3564984728130593}{16831531193226256} a^{5} + \frac{1047071602409811}{8415765596613128} a^{4} + \frac{2535334011564737}{8415765596613128} a^{3} - \frac{66348014672126}{1051970699576641} a^{2} - \frac{1125355402888325}{16831531193226256} a - \frac{3721165385039223}{16831531193226256}$, $\frac{1}{28181295799113236300085397456} a^{32} + \frac{49004071265}{14090647899556618150042698728} a^{31} + \frac{2}{1761330987444577268755337341} a^{30} - \frac{77594711743}{7045323949778309075021349364} a^{29} + \frac{29}{1761330987444577268755337341} a^{28} + \frac{677643040495}{28181295799113236300085397456} a^{27} + \frac{252}{1761330987444577268755337341} a^{26} + \frac{581508143370945}{28181295799113236300085397456} a^{25} + \frac{225}{270973998068396502885436514} a^{24} - \frac{51333044395705377531133}{28181295799113236300085397456} a^{23} - \frac{41338375130077704622537}{28181295799113236300085397456} a^{22} - \frac{21033979583588955921713}{14090647899556618150042698728} a^{21} + \frac{562568027139481394261}{265861281123709776415899976} a^{20} + \frac{6292070871665834894693}{28181295799113236300085397456} a^{19} - \frac{26562909621577293474317}{14090647899556618150042698728} a^{18} - \frac{17464866378513650299220289}{7045323949778309075021349364} a^{17} - \frac{9372300756193388567061669}{3522661974889154537510674682} a^{16} + \frac{3181072585868496477621471}{14090647899556618150042698728} a^{15} - \frac{74367138000674007340543303}{28181295799113236300085397456} a^{14} + \frac{3879961201273241640472587449}{28181295799113236300085397456} a^{13} + \frac{176704443401286648198416607}{14090647899556618150042698728} a^{12} - \frac{540889884573907566668722851}{2167791984547172023083492112} a^{11} - \frac{1116651613710545015141781559}{14090647899556618150042698728} a^{10} - \frac{89122803254354930863895963}{541947996136793005770873028} a^{9} + \frac{290100710464513953889908383}{7045323949778309075021349364} a^{8} + \frac{438553219452005865340992201}{2167791984547172023083492112} a^{7} + \frac{6003050093482729705737455951}{28181295799113236300085397456} a^{6} + \frac{2813570609665879805634375}{33994325451282552834843664} a^{5} + \frac{3109565038880677065928677307}{14090647899556618150042698728} a^{4} + \frac{13573552483461837012271206011}{28181295799113236300085397456} a^{3} - \frac{2105534087420116175027889835}{14090647899556618150042698728} a^{2} - \frac{1572425901392427197643921203}{14090647899556618150042698728} a + \frac{918228156765059657780680361}{28181295799113236300085397456}$, $\frac{1}{28181295799113236300085397456} a^{33} + \frac{33}{28181295799113236300085397456} a^{31} - \frac{49004071265}{14090647899556618150042698728} a^{30} + \frac{495}{28181295799113236300085397456} a^{29} + \frac{204193494751}{14090647899556618150042698728} a^{28} + \frac{2233}{14090647899556618150042698728} a^{27} + \frac{12679936610301}{7045323949778309075021349364} a^{26} + \frac{2079}{2167791984547172023083492112} a^{25} - \frac{3948696439321205867779}{2167791984547172023083492112} a^{24} + \frac{4455}{1083895992273586011541746056} a^{23} + \frac{12230733787264415634123}{7045323949778309075021349364} a^{22} + \frac{41338375130077705080007}{28181295799113236300085397456} a^{21} + \frac{3889173318820606095293}{7045323949778309075021349364} a^{20} + \frac{41353440290626442278559}{28181295799113236300085397456} a^{19} - \frac{8169075800407084115719}{7045323949778309075021349364} a^{18} - \frac{441786935897088140270592}{1761330987444577268755337341} a^{17} + \frac{17577793401570475320257737}{7045323949778309075021349364} a^{16} - \frac{15381639416250917706001853}{3522661974889154537510674682} a^{15} - \frac{3839449493486645386632924}{1761330987444577268755337341} a^{14} - \frac{3209070704553589603867544975}{14090647899556618150042698728} a^{13} - \frac{1208707428830786590182833751}{28181295799113236300085397456} a^{12} - \frac{5615549544380715308231382525}{28181295799113236300085397456} a^{11} + \frac{1152651737448083425174727309}{14090647899556618150042698728} a^{10} - \frac{1338453941358156020507241807}{28181295799113236300085397456} a^{9} + \frac{4793994053052476584228911855}{28181295799113236300085397456} a^{8} - \frac{342348262476166487233582381}{7045323949778309075021349364} a^{7} + \frac{3491972840572343178530481503}{28181295799113236300085397456} a^{6} - \frac{242203537639045163420217389}{28181295799113236300085397456} a^{5} - \frac{3873464778267560885040454441}{28181295799113236300085397456} a^{4} - \frac{892684450440615818446281609}{2167791984547172023083492112} a^{3} - \frac{3388228642188168560614956005}{28181295799113236300085397456} a^{2} + \frac{460795252485470094597980615}{3522661974889154537510674682} a + \frac{426513175418952714401668651}{3522661974889154537510674682}$, $\frac{1}{28181295799113236300085397456} a^{34} + \frac{8177209691}{14090647899556618150042698728} a^{31} - \frac{561}{28181295799113236300085397456} a^{30} + \frac{151248785843}{7045323949778309075021349364} a^{29} - \frac{5423}{14090647899556618150042698728} a^{28} - \frac{13229956381}{3522661974889154537510674682} a^{27} - \frac{106029}{28181295799113236300085397456} a^{26} - \frac{51333054652578326714113}{28181295799113236300085397456} a^{25} - \frac{25245}{1083895992273586011541746056} a^{24} + \frac{35025408768228371608287}{28181295799113236300085397456} a^{23} + \frac{4728105895839952652051}{14090647899556618150042698728} a^{22} - \frac{19438212371040115105017}{28181295799113236300085397456} a^{21} + \frac{24374257989391091290165}{28181295799113236300085397456} a^{20} + \frac{852827285216815928595}{541947996136793005770873028} a^{19} - \frac{70586609552155762901299}{28181295799113236300085397456} a^{18} + \frac{57215192202764498407448485}{28181295799113236300085397456} a^{17} - \frac{21138547628169568173261167}{14090647899556618150042698728} a^{16} + \frac{22431052636693419376440533}{14090647899556618150042698728} a^{15} + \frac{23424475743162052375904025}{14090647899556618150042698728} a^{14} + \frac{1536327614156136280129172455}{7045323949778309075021349364} a^{13} + \frac{2638886222997584206786293703}{28181295799113236300085397456} a^{12} - \frac{6733318255198763452557216385}{28181295799113236300085397456} a^{11} - \frac{653364389916320840687942427}{14090647899556618150042698728} a^{10} + \frac{6820233869207338944947130707}{28181295799113236300085397456} a^{9} + \frac{6122779023662928779392412955}{14090647899556618150042698728} a^{8} + \frac{49687383245585847556818775}{1761330987444577268755337341} a^{7} - \frac{1401934082869679850599748649}{7045323949778309075021349364} a^{6} - \frac{8631306636374624494281094389}{28181295799113236300085397456} a^{5} - \frac{12799776939530446051343427377}{28181295799113236300085397456} a^{4} + \frac{12085457852132375974767860689}{28181295799113236300085397456} a^{3} - \frac{11481971251664178661229185413}{28181295799113236300085397456} a^{2} + \frac{12049794785559187577800685411}{28181295799113236300085397456} a - \frac{4913584198135562470020112061}{14090647899556618150042698728}$, $\frac{1}{28181295799113236300085397456} a^{35} - \frac{595}{28181295799113236300085397456} a^{31} + \frac{20413430787}{7045323949778309075021349364} a^{30} - \frac{2975}{7045323949778309075021349364} a^{29} + \frac{677287919209}{28181295799113236300085397456} a^{28} - \frac{120785}{28181295799113236300085397456} a^{27} - \frac{25666527246243367623851}{14090647899556618150042698728} a^{26} - \frac{7497}{270973998068396502885436514} a^{25} + \frac{35025411097926283045107}{28181295799113236300085397456} a^{24} - \frac{133875}{1083895992273586011541746056} a^{23} + \frac{4499335449581581902769}{7045323949778309075021349364} a^{22} - \frac{50794586921757624000569}{28181295799113236300085397456} a^{21} + \frac{1075750016948458406623}{2167791984547172023083492112} a^{20} - \frac{10668906579093204325645}{7045323949778309075021349364} a^{19} - \frac{1430195113200823307677}{14090647899556618150042698728} a^{18} - \frac{10480578842414509254003367}{7045323949778309075021349364} a^{17} - \frac{62456686571589786056865355}{14090647899556618150042698728} a^{16} + \frac{31725299083784217891788631}{28181295799113236300085397456} a^{15} + \frac{597829775955390520389637}{1761330987444577268755337341} a^{14} + \frac{2753775016251440777370944033}{28181295799113236300085397456} a^{13} + \frac{2409675844415045664268348609}{28181295799113236300085397456} a^{12} - \frac{5662099471175816694324171351}{28181295799113236300085397456} a^{11} - \frac{1706794327905959693596035215}{28181295799113236300085397456} a^{10} + \frac{43651905070470709849670959}{14090647899556618150042698728} a^{9} - \frac{655854094149616843420275987}{7045323949778309075021349364} a^{8} - \frac{5285131299568670913151151575}{14090647899556618150042698728} a^{7} + \frac{3651495358936578185652846061}{14090647899556618150042698728} a^{6} + \frac{2309493945468104471462205655}{28181295799113236300085397456} a^{5} + \frac{2960050081835577698388127285}{14090647899556618150042698728} a^{4} + \frac{3335996272349692221723512323}{7045323949778309075021349364} a^{3} + \frac{3516518878580783135007183373}{14090647899556618150042698728} a^{2} + \frac{13400409724369768392104926861}{28181295799113236300085397456} a - \frac{1765385435327491014051992871}{14090647899556618150042698728}$

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:  \( -1 \) (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

$C_{36}$ (as 36T1):

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

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{9})^+\), 4.0.44217.1, 6.6.32234193.1, \(\Q(\zeta_{27})^+\), 12.0.45943373101939347033.1, 18.18.116781890125989356502353933497857.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{4}$ R $36$ $36$ $36$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{4}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ $36$ $36$ $36$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ $36$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{36}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{4}$

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
3Data not computed
$17$17.12.9.2$x^{12} - 34 x^{8} + 289 x^{4} - 44217$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
17.12.9.2$x^{12} - 34 x^{8} + 289 x^{4} - 44217$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
17.12.9.2$x^{12} - 34 x^{8} + 289 x^{4} - 44217$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$