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Properties

Label 36.0.145...377.1
Degree $36$
Signature $[0, 18]$
Discriminant $1.456\times 10^{76}$
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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Normalized defining polynomial

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

\( x^{36} + 36 x^{34} + 594 x^{32} + 5952 x^{30} + 40455 x^{28} - 14128 x^{27} + 197316 x^{26} + \cdots + 88739165414377 \) Copy content Toggle raw display

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

Invariants

Degree:  $36$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 18]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(14555723661975701129520414713591505717264293810334661879947459060497463940377\) \(\medspace = 3^{90}\cdot 17^{27}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(130.51\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{5/2}17^{3/4}\approx 130.5088094322623$
Ramified primes:   \(3\), \(17\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{17}) \)
$\card{ \Gal(K/\Q) }$:  $36$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
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.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{106}a^{14}-\frac{23}{106}a^{13}+\frac{7}{53}a^{12}+\frac{19}{106}a^{11}+\frac{12}{53}a^{10}-\frac{11}{106}a^{9}+\frac{51}{106}a^{8}+\frac{8}{53}a^{7}+\frac{29}{106}a^{6}-\frac{26}{53}a^{5}-\frac{8}{53}a^{4}+\frac{27}{106}a^{3}-\frac{2}{53}a^{2}-\frac{17}{53}a+\frac{1}{53}$, $\frac{1}{106}a^{15}+\frac{15}{106}a^{13}+\frac{23}{106}a^{12}-\frac{8}{53}a^{11}+\frac{11}{106}a^{10}+\frac{5}{53}a^{9}+\frac{23}{106}a^{8}-\frac{27}{106}a^{7}-\frac{21}{106}a^{6}+\frac{7}{106}a^{5}-\frac{23}{106}a^{4}+\frac{17}{53}a^{3}+\frac{33}{106}a^{2}+\frac{15}{106}a-\frac{7}{106}$, $\frac{1}{106}a^{16}-\frac{3}{106}a^{13}-\frac{7}{53}a^{12}-\frac{9}{106}a^{11}+\frac{21}{106}a^{10}-\frac{12}{53}a^{9}+\frac{3}{106}a^{8}+\frac{2}{53}a^{7}+\frac{49}{106}a^{6}+\frac{15}{106}a^{5}+\frac{9}{106}a^{4}+\frac{26}{53}a^{3}+\frac{11}{53}a^{2}-\frac{27}{106}a+\frac{23}{106}$, $\frac{1}{106}a^{17}+\frac{23}{106}a^{13}-\frac{10}{53}a^{12}+\frac{25}{106}a^{11}-\frac{5}{106}a^{10}+\frac{23}{106}a^{9}-\frac{1}{53}a^{8}+\frac{22}{53}a^{7}-\frac{2}{53}a^{6}-\frac{41}{106}a^{5}-\frac{49}{106}a^{4}-\frac{3}{106}a^{3}+\frac{7}{53}a^{2}+\frac{27}{106}a+\frac{3}{53}$, $\frac{1}{198008}a^{18}+\frac{9}{99004}a^{16}+\frac{135}{198008}a^{14}+\frac{273}{99004}a^{12}+\frac{1287}{198008}a^{10}-\frac{11573}{49502}a^{9}+\frac{891}{99004}a^{8}-\frac{5153}{49502}a^{7}+\frac{693}{99004}a^{6}-\frac{15459}{49502}a^{5}+\frac{135}{49502}a^{4}-\frac{338}{24751}a^{3}+\frac{81}{198008}a^{2}-\frac{5153}{49502}a-\frac{91317}{198008}$, $\frac{1}{198008}a^{19}+\frac{9}{99004}a^{17}+\frac{135}{198008}a^{15}+\frac{273}{99004}a^{13}+\frac{1287}{198008}a^{11}-\frac{11573}{49502}a^{10}+\frac{891}{99004}a^{9}-\frac{5153}{49502}a^{8}+\frac{693}{99004}a^{7}-\frac{15459}{49502}a^{6}+\frac{135}{49502}a^{5}-\frac{338}{24751}a^{4}+\frac{81}{198008}a^{3}-\frac{5153}{49502}a^{2}-\frac{91317}{198008}a$, $\frac{1}{198008}a^{20}-\frac{189}{198008}a^{16}-\frac{2}{24751}a^{14}-\frac{23}{106}a^{13}+\frac{17611}{198008}a^{12}-\frac{1350}{24751}a^{11}+\frac{2931}{24751}a^{10}+\frac{8}{24751}a^{9}+\frac{32289}{99004}a^{8}-\frac{14237}{49502}a^{7}+\frac{7441}{49502}a^{6}+\frac{2896}{24751}a^{5}-\frac{39527}{198008}a^{4}+\frac{9812}{24751}a^{3}+\frac{97761}{198008}a^{2}-\frac{11064}{24751}a+\frac{31689}{99004}$, $\frac{1}{198008}a^{21}-\frac{189}{198008}a^{17}-\frac{2}{24751}a^{15}+\frac{19479}{198008}a^{13}-\frac{416}{24751}a^{12}+\frac{11933}{49502}a^{11}+\frac{5145}{24751}a^{10}-\frac{6005}{99004}a^{9}-\frac{5484}{24751}a^{8}-\frac{18711}{49502}a^{7}+\frac{20269}{49502}a^{6}-\frac{95567}{198008}a^{5}-\frac{1863}{24751}a^{4}+\frac{69741}{198008}a^{3}-\frac{7795}{24751}a^{2}-\frac{107}{1868}a+\frac{23}{53}$, $\frac{1}{198008}a^{22}-\frac{175}{99004}a^{16}+\frac{81}{99004}a^{14}+\frac{6122}{24751}a^{13}-\frac{14201}{99004}a^{12}+\frac{1876}{24751}a^{11}-\frac{32155}{198008}a^{10}+\frac{886}{24751}a^{9}+\frac{21699}{99004}a^{8}-\frac{22917}{49502}a^{7}-\frac{29753}{198008}a^{6}-\frac{2663}{24751}a^{5}+\frac{63457}{198008}a^{4}+\frac{241}{24751}a^{3}+\frac{2099}{198008}a^{2}-\frac{1623}{49502}a+\frac{89203}{198008}$, $\frac{1}{198008}a^{23}-\frac{175}{99004}a^{17}+\frac{81}{99004}a^{15}+\frac{51}{24751}a^{14}-\frac{191}{99004}a^{13}+\frac{7021}{49502}a^{12}+\frac{35093}{198008}a^{11}+\frac{3688}{24751}a^{10}-\frac{8189}{99004}a^{9}+\frac{1367}{49502}a^{8}+\frac{84195}{198008}a^{7}+\frac{13821}{49502}a^{6}-\frac{84115}{198008}a^{5}-\frac{1627}{24751}a^{4}+\frac{76819}{198008}a^{3}-\frac{2557}{49502}a^{2}+\frac{57447}{198008}a-\frac{26}{53}$, $\frac{1}{198008}a^{24}+\frac{429}{99004}a^{16}+\frac{51}{24751}a^{15}+\frac{21}{24751}a^{14}+\frac{3744}{24751}a^{13}+\frac{46865}{198008}a^{12}-\frac{1916}{24751}a^{11}-\frac{3095}{49502}a^{10}-\frac{1227}{49502}a^{9}+\frac{91455}{198008}a^{8}-\frac{2053}{49502}a^{7}-\frac{39863}{198008}a^{6}+\frac{23377}{49502}a^{5}+\frac{71547}{198008}a^{4}-\frac{8459}{49502}a^{3}+\frac{50305}{198008}a^{2}+\frac{8875}{24751}a-\frac{3471}{99004}$, $\frac{1}{198008}a^{25}+\frac{429}{99004}a^{17}+\frac{51}{24751}a^{16}+\frac{21}{24751}a^{15}+\frac{8}{24751}a^{14}+\frac{41261}{198008}a^{13}-\frac{4718}{24751}a^{12}+\frac{3443}{49502}a^{11}-\frac{3649}{24751}a^{10}+\frac{24207}{198008}a^{9}-\frac{5930}{24751}a^{8}-\frac{23051}{198008}a^{7}-\frac{10027}{24751}a^{6}-\frac{57345}{198008}a^{5}-\frac{6331}{24751}a^{4}-\frac{63643}{198008}a^{3}-\frac{932}{24751}a^{2}-\frac{39897}{99004}a-\frac{16}{53}$, $\frac{1}{198008}a^{26}+\frac{51}{24751}a^{17}-\frac{83}{49502}a^{16}+\frac{8}{24751}a^{15}+\frac{151}{198008}a^{14}-\frac{2383}{24751}a^{13}-\frac{1731}{24751}a^{12}-\frac{7765}{49502}a^{11}+\frac{37025}{198008}a^{10}-\frac{5509}{49502}a^{9}-\frac{72551}{198008}a^{8}-\frac{12397}{49502}a^{7}+\frac{68539}{198008}a^{6}+\frac{4921}{24751}a^{5}-\frac{3923}{198008}a^{4}+\frac{14475}{49502}a^{3}-\frac{2550}{24751}a^{2}+\frac{3595}{24751}a+\frac{17977}{99004}$, $\frac{1}{16\!\cdots\!56}a^{27}+\frac{27}{16\!\cdots\!56}a^{25}+\frac{81}{42\!\cdots\!64}a^{23}+\frac{2277}{16\!\cdots\!56}a^{21}+\frac{10395}{16\!\cdots\!56}a^{19}+\frac{2059}{4267864432}a^{18}+\frac{32319}{16\!\cdots\!56}a^{17}+\frac{18531}{2133932216}a^{16}+\frac{8721}{21\!\cdots\!82}a^{15}+\frac{277965}{4267864432}a^{14}+\frac{26163}{42\!\cdots\!64}a^{13}+\frac{43239}{164148632}a^{12}+\frac{4131}{647366584354856}a^{11}+\frac{203841}{328297264}a^{10}+\frac{112018068637735}{647366584354856}a^{9}+\frac{1834569}{2133932216}a^{8}-\frac{143285275497077}{323683292177428}a^{7}+\frac{1426887}{2133932216}a^{6}-\frac{424690137261773}{12\!\cdots\!12}a^{5}+\frac{277965}{1066966108}a^{4}+\frac{247418274547303}{12\!\cdots\!12}a^{3}+\frac{166779}{4267864432}a^{2}-\frac{74\!\cdots\!65}{16\!\cdots\!56}a+\frac{72\!\cdots\!71}{16\!\cdots\!56}$, $\frac{1}{16\!\cdots\!56}a^{28}+\frac{27}{16\!\cdots\!56}a^{26}+\frac{81}{42\!\cdots\!64}a^{24}+\frac{2277}{16\!\cdots\!56}a^{22}+\frac{10395}{16\!\cdots\!56}a^{20}+\frac{2059}{4267864432}a^{19}+\frac{32319}{16\!\cdots\!56}a^{18}+\frac{18531}{2133932216}a^{17}+\frac{8721}{21\!\cdots\!82}a^{16}+\frac{277965}{4267864432}a^{15}+\frac{26163}{42\!\cdots\!64}a^{14}+\frac{43239}{164148632}a^{13}+\frac{4131}{647366584354856}a^{12}+\frac{203841}{328297264}a^{11}+\frac{112018068637735}{647366584354856}a^{10}+\frac{1834569}{2133932216}a^{9}-\frac{143285275497077}{323683292177428}a^{8}+\frac{1426887}{2133932216}a^{7}-\frac{424690137261773}{12\!\cdots\!12}a^{6}+\frac{277965}{1066966108}a^{5}+\frac{247418274547303}{12\!\cdots\!12}a^{4}+\frac{166779}{4267864432}a^{3}-\frac{74\!\cdots\!65}{16\!\cdots\!56}a^{2}+\frac{72\!\cdots\!71}{16\!\cdots\!56}a$, $\frac{1}{16\!\cdots\!56}a^{29}-\frac{405}{16\!\cdots\!56}a^{25}-\frac{6471}{16\!\cdots\!56}a^{23}-\frac{12771}{42\!\cdots\!64}a^{21}+\frac{2059}{4267864432}a^{20}-\frac{124173}{84\!\cdots\!28}a^{19}+\frac{3023}{4267864432}a^{18}-\frac{802845}{16\!\cdots\!56}a^{17}-\frac{25749}{328297264}a^{16}-\frac{444771}{42\!\cdots\!64}a^{15}-\frac{3471051}{4267864432}a^{14}-\frac{1359099}{84\!\cdots\!28}a^{13}-\frac{1225797}{328297264}a^{12}+\frac{56009034263099}{323683292177428}a^{11}-\frac{40139055}{4267864432}a^{10}+\frac{98110925153465}{647366584354856}a^{9}-\frac{14450931}{1066966108}a^{8}+\frac{25910591145887}{12\!\cdots\!12}a^{7}-\frac{23033097}{2133932216}a^{6}+\frac{152243645618859}{647366584354856}a^{5}-\frac{18214281}{4267864432}a^{4}+\frac{32\!\cdots\!95}{84\!\cdots\!28}a^{3}+\frac{278813661958649}{647366584354856}a^{2}+\frac{58\!\cdots\!27}{16\!\cdots\!56}a+\frac{66\!\cdots\!89}{16\!\cdots\!56}$, $\frac{1}{16\!\cdots\!56}a^{30}-\frac{405}{16\!\cdots\!56}a^{26}-\frac{6471}{16\!\cdots\!56}a^{24}-\frac{12771}{42\!\cdots\!64}a^{22}+\frac{2059}{4267864432}a^{21}-\frac{124173}{84\!\cdots\!28}a^{20}+\frac{3023}{4267864432}a^{19}-\frac{802845}{16\!\cdots\!56}a^{18}-\frac{25749}{328297264}a^{17}-\frac{444771}{42\!\cdots\!64}a^{16}-\frac{3471051}{4267864432}a^{15}-\frac{1359099}{84\!\cdots\!28}a^{14}-\frac{1225797}{328297264}a^{13}+\frac{56009034263099}{323683292177428}a^{12}-\frac{40139055}{4267864432}a^{11}+\frac{98110925153465}{647366584354856}a^{10}-\frac{14450931}{1066966108}a^{9}+\frac{25910591145887}{12\!\cdots\!12}a^{8}-\frac{23033097}{2133932216}a^{7}+\frac{152243645618859}{647366584354856}a^{6}-\frac{18214281}{4267864432}a^{5}+\frac{32\!\cdots\!95}{84\!\cdots\!28}a^{4}+\frac{278813661958649}{647366584354856}a^{3}+\frac{58\!\cdots\!27}{16\!\cdots\!56}a^{2}+\frac{66\!\cdots\!89}{16\!\cdots\!56}a$, $\frac{1}{16\!\cdots\!56}a^{31}+\frac{279}{10\!\cdots\!41}a^{25}+\frac{189}{39697007531194}a^{23}+\frac{2059}{4267864432}a^{22}+\frac{673839}{16\!\cdots\!56}a^{21}+\frac{3023}{4267864432}a^{20}+\frac{1703565}{84\!\cdots\!28}a^{19}+\frac{427}{533483054}a^{18}+\frac{11310111}{16\!\cdots\!56}a^{17}+\frac{2615703}{4267864432}a^{16}+\frac{12768921}{84\!\cdots\!28}a^{15}-\frac{5273789}{2133932216}a^{14}-\frac{231396384959759}{21\!\cdots\!82}a^{13}-\frac{419187725}{4267864432}a^{12}-\frac{4481619950781}{161841646088714}a^{11}-\frac{588909741}{4267864432}a^{10}+\frac{104507743462477}{12\!\cdots\!12}a^{9}-\frac{374221005}{1066966108}a^{8}+\frac{127282441772117}{647366584354856}a^{7}-\frac{717157511}{4267864432}a^{6}+\frac{35\!\cdots\!93}{16\!\cdots\!56}a^{5}+\frac{10\!\cdots\!11}{84\!\cdots\!28}a^{4}+\frac{25\!\cdots\!37}{84\!\cdots\!28}a^{3}-\frac{66348014672126}{10\!\cdots\!41}a^{2}-\frac{11\!\cdots\!25}{16\!\cdots\!56}a-\frac{37\!\cdots\!23}{16\!\cdots\!56}$, $\frac{1}{28\!\cdots\!56}a^{32}+\frac{49004071265}{14\!\cdots\!28}a^{31}+\frac{2}{17\!\cdots\!41}a^{30}-\frac{77594711743}{70\!\cdots\!64}a^{29}+\frac{29}{17\!\cdots\!41}a^{28}+\frac{677643040495}{28\!\cdots\!56}a^{27}+\frac{252}{17\!\cdots\!41}a^{26}+\frac{581508143370945}{28\!\cdots\!56}a^{25}+\frac{225}{27\!\cdots\!14}a^{24}-\frac{51\!\cdots\!33}{28\!\cdots\!56}a^{23}-\frac{41\!\cdots\!37}{28\!\cdots\!56}a^{22}-\frac{21\!\cdots\!13}{14\!\cdots\!28}a^{21}+\frac{56\!\cdots\!61}{26\!\cdots\!76}a^{20}+\frac{62\!\cdots\!93}{28\!\cdots\!56}a^{19}-\frac{26\!\cdots\!17}{14\!\cdots\!28}a^{18}-\frac{17\!\cdots\!89}{70\!\cdots\!64}a^{17}-\frac{93\!\cdots\!69}{35\!\cdots\!82}a^{16}+\frac{31\!\cdots\!71}{14\!\cdots\!28}a^{15}-\frac{74\!\cdots\!03}{28\!\cdots\!56}a^{14}+\frac{38\!\cdots\!49}{28\!\cdots\!56}a^{13}+\frac{17\!\cdots\!07}{14\!\cdots\!28}a^{12}-\frac{54\!\cdots\!51}{21\!\cdots\!12}a^{11}-\frac{11\!\cdots\!59}{14\!\cdots\!28}a^{10}-\frac{89\!\cdots\!63}{54\!\cdots\!28}a^{9}+\frac{29\!\cdots\!83}{70\!\cdots\!64}a^{8}+\frac{43\!\cdots\!01}{21\!\cdots\!12}a^{7}+\frac{60\!\cdots\!51}{28\!\cdots\!56}a^{6}+\frac{28\!\cdots\!75}{33\!\cdots\!64}a^{5}+\frac{31\!\cdots\!07}{14\!\cdots\!28}a^{4}+\frac{13\!\cdots\!11}{28\!\cdots\!56}a^{3}-\frac{21\!\cdots\!35}{14\!\cdots\!28}a^{2}-\frac{15\!\cdots\!03}{14\!\cdots\!28}a+\frac{91\!\cdots\!61}{28\!\cdots\!56}$, $\frac{1}{28\!\cdots\!56}a^{33}+\frac{33}{28\!\cdots\!56}a^{31}-\frac{49004071265}{14\!\cdots\!28}a^{30}+\frac{495}{28\!\cdots\!56}a^{29}+\frac{204193494751}{14\!\cdots\!28}a^{28}+\frac{2233}{14\!\cdots\!28}a^{27}+\frac{12679936610301}{70\!\cdots\!64}a^{26}+\frac{2079}{21\!\cdots\!12}a^{25}-\frac{39\!\cdots\!79}{21\!\cdots\!12}a^{24}+\frac{4455}{10\!\cdots\!56}a^{23}+\frac{12\!\cdots\!23}{70\!\cdots\!64}a^{22}+\frac{41\!\cdots\!07}{28\!\cdots\!56}a^{21}+\frac{38\!\cdots\!93}{70\!\cdots\!64}a^{20}+\frac{41\!\cdots\!59}{28\!\cdots\!56}a^{19}-\frac{81\!\cdots\!19}{70\!\cdots\!64}a^{18}-\frac{44\!\cdots\!92}{17\!\cdots\!41}a^{17}+\frac{17\!\cdots\!37}{70\!\cdots\!64}a^{16}-\frac{15\!\cdots\!53}{35\!\cdots\!82}a^{15}-\frac{38\!\cdots\!24}{17\!\cdots\!41}a^{14}-\frac{32\!\cdots\!75}{14\!\cdots\!28}a^{13}-\frac{12\!\cdots\!51}{28\!\cdots\!56}a^{12}-\frac{56\!\cdots\!25}{28\!\cdots\!56}a^{11}+\frac{11\!\cdots\!09}{14\!\cdots\!28}a^{10}-\frac{13\!\cdots\!07}{28\!\cdots\!56}a^{9}+\frac{47\!\cdots\!55}{28\!\cdots\!56}a^{8}-\frac{34\!\cdots\!81}{70\!\cdots\!64}a^{7}+\frac{34\!\cdots\!03}{28\!\cdots\!56}a^{6}-\frac{24\!\cdots\!89}{28\!\cdots\!56}a^{5}-\frac{38\!\cdots\!41}{28\!\cdots\!56}a^{4}-\frac{89\!\cdots\!09}{21\!\cdots\!12}a^{3}-\frac{33\!\cdots\!05}{28\!\cdots\!56}a^{2}+\frac{46\!\cdots\!15}{35\!\cdots\!82}a+\frac{42\!\cdots\!51}{35\!\cdots\!82}$, $\frac{1}{28\!\cdots\!56}a^{34}+\frac{8177209691}{14\!\cdots\!28}a^{31}-\frac{561}{28\!\cdots\!56}a^{30}+\frac{151248785843}{70\!\cdots\!64}a^{29}-\frac{5423}{14\!\cdots\!28}a^{28}-\frac{13229956381}{35\!\cdots\!82}a^{27}-\frac{106029}{28\!\cdots\!56}a^{26}-\frac{51\!\cdots\!13}{28\!\cdots\!56}a^{25}-\frac{25245}{10\!\cdots\!56}a^{24}+\frac{35\!\cdots\!87}{28\!\cdots\!56}a^{23}+\frac{47\!\cdots\!51}{14\!\cdots\!28}a^{22}-\frac{19\!\cdots\!17}{28\!\cdots\!56}a^{21}+\frac{24\!\cdots\!65}{28\!\cdots\!56}a^{20}+\frac{85\!\cdots\!95}{54\!\cdots\!28}a^{19}-\frac{70\!\cdots\!99}{28\!\cdots\!56}a^{18}+\frac{57\!\cdots\!85}{28\!\cdots\!56}a^{17}-\frac{21\!\cdots\!67}{14\!\cdots\!28}a^{16}+\frac{22\!\cdots\!33}{14\!\cdots\!28}a^{15}+\frac{23\!\cdots\!25}{14\!\cdots\!28}a^{14}+\frac{15\!\cdots\!55}{70\!\cdots\!64}a^{13}+\frac{26\!\cdots\!03}{28\!\cdots\!56}a^{12}-\frac{67\!\cdots\!85}{28\!\cdots\!56}a^{11}-\frac{65\!\cdots\!27}{14\!\cdots\!28}a^{10}+\frac{68\!\cdots\!07}{28\!\cdots\!56}a^{9}+\frac{61\!\cdots\!55}{14\!\cdots\!28}a^{8}+\frac{49\!\cdots\!75}{17\!\cdots\!41}a^{7}-\frac{14\!\cdots\!49}{70\!\cdots\!64}a^{6}-\frac{86\!\cdots\!89}{28\!\cdots\!56}a^{5}-\frac{12\!\cdots\!77}{28\!\cdots\!56}a^{4}+\frac{12\!\cdots\!89}{28\!\cdots\!56}a^{3}-\frac{11\!\cdots\!13}{28\!\cdots\!56}a^{2}+\frac{12\!\cdots\!11}{28\!\cdots\!56}a-\frac{49\!\cdots\!61}{14\!\cdots\!28}$, $\frac{1}{28\!\cdots\!56}a^{35}-\frac{595}{28\!\cdots\!56}a^{31}+\frac{20413430787}{70\!\cdots\!64}a^{30}-\frac{2975}{70\!\cdots\!64}a^{29}+\frac{677287919209}{28\!\cdots\!56}a^{28}-\frac{120785}{28\!\cdots\!56}a^{27}-\frac{25\!\cdots\!51}{14\!\cdots\!28}a^{26}-\frac{7497}{27\!\cdots\!14}a^{25}+\frac{35\!\cdots\!07}{28\!\cdots\!56}a^{24}-\frac{133875}{10\!\cdots\!56}a^{23}+\frac{44\!\cdots\!69}{70\!\cdots\!64}a^{22}-\frac{50\!\cdots\!69}{28\!\cdots\!56}a^{21}+\frac{10\!\cdots\!23}{21\!\cdots\!12}a^{20}-\frac{10\!\cdots\!45}{70\!\cdots\!64}a^{19}-\frac{14\!\cdots\!77}{14\!\cdots\!28}a^{18}-\frac{10\!\cdots\!67}{70\!\cdots\!64}a^{17}-\frac{62\!\cdots\!55}{14\!\cdots\!28}a^{16}+\frac{31\!\cdots\!31}{28\!\cdots\!56}a^{15}+\frac{59\!\cdots\!37}{17\!\cdots\!41}a^{14}+\frac{27\!\cdots\!33}{28\!\cdots\!56}a^{13}+\frac{24\!\cdots\!09}{28\!\cdots\!56}a^{12}-\frac{56\!\cdots\!51}{28\!\cdots\!56}a^{11}-\frac{17\!\cdots\!15}{28\!\cdots\!56}a^{10}+\frac{43\!\cdots\!59}{14\!\cdots\!28}a^{9}-\frac{65\!\cdots\!87}{70\!\cdots\!64}a^{8}-\frac{52\!\cdots\!75}{14\!\cdots\!28}a^{7}+\frac{36\!\cdots\!61}{14\!\cdots\!28}a^{6}+\frac{23\!\cdots\!55}{28\!\cdots\!56}a^{5}+\frac{29\!\cdots\!85}{14\!\cdots\!28}a^{4}+\frac{33\!\cdots\!23}{70\!\cdots\!64}a^{3}+\frac{35\!\cdots\!73}{14\!\cdots\!28}a^{2}+\frac{13\!\cdots\!61}{28\!\cdots\!56}a-\frac{17\!\cdots\!71}{14\!\cdots\!28}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.9.0.1}{9} }^{4}$ R $36$ $36$ $36$ ${\href{/padicField/13.9.0.1}{9} }^{4}$ R ${\href{/padicField/19.6.0.1}{6} }^{6}$ $36$ $36$ $36$ ${\href{/padicField/37.12.0.1}{12} }^{3}$ $36$ $18^{2}$ $18^{2}$ ${\href{/padicField/53.1.0.1}{1} }^{36}$ ${\href{/padicField/59.9.0.1}{9} }^{4}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display Deg $36$$18$$2$$90$
\(17\) Copy content Toggle raw display 17.12.9.2$x^{12} - 34 x^{8} + 289 x^{4} + 962948$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
17.12.9.2$x^{12} - 34 x^{8} + 289 x^{4} + 962948$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
17.12.9.2$x^{12} - 34 x^{8} + 289 x^{4} + 962948$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$