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Properties

Label 36.0.142...601.1
Degree $36$
Signature $[0, 18]$
Discriminant $1.421\times 10^{58}$
Root discriminant \(41.24\)
Ramified primes $3,7$
Class number not computed
Class group not computed
Galois group $C_2\times C_{18}$ (as 36T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 5*x^27 - 487*x^18 - 2560*x^9 + 262144)
 
gp: K = bnfinit(y^36 - 5*y^27 - 487*y^18 - 2560*y^9 + 262144, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 5*x^27 - 487*x^18 - 2560*x^9 + 262144);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 5*x^27 - 487*x^18 - 2560*x^9 + 262144)
 

\( x^{36} - 5x^{27} - 487x^{18} - 2560x^{9} + 262144 \) 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:   \(14212734556341031905549296191351828189377245025195450200601\) \(\medspace = 3^{90}\cdot 7^{18}\) 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:  \(41.24\)
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:  $3^{5/2}7^{1/2}\approx 41.24318125460256$
Ramified primes:   \(3\), \(7\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $36$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(189=3^{3}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{189}(1,·)$, $\chi_{189}(134,·)$, $\chi_{189}(8,·)$, $\chi_{189}(139,·)$, $\chi_{189}(13,·)$, $\chi_{189}(146,·)$, $\chi_{189}(20,·)$, $\chi_{189}(22,·)$, $\chi_{189}(155,·)$, $\chi_{189}(29,·)$, $\chi_{189}(160,·)$, $\chi_{189}(34,·)$, $\chi_{189}(167,·)$, $\chi_{189}(41,·)$, $\chi_{189}(43,·)$, $\chi_{189}(176,·)$, $\chi_{189}(50,·)$, $\chi_{189}(181,·)$, $\chi_{189}(55,·)$, $\chi_{189}(188,·)$, $\chi_{189}(62,·)$, $\chi_{189}(64,·)$, $\chi_{189}(71,·)$, $\chi_{189}(76,·)$, $\chi_{189}(83,·)$, $\chi_{189}(85,·)$, $\chi_{189}(92,·)$, $\chi_{189}(97,·)$, $\chi_{189}(104,·)$, $\chi_{189}(106,·)$, $\chi_{189}(113,·)$, $\chi_{189}(118,·)$, $\chi_{189}(169,·)$, $\chi_{189}(148,·)$, $\chi_{189}(125,·)$, $\chi_{189}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17}a^{18}+\frac{6}{17}a^{9}+\frac{2}{17}$, $\frac{1}{34}a^{19}-\frac{11}{34}a^{10}-\frac{15}{34}a$, $\frac{1}{68}a^{20}+\frac{23}{68}a^{11}-\frac{15}{68}a^{2}$, $\frac{1}{136}a^{21}-\frac{45}{136}a^{12}-\frac{15}{136}a^{3}$, $\frac{1}{272}a^{22}+\frac{91}{272}a^{13}+\frac{121}{272}a^{4}$, $\frac{1}{544}a^{23}+\frac{91}{544}a^{14}+\frac{121}{544}a^{5}$, $\frac{1}{1088}a^{24}-\frac{453}{1088}a^{15}-\frac{423}{1088}a^{6}$, $\frac{1}{2176}a^{25}+\frac{635}{2176}a^{16}+\frac{665}{2176}a^{7}$, $\frac{1}{4352}a^{26}-\frac{1541}{4352}a^{17}-\frac{1511}{4352}a^{8}$, $\frac{1}{4238848}a^{27}+\frac{205}{8704}a^{18}-\frac{2561}{8704}a^{9}+\frac{1943}{8279}$, $\frac{1}{8477696}a^{28}+\frac{205}{17408}a^{19}-\frac{2561}{17408}a^{10}-\frac{3168}{8279}a$, $\frac{1}{16955392}a^{29}+\frac{205}{34816}a^{20}-\frac{2561}{34816}a^{11}+\frac{5111}{16558}a^{2}$, $\frac{1}{33910784}a^{30}+\frac{205}{69632}a^{21}+\frac{32255}{69632}a^{12}+\frac{5111}{33116}a^{3}$, $\frac{1}{67821568}a^{31}+\frac{205}{139264}a^{22}-\frac{37377}{139264}a^{13}-\frac{28005}{66232}a^{4}$, $\frac{1}{135643136}a^{32}+\frac{205}{278528}a^{23}+\frac{101887}{278528}a^{14}+\frac{38227}{132464}a^{5}$, $\frac{1}{271286272}a^{33}+\frac{205}{557056}a^{24}-\frac{176641}{557056}a^{15}-\frac{94237}{264928}a^{6}$, $\frac{1}{542572544}a^{34}+\frac{205}{1114112}a^{25}-\frac{176641}{1114112}a^{16}+\frac{170691}{529856}a^{7}$, $\frac{1}{1085145088}a^{35}+\frac{205}{2228224}a^{26}+\frac{937471}{2228224}a^{17}-\frac{359165}{1059712}a^{8}$ 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:   \( -\frac{4049}{542572544} a^{34} - \frac{93}{1114112} a^{25} + \frac{4049}{1114112} a^{16} + \frac{20245}{1059712} a^{7} \)  (order $54$) 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 - 5*x^27 - 487*x^18 - 2560*x^9 + 262144)
 
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 - 5*x^27 - 487*x^18 - 2560*x^9 + 262144, 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 - 5*x^27 - 487*x^18 - 2560*x^9 + 262144);
 
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 - 5*x^27 - 487*x^18 - 2560*x^9 + 262144);
 
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

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

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)
 
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\zeta_{9})\), 6.0.2250423.1, 6.6.6751269.1, \(\Q(\zeta_{27})^+\), 12.0.45579633110361.1, \(\Q(\zeta_{27})\), 18.0.39739057971752889532465351767.1, 18.18.119217173915258668597396055301.1

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 $18^{2}$ R $18^{2}$ R $18^{2}$ $18^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{6}$ ${\href{/padicField/19.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{12}$ $18^{2}$ ${\href{/padicField/43.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{18}$ $18^{2}$

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$
\(7\) Copy content Toggle raw display 7.18.9.2$x^{18} + 63 x^{16} + 1764 x^{14} + 12 x^{13} + 28814 x^{12} - 504 x^{11} + 302370 x^{10} - 17044 x^{9} + 2112804 x^{8} - 150180 x^{7} + 9908221 x^{6} - 209592 x^{5} + 29960739 x^{4} + 1787108 x^{3} + 51556212 x^{2} + 7225224 x + 40408804$$2$$9$$9$$C_{18}$$[\ ]_{2}^{9}$
7.18.9.2$x^{18} + 63 x^{16} + 1764 x^{14} + 12 x^{13} + 28814 x^{12} - 504 x^{11} + 302370 x^{10} - 17044 x^{9} + 2112804 x^{8} - 150180 x^{7} + 9908221 x^{6} - 209592 x^{5} + 29960739 x^{4} + 1787108 x^{3} + 51556212 x^{2} + 7225224 x + 40408804$$2$$9$$9$$C_{18}$$[\ ]_{2}^{9}$