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Properties

Label 18.18.456...917.1
Degree $18$
Signature $[18, 0]$
Discriminant $4.565\times 10^{26}$
Root discriminant \(30.27\)
Ramified prime $37$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 17*x^16 + 16*x^15 + 120*x^14 - 105*x^13 - 455*x^12 + 364*x^11 + 1001*x^10 - 715*x^9 - 1287*x^8 + 792*x^7 + 924*x^6 - 462*x^5 - 330*x^4 + 120*x^3 + 45*x^2 - 9*x - 1)
 
gp: K = bnfinit(y^18 - y^17 - 17*y^16 + 16*y^15 + 120*y^14 - 105*y^13 - 455*y^12 + 364*y^11 + 1001*y^10 - 715*y^9 - 1287*y^8 + 792*y^7 + 924*y^6 - 462*y^5 - 330*y^4 + 120*y^3 + 45*y^2 - 9*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 - 17*x^16 + 16*x^15 + 120*x^14 - 105*x^13 - 455*x^12 + 364*x^11 + 1001*x^10 - 715*x^9 - 1287*x^8 + 792*x^7 + 924*x^6 - 462*x^5 - 330*x^4 + 120*x^3 + 45*x^2 - 9*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 - 17*x^16 + 16*x^15 + 120*x^14 - 105*x^13 - 455*x^12 + 364*x^11 + 1001*x^10 - 715*x^9 - 1287*x^8 + 792*x^7 + 924*x^6 - 462*x^5 - 330*x^4 + 120*x^3 + 45*x^2 - 9*x - 1)
 

\( x^{18} - x^{17} - 17 x^{16} + 16 x^{15} + 120 x^{14} - 105 x^{13} - 455 x^{12} + 364 x^{11} + 1001 x^{10} + \cdots - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[18, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(456487940826035155404146917\) \(\medspace = 37^{17}\) 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:  \(30.27\)
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:  $37^{17/18}\approx 30.27466917182413$
Ramified primes:   \(37\) 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:  \(\Q(\sqrt{37}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(37\)
Dirichlet character group:    $\lbrace$$\chi_{37}(1,·)$, $\chi_{37}(3,·)$, $\chi_{37}(4,·)$, $\chi_{37}(7,·)$, $\chi_{37}(9,·)$, $\chi_{37}(10,·)$, $\chi_{37}(11,·)$, $\chi_{37}(12,·)$, $\chi_{37}(16,·)$, $\chi_{37}(21,·)$, $\chi_{37}(25,·)$, $\chi_{37}(26,·)$, $\chi_{37}(27,·)$, $\chi_{37}(28,·)$, $\chi_{37}(30,·)$, $\chi_{37}(33,·)$, $\chi_{37}(34,·)$, $\chi_{37}(36,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$ Copy content Toggle raw display

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:   $a^{5}-5a^{3}+5a$, $a^{2}-2$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a-1$, $a^{6}-6a^{4}+9a^{2}-2$, $a$, $a^{3}-3a$, $a^{17}-16a^{15}+105a^{13}-364a^{11}+715a^{9}-793a^{7}+468a^{5}-a^{4}-130a^{3}+3a^{2}+13a-1$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+661a^{8}-680a^{6}+356a^{4}-80a^{2}+5$, $a^{17}-16a^{15}+105a^{13}-364a^{11}+715a^{9}-792a^{7}+462a^{5}-120a^{3}+9a$, $a^{14}-a^{13}-13a^{12}+12a^{11}+66a^{10}-55a^{9}-165a^{8}+120a^{7}+210a^{6}-126a^{5}-126a^{4}+57a^{3}+27a^{2}-9a$, $a^{17}-16a^{15}-a^{14}+104a^{13}+13a^{12}-352a^{11}-66a^{10}+661a^{9}+166a^{8}-680a^{7}-218a^{6}+356a^{5}+146a^{4}-80a^{3}-44a^{2}+5a+4$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+106a^{4}-40a^{2}+4$, $a^{14}-14a^{12}+77a^{10}-a^{9}-210a^{8}+9a^{7}+294a^{6}-27a^{5}-196a^{4}+30a^{3}+49a^{2}-9a-1$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}-a^{2}+13a+2$, $a^{4}-4a^{2}+2$ Copy content Toggle raw display (assuming GRH)
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:  \( 24199468.4506 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 24199468.4506 \cdot 1}{2\cdot\sqrt{456487940826035155404146917}}\cr\approx \mathstrut & 0.148457142970 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 17*x^16 + 16*x^15 + 120*x^14 - 105*x^13 - 455*x^12 + 364*x^11 + 1001*x^10 - 715*x^9 - 1287*x^8 + 792*x^7 + 924*x^6 - 462*x^5 - 330*x^4 + 120*x^3 + 45*x^2 - 9*x - 1)
 
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^18 - x^17 - 17*x^16 + 16*x^15 + 120*x^14 - 105*x^13 - 455*x^12 + 364*x^11 + 1001*x^10 - 715*x^9 - 1287*x^8 + 792*x^7 + 924*x^6 - 462*x^5 - 330*x^4 + 120*x^3 + 45*x^2 - 9*x - 1, 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^18 - x^17 - 17*x^16 + 16*x^15 + 120*x^14 - 105*x^13 - 455*x^12 + 364*x^11 + 1001*x^10 - 715*x^9 - 1287*x^8 + 792*x^7 + 924*x^6 - 462*x^5 - 330*x^4 + 120*x^3 + 45*x^2 - 9*x - 1);
 
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^18 - x^17 - 17*x^16 + 16*x^15 + 120*x^14 - 105*x^13 - 455*x^12 + 364*x^11 + 1001*x^10 - 715*x^9 - 1287*x^8 + 792*x^7 + 924*x^6 - 462*x^5 - 330*x^4 + 120*x^3 + 45*x^2 - 9*x - 1);
 
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

$C_{18}$ (as 18T1):

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 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.1369.1, 6.6.69343957.1, 9.9.3512479453921.1

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 $18$ ${\href{/padicField/3.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/7.9.0.1}{9} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{6}$ $18$ $18$ $18$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.2.0.1}{2} }^{9}$ R ${\href{/padicField/41.9.0.1}{9} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.9.0.1}{9} }^{2}$ $18$

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
\(37\) Copy content Toggle raw display 37.18.17.1$x^{18} + 37$$18$$1$$17$$C_{18}$$[\ ]_{18}$