↑

Properties

Label 18.0.71224061078...2303.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,19^{17}\cdot 37^{9}$
Root discriminant $98.13$
Ramified primes $19, 37$
Class number $928256$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 14, 518]$ (GRH)
Galois group $C_{18}$ (as 18T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27132279301, -19771290010, 19771290010, -7502974525, 7502974525, -1505131399, 1505131399, -172277371, 172277371, -11841238, 11841238, -497269, 497269, -12484, 12484, -172, 172, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 172*x^16 - 172*x^15 + 12484*x^14 - 12484*x^13 + 497269*x^12 - 497269*x^11 + 11841238*x^10 - 11841238*x^9 + 172277371*x^8 - 172277371*x^7 + 1505131399*x^6 - 1505131399*x^5 + 7502974525*x^4 - 7502974525*x^3 + 19771290010*x^2 - 19771290010*x + 27132279301)
 
gp: K = bnfinit(x^18 - x^17 + 172*x^16 - 172*x^15 + 12484*x^14 - 12484*x^13 + 497269*x^12 - 497269*x^11 + 11841238*x^10 - 11841238*x^9 + 172277371*x^8 - 172277371*x^7 + 1505131399*x^6 - 1505131399*x^5 + 7502974525*x^4 - 7502974525*x^3 + 19771290010*x^2 - 19771290010*x + 27132279301, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 172 x^{16} - 172 x^{15} + 12484 x^{14} - 12484 x^{13} + 497269 x^{12} - 497269 x^{11} + 11841238 x^{10} - 11841238 x^{9} + 172277371 x^{8} - 172277371 x^{7} + 1505131399 x^{6} - 1505131399 x^{5} + 7502974525 x^{4} - 7502974525 x^{3} + 19771290010 x^{2} - 19771290010 x + 27132279301 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-712240610787788121582600955510822303=-\,19^{17}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.13$
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:  $19, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(703=19\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{703}(1,·)$, $\chi_{703}(519,·)$, $\chi_{703}(334,·)$, $\chi_{703}(591,·)$, $\chi_{703}(593,·)$, $\chi_{703}(147,·)$, $\chi_{703}(149,·)$, $\chi_{703}(408,·)$, $\chi_{703}(221,·)$, $\chi_{703}(482,·)$, $\chi_{703}(295,·)$, $\chi_{703}(554,·)$, $\chi_{703}(556,·)$, $\chi_{703}(110,·)$, $\chi_{703}(112,·)$, $\chi_{703}(369,·)$, $\chi_{703}(184,·)$, $\chi_{703}(702,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4476859381} a^{10} - \frac{414545938}{4476859381} a^{9} + \frac{90}{4476859381} a^{8} + \frac{2236654070}{4476859381} a^{7} + \frac{2835}{4476859381} a^{6} + \frac{2190487937}{4476859381} a^{5} + \frac{36450}{4476859381} a^{4} - \frac{479417535}{4476859381} a^{3} + \frac{164025}{4476859381} a^{2} + \frac{944002346}{4476859381} a + \frac{118098}{4476859381}$, $\frac{1}{4476859381} a^{11} + \frac{99}{4476859381} a^{9} - \frac{745945939}{4476859381} a^{8} + \frac{3564}{4476859381} a^{7} + \frac{14204964}{4476859381} a^{6} + \frac{56133}{4476859381} a^{5} + \frac{319611690}{4476859381} a^{4} + \frac{360855}{4476859381} a^{3} - \frac{2175655213}{4476859381} a^{2} + \frac{649539}{4476859381} a - \frac{1888004692}{4476859381}$, $\frac{1}{4476859381} a^{12} + \frac{2367494}{4476859381} a^{9} - \frac{5346}{4476859381} a^{8} - \frac{2048438297}{4476859381} a^{7} - \frac{224532}{4476859381} a^{6} - \frac{1649443785}{4476859381} a^{5} - \frac{3247695}{4476859381} a^{4} + \frac{518086942}{4476859381} a^{3} - \frac{15588936}{4476859381} a^{2} - \frac{1330189945}{4476859381} a - \frac{11691702}{4476859381}$, $\frac{1}{4476859381} a^{13} - \frac{6318}{4476859381} a^{9} + \frac{2215346624}{4476859381} a^{8} - \frac{303264}{4476859381} a^{7} + \frac{592429487}{4476859381} a^{6} - \frac{5373459}{4476859381} a^{5} - \frac{716741119}{4476859381} a^{4} - \frac{36846576}{4476859381} a^{3} - \frac{171627148}{4476859381} a^{2} - \frac{69087330}{4476859381} a - \frac{2031024790}{4476859381}$, $\frac{1}{4476859381} a^{14} + \frac{2076848225}{4476859381} a^{9} + \frac{265356}{4476859381} a^{8} - \frac{1672222070}{4476859381} a^{7} + \frac{12538071}{4476859381} a^{6} + \frac{813698176}{4476859381} a^{5} + \frac{193444524}{4476859381} a^{4} + \frac{1702187659}{4476859381} a^{3} + \frac{967222620}{4476859381} a^{2} - \frac{1000898254}{4476859381} a + \frac{746143164}{4476859381}$, $\frac{1}{4476859381} a^{15} + \frac{331695}{4476859381} a^{9} - \frac{560468318}{4476859381} a^{8} + \frac{17911530}{4476859381} a^{7} + \frac{19066316}{4476859381} a^{6} + \frac{338527917}{4476859381} a^{5} - \frac{200340262}{4476859381} a^{4} - \frac{2058802831}{4476859381} a^{3} + \frac{1629874554}{4476859381} a^{2} + \frac{186535394}{4476859381} a + \frac{2073230797}{4476859381}$, $\frac{1}{4476859381} a^{16} - \frac{4591442}{4476859381} a^{9} - \frac{11941020}{4476859381} a^{8} + \frac{276499462}{4476859381} a^{7} - \frac{601827408}{4476859381} a^{6} - \frac{2203364082}{4476859381} a^{5} - \frac{718507438}{4476859381} a^{4} - \frac{492926122}{4476859381} a^{3} - \frac{497424409}{4476859381} a^{2} + \frac{1713900229}{4476859381} a + \frac{1119218319}{4476859381}$, $\frac{1}{4476859381} a^{17} - \frac{15615180}{4476859381} a^{9} + \frac{689729242}{4476859381} a^{8} - \frac{899434368}{4476859381} a^{7} + \frac{1859655226}{4476859381} a^{6} + \frac{199823404}{4476859381} a^{5} + \frac{1221337681}{4476859381} a^{4} - \frac{267834751}{4476859381} a^{3} - \frac{1764061110}{4476859381} a^{2} - \frac{947005233}{4476859381} a + \frac{540132215}{4476859381}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{14}\times C_{518}$, which has order $928256$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 22305.8950792 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-703}) \), 3.3.361.1, 6.0.125421842647.1, \(\Q(\zeta_{19})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ R $18$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
19Data not computed
$37$37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$