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Properties

Label 18.12.3772620972...3923.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,3^{18}\cdot 7^{15}\cdot 29^{5}$
Root discriminant $38.69$
Ramified primes $3, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T282

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6119, -696, -12528, -20764, 435, 47241, 23847, -37752, -24876, 15071, 11238, -3690, -2937, 549, 477, -26, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 - 26*x^15 + 477*x^14 + 549*x^13 - 2937*x^12 - 3690*x^11 + 11238*x^10 + 15071*x^9 - 24876*x^8 - 37752*x^7 + 23847*x^6 + 47241*x^5 + 435*x^4 - 20764*x^3 - 12528*x^2 - 696*x + 6119)
 
gp: K = bnfinit(x^18 - 36*x^16 - 26*x^15 + 477*x^14 + 549*x^13 - 2937*x^12 - 3690*x^11 + 11238*x^10 + 15071*x^9 - 24876*x^8 - 37752*x^7 + 23847*x^6 + 47241*x^5 + 435*x^4 - 20764*x^3 - 12528*x^2 - 696*x + 6119, 1)
 

Normalized defining polynomial

\( x^{18} - 36 x^{16} - 26 x^{15} + 477 x^{14} + 549 x^{13} - 2937 x^{12} - 3690 x^{11} + 11238 x^{10} + 15071 x^{9} - 24876 x^{8} - 37752 x^{7} + 23847 x^{6} + 47241 x^{5} + 435 x^{4} - 20764 x^{3} - 12528 x^{2} - 696 x + 6119 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-37726209720370552017864793923=-\,3^{18}\cdot 7^{15}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.69$
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:  $3, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{9} + \frac{2}{7} a^{8} + \frac{1}{7} a^{6} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{11} + \frac{2}{7} a^{10} + \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{14} + \frac{3}{7} a^{7} + \frac{3}{7}$, $\frac{1}{49} a^{15} + \frac{2}{49} a^{14} - \frac{3}{49} a^{13} - \frac{3}{49} a^{12} - \frac{17}{49} a^{11} - \frac{24}{49} a^{10} - \frac{10}{49} a^{9} + \frac{1}{7} a^{8} - \frac{4}{49} a^{7} + \frac{23}{49} a^{6} - \frac{12}{49} a^{5} + \frac{2}{49} a^{4} + \frac{2}{49} a^{3} + \frac{23}{49} a^{2} - \frac{9}{49} a + \frac{15}{49}$, $\frac{1}{637} a^{16} + \frac{5}{637} a^{15} + \frac{17}{637} a^{14} + \frac{30}{637} a^{13} + \frac{37}{637} a^{12} - \frac{194}{637} a^{11} + \frac{184}{637} a^{10} + \frac{222}{637} a^{9} - \frac{193}{637} a^{8} - \frac{101}{637} a^{7} + \frac{246}{637} a^{6} + \frac{183}{637} a^{5} - \frac{83}{637} a^{4} + \frac{288}{637} a^{3} - \frac{150}{637} a^{2} + \frac{135}{637} a + \frac{80}{637}$, $\frac{1}{54767852381818486001} a^{17} - \frac{38052350708198017}{54767852381818486001} a^{16} + \frac{97129033098370636}{54767852381818486001} a^{15} + \frac{201541647161859416}{4212911721678345077} a^{14} + \frac{52280066310616601}{1117711273098336449} a^{13} + \frac{553178108085503192}{7823978911688355143} a^{12} + \frac{2444884039222116260}{7823978911688355143} a^{11} + \frac{5245547083933040377}{54767852381818486001} a^{10} - \frac{66456851855039190}{54767852381818486001} a^{9} + \frac{15897392994303861912}{54767852381818486001} a^{8} + \frac{18295315454785408126}{54767852381818486001} a^{7} + \frac{1982004899017463077}{7823978911688355143} a^{6} - \frac{2371849003438427616}{7823978911688355143} a^{5} - \frac{3864865692403484976}{7823978911688355143} a^{4} + \frac{4723807986046339480}{54767852381818486001} a^{3} - \frac{359411411602677552}{54767852381818486001} a^{2} + \frac{12294134979868626791}{54767852381818486001} a - \frac{25390349300180441098}{54767852381818486001}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 134045160.39 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T282:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 56 conjugacy class representatives for t18n282 are not computed
Character table for t18n282 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.487403.1, 9.9.13632439166829.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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
$3$3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.6.5.2$x^{6} + 58$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$