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Properties

Label 18.18.4567297556...9733.1
Degree $18$
Signature $[18, 0]$
Discriminant $7^{12}\cdot 53^{9}$
Root discriminant $26.64$
Ramified primes $7, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 10, 35, -342, -66, 2861, -2609, -5183, 7364, 2028, -6472, 1242, 2016, -798, -219, 141, -1, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - x^16 + 141*x^15 - 219*x^14 - 798*x^13 + 2016*x^12 + 1242*x^11 - 6472*x^10 + 2028*x^9 + 7364*x^8 - 5183*x^7 - 2609*x^6 + 2861*x^5 - 66*x^4 - 342*x^3 + 35*x^2 + 10*x - 1)
 
gp: K = bnfinit(x^18 - 8*x^17 - x^16 + 141*x^15 - 219*x^14 - 798*x^13 + 2016*x^12 + 1242*x^11 - 6472*x^10 + 2028*x^9 + 7364*x^8 - 5183*x^7 - 2609*x^6 + 2861*x^5 - 66*x^4 - 342*x^3 + 35*x^2 + 10*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} - x^{16} + 141 x^{15} - 219 x^{14} - 798 x^{13} + 2016 x^{12} + 1242 x^{11} - 6472 x^{10} + 2028 x^{9} + 7364 x^{8} - 5183 x^{7} - 2609 x^{6} + 2861 x^{5} - 66 x^{4} - 342 x^{3} + 35 x^{2} + 10 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(45672975569536652017399733=7^{12}\cdot 53^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.64$
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:  $7, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{879} a^{15} - \frac{121}{879} a^{14} + \frac{358}{879} a^{13} + \frac{94}{879} a^{12} - \frac{42}{293} a^{11} - \frac{367}{879} a^{10} - \frac{127}{879} a^{9} + \frac{238}{879} a^{8} - \frac{275}{879} a^{7} - \frac{125}{879} a^{6} - \frac{146}{879} a^{5} - \frac{90}{293} a^{4} + \frac{338}{879} a^{3} + \frac{176}{879} a^{2} + \frac{67}{293} a - \frac{178}{879}$, $\frac{1}{2637} a^{16} + \frac{1}{2637} a^{15} - \frac{47}{2637} a^{14} + \frac{113}{2637} a^{13} - \frac{964}{2637} a^{12} + \frac{376}{2637} a^{11} + \frac{514}{2637} a^{10} - \frac{313}{2637} a^{9} + \frac{340}{2637} a^{8} - \frac{566}{2637} a^{7} + \frac{1012}{2637} a^{6} + \frac{107}{293} a^{5} - \frac{124}{879} a^{4} - \frac{487}{2637} a^{3} - \frac{1}{293} a^{2} - \frac{854}{2637} a + \frac{845}{2637}$, $\frac{1}{349302644721} a^{17} - \frac{12558044}{349302644721} a^{16} - \frac{145809296}{349302644721} a^{15} - \frac{26213133067}{349302644721} a^{14} - \frac{129468013330}{349302644721} a^{13} - \frac{147145878911}{349302644721} a^{12} + \frac{125821304923}{349302644721} a^{11} - \frac{134846312755}{349302644721} a^{10} - \frac{90820306562}{349302644721} a^{9} + \frac{150472048723}{349302644721} a^{8} + \frac{174202739866}{349302644721} a^{7} + \frac{55224318194}{116434214907} a^{6} + \frac{28124147144}{116434214907} a^{5} - \frac{41057063404}{349302644721} a^{4} + \frac{57083000902}{116434214907} a^{3} + \frac{106611416344}{349302644721} a^{2} - \frac{47199582475}{349302644721} a - \frac{9435969875}{38811404969}$

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:  $17$
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:  \( 6981069.98989 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{53}) \), 3.3.2597.1 x3, \(\Q(\zeta_{7})^+\), 6.6.357453677.2, 6.6.7294973.1 x2, 6.6.357453677.1, 9.9.17515230173.1 x3

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

Sibling fields

Degree 6 sibling: 6.6.7294973.1
Degree 9 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$53$53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$