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Properties

Label 18.0.13408515966...0583.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{15}$
Root discriminant $21.90$
Ramified primes $3, 7$
Class number $1$
Class group Trivial
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![343, 1029, 1323, 980, 525, 378, 498, 588, 450, 140, -108, -126, -6, 42, 9, -7, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 7*x^15 + 9*x^14 + 42*x^13 - 6*x^12 - 126*x^11 - 108*x^10 + 140*x^9 + 450*x^8 + 588*x^7 + 498*x^6 + 378*x^5 + 525*x^4 + 980*x^3 + 1323*x^2 + 1029*x + 343)
 
gp: K = bnfinit(x^18 - 3*x^16 - 7*x^15 + 9*x^14 + 42*x^13 - 6*x^12 - 126*x^11 - 108*x^10 + 140*x^9 + 450*x^8 + 588*x^7 + 498*x^6 + 378*x^5 + 525*x^4 + 980*x^3 + 1323*x^2 + 1029*x + 343, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} - 7 x^{15} + 9 x^{14} + 42 x^{13} - 6 x^{12} - 126 x^{11} - 108 x^{10} + 140 x^{9} + 450 x^{8} + 588 x^{7} + 498 x^{6} + 378 x^{5} + 525 x^{4} + 980 x^{3} + 1323 x^{2} + 1029 x + 343 \)

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:  \(-1340851596668237962730583=-\,3^{24}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.90$
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:  $3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{58} a^{11} - \frac{3}{29} a^{10} - \frac{1}{29} a^{9} + \frac{11}{58} a^{8} - \frac{4}{29} a^{7} - \frac{14}{29} a^{6} + \frac{6}{29} a^{5} + \frac{11}{58} a^{4} - \frac{13}{29} a^{3} + \frac{3}{29} a^{2} - \frac{1}{58} a - \frac{5}{29}$, $\frac{1}{58} a^{12} - \frac{9}{58} a^{10} - \frac{1}{58} a^{9} + \frac{11}{58} a^{7} + \frac{9}{29} a^{6} + \frac{25}{58} a^{5} - \frac{9}{29} a^{4} - \frac{5}{58} a^{3} - \frac{23}{58} a^{2} - \frac{8}{29} a + \frac{27}{58}$, $\frac{1}{58} a^{13} + \frac{3}{58} a^{10} + \frac{11}{58} a^{9} - \frac{3}{29} a^{8} + \frac{2}{29} a^{7} + \frac{5}{58} a^{6} - \frac{13}{29} a^{5} - \frac{11}{29} a^{4} - \frac{25}{58} a^{3} + \frac{9}{58} a^{2} + \frac{9}{29} a + \frac{13}{29}$, $\frac{1}{812} a^{14} + \frac{1}{203} a^{12} - \frac{38}{203} a^{10} + \frac{4}{29} a^{9} - \frac{3}{14} a^{8} - \frac{1}{58} a^{7} - \frac{69}{203} a^{6} - \frac{13}{29} a^{5} + \frac{11}{203} a^{4} - \frac{9}{29} a^{3} + \frac{93}{203} a^{2} - \frac{17}{58} a - \frac{1}{116}$, $\frac{1}{812} a^{15} + \frac{1}{203} a^{13} + \frac{1}{406} a^{11} - \frac{19}{203} a^{9} + \frac{2}{29} a^{8} + \frac{1}{7} a^{7} + \frac{7}{29} a^{6} + \frac{67}{203} a^{5} - \frac{13}{58} a^{4} - \frac{96}{203} a^{3} + \frac{10}{29} a^{2} - \frac{23}{116} a + \frac{3}{29}$, $\frac{1}{73892} a^{16} - \frac{1}{10556} a^{15} + \frac{9}{36946} a^{14} - \frac{41}{5278} a^{13} + \frac{13}{2842} a^{12} - \frac{19}{5278} a^{11} + \frac{4477}{36946} a^{10} - \frac{521}{2639} a^{9} + \frac{1681}{18473} a^{8} - \frac{111}{5278} a^{7} + \frac{11537}{36946} a^{6} + \frac{1077}{5278} a^{5} + \frac{17469}{36946} a^{4} + \frac{723}{5278} a^{3} + \frac{13}{812} a^{2} + \frac{515}{1508} a + \frac{48}{377}$, $\frac{1}{2142868} a^{17} - \frac{3}{535717} a^{16} - \frac{146}{535717} a^{15} - \frac{1119}{2142868} a^{14} + \frac{2076}{535717} a^{13} + \frac{2571}{1071434} a^{12} - \frac{8235}{1071434} a^{11} - \frac{6500}{41209} a^{10} - \frac{14919}{82418} a^{9} + \frac{105275}{535717} a^{8} + \frac{67}{535717} a^{7} - \frac{147832}{535717} a^{6} + \frac{21179}{1071434} a^{5} - \frac{117137}{1071434} a^{4} - \frac{105887}{306124} a^{3} + \frac{1993}{21866} a^{2} + \frac{3600}{10933} a + \frac{15199}{43732}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{32589}{535717} a^{17} + \frac{53511}{1071434} a^{16} + \frac{315727}{2142868} a^{15} + \frac{621603}{2142868} a^{14} - \frac{416109}{535717} a^{13} - \frac{1048345}{535717} a^{12} + \frac{2262231}{1071434} a^{11} + \frac{3163662}{535717} a^{10} + \frac{812215}{535717} a^{9} - \frac{823023}{82418} a^{8} - \frac{1561551}{82418} a^{7} - \frac{802056}{41209} a^{6} - \frac{7308396}{535717} a^{5} - \frac{12237327}{1071434} a^{4} - \frac{1711839}{76531} a^{3} - \frac{6283287}{153062} a^{2} - \frac{2017401}{43732} a - \frac{1014017}{43732} \) (order $14$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 725499.841066 \)
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

\(\Q(\sqrt{-7}) \), 3.1.567.1 x3, \(\Q(\zeta_{7})^+\), 6.0.2250423.3, 6.0.110270727.4 x2, \(\Q(\zeta_{7})\), 9.3.437664515463.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.110270727.4
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$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}$