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Properties

Label 18.18.7072037454...8048.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{24}\cdot 7^{8}\cdot 13^{9}$
Root discriminant $58.81$
Ramified primes $2, 3, 7, 13$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $He_3:C_2$ (as 18T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-377, 1794, 1836, -17236, 8760, 47538, -53993, -28836, 63708, -9916, -24564, 10464, 3125, -2394, 12, 208, -24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 24*x^16 + 208*x^15 + 12*x^14 - 2394*x^13 + 3125*x^12 + 10464*x^11 - 24564*x^10 - 9916*x^9 + 63708*x^8 - 28836*x^7 - 53993*x^6 + 47538*x^5 + 8760*x^4 - 17236*x^3 + 1836*x^2 + 1794*x - 377)
 
gp: K = bnfinit(x^18 - 6*x^17 - 24*x^16 + 208*x^15 + 12*x^14 - 2394*x^13 + 3125*x^12 + 10464*x^11 - 24564*x^10 - 9916*x^9 + 63708*x^8 - 28836*x^7 - 53993*x^6 + 47538*x^5 + 8760*x^4 - 17236*x^3 + 1836*x^2 + 1794*x - 377, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 24 x^{16} + 208 x^{15} + 12 x^{14} - 2394 x^{13} + 3125 x^{12} + 10464 x^{11} - 24564 x^{10} - 9916 x^{9} + 63708 x^{8} - 28836 x^{7} - 53993 x^{6} + 47538 x^{5} + 8760 x^{4} - 17236 x^{3} + 1836 x^{2} + 1794 x - 377 \)

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:  \(70720374548224359605005470978048=2^{12}\cdot 3^{24}\cdot 7^{8}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.81$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{3} a$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{12} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{5}{12} a^{2} + \frac{5}{12} a + \frac{5}{12}$, $\frac{1}{36} a^{15} - \frac{1}{12} a^{12} - \frac{1}{6} a^{11} - \frac{1}{9} a^{9} + \frac{1}{6} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{7}{36} a^{3} + \frac{1}{6} a^{2} - \frac{11}{36}$, $\frac{1}{36} a^{16} - \frac{1}{12} a^{13} - \frac{1}{9} a^{10} - \frac{1}{2} a^{7} + \frac{7}{36} a^{4} - \frac{1}{3} a^{3} - \frac{11}{36} a - \frac{1}{3}$, $\frac{1}{140694479944385580} a^{17} - \frac{1926399192169}{9379631996292372} a^{16} - \frac{1269323693212649}{140694479944385580} a^{15} - \frac{664992186716177}{46898159981461860} a^{14} - \frac{534954144221811}{15632719993820620} a^{13} + \frac{3241887869677769}{46898159981461860} a^{12} - \frac{3812124142353487}{35173619986096395} a^{11} - \frac{1648884417721739}{23449079990730930} a^{10} - \frac{4227911119651747}{35173619986096395} a^{9} + \frac{10904077949417491}{23449079990730930} a^{8} + \frac{1522515853036389}{3908179998455155} a^{7} + \frac{1355951312758431}{7816359996910310} a^{6} - \frac{4828423574431381}{28138895988877116} a^{5} - \frac{5128885894849463}{15632719993820620} a^{4} + \frac{1179768124935533}{3431572681570380} a^{3} + \frac{53574426735627817}{140694479944385580} a^{2} + \frac{6291624911931227}{15632719993820620} a - \frac{33759582879122153}{140694479944385580}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 11723192956.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$He_3:C_2$ (as 18T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $He_3:C_2$
Character table for $He_3:C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.4212.1 x3, 6.6.230632272.1, 9.9.2332386934323264.1 x3

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

Sibling fields

Degree 9 siblings: data not computed
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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$