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Properties

Label 18.0.31148751066...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{8}\cdot 5^{12}\cdot 7^{15}$
Root discriminant $38.28$
Ramified primes $2, 3, 5, 7$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_2\times C_3:S_3$ (as 18T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1408, -7584, 14984, -11598, 60, 1746, 6185, -7578, 2597, 180, 14, -438, 485, -336, 150, -48, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 17*x^16 - 48*x^15 + 150*x^14 - 336*x^13 + 485*x^12 - 438*x^11 + 14*x^10 + 180*x^9 + 2597*x^8 - 7578*x^7 + 6185*x^6 + 1746*x^5 + 60*x^4 - 11598*x^3 + 14984*x^2 - 7584*x + 1408)
 
gp: K = bnfinit(x^18 - 6*x^17 + 17*x^16 - 48*x^15 + 150*x^14 - 336*x^13 + 485*x^12 - 438*x^11 + 14*x^10 + 180*x^9 + 2597*x^8 - 7578*x^7 + 6185*x^6 + 1746*x^5 + 60*x^4 - 11598*x^3 + 14984*x^2 - 7584*x + 1408, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 17 x^{16} - 48 x^{15} + 150 x^{14} - 336 x^{13} + 485 x^{12} - 438 x^{11} + 14 x^{10} + 180 x^{9} + 2597 x^{8} - 7578 x^{7} + 6185 x^{6} + 1746 x^{5} + 60 x^{4} - 11598 x^{3} + 14984 x^{2} - 7584 x + 1408 \)

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:  \(-31148751066736023000000000000=-\,2^{12}\cdot 3^{8}\cdot 5^{12}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.28$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{3} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{4} a^{8} + \frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{10} - \frac{1}{6} a^{8} - \frac{5}{12} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3}$, $\frac{1}{204} a^{15} - \frac{5}{204} a^{14} - \frac{1}{34} a^{13} + \frac{1}{34} a^{12} + \frac{23}{204} a^{11} + \frac{7}{102} a^{10} + \frac{7}{204} a^{9} - \frac{49}{102} a^{8} - \frac{43}{102} a^{7} - \frac{35}{204} a^{6} + \frac{29}{204} a^{5} - \frac{55}{204} a^{4} + \frac{8}{17} a^{3} - \frac{13}{34} a^{2} - \frac{31}{102} a - \frac{20}{51}$, $\frac{1}{2448} a^{16} - \frac{1}{408} a^{15} - \frac{35}{2448} a^{14} + \frac{1}{204} a^{13} - \frac{1}{72} a^{12} - \frac{5}{204} a^{11} + \frac{43}{816} a^{10} - \frac{3}{136} a^{9} + \frac{193}{1224} a^{8} - \frac{1}{2} a^{7} + \frac{95}{816} a^{6} + \frac{1}{136} a^{5} - \frac{19}{2448} a^{4} - \frac{55}{136} a^{3} - \frac{47}{612} a^{2} + \frac{167}{408} a - \frac{109}{306}$, $\frac{1}{7003593185808140067456} a^{17} + \frac{149245888230187561}{1167265530968023344576} a^{16} + \frac{8898762477955793}{30058339853253820032} a^{15} - \frac{6576557705968043973}{194544255161337224096} a^{14} + \frac{42686451786987106147}{3501796592904070033728} a^{13} + \frac{11092105905335662073}{291816382742005836144} a^{12} - \frac{5439783941887512755}{45775118861491111552} a^{11} + \frac{67297944080471109113}{1167265530968023344576} a^{10} - \frac{763682901196573544597}{3501796592904070033728} a^{9} + \frac{51585498542015631135}{194544255161337224096} a^{8} + \frac{61440938172044435085}{778177020645348896384} a^{7} - \frac{204519568525574379725}{1167265530968023344576} a^{6} - \frac{41632751718149292607}{7003593185808140067456} a^{5} + \frac{296155635020613706733}{1167265530968023344576} a^{4} - \frac{232346240369880762719}{1750898296452035016864} a^{3} + \frac{420281826681310661243}{1167265530968023344576} a^{2} + \frac{7294997140661869814}{54715571764126094277} a + \frac{4234440041846084811}{24318031895167153012}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( 6910271.661602281 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3$ (as 18T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_2\times C_3:S_3$
Character table for $C_2\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.1.14700.1, 3.1.3675.1, 3.1.300.1, 3.1.588.1, 6.0.94539375.1, 6.0.1512630000.1, 6.0.30870000.1, 6.0.2420208.1, 9.1.9529569000000.1

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

Sibling fields

Galois closure: data not computed
Degree 18 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 R R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{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$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
7Data not computed