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Properties

Label 18.12.5354380198...1712.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{8}\cdot 3^{17}\cdot 503^{6}$
Root discriminant $30.54$
Ramified primes $2, 3, 503$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times (C_3\times A_4):S_3$ (as 18T156)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 17, 16, 100, -200, -1317, 282, 2977, -406, -2812, 374, 1431, -219, -424, 91, 61, -17, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 17*x^16 + 61*x^15 + 91*x^14 - 424*x^13 - 219*x^12 + 1431*x^11 + 374*x^10 - 2812*x^9 - 406*x^8 + 2977*x^7 + 282*x^6 - 1317*x^5 - 200*x^4 + 100*x^3 + 16*x^2 + 17*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - 17*x^16 + 61*x^15 + 91*x^14 - 424*x^13 - 219*x^12 + 1431*x^11 + 374*x^10 - 2812*x^9 - 406*x^8 + 2977*x^7 + 282*x^6 - 1317*x^5 - 200*x^4 + 100*x^3 + 16*x^2 + 17*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 17 x^{16} + 61 x^{15} + 91 x^{14} - 424 x^{13} - 219 x^{12} + 1431 x^{11} + 374 x^{10} - 2812 x^{9} - 406 x^{8} + 2977 x^{7} + 282 x^{6} - 1317 x^{5} - 200 x^{4} + 100 x^{3} + 16 x^{2} + 17 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-535438019832160469313491712=-\,2^{8}\cdot 3^{17}\cdot 503^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.54$
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:  $2, 3, 503$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{154} a^{16} - \frac{9}{77} a^{15} + \frac{13}{154} a^{14} - \frac{7}{22} a^{13} + \frac{8}{77} a^{12} - \frac{69}{154} a^{11} - \frac{3}{22} a^{10} - \frac{10}{77} a^{9} - \frac{61}{154} a^{8} + \frac{27}{77} a^{7} + \frac{13}{77} a^{6} + \frac{1}{7} a^{5} + \frac{13}{77} a^{4} - \frac{57}{154} a^{3} - \frac{41}{154} a^{2} + \frac{73}{154} a + \frac{30}{77}$, $\frac{1}{202923796196506} a^{17} + \frac{45358171797}{28989113742358} a^{16} + \frac{17105007322804}{101461898098253} a^{15} + \frac{43978015592285}{101461898098253} a^{14} - \frac{4826316469092}{101461898098253} a^{13} - \frac{4328129817255}{101461898098253} a^{12} - \frac{24258872764761}{101461898098253} a^{11} - \frac{25479267066977}{101461898098253} a^{10} - \frac{14365976576326}{101461898098253} a^{9} + \frac{25893603747647}{202923796196506} a^{8} + \frac{32230911447781}{202923796196506} a^{7} + \frac{4758394144074}{14494556871179} a^{6} - \frac{30515884801653}{101461898098253} a^{5} - \frac{51530239720437}{202923796196506} a^{4} - \frac{19707638595090}{101461898098253} a^{3} - \frac{12913248241207}{28989113742358} a^{2} - \frac{4492477210283}{101461898098253} a - \frac{69978088029015}{202923796196506}$

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:  \( 11210231.304 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times (C_3\times A_4):S_3$ (as 18T156):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 38 conjugacy class representatives for $C_2\times (C_3\times A_4):S_3$
Character table for $C_2\times (C_3\times A_4):S_3$ is not computed

Intermediate fields

3.3.1509.1, 6.4.6831243.1, 9.9.4453205336784.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.8.2$x^{12} - 8 x^{3} + 16$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.14.15$x^{12} + 9 x^{11} - 6 x^{10} + 6 x^{9} - 3 x^{8} + 9 x^{7} + 6 x^{6} - 9 x^{5} - 9 x^{4} - 9 x^{3} - 9 x^{2} + 9$$6$$2$$14$$C_6\times S_3$$[3/2]_{2}^{6}$
503Data not computed