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Properties

Label 18.18.5702674039...2992.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 3^{9}\cdot 7^{12}\cdot 41^{8}$
Root discriminant $66.04$
Ramified primes $2, 3, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $He_3:C_2$ (as 18T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -397, -2204, 9156, 57892, -20421, -192314, 176585, 51052, -112571, 28718, 13879, -7390, 154, 456, -68, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 68*x^16 + 456*x^15 + 154*x^14 - 7390*x^13 + 13879*x^12 + 28718*x^11 - 112571*x^10 + 51052*x^9 + 176585*x^8 - 192314*x^7 - 20421*x^6 + 57892*x^5 + 9156*x^4 - 2204*x^3 - 397*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 - 68*x^16 + 456*x^15 + 154*x^14 - 7390*x^13 + 13879*x^12 + 28718*x^11 - 112571*x^10 + 51052*x^9 + 176585*x^8 - 192314*x^7 - 20421*x^6 + 57892*x^5 + 9156*x^4 - 2204*x^3 - 397*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 68 x^{16} + 456 x^{15} + 154 x^{14} - 7390 x^{13} + 13879 x^{12} + 28718 x^{11} - 112571 x^{10} + 51052 x^{9} + 176585 x^{8} - 192314 x^{7} - 20421 x^{6} + 57892 x^{5} + 9156 x^{4} - 2204 x^{3} - 397 x^{2} - 4 x + 1 \)

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:  \(570267403955912384799325092052992=2^{18}\cdot 3^{9}\cdot 7^{12}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.04$
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:  $2, 3, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{2}$, $\frac{1}{17438593252439936662339279706} a^{17} - \frac{519376253679179259224170405}{8719296626219968331169639853} a^{16} - \frac{1307549066734168906947891726}{8719296626219968331169639853} a^{15} - \frac{3075675113738351021464079223}{17438593252439936662339279706} a^{14} + \frac{1993877773428837006656276454}{8719296626219968331169639853} a^{13} - \frac{3598800185939366744888000905}{17438593252439936662339279706} a^{12} - \frac{2044717682674380819462265718}{8719296626219968331169639853} a^{11} + \frac{2398319041430967597255646723}{17438593252439936662339279706} a^{10} + \frac{3261629564357567524186088780}{8719296626219968331169639853} a^{9} - \frac{2215834825450822659703818943}{17438593252439936662339279706} a^{8} - \frac{3540984576739641825815366291}{8719296626219968331169639853} a^{7} - \frac{4263938623541435838518576653}{17438593252439936662339279706} a^{6} - \frac{1638966461696630950049071824}{8719296626219968331169639853} a^{5} - \frac{715563810560368958586040235}{17438593252439936662339279706} a^{4} + \frac{2669916647624492180581887859}{17438593252439936662339279706} a^{3} + \frac{6546721230677430677684326753}{17438593252439936662339279706} a^{2} + \frac{776918958080644836707895015}{8719296626219968331169639853} a - \frac{583384966257250973045748584}{8719296626219968331169639853}$

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:  \( 67003902454.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$He_3:C_2$ (as 18T20):

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

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{7})^+\), 6.6.4148928.1, 9.9.574470067776192.1 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$