Properties

Label 18.0.435...963.1
Degree $18$
Signature $[0, 9]$
Discriminant $-4.356\times 10^{19}$
Root discriminant $12.33$
Ramified primes $3, 19$
Class number $1$
Class group trivial
Galois group $S_3 \times C_3$ (as 18T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 4*x^16 - x^15 + 12*x^14 - 9*x^13 + 12*x^12 - 18*x^11 + 26*x^10 - 12*x^9 - 4*x^8 - 5*x^7 + 31*x^6 - 17*x^5 - 3*x^4 + 5*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^18 - x^17 + 4*x^16 - x^15 + 12*x^14 - 9*x^13 + 12*x^12 - 18*x^11 + 26*x^10 - 12*x^9 - 4*x^8 - 5*x^7 + 31*x^6 - 17*x^5 - 3*x^4 + 5*x^2 - 3*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 5, 0, -3, -17, 31, -5, -4, -12, 26, -18, 12, -9, 12, -1, 4, -1, 1]);
 

\( x^{18} - x^{17} + 4 x^{16} - x^{15} + 12 x^{14} - 9 x^{13} + 12 x^{12} - 18 x^{11} + 26 x^{10} - 12 x^{9} - 4 x^{8} - 5 x^{7} + 31 x^{6} - 17 x^{5} - 3 x^{4} + 5 x^{2} - 3 x + 1 \)

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-43564677551979246963\)\(\medspace = -\,3^{9}\cdot 19^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $12.33$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 19$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $18$
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{78} a^{15} - \frac{3}{26} a^{14} + \frac{3}{26} a^{13} - \frac{1}{78} a^{12} + \frac{6}{13} a^{11} - \frac{17}{78} a^{10} + \frac{35}{78} a^{9} + \frac{1}{26} a^{8} + \frac{1}{39} a^{7} + \frac{5}{39} a^{6} + \frac{35}{78} a^{5} + \frac{11}{78} a^{4} - \frac{8}{39} a^{3} - \frac{23}{78} a^{2} + \frac{7}{39} a - \frac{7}{78}$, $\frac{1}{858} a^{16} - \frac{2}{429} a^{15} - \frac{19}{143} a^{14} - \frac{4}{429} a^{13} + \frac{83}{858} a^{12} + \frac{397}{858} a^{11} + \frac{9}{143} a^{10} - \frac{70}{143} a^{9} - \frac{133}{286} a^{8} - \frac{185}{429} a^{7} - \frac{23}{78} a^{6} - \frac{37}{429} a^{5} - \frac{1}{2} a^{4} - \frac{11}{26} a^{3} + \frac{107}{858} a^{2} - \frac{145}{858} a - \frac{347}{858}$, $\frac{1}{1570998} a^{17} - \frac{7}{40282} a^{16} - \frac{8135}{1570998} a^{15} - \frac{156595}{1570998} a^{14} - \frac{2286}{20141} a^{13} + \frac{256579}{1570998} a^{12} - \frac{616127}{1570998} a^{11} + \frac{11575}{1570998} a^{10} + \frac{289771}{785499} a^{9} + \frac{177115}{785499} a^{8} + \frac{56867}{523666} a^{7} - \frac{122825}{523666} a^{6} - \frac{29609}{261833} a^{5} - \frac{69749}{142818} a^{4} - \frac{177217}{785499} a^{3} - \frac{170047}{1570998} a^{2} - \frac{254560}{785499} a + \frac{9201}{261833}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{16945}{142818} a^{17} - \frac{23327}{71409} a^{16} + \frac{36250}{71409} a^{15} - \frac{21461}{23803} a^{14} + \frac{47967}{47606} a^{13} - \frac{549617}{142818} a^{12} + \frac{26899}{23803} a^{11} - \frac{324175}{71409} a^{10} + \frac{753937}{142818} a^{9} - \frac{346456}{71409} a^{8} - \frac{106321}{142818} a^{7} - \frac{17395}{71409} a^{6} + \frac{814865}{142818} a^{5} - \frac{345281}{47606} a^{4} - \frac{194975}{142818} a^{3} - \frac{34673}{142818} a^{2} + \frac{95847}{47606} a - \frac{6029}{23803} \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 527.323608131 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{9}\cdot 527.323608131 \cdot 1}{6\sqrt{43564677551979246963}}\approx 0.203225185150$

Galois group

$C_3\times S_3$ (as 18T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-3}) \), 3.1.1083.1 x3, 3.3.361.1, 6.0.3518667.2, 6.0.9747.1 x2, 6.0.3518667.1, 9.3.1270238787.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.9747.1
Degree 9 sibling: 9.3.1270238787.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$