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Properties

Label 18.0.20836299215...5488.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 23^{9}$
Root discriminant $32.94$
Ramified primes $2, 3, 23$
Class number $14$
Class group $[14]$
Galois group $C_3:S_3:S_4$ (as 18T155)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10104, 11898, 19827, -1689, 9138, -9465, 9585, -8646, 6096, -5050, 3306, -2142, 1077, -471, 195, -54, 21, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 21*x^16 - 54*x^15 + 195*x^14 - 471*x^13 + 1077*x^12 - 2142*x^11 + 3306*x^10 - 5050*x^9 + 6096*x^8 - 8646*x^7 + 9585*x^6 - 9465*x^5 + 9138*x^4 - 1689*x^3 + 19827*x^2 + 11898*x + 10104)
 
gp: K = bnfinit(x^18 - 3*x^17 + 21*x^16 - 54*x^15 + 195*x^14 - 471*x^13 + 1077*x^12 - 2142*x^11 + 3306*x^10 - 5050*x^9 + 6096*x^8 - 8646*x^7 + 9585*x^6 - 9465*x^5 + 9138*x^4 - 1689*x^3 + 19827*x^2 + 11898*x + 10104, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 21 x^{16} - 54 x^{15} + 195 x^{14} - 471 x^{13} + 1077 x^{12} - 2142 x^{11} + 3306 x^{10} - 5050 x^{9} + 6096 x^{8} - 8646 x^{7} + 9585 x^{6} - 9465 x^{5} + 9138 x^{4} - 1689 x^{3} + 19827 x^{2} + 11898 x + 10104 \)

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:  \(-2083629921519675807054655488=-\,2^{12}\cdot 3^{24}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.94$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4283395454547680616343213947677688892} a^{17} + \frac{287577816960569872341330591611433457}{2141697727273840308171606973838844446} a^{16} - \frac{2072969534805385100170636023189394805}{4283395454547680616343213947677688892} a^{15} - \frac{1848185468169169335882860563273847443}{4283395454547680616343213947677688892} a^{14} - \frac{379892419790546185574897332605385153}{1070848863636920154085803486919422223} a^{13} + \frac{2051136251543440271563384729359369657}{4283395454547680616343213947677688892} a^{12} - \frac{4883710712915441527011036010421119}{36299961479217632341891643624387194} a^{11} - \frac{351981023625352852251647678756411075}{1070848863636920154085803486919422223} a^{10} + \frac{52524974667430168371019947875416797}{2141697727273840308171606973838844446} a^{9} - \frac{138238114359975967666244005803431669}{1070848863636920154085803486919422223} a^{8} - \frac{504056960636562406903950602593764982}{1070848863636920154085803486919422223} a^{7} + \frac{660148161246374462445380804607107605}{2141697727273840308171606973838844446} a^{6} - \frac{2055215344740168193259524310908841765}{4283395454547680616343213947677688892} a^{5} - \frac{272593066021105788331347223210432185}{2141697727273840308171606973838844446} a^{4} + \frac{372156150133519975736355881596310240}{1070848863636920154085803486919422223} a^{3} - \frac{946564190310012873898229837002289837}{4283395454547680616343213947677688892} a^{2} - \frac{11319194468119859468130131613509549}{36299961479217632341891643624387194} a - \frac{310111471409599065288292131026603111}{1070848863636920154085803486919422223}$

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

Class group and class number

$C_{14}$, which has order $14$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1089567.33033 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T155):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.3.621.1, 6.0.141915888.3, 9.3.6466042647.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.9.12.22$x^{9} + 6 x^{4} + 6 x^{3} + 3$$9$$1$$12$$C_3^2 : C_6$$[3/2, 3/2]_{2}^{3}$
3.9.12.22$x^{9} + 6 x^{4} + 6 x^{3} + 3$$9$$1$$12$$C_3^2 : C_6$$[3/2, 3/2]_{2}^{3}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.8.6.2$x^{8} - 1633 x^{4} + 1270129$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$