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Properties

Label 18.0.75212941670...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 5^{9}\cdot 7^{9}\cdot 13^{12}$
Root discriminant $51.92$
Ramified primes $2, 5, 7, 13$
Class number $1176$ (GRH)
Class group $[2, 14, 42]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![52025, -217450, 416495, -344955, -20211, 233575, -119189, -31209, 47263, -10508, -5496, 3323, 87, -449, 39, 46, -9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 9*x^16 + 46*x^15 + 39*x^14 - 449*x^13 + 87*x^12 + 3323*x^11 - 5496*x^10 - 10508*x^9 + 47263*x^8 - 31209*x^7 - 119189*x^6 + 233575*x^5 - 20211*x^4 - 344955*x^3 + 416495*x^2 - 217450*x + 52025)
 
gp: K = bnfinit(x^18 - 3*x^17 - 9*x^16 + 46*x^15 + 39*x^14 - 449*x^13 + 87*x^12 + 3323*x^11 - 5496*x^10 - 10508*x^9 + 47263*x^8 - 31209*x^7 - 119189*x^6 + 233575*x^5 - 20211*x^4 - 344955*x^3 + 416495*x^2 - 217450*x + 52025, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 9 x^{16} + 46 x^{15} + 39 x^{14} - 449 x^{13} + 87 x^{12} + 3323 x^{11} - 5496 x^{10} - 10508 x^{9} + 47263 x^{8} - 31209 x^{7} - 119189 x^{6} + 233575 x^{5} - 20211 x^{4} - 344955 x^{3} + 416495 x^{2} - 217450 x + 52025 \)

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:  \(-7521294167081161111736000000000=-\,2^{12}\cdot 5^{9}\cdot 7^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.92$
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:  $2, 5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{25} a^{12} - \frac{1}{25} a^{11} - \frac{2}{5} a^{9} - \frac{2}{25} a^{8} + \frac{2}{25} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{25} a^{4} - \frac{1}{25} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{75} a^{13} - \frac{2}{25} a^{11} - \frac{1}{15} a^{10} - \frac{37}{75} a^{9} - \frac{1}{25} a^{7} + \frac{2}{15} a^{6} + \frac{12}{25} a^{5} - \frac{1}{3} a^{4} - \frac{26}{75} a^{3} + \frac{4}{15} a^{2} - \frac{1}{5} a - \frac{1}{3}$, $\frac{1}{75} a^{14} + \frac{4}{75} a^{11} - \frac{7}{75} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{8}{75} a^{7} + \frac{12}{25} a^{6} + \frac{7}{15} a^{5} - \frac{4}{15} a^{4} + \frac{29}{75} a^{3} - \frac{2}{5} a^{2} + \frac{7}{15} a$, $\frac{1}{75} a^{15} + \frac{1}{75} a^{12} - \frac{4}{75} a^{11} + \frac{1}{5} a^{9} - \frac{2}{75} a^{8} + \frac{2}{5} a^{7} + \frac{7}{15} a^{6} + \frac{1}{3} a^{5} + \frac{26}{75} a^{4} - \frac{9}{25} a^{3} + \frac{1}{15} a^{2} - \frac{2}{5} a$, $\frac{1}{3240189098625} a^{16} - \frac{7134607532}{1080063032875} a^{15} - \frac{21448182242}{3240189098625} a^{14} - \frac{2667742584}{1080063032875} a^{13} - \frac{64379569438}{3240189098625} a^{12} - \frac{12969884393}{3240189098625} a^{11} - \frac{264579266456}{3240189098625} a^{10} - \frac{262332140206}{3240189098625} a^{9} - \frac{16348767224}{1080063032875} a^{8} - \frac{11600114707}{1080063032875} a^{7} + \frac{415624086851}{1080063032875} a^{6} - \frac{684824678207}{3240189098625} a^{5} + \frac{1086726363554}{3240189098625} a^{4} - \frac{30318216158}{216012606575} a^{3} + \frac{10852178504}{648037819725} a^{2} + \frac{1365889196}{129607563945} a + \frac{6526672906}{129607563945}$, $\frac{1}{13133807101602779056875} a^{17} - \frac{670614737}{13133807101602779056875} a^{16} - \frac{74171691431008115141}{13133807101602779056875} a^{15} - \frac{12685452239415135029}{2626761420320555811375} a^{14} - \frac{19195870622630907376}{13133807101602779056875} a^{13} + \frac{17135284874437906726}{875587140106851937125} a^{12} - \frac{364362958841133510101}{4377935700534259685625} a^{11} - \frac{15259215289239536411}{2626761420320555811375} a^{10} - \frac{6547456612890805992586}{13133807101602779056875} a^{9} - \frac{591610212338378167841}{1459311900178086561875} a^{8} - \frac{1659557004480924468011}{13133807101602779056875} a^{7} + \frac{400426657899267536081}{2626761420320555811375} a^{6} + \frac{5620493510021414624396}{13133807101602779056875} a^{5} - \frac{1935817807655083171303}{4377935700534259685625} a^{4} + \frac{180174939297732002239}{875587140106851937125} a^{3} + \frac{13684753960020337877}{875587140106851937125} a^{2} + \frac{207047811752924169662}{525352284064111162275} a + \frac{73678903930593000124}{525352284064111162275}$

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

Class group and class number

$C_{2}\times C_{14}\times C_{42}$, which has order $1176$ (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:  \( 502394.782618 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

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

\(\Q(\sqrt{-35}) \), 3.1.140.1 x3, 3.3.169.1, 6.0.686000.1, 6.0.1224552875.2, 6.0.19592846000.3 x2, 9.3.13244763896000.4 x3

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

Sibling fields

Degree 6 sibling: data not computed
Degree 9 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$