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Properties

Label 18.0.18294428062...7584.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 7^{12}\cdot 19^{9}$
Root discriminant $25.32$
Ramified primes $2, 7, 19$
Class number $3$
Class group $[3]$
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![1, -10, 35, 20, -100, -257, 704, -100, -300, -845, 1059, 484, -387, -25, 85, -19, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 2*x^16 - 19*x^15 + 85*x^14 - 25*x^13 - 387*x^12 + 484*x^11 + 1059*x^10 - 845*x^9 - 300*x^8 - 100*x^7 + 704*x^6 - 257*x^5 - 100*x^4 + 20*x^3 + 35*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^18 - x^17 - 2*x^16 - 19*x^15 + 85*x^14 - 25*x^13 - 387*x^12 + 484*x^11 + 1059*x^10 - 845*x^9 - 300*x^8 - 100*x^7 + 704*x^6 - 257*x^5 - 100*x^4 + 20*x^3 + 35*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 2 x^{16} - 19 x^{15} + 85 x^{14} - 25 x^{13} - 387 x^{12} + 484 x^{11} + 1059 x^{10} - 845 x^{9} - 300 x^{8} - 100 x^{7} + 704 x^{6} - 257 x^{5} - 100 x^{4} + 20 x^{3} + 35 x^{2} - 10 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-18294428062468623673667584=-\,2^{12}\cdot 7^{12}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.32$
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:  $2, 7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{11} - \frac{1}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{2}{7} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} + \frac{2}{7} a^{11} + \frac{2}{7} a^{10} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{2}{7}$, $\frac{1}{7} a^{14} - \frac{1}{7} a^{11} + \frac{2}{7} a^{10} - \frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{4} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{245} a^{15} + \frac{2}{245} a^{14} + \frac{3}{49} a^{13} + \frac{2}{35} a^{12} - \frac{3}{35} a^{11} + \frac{46}{245} a^{10} + \frac{24}{245} a^{9} + \frac{61}{245} a^{8} - \frac{58}{245} a^{7} + \frac{61}{245} a^{6} + \frac{111}{245} a^{5} - \frac{64}{245} a^{4} - \frac{96}{245} a^{3} + \frac{122}{245} a^{2} - \frac{12}{35} a - \frac{36}{245}$, $\frac{1}{10535} a^{16} - \frac{4}{2107} a^{15} - \frac{414}{10535} a^{14} + \frac{209}{10535} a^{13} + \frac{18}{1505} a^{12} - \frac{2397}{10535} a^{11} + \frac{4962}{10535} a^{10} - \frac{467}{10535} a^{9} + \frac{76}{301} a^{8} + \frac{691}{1505} a^{7} + \frac{1989}{10535} a^{6} + \frac{682}{1505} a^{5} - \frac{158}{10535} a^{4} - \frac{4626}{10535} a^{3} - \frac{3188}{10535} a^{2} - \frac{4103}{10535} a - \frac{2568}{10535}$, $\frac{1}{676519557436503445} a^{17} + \frac{8942239959764}{676519557436503445} a^{16} - \frac{490793146655324}{676519557436503445} a^{15} - \frac{45537615661689967}{676519557436503445} a^{14} - \frac{31473801622768523}{676519557436503445} a^{13} - \frac{46981982465730238}{676519557436503445} a^{12} + \frac{321698457300654124}{676519557436503445} a^{11} - \frac{3180426493757239}{676519557436503445} a^{10} - \frac{117429492737356598}{676519557436503445} a^{9} - \frac{138794772361708493}{676519557436503445} a^{8} - \frac{129116072791368153}{676519557436503445} a^{7} - \frac{41994335786936510}{135303911487300689} a^{6} - \frac{183801229821900962}{676519557436503445} a^{5} + \frac{233839149775778072}{676519557436503445} a^{4} + \frac{257697229475469053}{676519557436503445} a^{3} - \frac{9166487411462749}{135303911487300689} a^{2} + \frac{45250538563480616}{135303911487300689} a - \frac{146922164614550272}{676519557436503445}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( 31709.490295 \)
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

\(\Q(\sqrt{-19}) \), 3.1.3724.1 x3, \(\Q(\zeta_{7})^+\), 6.0.263495344.1, 6.0.5377456.1 x2, 6.0.16468459.1, 9.3.51645087424.3 x3

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

Sibling fields

Degree 6 sibling: 6.0.5377456.1
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$19$19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$