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Properties

Label 18.0.18294428062...7584.3
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![973, -322, 5432, -3620, 12826, -7127, 13693, -5436, 6756, -1124, 1327, 366, -77, 210, -24, 7, 11, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 11*x^16 + 7*x^15 - 24*x^14 + 210*x^13 - 77*x^12 + 366*x^11 + 1327*x^10 - 1124*x^9 + 6756*x^8 - 5436*x^7 + 13693*x^6 - 7127*x^5 + 12826*x^4 - 3620*x^3 + 5432*x^2 - 322*x + 973)
 
gp: K = bnfinit(x^18 - 3*x^17 + 11*x^16 + 7*x^15 - 24*x^14 + 210*x^13 - 77*x^12 + 366*x^11 + 1327*x^10 - 1124*x^9 + 6756*x^8 - 5436*x^7 + 13693*x^6 - 7127*x^5 + 12826*x^4 - 3620*x^3 + 5432*x^2 - 322*x + 973, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 11 x^{16} + 7 x^{15} - 24 x^{14} + 210 x^{13} - 77 x^{12} + 366 x^{11} + 1327 x^{10} - 1124 x^{9} + 6756 x^{8} - 5436 x^{7} + 13693 x^{6} - 7127 x^{5} + 12826 x^{4} - 3620 x^{3} + 5432 x^{2} - 322 x + 973 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{35} a^{15} - \frac{6}{35} a^{14} + \frac{12}{35} a^{13} + \frac{16}{35} a^{12} - \frac{4}{35} a^{11} + \frac{1}{35} a^{10} - \frac{2}{5} a^{9} - \frac{11}{35} a^{8} + \frac{1}{35} a^{7} - \frac{2}{35} a^{6} - \frac{13}{35} a^{5} + \frac{1}{7} a^{4} + \frac{13}{35} a^{3} + \frac{2}{5}$, $\frac{1}{63245} a^{16} + \frac{148}{63245} a^{15} + \frac{16973}{63245} a^{14} + \frac{5784}{63245} a^{13} + \frac{4363}{12649} a^{12} - \frac{5387}{12649} a^{11} - \frac{635}{1807} a^{10} + \frac{4133}{63245} a^{9} - \frac{15133}{63245} a^{8} + \frac{30462}{63245} a^{7} - \frac{19606}{63245} a^{6} + \frac{6613}{63245} a^{5} + \frac{20663}{63245} a^{4} - \frac{1594}{9035} a^{3} - \frac{469}{1807} a^{2} + \frac{1852}{9035} a - \frac{3}{65}$, $\frac{1}{1241205891621317761533344405} a^{17} - \frac{4775068737667124198894}{1241205891621317761533344405} a^{16} - \frac{378778358831172330090159}{95477376278562904733334185} a^{15} + \frac{77194875939977980451360022}{1241205891621317761533344405} a^{14} + \frac{41630949199592438584327648}{95477376278562904733334185} a^{13} - \frac{289446324557706644908080954}{1241205891621317761533344405} a^{12} + \frac{543170340103188298925344576}{1241205891621317761533344405} a^{11} + \frac{328252088487599470738627724}{1241205891621317761533344405} a^{10} + \frac{65690428324342088929478631}{177315127374473965933334915} a^{9} + \frac{87789668316726055885118357}{1241205891621317761533344405} a^{8} + \frac{355559153957073230223815341}{1241205891621317761533344405} a^{7} + \frac{427821958644919106107352728}{1241205891621317761533344405} a^{6} - \frac{524346324485423080451624916}{1241205891621317761533344405} a^{5} + \frac{303379318314555962111971691}{1241205891621317761533344405} a^{4} + \frac{23540539310895656467517898}{95477376278562904733334185} a^{3} - \frac{15888430855809991695505028}{177315127374473965933334915} a^{2} - \frac{28648105740317293757099646}{177315127374473965933334915} a + \frac{178916883389110691923589}{1275648398377510546282985}$

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:  \( 56079.0358968 \)
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

\(\Q(\sqrt{-19}) \), 3.1.3724.2 x3, \(\Q(\zeta_{7})^+\), 6.0.263495344.2, 6.0.5377456.2 x2, 6.0.16468459.1, 9.3.51645087424.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.5377456.2
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 ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{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}$