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Properties

Label 18.0.34729417703...0224.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{18}\cdot 7^{14}\cdot 19^{9}$
Root discriminant $94.29$
Ramified primes $2, 3, 7, 19$
Class number $56700$ (GRH)
Class group $[3, 15, 1260]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![260439319, -245213451, 194857845, -89051570, 41336448, -16057650, 7914304, -3139962, 1175109, -332479, 109749, -36498, 12354, -2856, 600, -140, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 140*x^15 + 600*x^14 - 2856*x^13 + 12354*x^12 - 36498*x^11 + 109749*x^10 - 332479*x^9 + 1175109*x^8 - 3139962*x^7 + 7914304*x^6 - 16057650*x^5 + 41336448*x^4 - 89051570*x^3 + 194857845*x^2 - 245213451*x + 260439319)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 140*x^15 + 600*x^14 - 2856*x^13 + 12354*x^12 - 36498*x^11 + 109749*x^10 - 332479*x^9 + 1175109*x^8 - 3139962*x^7 + 7914304*x^6 - 16057650*x^5 + 41336448*x^4 - 89051570*x^3 + 194857845*x^2 - 245213451*x + 260439319, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 140 x^{15} + 600 x^{14} - 2856 x^{13} + 12354 x^{12} - 36498 x^{11} + 109749 x^{10} - 332479 x^{9} + 1175109 x^{8} - 3139962 x^{7} + 7914304 x^{6} - 16057650 x^{5} + 41336448 x^{4} - 89051570 x^{3} + 194857845 x^{2} - 245213451 x + 260439319 \)

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:  \(-347294177030908919809654319019700224=-\,2^{12}\cdot 3^{18}\cdot 7^{14}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.29$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{8842567521018007472890326710928420268495274639332388} a^{17} - \frac{130435728170396617613253945479563012541448908425600}{2210641880254501868222581677732105067123818659833097} a^{16} - \frac{1098554579830046198452833856676631777004027208787463}{8842567521018007472890326710928420268495274639332388} a^{15} - \frac{173778706240653120624026093021630003483328098760915}{8842567521018007472890326710928420268495274639332388} a^{14} - \frac{481816076210821066636498142873303634251799096002825}{8842567521018007472890326710928420268495274639332388} a^{13} - \frac{80022787823074155022152581835536586526792075866691}{4421283760509003736445163355464210134247637319666194} a^{12} + \frac{723050305044440840378326145006876811242964550702533}{4421283760509003736445163355464210134247637319666194} a^{11} - \frac{1943183698140225996711099251408864535351054173973381}{8842567521018007472890326710928420268495274639332388} a^{10} + \frac{381968440509240248413851616482727444178170843945083}{2210641880254501868222581677732105067123818659833097} a^{9} - \frac{1513427858036849233208116444669278157850943870801709}{8842567521018007472890326710928420268495274639332388} a^{8} - \frac{507080866953134170485556292663058462985091362171163}{2210641880254501868222581677732105067123818659833097} a^{7} + \frac{1298908666566365019517868057882876964408821162398099}{8842567521018007472890326710928420268495274639332388} a^{6} - \frac{3678039567894541167008209209940922713709394422775467}{8842567521018007472890326710928420268495274639332388} a^{5} + \frac{899145219449462798393103727978107406888224112579597}{8842567521018007472890326710928420268495274639332388} a^{4} + \frac{1055329589043203979176210221778715146190799728983667}{8842567521018007472890326710928420268495274639332388} a^{3} - \frac{218118226728913117270143635031454168681108703060664}{2210641880254501868222581677732105067123818659833097} a^{2} - \frac{1538546754394020933859407322958378293105321981247243}{8842567521018007472890326710928420268495274639332388} a + \frac{2617223520219008044211862882134623683955351002886991}{8842567521018007472890326710928420268495274639332388}$

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

Class group and class number

$C_{3}\times C_{15}\times C_{1260}$, which has order $56700$ (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:  \( 296124.35954857944 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-19}) \), \(\Q(\zeta_{7})^+\), 3.3.756.1, 6.0.16468459.1, 6.0.3920165424.4, 9.9.1037426999616.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data 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}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$19$19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.12.6.1$x^{12} + 41154 x^{6} - 2476099 x^{2} + 423412929$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$