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Properties

Label 18.0.11151412592...1504.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{30}\cdot 7^{9}$
Root discriminant $46.70$
Ramified primes $2, 3, 7$
Class number $144$ (GRH)
Class group $[2, 6, 12]$ (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![298719, -1452222, 3278853, -4605120, 4799817, -3956256, 2750907, -1613178, 818277, -356156, 134040, -42762, 12243, -2964, 741, -156, 39, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 39*x^16 - 156*x^15 + 741*x^14 - 2964*x^13 + 12243*x^12 - 42762*x^11 + 134040*x^10 - 356156*x^9 + 818277*x^8 - 1613178*x^7 + 2750907*x^6 - 3956256*x^5 + 4799817*x^4 - 4605120*x^3 + 3278853*x^2 - 1452222*x + 298719)
 
gp: K = bnfinit(x^18 - 6*x^17 + 39*x^16 - 156*x^15 + 741*x^14 - 2964*x^13 + 12243*x^12 - 42762*x^11 + 134040*x^10 - 356156*x^9 + 818277*x^8 - 1613178*x^7 + 2750907*x^6 - 3956256*x^5 + 4799817*x^4 - 4605120*x^3 + 3278853*x^2 - 1452222*x + 298719, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 39 x^{16} - 156 x^{15} + 741 x^{14} - 2964 x^{13} + 12243 x^{12} - 42762 x^{11} + 134040 x^{10} - 356156 x^{9} + 818277 x^{8} - 1613178 x^{7} + 2750907 x^{6} - 3956256 x^{5} + 4799817 x^{4} - 4605120 x^{3} + 3278853 x^{2} - 1452222 x + 298719 \)

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:  \(-1115141259295002959559738261504=-\,2^{27}\cdot 3^{30}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.70$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{7}$, $\frac{1}{731134803285960499375029083468347854622779946971} a^{17} + \frac{2665694354884892906447196621172772002357531053}{243711601095320166458343027822782618207593315657} a^{16} - \frac{13006515382944505595641381471323925014600816440}{243711601095320166458343027822782618207593315657} a^{15} - \frac{66439694329728224585861538565206191260627993484}{731134803285960499375029083468347854622779946971} a^{14} + \frac{18736657388446287037579966333026153538991859122}{243711601095320166458343027822782618207593315657} a^{13} + \frac{36315582327238669239627036555034791358256075063}{243711601095320166458343027822782618207593315657} a^{12} + \frac{46912227185478099894054130967579513685075354229}{731134803285960499375029083468347854622779946971} a^{11} + \frac{83261021312459892159796847014818382706818832600}{731134803285960499375029083468347854622779946971} a^{10} + \frac{101152195560193437890083020041192026481353869523}{731134803285960499375029083468347854622779946971} a^{9} + \frac{110182585363330230337969795883590561967822313878}{731134803285960499375029083468347854622779946971} a^{8} - \frac{161063555185371170462846056150342166233606259317}{731134803285960499375029083468347854622779946971} a^{7} + \frac{177355362676560212053097292817347829900100557636}{731134803285960499375029083468347854622779946971} a^{6} + \frac{190523164823976620239262720023311108974069195962}{731134803285960499375029083468347854622779946971} a^{5} + \frac{12337538180409418324487964076581594268495668649}{243711601095320166458343027822782618207593315657} a^{4} + \frac{13145663449191389004616333027274840125151564225}{243711601095320166458343027822782618207593315657} a^{3} + \frac{68285443533273916140502840297078869245740955263}{243711601095320166458343027822782618207593315657} a^{2} - \frac{115637149157108997929297806494767589845462477721}{243711601095320166458343027822782618207593315657} a + \frac{34472320281371673749254233008013338512512873937}{243711601095320166458343027822782618207593315657}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{12}$, which has order $144$ (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:  \( 476246.2180069754 \) (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

\(\Q(\sqrt{-14}) \), 3.1.216.1, \(\Q(\zeta_{9})^+\), 6.0.128024064.1, 6.0.1152216576.2, 9.3.7346640384.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.9.15.13$x^{9} + 3 x^{7} + 6 x^{6} + 6 x^{3} + 3$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
3.9.15.13$x^{9} + 3 x^{7} + 6 x^{6} + 6 x^{3} + 3$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$