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Properties

Label 18.0.12488057521...3503.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 138041^{3}$
Root discriminant $36.38$
Ramified primes $7, 138041$
Class number $112$ (GRH)
Class group $[2, 56]$ (GRH)
Galois group 18T285

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44507, -71712, 141287, -226521, 303777, -296751, 221321, -119075, 43736, -6263, -3290, 2535, -409, -355, 337, -148, 44, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 44*x^16 - 148*x^15 + 337*x^14 - 355*x^13 - 409*x^12 + 2535*x^11 - 3290*x^10 - 6263*x^9 + 43736*x^8 - 119075*x^7 + 221321*x^6 - 296751*x^5 + 303777*x^4 - 226521*x^3 + 141287*x^2 - 71712*x + 44507)
 
gp: K = bnfinit(x^18 - 8*x^17 + 44*x^16 - 148*x^15 + 337*x^14 - 355*x^13 - 409*x^12 + 2535*x^11 - 3290*x^10 - 6263*x^9 + 43736*x^8 - 119075*x^7 + 221321*x^6 - 296751*x^5 + 303777*x^4 - 226521*x^3 + 141287*x^2 - 71712*x + 44507, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 44 x^{16} - 148 x^{15} + 337 x^{14} - 355 x^{13} - 409 x^{12} + 2535 x^{11} - 3290 x^{10} - 6263 x^{9} + 43736 x^{8} - 119075 x^{7} + 221321 x^{6} - 296751 x^{5} + 303777 x^{4} - 226521 x^{3} + 141287 x^{2} - 71712 x + 44507 \)

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:  \(-12488057521927227046014543503=-\,7^{15}\cdot 138041^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.38$
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:  $7, 138041$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{1693040551720163860810071518111673069} a^{17} + \frac{71989274121427376275098525370320458}{564346850573387953603357172703891023} a^{16} + \frac{35177094230577128681715759205104873}{564346850573387953603357172703891023} a^{15} - \frac{186793848739559706018245986660548592}{564346850573387953603357172703891023} a^{14} - \frac{187990557613581961446827734988569243}{564346850573387953603357172703891023} a^{13} + \frac{106880811858113688449720881770577024}{564346850573387953603357172703891023} a^{12} - \frac{762341836644965867834969659217561729}{1693040551720163860810071518111673069} a^{11} - \frac{143234426943394757590303953037725710}{564346850573387953603357172703891023} a^{10} - \frac{6876370287249465953940447745760957}{1693040551720163860810071518111673069} a^{9} + \frac{151769204804334520848779172021844688}{1693040551720163860810071518111673069} a^{8} + \frac{485621725383788961084624165237491294}{1693040551720163860810071518111673069} a^{7} - \frac{85065641336763211099383576078350984}{1693040551720163860810071518111673069} a^{6} + \frac{547080186010193983377716658566999678}{1693040551720163860810071518111673069} a^{5} + \frac{330231528491885611444198314048530423}{1693040551720163860810071518111673069} a^{4} + \frac{734804197879389576233609482011317905}{1693040551720163860810071518111673069} a^{3} + \frac{555848373726435231334794232775414897}{1693040551720163860810071518111673069} a^{2} + \frac{617687202566743480138624651763790386}{1693040551720163860810071518111673069} a - \frac{25035986921719525613510474314768003}{564346850573387953603357172703891023}$

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

Class group and class number

$C_{2}\times C_{56}$, which has order $112$ (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:  \( 22017.7641526 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T285:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 34 conjugacy class representatives for t18n285
Character table for t18n285 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.0.2320055087.1, 9.9.16240385609.1

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

Sibling fields

Degree 18 siblings: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
7Data not computed
138041Data not computed