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Properties

Label 18.0.82756087703...4224.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 7^{12}\cdot 97^{6}\cdot 1399^{3}$
Root discriminant $112.46$
Ramified primes $2, 7, 97, 1399$
Class number $2208$ (GRH)
Class group $[2, 4, 276]$ (GRH)
Galois group 18T401

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1165327834895671, 0, 40489309311755, 0, -2977593837154, 0, 89577029604, 0, -443361489, 0, -132405, 0, 560850, 0, -486, 0, 44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 44*x^16 - 486*x^14 + 560850*x^12 - 132405*x^10 - 443361489*x^8 + 89577029604*x^6 - 2977593837154*x^4 + 40489309311755*x^2 + 1165327834895671)
 
gp: K = bnfinit(x^18 + 44*x^16 - 486*x^14 + 560850*x^12 - 132405*x^10 - 443361489*x^8 + 89577029604*x^6 - 2977593837154*x^4 + 40489309311755*x^2 + 1165327834895671, 1)
 

Normalized defining polynomial

\( x^{18} + 44 x^{16} - 486 x^{14} + 560850 x^{12} - 132405 x^{10} - 443361489 x^{8} + 89577029604 x^{6} - 2977593837154 x^{4} + 40489309311755 x^{2} + 1165327834895671 \)

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:  \(-8275608770373144126499153648830644224=-\,2^{18}\cdot 7^{12}\cdot 97^{6}\cdot 1399^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.46$
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:  $2, 7, 97, 1399$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{97} a^{8} + \frac{44}{97} a^{6} - \frac{1}{97} a^{4} - \frac{4}{97} a^{2}$, $\frac{1}{97} a^{9} + \frac{44}{97} a^{7} - \frac{1}{97} a^{5} - \frac{4}{97} a^{3}$, $\frac{1}{9409} a^{10} + \frac{44}{9409} a^{8} - \frac{486}{9409} a^{6} - \frac{3690}{9409} a^{4} - \frac{7}{97} a^{2}$, $\frac{1}{9409} a^{11} + \frac{44}{9409} a^{9} - \frac{486}{9409} a^{7} - \frac{3690}{9409} a^{5} - \frac{7}{97} a^{3}$, $\frac{1}{912673} a^{12} + \frac{44}{912673} a^{10} - \frac{486}{912673} a^{8} - \frac{351823}{912673} a^{6} - \frac{1365}{9409} a^{4} + \frac{21}{97} a^{2}$, $\frac{1}{912673} a^{13} + \frac{44}{912673} a^{11} - \frac{486}{912673} a^{9} - \frac{351823}{912673} a^{7} - \frac{1365}{9409} a^{5} + \frac{21}{97} a^{3}$, $\frac{1}{88529281} a^{14} + \frac{44}{88529281} a^{12} - \frac{486}{88529281} a^{10} - \frac{351823}{88529281} a^{8} - \frac{415361}{912673} a^{6} + \frac{21}{9409} a^{4} - \frac{12}{97} a^{2}$, $\frac{1}{88529281} a^{15} + \frac{44}{88529281} a^{13} - \frac{486}{88529281} a^{11} - \frac{351823}{88529281} a^{9} - \frac{415361}{912673} a^{7} + \frac{21}{9409} a^{5} - \frac{12}{97} a^{3}$, $\frac{1}{508387518986143772563140052556556823023259} a^{16} + \frac{1602054915003718378814435993625883}{508387518986143772563140052556556823023259} a^{14} - \frac{84870608443986495843863649518593503}{508387518986143772563140052556556823023259} a^{12} - \frac{6090550887575574431424105057846219926}{508387518986143772563140052556556823023259} a^{10} - \frac{9422462251322521748237517803765388451}{5241108443156121366630309820170688897147} a^{8} + \frac{17049665373520014411372554318303619300}{54032045805733209965260925981141122651} a^{6} + \frac{56625058092484930896768965194791100}{557031400059105257373823979187021883} a^{4} + \frac{2029465152964613826182607249403611}{5742591753186652137874474012237339} a^{2} - \frac{28225365171394171935014283250796}{59201976836975795235819319713787}$, $\frac{1}{508387518986143772563140052556556823023259} a^{17} + \frac{1602054915003718378814435993625883}{508387518986143772563140052556556823023259} a^{15} - \frac{84870608443986495843863649518593503}{508387518986143772563140052556556823023259} a^{13} - \frac{6090550887575574431424105057846219926}{508387518986143772563140052556556823023259} a^{11} - \frac{9422462251322521748237517803765388451}{5241108443156121366630309820170688897147} a^{9} + \frac{17049665373520014411372554318303619300}{54032045805733209965260925981141122651} a^{7} + \frac{56625058092484930896768965194791100}{557031400059105257373823979187021883} a^{5} + \frac{2029465152964613826182607249403611}{5742591753186652137874474012237339} a^{3} - \frac{28225365171394171935014283250796}{59201976836975795235819319713787} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{276}$, which has order $2208$ (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:  \( 31231444.8002 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T401:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2592
The 44 conjugacy class representatives for t18n401
Character table for t18n401 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.0.2022708581824.2, 9.3.164590951.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
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}$
97Data not computed
1399Data not computed