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Properties

Label 18.0.24418467389...8663.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 22679^{2}$
Root discriminant $15.42$
Ramified primes $7, 22679$
Class number $1$
Class group Trivial
Galois group 18T286

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 28, -65, 105, -112, 67, -14, -7, -29, 63, -42, 44, -14, 21, -4, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 7*x^16 - 4*x^15 + 21*x^14 - 14*x^13 + 44*x^12 - 42*x^11 + 63*x^10 - 29*x^9 - 7*x^8 - 14*x^7 + 67*x^6 - 112*x^5 + 105*x^4 - 65*x^3 + 28*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^18 + 7*x^16 - 4*x^15 + 21*x^14 - 14*x^13 + 44*x^12 - 42*x^11 + 63*x^10 - 29*x^9 - 7*x^8 - 14*x^7 + 67*x^6 - 112*x^5 + 105*x^4 - 65*x^3 + 28*x^2 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 7 x^{16} - 4 x^{15} + 21 x^{14} - 14 x^{13} + 44 x^{12} - 42 x^{11} + 63 x^{10} - 29 x^{9} - 7 x^{8} - 14 x^{7} + 67 x^{6} - 112 x^{5} + 105 x^{4} - 65 x^{3} + 28 x^{2} - 7 x + 1 \)

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:  \(-2441846738989574698663=-\,7^{15}\cdot 22679^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.42$
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:  $7, 22679$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{28823118226979} a^{17} - \frac{7887672244992}{28823118226979} a^{16} - \frac{13226615622418}{28823118226979} a^{15} - \frac{13421637667163}{28823118226979} a^{14} - \frac{10805348651651}{28823118226979} a^{13} - \frac{2426929238191}{28823118226979} a^{12} + \frac{7768874730856}{28823118226979} a^{11} + \frac{6839999844043}{28823118226979} a^{10} - \frac{13925585473101}{28823118226979} a^{9} + \frac{4752995374337}{28823118226979} a^{8} + \frac{2232325532181}{28823118226979} a^{7} - \frac{9692681197613}{28823118226979} a^{6} - \frac{3076363707937}{28823118226979} a^{5} + \frac{14308355022484}{28823118226979} a^{4} + \frac{14051406174313}{28823118226979} a^{3} + \frac{14201598272359}{28823118226979} a^{2} + \frac{1278616740366}{28823118226979} a - \frac{6111844114617}{28823118226979}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{39236251995471}{28823118226979} a^{17} - \frac{13550143165583}{28823118226979} a^{16} - \frac{278892668479602}{28823118226979} a^{15} + \frac{61373112717544}{28823118226979} a^{14} - \frac{798328628506907}{28823118226979} a^{13} + \frac{277186195462692}{28823118226979} a^{12} - \frac{1616111860465958}{28823118226979} a^{11} + \frac{1093600635191647}{28823118226979} a^{10} - \frac{2065905829327401}{28823118226979} a^{9} + \frac{419853805631494}{28823118226979} a^{8} + \frac{453591459307878}{28823118226979} a^{7} + \frac{689352358791246}{28823118226979} a^{6} - \frac{2372097723796100}{28823118226979} a^{5} + \frac{3552362317552375}{28823118226979} a^{4} - \frac{2921644430980633}{28823118226979} a^{3} + \frac{1518729123254790}{28823118226979} a^{2} - \frac{501234009956309}{28823118226979} a + \frac{63661091866187}{28823118226979} \) (order $14$)
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:  \( 8111.04910311 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T286:

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 t18n286
Character table for t18n286 is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 9.7.2668161671.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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
22679Data not computed