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Properties

Label 18.6.18239135896...4801.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{14}\cdot 769^{5}$
Root discriminant $28.77$
Ramified primes $7, 769$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T696

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -19, 98, -248, 290, 338, -2149, 4295, -5593, 5727, -4515, 2851, -1393, 508, -114, -3, 14, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 14*x^16 - 3*x^15 - 114*x^14 + 508*x^13 - 1393*x^12 + 2851*x^11 - 4515*x^10 + 5727*x^9 - 5593*x^8 + 4295*x^7 - 2149*x^6 + 338*x^5 + 290*x^4 - 248*x^3 + 98*x^2 - 19*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 14*x^16 - 3*x^15 - 114*x^14 + 508*x^13 - 1393*x^12 + 2851*x^11 - 4515*x^10 + 5727*x^9 - 5593*x^8 + 4295*x^7 - 2149*x^6 + 338*x^5 + 290*x^4 - 248*x^3 + 98*x^2 - 19*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 14 x^{16} - 3 x^{15} - 114 x^{14} + 508 x^{13} - 1393 x^{12} + 2851 x^{11} - 4515 x^{10} + 5727 x^{9} - 5593 x^{8} + 4295 x^{7} - 2149 x^{6} + 338 x^{5} + 290 x^{4} - 248 x^{3} + 98 x^{2} - 19 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(182391358969169712549694801=7^{14}\cdot 769^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.77$
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:  $7, 769$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{720226514428363920551} a^{17} - \frac{142373794395019461}{102889502061194845793} a^{16} - \frac{38065111925301504499}{102889502061194845793} a^{15} - \frac{178317443187643473843}{720226514428363920551} a^{14} - \frac{157124105579791843548}{720226514428363920551} a^{13} + \frac{19202935100006885629}{720226514428363920551} a^{12} + \frac{322103443143559306835}{720226514428363920551} a^{11} + \frac{203109682333621826677}{720226514428363920551} a^{10} + \frac{191889817179490137108}{720226514428363920551} a^{9} + \frac{19950947297368862928}{720226514428363920551} a^{8} - \frac{296244182198967119592}{720226514428363920551} a^{7} + \frac{57144366224349216130}{720226514428363920551} a^{6} + \frac{161833434076381026068}{720226514428363920551} a^{5} + \frac{13664956014024068640}{720226514428363920551} a^{4} - \frac{133661476881081385833}{720226514428363920551} a^{3} + \frac{87439151674318983517}{720226514428363920551} a^{2} + \frac{334550236631562672792}{720226514428363920551} a + \frac{263272845576525316428}{720226514428363920551}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 1887943.11562 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T696:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 192 conjugacy class representatives for t18n696 are not computed
Character table for t18n696 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.69573030289.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R $18$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ $18$

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
$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$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
769Data not computed