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Properties

Label 18.4.35659322871...3591.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,7^{12}\cdot 97^{2}\cdot 1399^{3}$
Root discriminant $20.34$
Ramified primes $7, 97, 1399$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T840

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![841, -116, -1401, 588, 635, -1026, 931, 119, -754, 424, -96, -83, 91, -5, -11, 3, 1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + x^16 + 3*x^15 - 11*x^14 - 5*x^13 + 91*x^12 - 83*x^11 - 96*x^10 + 424*x^9 - 754*x^8 + 119*x^7 + 931*x^6 - 1026*x^5 + 635*x^4 + 588*x^3 - 1401*x^2 - 116*x + 841)
 
gp: K = bnfinit(x^18 - 3*x^17 + x^16 + 3*x^15 - 11*x^14 - 5*x^13 + 91*x^12 - 83*x^11 - 96*x^10 + 424*x^9 - 754*x^8 + 119*x^7 + 931*x^6 - 1026*x^5 + 635*x^4 + 588*x^3 - 1401*x^2 - 116*x + 841, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + x^{16} + 3 x^{15} - 11 x^{14} - 5 x^{13} + 91 x^{12} - 83 x^{11} - 96 x^{10} + 424 x^{9} - 754 x^{8} + 119 x^{7} + 931 x^{6} - 1026 x^{5} + 635 x^{4} + 588 x^{3} - 1401 x^{2} - 116 x + 841 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-356593228716323827483591=-\,7^{12}\cdot 97^{2}\cdot 1399^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.34$
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:  $7, 97, 1399$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{106726000143058855532669191} a^{17} + \frac{3911882352735534244202026}{106726000143058855532669191} a^{16} + \frac{8782967874305753263943965}{106726000143058855532669191} a^{15} - \frac{18001709372379779128563738}{106726000143058855532669191} a^{14} - \frac{13189392625240334797561566}{106726000143058855532669191} a^{13} + \frac{11322715727881831788375698}{106726000143058855532669191} a^{12} + \frac{25528739280825052052741596}{106726000143058855532669191} a^{11} + \frac{17500911990434161950342929}{106726000143058855532669191} a^{10} - \frac{20812356775446460204085302}{106726000143058855532669191} a^{9} + \frac{30212014752747641436314743}{106726000143058855532669191} a^{8} - \frac{1452548710293219458175118}{3680206901484788121816179} a^{7} - \frac{13894780631848945868742484}{106726000143058855532669191} a^{6} - \frac{39974259756730783612587518}{106726000143058855532669191} a^{5} - \frac{11656845800869792806012674}{106726000143058855532669191} a^{4} - \frac{15218748250791920941094883}{106726000143058855532669191} a^{3} - \frac{31992856592039504895125788}{106726000143058855532669191} a^{2} + \frac{30091156123572413515853245}{106726000143058855532669191} a + \frac{1146104110581750989110765}{3680206901484788121816179}$

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:  $10$
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:  \( 36324.4403692 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T840:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 180 conjugacy class representatives for t18n840 are not computed
Character table for t18n840 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$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