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Properties

Label 18.4.26055908424...3543.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,7^{13}\cdot 769^{5}$
Root discriminant $25.82$
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![7, -14, -21, 62, -28, 25, -294, 726, -793, 680, -816, 787, -549, 325, -106, 7, 13, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 13*x^16 + 7*x^15 - 106*x^14 + 325*x^13 - 549*x^12 + 787*x^11 - 816*x^10 + 680*x^9 - 793*x^8 + 726*x^7 - 294*x^6 + 25*x^5 - 28*x^4 + 62*x^3 - 21*x^2 - 14*x + 7)
 
gp: K = bnfinit(x^18 - 5*x^17 + 13*x^16 + 7*x^15 - 106*x^14 + 325*x^13 - 549*x^12 + 787*x^11 - 816*x^10 + 680*x^9 - 793*x^8 + 726*x^7 - 294*x^6 + 25*x^5 - 28*x^4 + 62*x^3 - 21*x^2 - 14*x + 7, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 13 x^{16} + 7 x^{15} - 106 x^{14} + 325 x^{13} - 549 x^{12} + 787 x^{11} - 816 x^{10} + 680 x^{9} - 793 x^{8} + 726 x^{7} - 294 x^{6} + 25 x^{5} - 28 x^{4} + 62 x^{3} - 21 x^{2} - 14 x + 7 \)

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:  \(-26055908424167101792813543=-\,7^{13}\cdot 769^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.82$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3725695487427979071007} a^{17} - \frac{808478843990610198812}{3725695487427979071007} a^{16} - \frac{711146117648791535354}{3725695487427979071007} a^{15} - \frac{1415357448804052950734}{3725695487427979071007} a^{14} - \frac{598922495529382052423}{3725695487427979071007} a^{13} - \frac{431696783823307159461}{3725695487427979071007} a^{12} - \frac{993793164124048975337}{3725695487427979071007} a^{11} - \frac{255969454674168806114}{3725695487427979071007} a^{10} - \frac{893779161673993737862}{3725695487427979071007} a^{9} - \frac{1394365461885145707403}{3725695487427979071007} a^{8} + \frac{1663866741118783073703}{3725695487427979071007} a^{7} + \frac{304044315060884700414}{3725695487427979071007} a^{6} - \frac{1075623427678363395416}{3725695487427979071007} a^{5} + \frac{1474263165238664029065}{3725695487427979071007} a^{4} - \frac{218241623999209512776}{3725695487427979071007} a^{3} - \frac{1766474908908048877705}{3725695487427979071007} a^{2} - \frac{1045307767528603107059}{3725695487427979071007} a + \frac{552950192536858063202}{3725695487427979071007}$

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:  \( 389903.150424 \) (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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R $18$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ $18$ $18$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ $18$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$7.3.2.2$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$$[\ ]_{6}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
769Data not computed