Properties

Label 18.2.71828908288...3137.1
Degree $18$
Signature $[2, 8]$
Discriminant $7^{12}\cdot 113\cdot 2143^{2}$
Root discriminant $11.16$
Ramified primes $7, 113, 2143$
Class number $1$
Class group Trivial
Galois Group 18T879

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 10, -15, 29, -44, 53, -71, 91, -98, 99, -97, 85, -63, 39, -22, 12, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 12*x^16 - 22*x^15 + 39*x^14 - 63*x^13 + 85*x^12 - 97*x^11 + 99*x^10 - 98*x^9 + 91*x^8 - 71*x^7 + 53*x^6 - 44*x^5 + 29*x^4 - 15*x^3 + 10*x^2 - 4*x + 1)
gp: K = bnfinit(x^18 - 5*x^17 + 12*x^16 - 22*x^15 + 39*x^14 - 63*x^13 + 85*x^12 - 97*x^11 + 99*x^10 - 98*x^9 + 91*x^8 - 71*x^7 + 53*x^6 - 44*x^5 + 29*x^4 - 15*x^3 + 10*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 5 x^{17} \) \(\mathstrut +\mathstrut 12 x^{16} \) \(\mathstrut -\mathstrut 22 x^{15} \) \(\mathstrut +\mathstrut 39 x^{14} \) \(\mathstrut -\mathstrut 63 x^{13} \) \(\mathstrut +\mathstrut 85 x^{12} \) \(\mathstrut -\mathstrut 97 x^{11} \) \(\mathstrut +\mathstrut 99 x^{10} \) \(\mathstrut -\mathstrut 98 x^{9} \) \(\mathstrut +\mathstrut 91 x^{8} \) \(\mathstrut -\mathstrut 71 x^{7} \) \(\mathstrut +\mathstrut 53 x^{6} \) \(\mathstrut -\mathstrut 44 x^{5} \) \(\mathstrut +\mathstrut 29 x^{4} \) \(\mathstrut -\mathstrut 15 x^{3} \) \(\mathstrut +\mathstrut 10 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[2, 8]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(7182890828838813137=7^{12}\cdot 113\cdot 2143^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.16$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 113, 2143$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{639241} a^{17} + \frac{230257}{639241} a^{16} + \frac{149565}{639241} a^{15} + \frac{27133}{639241} a^{14} - \frac{242649}{639241} a^{13} + \frac{15504}{639241} a^{12} - \frac{178852}{639241} a^{11} + \frac{282104}{639241} a^{10} + \frac{78650}{639241} a^{9} - \frac{230569}{639241} a^{8} + \frac{243027}{639241} a^{7} + \frac{86622}{639241} a^{6} + \frac{157335}{639241} a^{5} - \frac{72708}{639241} a^{4} - \frac{167677}{639241} a^{3} - \frac{124230}{639241} a^{2} - \frac{52741}{639241} a + \frac{52372}{639241}$

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

Class group and class number

Trivial Abelian group, 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:  $9$
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:  \( \frac{80431}{639241} a^{17} - \frac{289485}{639241} a^{16} + \frac{425377}{639241} a^{15} - \frac{673692}{639241} a^{14} + \frac{1443734}{639241} a^{13} - \frac{2074690}{639241} a^{12} + \frac{2151975}{639241} a^{11} - \frac{2509435}{639241} a^{10} + \frac{3165419}{639241} a^{9} - \frac{3710034}{639241} a^{8} + \frac{3389544}{639241} a^{7} - \frac{2550541}{639241} a^{6} + \frac{3392754}{639241} a^{5} - \frac{2757444}{639241} a^{4} + \frac{277831}{639241} a^{3} - \frac{606300}{639241} a^{2} + \frac{1270387}{639241} a + \frac{373383}{639241} \),  \( \frac{418297}{639241} a^{17} - \frac{1825346}{639241} a^{16} + \frac{3909581}{639241} a^{15} - \frac{7103105}{639241} a^{14} + \frac{12761288}{639241} a^{13} - \frac{19639728}{639241} a^{12} + \frac{25245790}{639241} a^{11} - \frac{28119075}{639241} a^{10} + \frac{28008348}{639241} a^{9} - \frac{27043999}{639241} a^{8} + \frac{23259947}{639241} a^{7} - \frac{16995135}{639241} a^{6} + \frac{14403883}{639241} a^{5} - \frac{11236316}{639241} a^{4} + \frac{5128861}{639241} a^{3} - \frac{3692384}{639241} a^{2} + \frac{2640279}{639241} a - \frac{377827}{639241} \),  \( a \),  \( \frac{152543}{639241} a^{17} - \frac{920917}{639241} a^{16} + \frac{2500228}{639241} a^{15} - \frac{4610943}{639241} a^{14} + \frac{7875349}{639241} a^{13} - \frac{12949848}{639241} a^{12} + \frac{18083992}{639241} a^{11} - \frac{20530119}{639241} a^{10} + \frac{20048333}{639241} a^{9} - \frac{19185136}{639241} a^{8} + \frac{18463096}{639241} a^{7} - \frac{14873508}{639241} a^{6} + \frac{9638175}{639241} a^{5} - \frac{7296745}{639241} a^{4} + \frac{5750691}{639241} a^{3} - \frac{2674409}{639241} a^{2} + \frac{216863}{639241} a - \frac{252022}{639241} \),  \( \frac{43345}{639241} a^{17} + \frac{19932}{639241} a^{16} - \frac{287297}{639241} a^{15} + \frac{515686}{639241} a^{14} - \frac{827973}{639241} a^{13} + \frac{1457071}{639241} a^{12} - \frac{2182056}{639241} a^{11} + \frac{2313755}{639241} a^{10} - \frac{1266485}{639241} a^{9} - \frac{119511}{639241} a^{8} + \frac{592117}{639241} a^{7} - \frac{1549526}{639241} a^{6} + \frac{2180310}{639241} a^{5} - \frac{1987853}{639241} a^{4} + \frac{1489087}{639241} a^{3} - \frac{1061648}{639241} a^{2} + \frac{1145653}{639241} a - \frac{519692}{639241} \),  \( \frac{141184}{639241} a^{17} - \frac{636008}{639241} a^{16} + \frac{1415489}{639241} a^{15} - \frac{2782805}{639241} a^{14} + \frac{5161184}{639241} a^{13} - \frac{8154581}{639241} a^{12} + \frac{11124311}{639241} a^{11} - \frac{13402671}{639241} a^{10} + \frac{14568732}{639241} a^{9} - \frac{14647555}{639241} a^{8} + \frac{13048113}{639241} a^{7} - \frac{10546220}{639241} a^{6} + \frac{9148505}{639241} a^{5} - \frac{6666704}{639241} a^{4} + \frac{4177072}{639241} a^{3} - \frac{2989967}{639241} a^{2} + \frac{2250788}{639241} a - \frac{651440}{639241} \),  \( \frac{374956}{639241} a^{17} - \frac{1563491}{639241} a^{16} + \frac{2877415}{639241} a^{15} - \frac{4314054}{639241} a^{14} + \frac{7504737}{639241} a^{13} - \frac{11446168}{639241} a^{12} + \frac{12649136}{639241} a^{11} - \frac{10766425}{639241} a^{10} + \frac{9133721}{639241} a^{9} - \frac{10587257}{639241} a^{8} + \frac{10215877}{639241} a^{7} - \frac{4871265}{639241} a^{6} + \frac{3264298}{639241} a^{5} - \frac{5064608}{639241} a^{4} + \frac{2329825}{639241} a^{3} + \frac{68549}{639241} a^{2} + \frac{1283662}{639241} a - \frac{287888}{639241} \),  \( \frac{248192}{639241} a^{17} - \frac{839297}{639241} a^{16} + \frac{1390092}{639241} a^{15} - \frac{2767363}{639241} a^{14} + \frac{5746412}{639241} a^{13} - \frac{8572185}{639241} a^{12} + \frac{10765794}{639241} a^{11} - \frac{13534823}{639241} a^{10} + \frac{16418649}{639241} a^{9} - \frac{17786435}{639241} a^{8} + \frac{14557449}{639241} a^{7} - \frac{10932985}{639241} a^{6} + \frac{12118932}{639241} a^{5} - \frac{9359121}{639241} a^{4} + \frac{3613044}{639241} a^{3} - \frac{3577212}{639241} a^{2} + \frac{3039890}{639241} a - \frac{654311}{639241} \),  \( \frac{256051}{639241} a^{17} - \frac{940805}{639241} a^{16} + \frac{1896469}{639241} a^{15} - \frac{3674851}{639241} a^{14} + \frac{6902306}{639241} a^{13} - \frac{10739003}{639241} a^{12} + \frac{14055090}{639241} a^{11} - \frac{16563480}{639241} a^{10} + \frac{18300675}{639241} a^{9} - \frac{18858453}{639241} a^{8} + \frac{15733016}{639241} a^{7} - \frac{11601593}{639241} a^{6} + \frac{9765639}{639241} a^{5} - \frac{8011357}{639241} a^{4} + \frac{4593684}{639241} a^{3} - \frac{2501293}{639241} a^{2} + \frac{2137298}{639241} a - \frac{733967}{639241} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 84.7255286435 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T879:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 331776
The 360 conjugacy class representatives for t18n879 are not computed
Character table for t18n879 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.3.252121807.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ $18$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$113$$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
2143Data not computed