Properties

Label 18.0.27455467305...1447.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,23^{6}\cdot 2647^{3}$
Root discriminant $10.58$
Ramified primes $23, 2647$
Class number $1$
Class group Trivial
Galois Group 18T314

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 19, -40, 70, -102, 137, -173, 188, -168, 111, -46, 8, -2, 12, -17, 12, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 12*x^16 - 17*x^15 + 12*x^14 - 2*x^13 + 8*x^12 - 46*x^11 + 111*x^10 - 168*x^9 + 188*x^8 - 173*x^7 + 137*x^6 - 102*x^5 + 70*x^4 - 40*x^3 + 19*x^2 - 5*x + 1)
gp: K = bnfinit(x^18 - 5*x^17 + 12*x^16 - 17*x^15 + 12*x^14 - 2*x^13 + 8*x^12 - 46*x^11 + 111*x^10 - 168*x^9 + 188*x^8 - 173*x^7 + 137*x^6 - 102*x^5 + 70*x^4 - 40*x^3 + 19*x^2 - 5*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 5 x^{17} \) \(\mathstrut +\mathstrut 12 x^{16} \) \(\mathstrut -\mathstrut 17 x^{15} \) \(\mathstrut +\mathstrut 12 x^{14} \) \(\mathstrut -\mathstrut 2 x^{13} \) \(\mathstrut +\mathstrut 8 x^{12} \) \(\mathstrut -\mathstrut 46 x^{11} \) \(\mathstrut +\mathstrut 111 x^{10} \) \(\mathstrut -\mathstrut 168 x^{9} \) \(\mathstrut +\mathstrut 188 x^{8} \) \(\mathstrut -\mathstrut 173 x^{7} \) \(\mathstrut +\mathstrut 137 x^{6} \) \(\mathstrut -\mathstrut 102 x^{5} \) \(\mathstrut +\mathstrut 70 x^{4} \) \(\mathstrut -\mathstrut 40 x^{3} \) \(\mathstrut +\mathstrut 19 x^{2} \) \(\mathstrut -\mathstrut 5 x \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 9]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-2745546730527991447=-\,23^{6}\cdot 2647^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.58$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $23, 2647$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{107897} a^{17} + \frac{40872}{107897} a^{16} + \frac{47608}{107897} a^{15} + \frac{41907}{107897} a^{14} - \frac{48218}{107897} a^{13} - \frac{52689}{107897} a^{12} - \frac{36228}{107897} a^{11} - \frac{5677}{107897} a^{10} + \frac{27829}{107897} a^{9} + \frac{7794}{107897} a^{8} - \frac{24315}{107897} a^{7} + \frac{22736}{107897} a^{6} - \frac{45149}{107897} a^{5} + \frac{22410}{107897} a^{4} + \frac{8110}{107897} a^{3} + \frac{52846}{107897} a^{2} - \frac{19876}{107897} a - \frac{6847}{107897}$

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:  $8$
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:  \( \frac{22226}{107897} a^{17} - \frac{179565}{107897} a^{16} + \frac{636911}{107897} a^{15} - \frac{1128789}{107897} a^{14} + \frac{1018706}{107897} a^{13} - \frac{59573}{107897} a^{12} - \frac{399805}{107897} a^{11} - \frac{1879658}{107897} a^{10} + \frac{6211879}{107897} a^{9} - \frac{10303353}{107897} a^{8} + \frac{11791656}{107897} a^{7} - \frac{9985736}{107897} a^{6} + \frac{7944804}{107897} a^{5} - \frac{5794330}{107897} a^{4} + \frac{3841265}{107897} a^{3} - \frac{2493177}{107897} a^{2} + \frac{721621}{107897} a - \frac{262446}{107897} \),  \( \frac{68176}{107897} a^{17} - \frac{274244}{107897} a^{16} + \frac{612836}{107897} a^{15} - \frac{816207}{107897} a^{14} + \frac{527016}{107897} a^{13} - \frac{126237}{107897} a^{12} + \frac{421687}{107897} a^{11} - \frac{2382347}{107897} a^{10} + \frac{5727597}{107897} a^{9} - \frac{8444947}{107897} a^{8} + \frac{9309210}{107897} a^{7} - \frac{7875247}{107897} a^{6} + \frac{5725933}{107897} a^{5} - \frac{4205343}{107897} a^{4} + \frac{2632660}{107897} a^{3} - \frac{1578286}{107897} a^{2} + \frac{443835}{107897} a + \frac{69247}{107897} \),  \( \frac{6437}{107897} a^{17} - \frac{175616}{107897} a^{16} + \frac{672598}{107897} a^{15} - \frac{1281905}{107897} a^{14} + \frac{1227270}{107897} a^{13} - \frac{38822}{107897} a^{12} - \frac{573704}{107897} a^{11} - \frac{1692118}{107897} a^{10} + \frac{6715867}{107897} a^{9} - \frac{11655003}{107897} a^{8} + \frac{13098429}{107897} a^{7} - \frac{11393782}{107897} a^{6} + \frac{8790062}{107897} a^{5} - \frac{6478939}{107897} a^{4} + \frac{4729390}{107897} a^{3} - \frac{2619067}{107897} a^{2} + \frac{995103}{107897} a - \frac{160060}{107897} \),  \( \frac{70645}{107897} a^{17} - \frac{244971}{107897} a^{16} + \frac{225567}{107897} a^{15} + \frac{257923}{107897} a^{14} - \frac{915496}{107897} a^{13} + \frac{447889}{107897} a^{12} + \frac{1500338}{107897} a^{11} - \frac{2480147}{107897} a^{10} + \frac{1283232}{107897} a^{9} + \frac{1411400}{107897} a^{8} - \frac{3573536}{107897} a^{7} + \frac{3051094}{107897} a^{6} - \frac{2165828}{107897} a^{5} + \frac{1708121}{107897} a^{4} - \frac{1296884}{107897} a^{3} + \frac{1256337}{107897} a^{2} - \frac{292153}{107897} a - \frac{4064}{107897} \),  \( \frac{48842}{107897} a^{17} - \frac{255864}{107897} a^{16} + \frac{521174}{107897} a^{15} - \frac{523881}{107897} a^{14} + \frac{4263}{107897} a^{13} + \frac{446797}{107897} a^{12} + \frac{494412}{107897} a^{11} - \frac{2570272}{107897} a^{10} + \frac{4685080}{107897} a^{9} - \frac{5380918}{107897} a^{8} + \frac{4021238}{107897} a^{7} - \frac{2809534}{107897} a^{6} + \frac{1865677}{107897} a^{5} - \frac{1252712}{107897} a^{4} + \frac{881909}{107897} a^{3} + \frac{100195}{107897} a^{2} - \frac{142180}{107897} a + \frac{167423}{107897} \),  \( \frac{61558}{107897} a^{17} - \frac{159464}{107897} a^{16} + \frac{62847}{107897} a^{15} + \frac{325424}{107897} a^{14} - \frac{712453}{107897} a^{13} + \frac{170152}{107897} a^{12} + \frac{1078839}{107897} a^{11} - \frac{1281147}{107897} a^{10} + \frac{232707}{107897} a^{9} + \frac{1907239}{107897} a^{8} - \frac{3488290}{107897} a^{7} + \frac{3071817}{107897} a^{6} - \frac{2876538}{107897} a^{5} + \frac{2317472}{107897} a^{4} - \frac{1730391}{107897} a^{3} + \frac{1402179}{107897} a^{2} - \frac{298519}{107897} a + \frac{173850}{107897} \),  \( \frac{115087}{107897} a^{17} - \frac{581233}{107897} a^{16} + \frac{1239103}{107897} a^{15} - \frac{1339755}{107897} a^{14} + \frac{201435}{107897} a^{13} + \frac{855633}{107897} a^{12} + \frac{1063008}{107897} a^{11} - \frac{5966899}{107897} a^{10} + \frac{11162863}{107897} a^{9} - \frac{13231014}{107897} a^{8} + \frac{11297475}{107897} a^{7} - \frac{8623875}{107897} a^{6} + \frac{6082995}{107897} a^{5} - \frac{4170304}{107897} a^{4} + \frac{2636048}{107897} a^{3} - \frac{697876}{107897} a^{2} + \frac{162982}{107897} a + \frac{78999}{107897} \),  \( \frac{83016}{107897} a^{17} - \frac{438595}{107897} a^{16} + \frac{1037588}{107897} a^{15} - \frac{1374120}{107897} a^{14} + \frac{760594}{107897} a^{13} + \frac{222253}{107897} a^{12} + \frac{556815}{107897} a^{11} - \frac{4195719}{107897} a^{10} + \frac{9564533}{107897} a^{9} - \frac{13518730}{107897} a^{8} + \frac{13813852}{107897} a^{7} - \frac{11643321}{107897} a^{6} + \frac{8776259}{107897} a^{5} - \frac{6337437}{107897} a^{4} + \frac{4082566}{107897} a^{3} - \frac{1970630}{107897} a^{2} + \frac{690187}{107897} a - \frac{9156}{107897} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 35.6314081744 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T314:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 1296
The 22 conjugacy class representatives for t18n314
Character table for t18n314 is not computed

Intermediate fields

3.1.23.1, 6.0.1400263.1, 9.1.32206049.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
2647Data not computed