Properties

Label 18.2.89757659368...8489.1
Degree $18$
Signature $[2, 8]$
Discriminant $7^{12}\cdot 41^{3}\cdot 97^{2}$
Root discriminant $11.30$
Ramified primes $7, 41, 97$
Class number $1$
Class group Trivial
Galois Group 18T472

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 0, 3, -1, 1, 7, 2, 3, 11, 3, 2, 7, 1, -1, 3, 0, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 3*x^15 - x^14 + x^13 + 7*x^12 + 2*x^11 + 3*x^10 + 11*x^9 + 3*x^8 + 2*x^7 + 7*x^6 + x^5 - x^4 + 3*x^3 - x + 1)
gp: K = bnfinit(x^18 - x^17 + 3*x^15 - x^14 + x^13 + 7*x^12 + 2*x^11 + 3*x^10 + 11*x^9 + 3*x^8 + 2*x^7 + 7*x^6 + x^5 - x^4 + 3*x^3 - x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut x^{17} \) \(\mathstrut +\mathstrut 3 x^{15} \) \(\mathstrut -\mathstrut x^{14} \) \(\mathstrut +\mathstrut x^{13} \) \(\mathstrut +\mathstrut 7 x^{12} \) \(\mathstrut +\mathstrut 2 x^{11} \) \(\mathstrut +\mathstrut 3 x^{10} \) \(\mathstrut +\mathstrut 11 x^{9} \) \(\mathstrut +\mathstrut 3 x^{8} \) \(\mathstrut +\mathstrut 2 x^{7} \) \(\mathstrut +\mathstrut 7 x^{6} \) \(\mathstrut +\mathstrut x^{5} \) \(\mathstrut -\mathstrut x^{4} \) \(\mathstrut +\mathstrut 3 x^{3} \) \(\mathstrut -\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

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:  \(8975765936889758489=7^{12}\cdot 41^{3}\cdot 97^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.30$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$

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:  \( a \),  \( 2 a^{17} - 4 a^{16} + 3 a^{15} + 7 a^{14} - 13 a^{13} + 13 a^{12} + 11 a^{11} - 15 a^{10} + 21 a^{9} + 15 a^{8} - 13 a^{7} + 19 a^{6} + 13 a^{5} - 14 a^{4} + 13 a^{3} + 3 a^{2} - 6 a + 4 \),  \( 9 a^{17} - 20 a^{16} + 21 a^{15} + 10 a^{14} - 34 a^{13} + 55 a^{12} + 5 a^{11} - 18 a^{10} + 64 a^{9} + 18 a^{8} - 27 a^{7} + 59 a^{6} - 5 a^{5} - 17 a^{4} + 31 a^{3} - 11 a^{2} + 4 \),  \( 3 a^{17} + a^{16} - 13 a^{15} + 29 a^{14} - 12 a^{13} - 11 a^{12} + 59 a^{11} - 21 a^{10} + 12 a^{9} + 63 a^{8} - 11 a^{7} + 56 a^{5} - 28 a^{4} + 6 a^{3} + 22 a^{2} - 19 a + 8 \),  \( 9 a^{17} - 12 a^{16} - 2 a^{15} + 43 a^{14} - 43 a^{13} + 24 a^{12} + 77 a^{11} - 52 a^{10} + 55 a^{9} + 88 a^{8} - 47 a^{7} + 34 a^{6} + 67 a^{5} - 58 a^{4} + 27 a^{3} + 21 a^{2} - 27 a + 13 \),  \( 3 a^{17} - 4 a^{16} + a^{15} + 9 a^{14} - 8 a^{13} + 9 a^{12} + 15 a^{11} - 4 a^{10} + 17 a^{9} + 18 a^{8} - 5 a^{7} + 12 a^{6} + 8 a^{5} - 9 a^{4} + 7 a^{3} + a^{2} - 4 a + 3 \),  \( 11 a^{17} - 16 a^{16} + 4 a^{15} + 41 a^{14} - 45 a^{13} + 37 a^{12} + 73 a^{11} - 44 a^{10} + 70 a^{9} + 90 a^{8} - 41 a^{7} + 49 a^{6} + 60 a^{5} - 53 a^{4} + 30 a^{3} + 16 a^{2} - 24 a + 12 \),  \( a^{16} - 2 a^{15} + 2 a^{14} + a^{13} - 3 a^{12} + 5 a^{11} + 2 a^{10} - 2 a^{9} + 5 a^{8} + 4 a^{7} - 4 a^{6} + 3 a^{5} - 3 a^{3} \),  \( a^{17} - 2 a^{15} + 5 a^{14} - 2 a^{12} + 12 a^{11} + 4 a^{10} + a^{9} + 16 a^{8} + 7 a^{7} - 2 a^{6} + 10 a^{5} + a^{4} - 4 a^{3} + 3 a^{2} - 2 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 95.2465332123 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T472:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 5184
The 88 conjugacy class representatives for t18n472 are not computed
Character table for t18n472 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.2.98441.1, 9.5.467890073.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ $18$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
7Data not computed
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
41.6.0.1$x^{6} - x + 7$$1$$6$$0$$C_6$$[\ ]^{6}$
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$