Properties

Label 18.0.10383286663...3863.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{8}\cdot 23^{9}$
Root discriminant $11.39$
Ramified primes $7, 23$
Class number $1$
Class group Trivial
Galois Group $D_9:C_3$ (as 18T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 11, -22, 36, -53, 80, -116, 154, -173, 154, -116, 80, -53, 36, -22, 11, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 11*x^16 - 22*x^15 + 36*x^14 - 53*x^13 + 80*x^12 - 116*x^11 + 154*x^10 - 173*x^9 + 154*x^8 - 116*x^7 + 80*x^6 - 53*x^5 + 36*x^4 - 22*x^3 + 11*x^2 - 4*x + 1)
gp: K = bnfinit(x^18 - 4*x^17 + 11*x^16 - 22*x^15 + 36*x^14 - 53*x^13 + 80*x^12 - 116*x^11 + 154*x^10 - 173*x^9 + 154*x^8 - 116*x^7 + 80*x^6 - 53*x^5 + 36*x^4 - 22*x^3 + 11*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 4 x^{17} \) \(\mathstrut +\mathstrut 11 x^{16} \) \(\mathstrut -\mathstrut 22 x^{15} \) \(\mathstrut +\mathstrut 36 x^{14} \) \(\mathstrut -\mathstrut 53 x^{13} \) \(\mathstrut +\mathstrut 80 x^{12} \) \(\mathstrut -\mathstrut 116 x^{11} \) \(\mathstrut +\mathstrut 154 x^{10} \) \(\mathstrut -\mathstrut 173 x^{9} \) \(\mathstrut +\mathstrut 154 x^{8} \) \(\mathstrut -\mathstrut 116 x^{7} \) \(\mathstrut +\mathstrut 80 x^{6} \) \(\mathstrut -\mathstrut 53 x^{5} \) \(\mathstrut +\mathstrut 36 x^{4} \) \(\mathstrut -\mathstrut 22 x^{3} \) \(\mathstrut +\mathstrut 11 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 9]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-10383286663954563863=-\,7^{8}\cdot 23^{9}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.39$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{1597} a^{16} + \frac{210}{1597} a^{15} + \frac{234}{1597} a^{14} + \frac{337}{1597} a^{13} + \frac{55}{1597} a^{12} + \frac{201}{1597} a^{11} - \frac{80}{1597} a^{10} + \frac{130}{1597} a^{9} - \frac{692}{1597} a^{8} + \frac{130}{1597} a^{7} - \frac{80}{1597} a^{6} + \frac{201}{1597} a^{5} + \frac{55}{1597} a^{4} + \frac{337}{1597} a^{3} + \frac{234}{1597} a^{2} + \frac{210}{1597} a + \frac{1}{1597}$, $\frac{1}{11179} a^{17} + \frac{2}{11179} a^{16} - \frac{327}{11179} a^{15} + \frac{2769}{11179} a^{14} + \frac{1824}{11179} a^{13} + \frac{4731}{11179} a^{12} - \frac{1963}{11179} a^{11} - \frac{3991}{11179} a^{10} + \frac{4208}{11179} a^{9} + \frac{1933}{11179} a^{8} - \frac{3165}{11179} a^{7} + \frac{2468}{11179} a^{6} - \frac{1828}{11179} a^{5} - \frac{3118}{11179} a^{4} - \frac{4385}{11179} a^{3} + \frac{1045}{11179} a^{2} + \frac{2634}{11179} a - \frac{1805}{11179}$

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:  \( \frac{5140}{11179} a^{17} - \frac{22942}{11179} a^{16} + \frac{62220}{11179} a^{15} - \frac{125719}{11179} a^{14} + \frac{202945}{11179} a^{13} - \frac{292672}{11179} a^{12} + \frac{437039}{11179} a^{11} - \frac{640320}{11179} a^{10} + \frac{843593}{11179} a^{9} - \frac{935947}{11179} a^{8} + \frac{798467}{11179} a^{7} - \frac{530900}{11179} a^{6} + \frac{348409}{11179} a^{5} - \frac{213268}{11179} a^{4} + \frac{137656}{11179} a^{3} - \frac{99774}{11179} a^{2} + \frac{33604}{11179} a - \frac{9994}{11179} \),  \( \frac{5956}{11179} a^{17} - \frac{19203}{11179} a^{16} + \frac{47815}{11179} a^{15} - \frac{78495}{11179} a^{14} + \frac{109693}{11179} a^{13} - \frac{139509}{11179} a^{12} + \frac{220132}{11179} a^{11} - \frac{309392}{11179} a^{10} + \frac{370285}{11179} a^{9} - \frac{280071}{11179} a^{8} + \frac{77155}{11179} a^{7} + \frac{62380}{11179} a^{6} - \frac{60171}{11179} a^{5} - \frac{3427}{11179} a^{4} - \frac{13948}{11179} a^{3} + \frac{38652}{11179} a^{2} - \frac{24005}{11179} a + \frac{17219}{11179} \),  \( \frac{1478}{11179} a^{17} - \frac{11380}{11179} a^{16} + \frac{27519}{11179} a^{15} - \frac{55751}{11179} a^{14} + \frac{78082}{11179} a^{13} - \frac{101018}{11179} a^{12} + \frac{142020}{11179} a^{11} - \frac{235502}{11179} a^{10} + \frac{286588}{11179} a^{9} - \frac{279586}{11179} a^{8} + \frac{132313}{11179} a^{7} + \frac{32330}{11179} a^{6} - \frac{16178}{11179} a^{5} - \frac{30964}{11179} a^{4} + \frac{23244}{11179} a^{3} + \frac{884}{11179} a^{2} - \frac{23007}{11179} a + \frac{12013}{11179} \),  \( \frac{2878}{11179} a^{17} - \frac{11436}{11179} a^{16} + \frac{43143}{11179} a^{15} - \frac{100544}{11179} a^{14} + \frac{193582}{11179} a^{13} - \frac{308561}{11179} a^{12} + \frac{452939}{11179} a^{11} - \frac{642125}{11179} a^{10} + \frac{932464}{11179} a^{9} - \frac{1186534}{11179} a^{8} + \frac{1266082}{11179} a^{7} - \frac{1090960}{11179} a^{6} + \frac{662605}{11179} a^{5} - \frac{405835}{11179} a^{4} + \frac{311091}{11179} a^{3} - \frac{199351}{11179} a^{2} + \frac{113577}{11179} a - \frac{36105}{11179} \),  \( \frac{27714}{11179} a^{17} - \frac{96423}{11179} a^{16} + \frac{243427}{11179} a^{15} - \frac{446022}{11179} a^{14} + \frac{673313}{11179} a^{13} - \frac{954864}{11179} a^{12} + \frac{1477858}{11179} a^{11} - \frac{2106693}{11179} a^{10} + \frac{2640912}{11179} a^{9} - \frac{2672095}{11179} a^{8} + \frac{1953433}{11179} a^{7} - \frac{1306406}{11179} a^{6} + \frac{937691}{11179} a^{5} - \frac{603340}{11179} a^{4} + \frac{407138}{11179} a^{3} - \frac{211153}{11179} a^{2} + \frac{71857}{11179} a - \frac{26627}{11179} \),  \( \frac{6873}{11179} a^{17} - \frac{26588}{11179} a^{16} + \frac{81303}{11179} a^{15} - \frac{166086}{11179} a^{14} + \frac{285274}{11179} a^{13} - \frac{422099}{11179} a^{12} + \frac{625019}{11179} a^{11} - \frac{895208}{11179} a^{10} + \frac{1241911}{11179} a^{9} - \frac{1428890}{11179} a^{8} + \frac{1331121}{11179} a^{7} - \frac{994921}{11179} a^{6} + \frac{580301}{11179} a^{5} - \frac{407243}{11179} a^{4} + \frac{325776}{11179} a^{3} - \frac{187756}{11179} a^{2} + \frac{97655}{11179} a - \frac{26230}{11179} \),  \( \frac{6815}{11179} a^{17} - \frac{15945}{11179} a^{16} + \frac{34427}{11179} a^{15} - \frac{44882}{11179} a^{14} + \frac{49121}{11179} a^{13} - \frac{60036}{11179} a^{12} + \frac{117861}{11179} a^{11} - \frac{138254}{11179} a^{10} + \frac{116001}{11179} a^{9} + \frac{12863}{11179} a^{8} - \frac{183222}{11179} a^{7} + \frac{136404}{11179} a^{6} - \frac{57656}{11179} a^{5} + \frac{29976}{11179} a^{4} - \frac{19773}{11179} a^{3} + \frac{55798}{11179} a^{2} - \frac{31527}{11179} a + \frac{10966}{11179} \),  \( \frac{8937}{11179} a^{17} - \frac{34570}{11179} a^{16} + \frac{82827}{11179} a^{15} - \frac{146434}{11179} a^{14} + \frac{203699}{11179} a^{13} - \frac{266565}{11179} a^{12} + \frac{410696}{11179} a^{11} - \frac{606828}{11179} a^{10} + \frac{717666}{11179} a^{9} - \frac{618144}{11179} a^{8} + \frac{255852}{11179} a^{7} + \frac{26102}{11179} a^{6} + \frac{7415}{11179} a^{5} - \frac{52452}{11179} a^{4} + \frac{16379}{11179} a^{3} + \frac{29704}{11179} a^{2} - \frac{38378}{11179} a + \frac{25821}{11179} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 77.8161636269 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_9:C_3$ (as 18T18):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 54
The 10 conjugacy class representatives for $D_9:C_3$
Character table for $D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.12167.1, 9.1.671898241.1 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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$