Properties

Label 18.2.476...777.1
Degree $18$
Signature $[2, 8]$
Discriminant $4.769\times 10^{27}$
Root discriminant $34.49$
Ramified primes $7, 17$
Class number $1$
Class group trivial
Galois group $D_{18}$ (as 18T13)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225)
 
gp: K = bnfinit(x^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![225, -750, 17, 3264, -3451, 1972, 6936, -3502, -3638, 1632, 493, -476, 119, 51, -17, 17, 0, -2, 1]);
 

\( x^{18} - 2 x^{17} + 17 x^{15} - 17 x^{14} + 51 x^{13} + 119 x^{12} - 476 x^{11} + 493 x^{10} + 1632 x^{9} - 3638 x^{8} - 3502 x^{7} + 6936 x^{6} + 1972 x^{5} - 3451 x^{4} + 3264 x^{3} + 17 x^{2} - 750 x + 225 \)

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(4768875488962616064464333777\)\(\medspace = 7^{8}\cdot 17^{17}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $34.49$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{15} a^{9} + \frac{1}{15} a^{8} - \frac{2}{15} a^{7} + \frac{7}{15} a^{6} + \frac{1}{15} a^{5} - \frac{1}{3} a^{4} + \frac{7}{15} a^{3} + \frac{4}{15} a^{2} + \frac{1}{5} a$, $\frac{1}{15} a^{10} + \frac{2}{15} a^{8} - \frac{1}{15} a^{7} - \frac{1}{15} a^{6} - \frac{1}{15} a^{5} + \frac{2}{15} a^{4} + \frac{2}{15} a^{3} + \frac{4}{15} a^{2} + \frac{2}{15} a$, $\frac{1}{15} a^{11} + \frac{2}{15} a^{8} - \frac{7}{15} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{2}{15} a^{4} - \frac{1}{3} a^{3} - \frac{1}{15} a^{2} - \frac{1}{15} a$, $\frac{1}{195} a^{12} + \frac{4}{195} a^{11} + \frac{1}{39} a^{10} + \frac{4}{195} a^{9} - \frac{4}{65} a^{8} + \frac{11}{65} a^{7} - \frac{17}{65} a^{6} + \frac{18}{65} a^{5} + \frac{38}{195} a^{4} + \frac{8}{195} a^{3} - \frac{62}{195} a^{2} - \frac{73}{195} a - \frac{3}{13}$, $\frac{1}{195} a^{13} + \frac{2}{195} a^{11} - \frac{1}{65} a^{10} - \frac{2}{195} a^{9} + \frac{29}{195} a^{8} - \frac{79}{195} a^{7} - \frac{28}{195} a^{6} - \frac{7}{39} a^{5} + \frac{38}{195} a^{4} - \frac{27}{65} a^{3} - \frac{7}{195} a^{2} + \frac{1}{15} a - \frac{1}{13}$, $\frac{1}{975} a^{14} - \frac{1}{975} a^{12} - \frac{1}{65} a^{11} - \frac{2}{65} a^{10} - \frac{3}{325} a^{9} - \frac{32}{195} a^{8} + \frac{68}{975} a^{7} + \frac{79}{975} a^{6} - \frac{397}{975} a^{5} - \frac{4}{25} a^{4} + \frac{476}{975} a^{3} - \frac{217}{975} a^{2} + \frac{7}{195} a - \frac{6}{13}$, $\frac{1}{4875} a^{15} + \frac{2}{4875} a^{14} - \frac{1}{4875} a^{13} + \frac{1}{1625} a^{12} + \frac{4}{975} a^{11} + \frac{32}{1625} a^{10} + \frac{97}{4875} a^{9} - \frac{167}{4875} a^{8} - \frac{102}{325} a^{7} + \frac{41}{4875} a^{6} + \frac{34}{75} a^{5} - \frac{896}{4875} a^{4} + \frac{283}{975} a^{3} + \frac{376}{4875} a^{2} - \frac{28}{65} a - \frac{11}{65}$, $\frac{1}{102375} a^{16} - \frac{1}{14625} a^{15} + \frac{2}{34125} a^{14} - \frac{188}{102375} a^{13} - \frac{1}{14625} a^{12} + \frac{397}{34125} a^{11} - \frac{467}{102375} a^{10} + \frac{562}{20475} a^{9} + \frac{103}{1125} a^{8} + \frac{199}{2625} a^{7} + \frac{341}{102375} a^{6} + \frac{21514}{102375} a^{5} + \frac{1249}{4875} a^{4} - \frac{25009}{102375} a^{3} + \frac{18841}{102375} a^{2} - \frac{179}{1365} a + \frac{6}{35}$, $\frac{1}{313358232607875} a^{17} + \frac{1404856129}{313358232607875} a^{16} + \frac{25409243719}{313358232607875} a^{15} + \frac{155108268703}{313358232607875} a^{14} + \frac{7330690499}{4820895886275} a^{13} - \frac{209454488186}{313358232607875} a^{12} + \frac{41219692024}{3443497061625} a^{11} - \frac{2913813250412}{313358232607875} a^{10} - \frac{739245317434}{24104479431375} a^{9} - \frac{1195848720691}{313358232607875} a^{8} + \frac{61169396576297}{313358232607875} a^{7} - \frac{3361738081379}{8953092360225} a^{6} + \frac{186254065181}{777563852625} a^{5} + \frac{4692836456396}{12534329304315} a^{4} + \frac{11845255351616}{44765461801125} a^{3} + \frac{6362826393968}{44765461801125} a^{2} + \frac{39793089398}{134777734455} a + \frac{351822543028}{1392703256035}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 10997947.525 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{8}\cdot 10997947.525 \cdot 1}{2\sqrt{4768875488962616064464333777}}\approx 0.77369999480$

Galois group

$D_{18}$ (as 18T13):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 36
The 12 conjugacy class representatives for $D_{18}$
Character table for $D_{18}$

Intermediate fields

\(\Q(\sqrt{17}) \), 3.1.2023.1, 6.2.69572993.1, 9.1.16748793615841.1

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

Sibling fields

Galois closure: data not computed
Degree 18 sibling: 18.0.33382128422738312451250336439.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ $18$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

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$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17Data not computed

Additional information

This field is associated with the 17-torsion points on the elliptic curves from the isogeny class 49.a1, a curve with complex multiplication by $\Q(\sqrt{-7})$.