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Properties

Label 18.10.3449253460...7296.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{6}\cdot 3^{18}\cdot 7^{14}\cdot 29^{5}$
Root discriminant $43.75$
Ramified primes $2, 3, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -72, -3, 678, 339, -2252, -1986, 3009, 2638, -2283, -378, 1005, -345, -87, 96, -18, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 18*x^16 + 96*x^15 - 87*x^14 - 345*x^13 + 1005*x^12 - 378*x^11 - 2283*x^10 + 2638*x^9 + 3009*x^8 - 1986*x^7 - 2252*x^6 + 339*x^5 + 678*x^4 - 3*x^3 - 72*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - 18*x^16 + 96*x^15 - 87*x^14 - 345*x^13 + 1005*x^12 - 378*x^11 - 2283*x^10 + 2638*x^9 + 3009*x^8 - 1986*x^7 - 2252*x^6 + 339*x^5 + 678*x^4 - 3*x^3 - 72*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 18 x^{16} + 96 x^{15} - 87 x^{14} - 345 x^{13} + 1005 x^{12} - 378 x^{11} - 2283 x^{10} + 2638 x^{9} + 3009 x^{8} - 1986 x^{7} - 2252 x^{6} + 339 x^{5} + 678 x^{4} - 3 x^{3} - 72 x^{2} + 3 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(344925346014816475591906687296=2^{6}\cdot 3^{18}\cdot 7^{14}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.75$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{11} - \frac{2}{7} a^{10} - \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{3}{7} a^{5} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{14} - \frac{3}{7} a^{10} + \frac{3}{7} a^{9} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{49} a^{15} - \frac{1}{49} a^{12} + \frac{8}{49} a^{11} + \frac{4}{49} a^{10} - \frac{16}{49} a^{9} - \frac{3}{49} a^{8} + \frac{3}{7} a^{7} - \frac{15}{49} a^{6} + \frac{1}{7} a^{5} - \frac{9}{49} a^{4} - \frac{12}{49} a^{3} + \frac{1}{49} a^{2} + \frac{15}{49} a - \frac{1}{49}$, $\frac{1}{49} a^{16} - \frac{1}{49} a^{13} + \frac{1}{49} a^{12} - \frac{17}{49} a^{11} - \frac{9}{49} a^{10} - \frac{17}{49} a^{9} - \frac{3}{7} a^{8} - \frac{22}{49} a^{7} + \frac{2}{7} a^{6} + \frac{12}{49} a^{5} + \frac{23}{49} a^{4} + \frac{8}{49} a^{3} + \frac{1}{49} a^{2} - \frac{15}{49} a - \frac{1}{7}$, $\frac{1}{5182511451477867913} a^{17} + \frac{34891450583842144}{5182511451477867913} a^{16} + \frac{10176644219744507}{5182511451477867913} a^{15} - \frac{222803992921880714}{5182511451477867913} a^{14} + \frac{14659760598821537}{5182511451477867913} a^{13} - \frac{162786184949942560}{5182511451477867913} a^{12} + \frac{340149587838976872}{5182511451477867913} a^{11} + \frac{468781394511659031}{5182511451477867913} a^{10} + \frac{2212266479508866498}{5182511451477867913} a^{9} - \frac{1695196335275252247}{5182511451477867913} a^{8} - \frac{452478246969787764}{5182511451477867913} a^{7} + \frac{1035111575702153891}{5182511451477867913} a^{6} - \frac{1978433034555268978}{5182511451477867913} a^{5} + \frac{1371243181486035223}{5182511451477867913} a^{4} + \frac{928857375100440689}{5182511451477867913} a^{3} + \frac{75738360423452116}{5182511451477867913} a^{2} + \frac{1008068820127193177}{5182511451477867913} a - \frac{1783378454342726851}{5182511451477867913}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
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:  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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 268168864.87 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T766:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 144 conjugacy class representatives for t18n766 are not computed
Character table for t18n766 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

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 R R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$2$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
3Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.5.2$x^{6} + 58$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$