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Properties

Label 18.6.16424356675...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{36}\cdot 5^{12}\cdot 7^{12}\cdot 29^{4}$
Root discriminant $90.45$
Ramified primes $2, 5, 7, 29$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 18T463

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-707281, 0, -2609623, 0, -736716, 0, 275906, 0, 93715, 0, -3079, 0, -2463, 0, -85, 0, 17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 17*x^16 - 85*x^14 - 2463*x^12 - 3079*x^10 + 93715*x^8 + 275906*x^6 - 736716*x^4 - 2609623*x^2 - 707281)
 
gp: K = bnfinit(x^18 + 17*x^16 - 85*x^14 - 2463*x^12 - 3079*x^10 + 93715*x^8 + 275906*x^6 - 736716*x^4 - 2609623*x^2 - 707281, 1)
 

Normalized defining polynomial

\( x^{18} + 17 x^{16} - 85 x^{14} - 2463 x^{12} - 3079 x^{10} + 93715 x^{8} + 275906 x^{6} - 736716 x^{4} - 2609623 x^{2} - 707281 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(164243566750563246800896000000000000=2^{36}\cdot 5^{12}\cdot 7^{12}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.45$
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:  $2, 5, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{29} a^{12} - \frac{12}{29} a^{10} + \frac{2}{29} a^{8} + \frac{2}{29} a^{6} - \frac{5}{29} a^{4} - \frac{13}{29} a^{2}$, $\frac{1}{29} a^{13} - \frac{12}{29} a^{11} + \frac{2}{29} a^{9} + \frac{2}{29} a^{7} - \frac{5}{29} a^{5} - \frac{13}{29} a^{3}$, $\frac{1}{1682} a^{14} + \frac{17}{1682} a^{12} + \frac{378}{841} a^{10} + \frac{30}{841} a^{8} + \frac{285}{1682} a^{6} - \frac{477}{1682} a^{4} - \frac{27}{58} a^{2} - \frac{1}{2}$, $\frac{1}{1682} a^{15} + \frac{17}{1682} a^{13} + \frac{378}{841} a^{11} + \frac{30}{841} a^{9} + \frac{285}{1682} a^{7} - \frac{477}{1682} a^{5} - \frac{27}{58} a^{3} - \frac{1}{2} a$, $\frac{1}{57043819179843384556} a^{16} - \frac{4176799639881699}{14260954794960846139} a^{14} + \frac{362664066650760087}{57043819179843384556} a^{12} + \frac{2039701412244929417}{28521909589921692278} a^{10} + \frac{4689275035012965707}{57043819179843384556} a^{8} + \frac{4709011569590323628}{14260954794960846139} a^{6} - \frac{256150471496477917}{983514123790403182} a^{4} - \frac{14109402491341995}{33914280130703558} a^{2} - \frac{629986741070909}{2338915871083004}$, $\frac{1}{57043819179843384556} a^{17} - \frac{4176799639881699}{14260954794960846139} a^{15} + \frac{362664066650760087}{57043819179843384556} a^{13} + \frac{2039701412244929417}{28521909589921692278} a^{11} + \frac{4689275035012965707}{57043819179843384556} a^{9} + \frac{4709011569590323628}{14260954794960846139} a^{7} - \frac{256150471496477917}{983514123790403182} a^{5} - \frac{14109402491341995}{33914280130703558} a^{3} - \frac{629986741070909}{2338915871083004} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 13491148085.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T463:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 48 conjugacy class representatives for t18n463
Character table for t18n463 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.9800.1, 9.9.941192000000.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
$2$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.12.30.174$x^{12} - 2 x^{10} - 5 x^{8} - 4 x^{6} + 19 x^{4} - 2 x^{2} - 15$$4$$3$$30$12T87$[2, 2, 2, 3, 7/2, 7/2]^{3}$
$5$5.9.6.1$x^{9} - 25 x^{3} + 250$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
5.9.6.1$x^{9} - 25 x^{3} + 250$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{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$$[\ ]^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$