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Properties

Label 18.10.2682996055...6592.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{6}\cdot 7^{12}\cdot 13^{13}$
Root discriminant $29.39$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -29, 267, -876, 511, 1391, -2314, 1076, 489, -880, 376, 41, -11, -53, -14, 34, -5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 5*x^16 + 34*x^15 - 14*x^14 - 53*x^13 - 11*x^12 + 41*x^11 + 376*x^10 - 880*x^9 + 489*x^8 + 1076*x^7 - 2314*x^6 + 1391*x^5 + 511*x^4 - 876*x^3 + 267*x^2 - 29*x + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 - 5*x^16 + 34*x^15 - 14*x^14 - 53*x^13 - 11*x^12 + 41*x^11 + 376*x^10 - 880*x^9 + 489*x^8 + 1076*x^7 - 2314*x^6 + 1391*x^5 + 511*x^4 - 876*x^3 + 267*x^2 - 29*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 5 x^{16} + 34 x^{15} - 14 x^{14} - 53 x^{13} - 11 x^{12} + 41 x^{11} + 376 x^{10} - 880 x^{9} + 489 x^{8} + 1076 x^{7} - 2314 x^{6} + 1391 x^{5} + 511 x^{4} - 876 x^{3} + 267 x^{2} - 29 x + 1 \)

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:  \(268299605528119179177846592=2^{6}\cdot 7^{12}\cdot 13^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.39$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{545810125607227281893} a^{17} + \frac{240018468762964168853}{545810125607227281893} a^{16} + \frac{16711958545160686655}{545810125607227281893} a^{15} - \frac{149557277986585137942}{545810125607227281893} a^{14} + \frac{247957231043615075657}{545810125607227281893} a^{13} + \frac{201343614110491144477}{545810125607227281893} a^{12} - \frac{172830225463215928612}{545810125607227281893} a^{11} + \frac{137384369293654124182}{545810125607227281893} a^{10} + \frac{233846449595659065438}{545810125607227281893} a^{9} + \frac{44768488492780367072}{545810125607227281893} a^{8} + \frac{90505908953723107033}{545810125607227281893} a^{7} - \frac{260851661424465214760}{545810125607227281893} a^{6} + \frac{4732938458276328219}{545810125607227281893} a^{5} + \frac{85146809446452882226}{545810125607227281893} a^{4} - \frac{113134845933795735045}{545810125607227281893} a^{3} + \frac{49107452059182636493}{545810125607227281893} a^{2} - \frac{130935178922243473827}{545810125607227281893} a + \frac{164530087856947142862}{545810125607227281893}$

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:  \( 4368861.40669 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 48 conjugacy class representatives for t18n263
Character table for t18n263 is not computed

Intermediate fields

3.3.169.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 9.9.567869252041.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
7Data not computed
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$