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Properties

Label 18.8.53649552381...4416.2
Degree $18$
Signature $[8, 5]$
Discriminant $-\,2^{18}\cdot 7^{12}\cdot 52879^{3}$
Root discriminant $44.84$
Ramified primes $2, 7, 52879$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T401

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![52879, 0, 40631, 0, -81732, 0, -2176, 0, 13606, 0, 2083, 0, -554, 0, -129, 0, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - 129*x^14 - 554*x^12 + 2083*x^10 + 13606*x^8 - 2176*x^6 - 81732*x^4 + 40631*x^2 + 52879)
 
gp: K = bnfinit(x^18 + 2*x^16 - 129*x^14 - 554*x^12 + 2083*x^10 + 13606*x^8 - 2176*x^6 - 81732*x^4 + 40631*x^2 + 52879, 1)
 

Normalized defining polynomial

\( x^{18} + 2 x^{16} - 129 x^{14} - 554 x^{12} + 2083 x^{10} + 13606 x^{8} - 2176 x^{6} - 81732 x^{4} + 40631 x^{2} + 52879 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-536495523810946581168977084416=-\,2^{18}\cdot 7^{12}\cdot 52879^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.84$
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:  $2, 7, 52879$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{3}{7} a^{10} - \frac{1}{7} a^{8} - \frac{1}{7} a^{4} - \frac{3}{7} a^{2} + \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{11} - \frac{1}{7} a^{9} - \frac{1}{7} a^{5} - \frac{3}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{7} a^{14} - \frac{3}{7} a^{10} - \frac{3}{7} a^{8} - \frac{1}{7} a^{6} + \frac{1}{7} a^{4} - \frac{1}{7} a^{2} + \frac{3}{7}$, $\frac{1}{7} a^{15} - \frac{3}{7} a^{11} - \frac{3}{7} a^{9} - \frac{1}{7} a^{7} + \frac{1}{7} a^{5} - \frac{1}{7} a^{3} + \frac{3}{7} a$, $\frac{1}{370515703629615229931} a^{16} + \frac{20441932479689837373}{370515703629615229931} a^{14} - \frac{3750005938555205810}{52930814804230747133} a^{12} - \frac{92904294410712695686}{370515703629615229931} a^{10} + \frac{175833012697128356033}{370515703629615229931} a^{8} + \frac{119467133269501835338}{370515703629615229931} a^{6} + \frac{109113912213779447213}{370515703629615229931} a^{4} + \frac{19674857763283535354}{370515703629615229931} a^{2} - \frac{74702246328811781249}{370515703629615229931}$, $\frac{1}{370515703629615229931} a^{17} + \frac{20441932479689837373}{370515703629615229931} a^{15} - \frac{3750005938555205810}{52930814804230747133} a^{13} - \frac{92904294410712695686}{370515703629615229931} a^{11} + \frac{175833012697128356033}{370515703629615229931} a^{9} + \frac{119467133269501835338}{370515703629615229931} a^{7} + \frac{109113912213779447213}{370515703629615229931} a^{5} + \frac{19674857763283535354}{370515703629615229931} a^{3} - \frac{74702246328811781249}{370515703629615229931} a$

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:  $12$
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:  \( 124435811.904 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T401:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2592
The 44 conjugacy class representatives for t18n401
Character table for t18n401 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.8125598656.2, 9.7.6221161471.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$7$7.6.4.3$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}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
52879Data not computed