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Properties

Label 18.6.37222726320...7649.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 769^{5}$
Root discriminant $23.18$
Ramified primes $7, 769$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T696

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, 77, 343, 860, 1442, 1662, 1242, 619, 314, 133, -12, 8, 5, -27, -5, 6, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 2*x^16 + 6*x^15 - 5*x^14 - 27*x^13 + 5*x^12 + 8*x^11 - 12*x^10 + 133*x^9 + 314*x^8 + 619*x^7 + 1242*x^6 + 1662*x^5 + 1442*x^4 + 860*x^3 + 343*x^2 + 77*x + 7)
 
gp: K = bnfinit(x^18 - 2*x^17 - 2*x^16 + 6*x^15 - 5*x^14 - 27*x^13 + 5*x^12 + 8*x^11 - 12*x^10 + 133*x^9 + 314*x^8 + 619*x^7 + 1242*x^6 + 1662*x^5 + 1442*x^4 + 860*x^3 + 343*x^2 + 77*x + 7, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 2 x^{16} + 6 x^{15} - 5 x^{14} - 27 x^{13} + 5 x^{12} + 8 x^{11} - 12 x^{10} + 133 x^{9} + 314 x^{8} + 619 x^{7} + 1242 x^{6} + 1662 x^{5} + 1442 x^{4} + 860 x^{3} + 343 x^{2} + 77 x + 7 \)

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:  \(3722272632023871684687649=7^{12}\cdot 769^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.18$
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:  $7, 769$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} - \frac{1}{7} a^{13} + \frac{3}{7} a^{12} - \frac{2}{7} a^{10} - \frac{2}{7} a^{9} + \frac{2}{7} a^{8} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3}$, $\frac{1}{49} a^{16} - \frac{3}{49} a^{15} + \frac{6}{49} a^{14} - \frac{8}{49} a^{13} - \frac{16}{49} a^{12} - \frac{9}{49} a^{11} + \frac{4}{49} a^{10} + \frac{8}{49} a^{9} - \frac{2}{7} a^{8} - \frac{23}{49} a^{7} - \frac{13}{49} a^{6} + \frac{20}{49} a^{5} - \frac{5}{49} a^{4} + \frac{3}{49} a^{3} - \frac{2}{7} a^{2} - \frac{3}{7}$, $\frac{1}{1792396311480131} a^{17} - \frac{3306576990253}{1792396311480131} a^{16} - \frac{89860067579956}{1792396311480131} a^{15} - \frac{458804154455723}{1792396311480131} a^{14} + \frac{31793923514658}{256056615925733} a^{13} + \frac{116848323675904}{1792396311480131} a^{12} + \frac{259505398648847}{1792396311480131} a^{11} - \frac{64149418506919}{256056615925733} a^{10} - \frac{621507384469715}{1792396311480131} a^{9} - \frac{700430447777881}{1792396311480131} a^{8} + \frac{557753533369765}{1792396311480131} a^{7} - \frac{789885969597966}{1792396311480131} a^{6} - \frac{613541932542493}{1792396311480131} a^{5} - \frac{858190512648190}{1792396311480131} a^{4} - \frac{159701373225274}{1792396311480131} a^{3} + \frac{9647661096504}{256056615925733} a^{2} - \frac{65885530679713}{256056615925733} a + \frac{23862307221640}{256056615925733}$

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:  $11$
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:  \( 263528.137952 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T696:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 192 conjugacy class representatives for t18n696 are not computed
Character table for t18n696 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.69573030289.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R $18$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ $18$

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
$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}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
769Data not computed