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Properties

Label 18.2.24724476901...3873.1
Degree $18$
Signature $[2, 8]$
Discriminant $7^{12}\cdot 97^{3}\cdot 1399^{2}$
Root discriminant $17.54$
Ramified primes $7, 97, 1399$
Class number $1$
Class group Trivial
Galois group 18T472

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 5, -17, 86, -101, 164, -128, 130, -78, 21, 8, -23, 35, -27, 17, -12, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 5*x^16 - 12*x^15 + 17*x^14 - 27*x^13 + 35*x^12 - 23*x^11 + 8*x^10 + 21*x^9 - 78*x^8 + 130*x^7 - 128*x^6 + 164*x^5 - 101*x^4 + 86*x^3 - 17*x^2 + 5*x - 1)
 
gp: K = bnfinit(x^18 - 2*x^17 + 5*x^16 - 12*x^15 + 17*x^14 - 27*x^13 + 35*x^12 - 23*x^11 + 8*x^10 + 21*x^9 - 78*x^8 + 130*x^7 - 128*x^6 + 164*x^5 - 101*x^4 + 86*x^3 - 17*x^2 + 5*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 5 x^{16} - 12 x^{15} + 17 x^{14} - 27 x^{13} + 35 x^{12} - 23 x^{11} + 8 x^{10} + 21 x^{9} - 78 x^{8} + 130 x^{7} - 128 x^{6} + 164 x^{5} - 101 x^{4} + 86 x^{3} - 17 x^{2} + 5 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24724476901703653513873=7^{12}\cdot 97^{3}\cdot 1399^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.54$
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:  $7, 97, 1399$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{61020000005467} a^{17} + \frac{26031844926228}{61020000005467} a^{16} + \frac{3961065776314}{61020000005467} a^{15} + \frac{8058465393183}{61020000005467} a^{14} + \frac{8673970383647}{61020000005467} a^{13} - \frac{22376813516661}{61020000005467} a^{12} - \frac{19245888912759}{61020000005467} a^{11} - \frac{786579311895}{61020000005467} a^{10} + \frac{23427538254562}{61020000005467} a^{9} + \frac{15084785340}{1419069767569} a^{8} - \frac{2637019923124}{61020000005467} a^{7} + \frac{27080790843972}{61020000005467} a^{6} + \frac{2076438095630}{61020000005467} a^{5} - \frac{6328617343319}{61020000005467} a^{4} - \frac{14326044649764}{61020000005467} a^{3} + \frac{16293656661954}{61020000005467} a^{2} + \frac{25344660197652}{61020000005467} a - \frac{23292854862734}{61020000005467}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6378.5311844 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T472:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5184
The 88 conjugacy class representatives for t18n472 are not computed
Character table for t18n472 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.2.232897.1, 9.3.164590951.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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
7Data not computed
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
97.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1399Data not computed