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Properties

Label 18.8.28967372723...9343.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,7^{15}\cdot 41^{3}\cdot 97^{4}$
Root discriminant $25.97$
Ramified primes $7, 41, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3569, -6125, 22326, -6698, -13459, 15315, -9044, -274, 4752, -4448, 1564, 186, -456, 260, -58, 2, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^16 + 2*x^15 - 58*x^14 + 260*x^13 - 456*x^12 + 186*x^11 + 1564*x^10 - 4448*x^9 + 4752*x^8 - 274*x^7 - 9044*x^6 + 15315*x^5 - 13459*x^4 - 6698*x^3 + 22326*x^2 - 6125*x - 3569)
 
gp: K = bnfinit(x^18 - 3*x^17 + 9*x^16 + 2*x^15 - 58*x^14 + 260*x^13 - 456*x^12 + 186*x^11 + 1564*x^10 - 4448*x^9 + 4752*x^8 - 274*x^7 - 9044*x^6 + 15315*x^5 - 13459*x^4 - 6698*x^3 + 22326*x^2 - 6125*x - 3569, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 9 x^{16} + 2 x^{15} - 58 x^{14} + 260 x^{13} - 456 x^{12} + 186 x^{11} + 1564 x^{10} - 4448 x^{9} + 4752 x^{8} - 274 x^{7} - 9044 x^{6} + 15315 x^{5} - 13459 x^{4} - 6698 x^{3} + 22326 x^{2} - 6125 x - 3569 \)

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:  \(-28967372723167138004689343=-\,7^{15}\cdot 41^{3}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.97$
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:  $7, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{449583923536176760080836374868900931521} a^{17} - \frac{77291943025444267121810706741126602173}{449583923536176760080836374868900931521} a^{16} + \frac{22939788127658413786050477281727620495}{449583923536176760080836374868900931521} a^{15} + \frac{111818295209416853268470881211162432070}{449583923536176760080836374868900931521} a^{14} + \frac{26248965211736602358061583204421806974}{449583923536176760080836374868900931521} a^{13} - \frac{4377034165427661154639090619286363362}{449583923536176760080836374868900931521} a^{12} - \frac{193830571826968778335380026812547453737}{449583923536176760080836374868900931521} a^{11} + \frac{87182033275364735391473909125135580006}{449583923536176760080836374868900931521} a^{10} + \frac{41128127528641272791167354989797485820}{449583923536176760080836374868900931521} a^{9} + \frac{69718274099037527728792189670917584495}{449583923536176760080836374868900931521} a^{8} + \frac{93807651031178536388648556898161783145}{449583923536176760080836374868900931521} a^{7} + \frac{68636903508045754564186890242182310300}{449583923536176760080836374868900931521} a^{6} + \frac{122110602469489229789242592459116711}{449583923536176760080836374868900931521} a^{5} - \frac{186166016416682302635697586623691186353}{449583923536176760080836374868900931521} a^{4} + \frac{125685553978223021535650630800374647039}{449583923536176760080836374868900931521} a^{3} - \frac{95198117914145975180317511034979974195}{449583923536176760080836374868900931521} a^{2} - \frac{93660382103415137795204377185752296019}{449583923536176760080836374868900931521} a - \frac{13154073542665913448131625413024386700}{449583923536176760080836374868900931521}$

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:  \( 753523.40911 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n765 are not computed
Character table for t18n765 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.5.467890073.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
7Data not computed
41Data not computed
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$