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Properties

Label 18.4.36535393449...8416.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,2^{18}\cdot 3^{18}\cdot 7^{13}\cdot 13^{5}$
Root discriminant $49.88$
Ramified primes $2, 3, 7, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![88829, 295230, 268086, 121238, 60840, 20826, -3967, -19110, -12477, 2754, 2553, -642, -241, 234, 102, -32, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 15*x^16 - 32*x^15 + 102*x^14 + 234*x^13 - 241*x^12 - 642*x^11 + 2553*x^10 + 2754*x^9 - 12477*x^8 - 19110*x^7 - 3967*x^6 + 20826*x^5 + 60840*x^4 + 121238*x^3 + 268086*x^2 + 295230*x + 88829)
 
gp: K = bnfinit(x^18 - 15*x^16 - 32*x^15 + 102*x^14 + 234*x^13 - 241*x^12 - 642*x^11 + 2553*x^10 + 2754*x^9 - 12477*x^8 - 19110*x^7 - 3967*x^6 + 20826*x^5 + 60840*x^4 + 121238*x^3 + 268086*x^2 + 295230*x + 88829, 1)
 

Normalized defining polynomial

\( x^{18} - 15 x^{16} - 32 x^{15} + 102 x^{14} + 234 x^{13} - 241 x^{12} - 642 x^{11} + 2553 x^{10} + 2754 x^{9} - 12477 x^{8} - 19110 x^{7} - 3967 x^{6} + 20826 x^{5} + 60840 x^{4} + 121238 x^{3} + 268086 x^{2} + 295230 x + 88829 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3653539344977096693261726908416=-\,2^{18}\cdot 3^{18}\cdot 7^{13}\cdot 13^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.88$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{88145834618095494746075206422675138699978412} a^{17} - \frac{6489262857424308539919924530732572901260149}{88145834618095494746075206422675138699978412} a^{16} - \frac{7587009952919069085282553161853484075254997}{88145834618095494746075206422675138699978412} a^{15} + \frac{17007360211842246291583323502787941925715553}{88145834618095494746075206422675138699978412} a^{14} + \frac{6458397148395080717969019855230788962835891}{88145834618095494746075206422675138699978412} a^{13} - \frac{176752665601350073230940450740077260574443}{22036458654523873686518801605668784674994603} a^{12} - \frac{10093536276807500924514465783118778626918019}{88145834618095494746075206422675138699978412} a^{11} + \frac{5713463891304665129476292143011938258932629}{44072917309047747373037603211337569349989206} a^{10} - \frac{19968013383264879120336925719818460863092791}{88145834618095494746075206422675138699978412} a^{9} - \frac{7839498324506778806417996747713913300061532}{22036458654523873686518801605668784674994603} a^{8} + \frac{24879847936988905325392577928623109264294203}{88145834618095494746075206422675138699978412} a^{7} + \frac{355609554375732772870286733903590345809866}{22036458654523873686518801605668784674994603} a^{6} - \frac{12130063100952167169985096713066960530861425}{88145834618095494746075206422675138699978412} a^{5} - \frac{3864152991565646718022477246277359610921517}{22036458654523873686518801605668784674994603} a^{4} + \frac{10420958880159892831406639585699779643128309}{22036458654523873686518801605668784674994603} a^{3} - \frac{27573430320730313764220719576634228364701535}{88145834618095494746075206422675138699978412} a^{2} - \frac{8572286619139206911782592473056302358447387}{44072917309047747373037603211337569349989206} a - \frac{25177359806188742305910358583637253241724969}{88145834618095494746075206422675138699978412}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $10$
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:  \( 86397036.0612 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T766:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 144 conjugacy class representatives for t18n766 are not computed
Character table for t18n766 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.25046451847872.1

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R $18$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$$[2, 2]^{6}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.5.1$x^{6} - 52$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$