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Properties

Label 18.2.34492534601...7296.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{6}\cdot 3^{18}\cdot 7^{14}\cdot 29^{5}$
Root discriminant $43.75$
Ramified primes $2, 3, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1757, -12243, 19278, -18689, 77940, -33933, 6029, 858, 12999, 4068, 714, -492, 19, -57, -3, -27, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 27*x^15 - 3*x^14 - 57*x^13 + 19*x^12 - 492*x^11 + 714*x^10 + 4068*x^9 + 12999*x^8 + 858*x^7 + 6029*x^6 - 33933*x^5 + 77940*x^4 - 18689*x^3 + 19278*x^2 - 12243*x + 1757)
 
gp: K = bnfinit(x^18 - 27*x^15 - 3*x^14 - 57*x^13 + 19*x^12 - 492*x^11 + 714*x^10 + 4068*x^9 + 12999*x^8 + 858*x^7 + 6029*x^6 - 33933*x^5 + 77940*x^4 - 18689*x^3 + 19278*x^2 - 12243*x + 1757, 1)
 

Normalized defining polynomial

\( x^{18} - 27 x^{15} - 3 x^{14} - 57 x^{13} + 19 x^{12} - 492 x^{11} + 714 x^{10} + 4068 x^{9} + 12999 x^{8} + 858 x^{7} + 6029 x^{6} - 33933 x^{5} + 77940 x^{4} - 18689 x^{3} + 19278 x^{2} - 12243 x + 1757 \)

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:  \(344925346014816475591906687296=2^{6}\cdot 3^{18}\cdot 7^{14}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.75$
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:  $2, 3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{188033159921269156020620600403478195959931501} a^{17} + \frac{15335500360196369686974354199831782108747458}{188033159921269156020620600403478195959931501} a^{16} + \frac{39979910922070335584743832971627073993570296}{188033159921269156020620600403478195959931501} a^{15} - \frac{72161329673207405966343243821356086559604925}{188033159921269156020620600403478195959931501} a^{14} + \frac{65563922869382930239276395420455224213912737}{188033159921269156020620600403478195959931501} a^{13} - \frac{55413895204525961951966483733499324267020186}{188033159921269156020620600403478195959931501} a^{12} - \frac{93691305632469685007641788569392413297959437}{188033159921269156020620600403478195959931501} a^{11} - \frac{27945133556360835433442689868275387312963018}{188033159921269156020620600403478195959931501} a^{10} + \frac{82062239255817253516383017321065074670444955}{188033159921269156020620600403478195959931501} a^{9} + \frac{43718808327456876412551660671109352939590883}{188033159921269156020620600403478195959931501} a^{8} + \frac{87236232092804079042874935801393689429176558}{188033159921269156020620600403478195959931501} a^{7} + \frac{5280119742471952405295666333700953282806802}{188033159921269156020620600403478195959931501} a^{6} + \frac{32551041993920159555151399272485867594199479}{188033159921269156020620600403478195959931501} a^{5} - \frac{45318137406037281294065419306505919644098792}{188033159921269156020620600403478195959931501} a^{4} - \frac{12747224546572697026213307721026586130722990}{188033159921269156020620600403478195959931501} a^{3} + \frac{23443219245449927758249881820173875623621483}{188033159921269156020620600403478195959931501} a^{2} - \frac{79287397736180734997549318243930574998032831}{188033159921269156020620600403478195959931501} a - \frac{58665586514206207716427821359688717546245218}{188033159921269156020620600403478195959931501}$

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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 47974020.3852 \) (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

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
3Data not computed
$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}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$