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Properties

Label 18.18.2251067121...0000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 5^{10}\cdot 17^{11}\cdot 37^{6}$
Root discriminant $92.05$
Ramified primes $2, 5, 17, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times (C_3\times A_4):S_3$ (as 18T156)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![284953, -1514508, 2007202, 2750688, -10063201, 8772146, 1171173, -6602254, 3537283, 245930, -794484, 180264, 45054, -21374, 463, 742, -69, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 69*x^16 + 742*x^15 + 463*x^14 - 21374*x^13 + 45054*x^12 + 180264*x^11 - 794484*x^10 + 245930*x^9 + 3537283*x^8 - 6602254*x^7 + 1171173*x^6 + 8772146*x^5 - 10063201*x^4 + 2750688*x^3 + 2007202*x^2 - 1514508*x + 284953)
 
gp: K = bnfinit(x^18 - 8*x^17 - 69*x^16 + 742*x^15 + 463*x^14 - 21374*x^13 + 45054*x^12 + 180264*x^11 - 794484*x^10 + 245930*x^9 + 3537283*x^8 - 6602254*x^7 + 1171173*x^6 + 8772146*x^5 - 10063201*x^4 + 2750688*x^3 + 2007202*x^2 - 1514508*x + 284953, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} - 69 x^{16} + 742 x^{15} + 463 x^{14} - 21374 x^{13} + 45054 x^{12} + 180264 x^{11} - 794484 x^{10} + 245930 x^{9} + 3537283 x^{8} - 6602254 x^{7} + 1171173 x^{6} + 8772146 x^{5} - 10063201 x^{4} + 2750688 x^{3} + 2007202 x^{2} - 1514508 x + 284953 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(225106712174089155532536320000000000=2^{18}\cdot 5^{10}\cdot 17^{11}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.05$
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:  $2, 5, 17, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{117731799579235795209320833044197623540703311} a^{17} - \frac{15130707714639101679606099954757239602746020}{117731799579235795209320833044197623540703311} a^{16} + \frac{53573914533562660060189909930579380192597175}{117731799579235795209320833044197623540703311} a^{15} + \frac{57700989851939008082298274477545868845442397}{117731799579235795209320833044197623540703311} a^{14} - \frac{31952913241914310233875373960628006693285346}{117731799579235795209320833044197623540703311} a^{13} + \frac{48409990628673895090058617945392445117537952}{117731799579235795209320833044197623540703311} a^{12} + \frac{49703179362811733969100879310375463211502864}{117731799579235795209320833044197623540703311} a^{11} - \frac{5202547233768617313345662448821437381046791}{117731799579235795209320833044197623540703311} a^{10} + \frac{31147145927199173604386322385336547240040362}{117731799579235795209320833044197623540703311} a^{9} + \frac{56754063636842848180404281072141609591396121}{117731799579235795209320833044197623540703311} a^{8} - \frac{40558361942818247934428422037889543564236593}{117731799579235795209320833044197623540703311} a^{7} - \frac{30223594209787004162567713632323415478827920}{117731799579235795209320833044197623540703311} a^{6} + \frac{37460827274100855347251751846026280839770375}{117731799579235795209320833044197623540703311} a^{5} + \frac{29613468848865709088750534151582610463208076}{117731799579235795209320833044197623540703311} a^{4} + \frac{2925904815722892031011826031422698246378735}{117731799579235795209320833044197623540703311} a^{3} - \frac{1898495674463631273544872435040593119892439}{117731799579235795209320833044197623540703311} a^{2} + \frac{5725528084667965864322014298485841664182279}{117731799579235795209320833044197623540703311} a + \frac{31073277626389867753580175664569513848015885}{117731799579235795209320833044197623540703311}$

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:  $17$
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:  \( 882349040481 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times (C_3\times A_4):S_3$ (as 18T156):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 38 conjugacy class representatives for $C_2\times (C_3\times A_4):S_3$
Character table for $C_2\times (C_3\times A_4):S_3$ is not computed

Intermediate fields

3.3.148.1, 6.6.37236800.1, 9.9.169223568520000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
2Data not computed
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.12.10.2$x^{12} + 15 x^{6} + 100$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$
$17$17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.12.8.2$x^{12} - 4913 x^{3} + 918731$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$
$37$37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.0.1$x^{6} - x + 20$$1$$6$$0$$C_6$$[\ ]^{6}$