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Properties

Label 18.16.1800838078...0000.1
Degree $18$
Signature $[16, 1]$
Discriminant $-\,2^{12}\cdot 5^{9}\cdot 7^{12}\cdot 379^{2}\cdot 106405699^{2}$
Root discriminant $195.87$
Ramified primes $2, 5, 7, 379, 106405699$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 18T857

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![295064741, 1249636670, 2281855540, 2274145760, 1219578806, 187792795, -171939012, -101964173, -6884230, 10108718, 2428094, -427804, -182721, 7297, 6745, 3, -128, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 128*x^16 + 3*x^15 + 6745*x^14 + 7297*x^13 - 182721*x^12 - 427804*x^11 + 2428094*x^10 + 10108718*x^9 - 6884230*x^8 - 101964173*x^7 - 171939012*x^6 + 187792795*x^5 + 1219578806*x^4 + 2274145760*x^3 + 2281855540*x^2 + 1249636670*x + 295064741)
 
gp: K = bnfinit(x^18 - x^17 - 128*x^16 + 3*x^15 + 6745*x^14 + 7297*x^13 - 182721*x^12 - 427804*x^11 + 2428094*x^10 + 10108718*x^9 - 6884230*x^8 - 101964173*x^7 - 171939012*x^6 + 187792795*x^5 + 1219578806*x^4 + 2274145760*x^3 + 2281855540*x^2 + 1249636670*x + 295064741, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 128 x^{16} + 3 x^{15} + 6745 x^{14} + 7297 x^{13} - 182721 x^{12} - 427804 x^{11} + 2428094 x^{10} + 10108718 x^{9} - 6884230 x^{8} - 101964173 x^{7} - 171939012 x^{6} + 187792795 x^{5} + 1219578806 x^{4} + 2274145760 x^{3} + 2281855540 x^{2} + 1249636670 x + 295064741 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-180083807836107946968856890731528000000000=-\,2^{12}\cdot 5^{9}\cdot 7^{12}\cdot 379^{2}\cdot 106405699^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $195.87$
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:  $2, 5, 7, 379, 106405699$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2778003744057559441633792559423481719} a^{17} - \frac{176670918619918788160864234241988752}{2778003744057559441633792559423481719} a^{16} + \frac{1305026571862508685603542011798774373}{2778003744057559441633792559423481719} a^{15} + \frac{47250211850948414266405044996967326}{95793232553708946263234226187016611} a^{14} + \frac{134918095046374932817580418795680587}{2778003744057559441633792559423481719} a^{13} + \frac{733625792552062404464860942234559310}{2778003744057559441633792559423481719} a^{12} + \frac{1172985502473347139518112873158420539}{2778003744057559441633792559423481719} a^{11} - \frac{1307492834459827255415906879669832304}{2778003744057559441633792559423481719} a^{10} - \frac{1090010235203045500916480229896606392}{2778003744057559441633792559423481719} a^{9} - \frac{1080748030748162279798314470286091584}{2778003744057559441633792559423481719} a^{8} + \frac{498536498117103549123935219789011979}{2778003744057559441633792559423481719} a^{7} - \frac{141166948711512956595623713866204764}{2778003744057559441633792559423481719} a^{6} - \frac{846229195847964480352763075572183609}{2778003744057559441633792559423481719} a^{5} - \frac{1367890294907911325309821644302536528}{2778003744057559441633792559423481719} a^{4} + \frac{1003321105313368567304326736691626514}{2778003744057559441633792559423481719} a^{3} - \frac{403351218238854414676512071205946713}{2778003744057559441633792559423481719} a^{2} + \frac{278743912591671457488773646215323594}{2778003744057559441633792559423481719} a + \frac{773379466079450642443620138259980126}{2778003744057559441633792559423481719}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $16$
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:  \( 127101380105000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T857:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 159 conjugacy class representatives for t18n857 are not computed
Character table for t18n857 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 6.6.300125.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $18$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ $18$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.7$x^{12} - 48 x^{10} + 53 x^{8} + 40 x^{6} + 27 x^{4} - 56 x^{2} + 47$$2$$6$$12$12T134$[2, 2, 2, 2, 2, 2]^{6}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
379Data not computed
106405699Data not computed