↑

Properties

Label 18.0.41716020420...4144.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{9}\cdot 7^{12}\cdot 97^{3}$
Root discriminant $34.23$
Ramified primes $2, 3, 7, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 18T176

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1085323, 2436132, 2252001, 1130120, 420755, 223408, 126940, 27200, -15270, -10134, -551, 1178, 339, -48, -17, -2, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - 2*x^15 - 17*x^14 - 48*x^13 + 339*x^12 + 1178*x^11 - 551*x^10 - 10134*x^9 - 15270*x^8 + 27200*x^7 + 126940*x^6 + 223408*x^5 + 420755*x^4 + 1130120*x^3 + 2252001*x^2 + 2436132*x + 1085323)
 
gp: K = bnfinit(x^18 + 2*x^16 - 2*x^15 - 17*x^14 - 48*x^13 + 339*x^12 + 1178*x^11 - 551*x^10 - 10134*x^9 - 15270*x^8 + 27200*x^7 + 126940*x^6 + 223408*x^5 + 420755*x^4 + 1130120*x^3 + 2252001*x^2 + 2436132*x + 1085323, 1)
 

Normalized defining polynomial

\( x^{18} + 2 x^{16} - 2 x^{15} - 17 x^{14} - 48 x^{13} + 339 x^{12} + 1178 x^{11} - 551 x^{10} - 10134 x^{9} - 15270 x^{8} + 27200 x^{7} + 126940 x^{6} + 223408 x^{5} + 420755 x^{4} + 1130120 x^{3} + 2252001 x^{2} + 2436132 x + 1085323 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-4171602042088091202978054144=-\,2^{24}\cdot 3^{9}\cdot 7^{12}\cdot 97^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.23$
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:  $2, 3, 7, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{667} a^{16} - \frac{182}{667} a^{15} - \frac{18}{667} a^{14} - \frac{201}{667} a^{13} + \frac{6}{23} a^{12} + \frac{248}{667} a^{11} - \frac{282}{667} a^{10} + \frac{205}{667} a^{9} + \frac{95}{667} a^{8} + \frac{132}{667} a^{7} + \frac{286}{667} a^{6} - \frac{328}{667} a^{5} + \frac{96}{667} a^{4} + \frac{13}{29} a^{3} - \frac{80}{667} a^{2} - \frac{329}{667} a + \frac{252}{667}$, $\frac{1}{1479441713002371912699304660130285310177095510219} a^{17} + \frac{577321948084691739122882751837648140124063663}{1479441713002371912699304660130285310177095510219} a^{16} - \frac{325155323836184077716724786308974534819856850302}{1479441713002371912699304660130285310177095510219} a^{15} - \frac{621281646095035103256718967126676690288394223487}{1479441713002371912699304660130285310177095510219} a^{14} + \frac{120929263950735250093081084990426204489662318650}{1479441713002371912699304660130285310177095510219} a^{13} - \frac{707452527565736714001455870858732769673993898941}{1479441713002371912699304660130285310177095510219} a^{12} - \frac{666185801567090884254408763996317213978467638072}{1479441713002371912699304660130285310177095510219} a^{11} - \frac{589544306617583261272730390557007316531613488872}{1479441713002371912699304660130285310177095510219} a^{10} - \frac{63974889660102737681590806804981779333915373}{51015231482840410782734643452768458971623983111} a^{9} + \frac{715135979838373626394981829784183338056303300765}{1479441713002371912699304660130285310177095510219} a^{8} - \frac{25147780795501362720923712462429712519503203268}{1479441713002371912699304660130285310177095510219} a^{7} + \frac{419331064040501602773154817055017486549874518584}{1479441713002371912699304660130285310177095510219} a^{6} - \frac{384908100782825778368463190136455473538252839260}{1479441713002371912699304660130285310177095510219} a^{5} - \frac{615262961377461991959480519188459998510564370031}{1479441713002371912699304660130285310177095510219} a^{4} - \frac{4351128617831517391184274050519887165233139234}{51015231482840410782734643452768458971623983111} a^{3} + \frac{202191709609210040051374923968083073170226337578}{1479441713002371912699304660130285310177095510219} a^{2} + \frac{507866764446814724892750122335554730708381020421}{1479441713002371912699304660130285310177095510219} a + \frac{624274081103069720077923951245277048648676869576}{1479441713002371912699304660130285310177095510219}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $8$
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:  \( 411762.512234 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T176:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 40 conjugacy class representatives for t18n176
Character table for t18n176 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.1.588.1, 9.3.203297472.1

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 R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.9.2$x^{12} - 9 x^{4} + 27$$4$$3$$9$$D_4 \times C_3$$[\ ]_{4}^{6}$
7Data not computed
97Data not computed