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Properties

Label 18.14.6214900093...5625.1
Degree $18$
Signature $[14, 2]$
Discriminant $5^{9}\cdot 7^{12}\cdot 701\cdot 2689^{2}\cdot 67346341^{2}$
Root discriminant $209.83$
Ramified primes $5, 7, 701, 2689, 67346341$
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![1656514609, 7530492799, 14049712931, 13917309913, 7472596864, 1502886480, -585458473, -416975522, -52542708, 25467429, 7949527, -460550, -403555, -9911, 10751, 486, -157, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - 157*x^16 + 486*x^15 + 10751*x^14 - 9911*x^13 - 403555*x^12 - 460550*x^11 + 7949527*x^10 + 25467429*x^9 - 52542708*x^8 - 416975522*x^7 - 585458473*x^6 + 1502886480*x^5 + 7472596864*x^4 + 13917309913*x^3 + 14049712931*x^2 + 7530492799*x + 1656514609)
 
gp: K = bnfinit(x^18 - 5*x^17 - 157*x^16 + 486*x^15 + 10751*x^14 - 9911*x^13 - 403555*x^12 - 460550*x^11 + 7949527*x^10 + 25467429*x^9 - 52542708*x^8 - 416975522*x^7 - 585458473*x^6 + 1502886480*x^5 + 7472596864*x^4 + 13917309913*x^3 + 14049712931*x^2 + 7530492799*x + 1656514609, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - 157 x^{16} + 486 x^{15} + 10751 x^{14} - 9911 x^{13} - 403555 x^{12} - 460550 x^{11} + 7949527 x^{10} + 25467429 x^{9} - 52542708 x^{8} - 416975522 x^{7} - 585458473 x^{6} + 1502886480 x^{5} + 7472596864 x^{4} + 13917309913 x^{3} + 14049712931 x^{2} + 7530492799 x + 1656514609 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(621490009368905226882279513578805666015625=5^{9}\cdot 7^{12}\cdot 701\cdot 2689^{2}\cdot 67346341^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $209.83$
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:  $5, 7, 701, 2689, 67346341$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{29} a^{16} + \frac{1}{29} a^{15} + \frac{7}{29} a^{14} - \frac{10}{29} a^{13} - \frac{6}{29} a^{12} - \frac{14}{29} a^{11} - \frac{8}{29} a^{10} + \frac{1}{29} a^{9} + \frac{7}{29} a^{8} + \frac{3}{29} a^{7} + \frac{7}{29} a^{6} - \frac{14}{29} a^{5} + \frac{1}{29} a^{4} - \frac{10}{29} a^{3} + \frac{9}{29} a^{2} - \frac{7}{29} a + \frac{8}{29}$, $\frac{1}{109446360694866576830501994125239194780974167159238266963891} a^{17} + \frac{1120115804724193051290376926418847720615504070480431138749}{109446360694866576830501994125239194780974167159238266963891} a^{16} + \frac{6855243868966759511039651487188555301721755406577798377622}{109446360694866576830501994125239194780974167159238266963891} a^{15} + \frac{16735089457347722196369236298178027937177229520228225336475}{109446360694866576830501994125239194780974167159238266963891} a^{14} + \frac{47719996503179542174600815927718687933140131538000346014354}{109446360694866576830501994125239194780974167159238266963891} a^{13} + \frac{34587365443327532650731712522335251382041138118228024735046}{109446360694866576830501994125239194780974167159238266963891} a^{12} + \frac{39400861584421681918710968769021033994998843160002370676308}{109446360694866576830501994125239194780974167159238266963891} a^{11} + \frac{10302151870554962176710394243708384022134442469275337740206}{109446360694866576830501994125239194780974167159238266963891} a^{10} + \frac{38001106812074144968197799803957520531624527350714633378502}{109446360694866576830501994125239194780974167159238266963891} a^{9} - \frac{39924339201502868626985850394699195812170450262295191911447}{109446360694866576830501994125239194780974167159238266963891} a^{8} - \frac{14884051849609638600809662989744753522238641859585319059921}{109446360694866576830501994125239194780974167159238266963891} a^{7} - \frac{25380946912849399468868753489512897630408126128943064049838}{109446360694866576830501994125239194780974167159238266963891} a^{6} - \frac{26724544406002232746271578553004270117879069985077671191590}{109446360694866576830501994125239194780974167159238266963891} a^{5} + \frac{49689088577867507501387719575261127361676743298459530636733}{109446360694866576830501994125239194780974167159238266963891} a^{4} - \frac{23747215853099332964364984982464499427451128998908165613865}{109446360694866576830501994125239194780974167159238266963891} a^{3} - \frac{39843295992406061368769273099547892301869738048629879985037}{109446360694866576830501994125239194780974167159238266963891} a^{2} - \frac{49817816766584106821439297625027626955191583715100611453667}{109446360694866576830501994125239194780974167159238266963891} a - \frac{43761366134260189913842365926121567967560765820943908630194}{109446360694866576830501994125239194780974167159238266963891}$

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:  $15$
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:  \( 188558726092000 \) (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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $18$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
5Data not computed
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
701Data not computed
2689Data not computed
67346341Data not computed