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Properties

Label 18.10.6867244576...5625.1
Degree $18$
Signature $[10, 4]$
Discriminant $5^{9}\cdot 7^{12}\cdot 181\cdot 37462668601^{2}$
Root discriminant $163.36$
Ramified primes $5, 7, 181, 37462668601$
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![1353125341, -4710371951, 6735093618, -4711668497, 1137717969, 585728929, -470393141, 64149377, 41375910, -15079175, -752532, 1072913, -79332, -36841, 5114, 618, -118, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 118*x^16 + 618*x^15 + 5114*x^14 - 36841*x^13 - 79332*x^12 + 1072913*x^11 - 752532*x^10 - 15079175*x^9 + 41375910*x^8 + 64149377*x^7 - 470393141*x^6 + 585728929*x^5 + 1137717969*x^4 - 4711668497*x^3 + 6735093618*x^2 - 4710371951*x + 1353125341)
 
gp: K = bnfinit(x^18 - 4*x^17 - 118*x^16 + 618*x^15 + 5114*x^14 - 36841*x^13 - 79332*x^12 + 1072913*x^11 - 752532*x^10 - 15079175*x^9 + 41375910*x^8 + 64149377*x^7 - 470393141*x^6 + 585728929*x^5 + 1137717969*x^4 - 4711668497*x^3 + 6735093618*x^2 - 4710371951*x + 1353125341, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 118 x^{16} + 618 x^{15} + 5114 x^{14} - 36841 x^{13} - 79332 x^{12} + 1072913 x^{11} - 752532 x^{10} - 15079175 x^{9} + 41375910 x^{8} + 64149377 x^{7} - 470393141 x^{6} + 585728929 x^{5} + 1137717969 x^{4} - 4711668497 x^{3} + 6735093618 x^{2} - 4710371951 x + 1353125341 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6867244576973684096872169267219884765625=5^{9}\cdot 7^{12}\cdot 181\cdot 37462668601^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $163.36$
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:  $5, 7, 181, 37462668601$
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}{99862348934881197922923136155009141739414144520214143} a^{17} - \frac{29902549281379222186469268689783036067771235683995255}{99862348934881197922923136155009141739414144520214143} a^{16} + \frac{39102034548783135935146411405337278065259624158692931}{99862348934881197922923136155009141739414144520214143} a^{15} - \frac{48093102593737906890867768177996914846415101780003913}{99862348934881197922923136155009141739414144520214143} a^{14} - \frac{40187782886270423252824540319618442380956853976456951}{99862348934881197922923136155009141739414144520214143} a^{13} - \frac{39767856807500375409460347160884948599169238668433353}{99862348934881197922923136155009141739414144520214143} a^{12} - \frac{14078643928569554309304071208376975326503233217458081}{99862348934881197922923136155009141739414144520214143} a^{11} + \frac{35266054482723859093031262764675423034077785714370342}{99862348934881197922923136155009141739414144520214143} a^{10} - \frac{2635851853013495765835761923610963590827202512182796}{99862348934881197922923136155009141739414144520214143} a^{9} - \frac{2310319073057418698209292747291030989867960263877094}{99862348934881197922923136155009141739414144520214143} a^{8} - \frac{1577663769895552741144024170248419136666566597860288}{99862348934881197922923136155009141739414144520214143} a^{7} + \frac{32209301179414595055470514242110989130728314558119237}{99862348934881197922923136155009141739414144520214143} a^{6} - \frac{10482832037985247330925400367995665189593239370587920}{99862348934881197922923136155009141739414144520214143} a^{5} + \frac{47055704627348916892142223497245434642971973721626266}{99862348934881197922923136155009141739414144520214143} a^{4} - \frac{5579156300606866748049813805496505113260582647033487}{99862348934881197922923136155009141739414144520214143} a^{3} - \frac{9312319789132980830901391290554375467531281209006074}{99862348934881197922923136155009141739414144520214143} a^{2} - \frac{45571512776704453506038913292730722101668760449078990}{99862348934881197922923136155009141739414144520214143} a - \frac{2654467892599571753004070359570571372741365453258038}{99862348934881197922923136155009141739414144520214143}$

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:  $13$
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:  \( 5551249345200 \) (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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $18$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
5Data not computed
$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}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
37462668601Data not computed