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Properties

Label 18.18.1736353743...8125.1
Degree $18$
Signature $[18, 0]$
Discriminant $5^{9}\cdot 7^{12}\cdot 41\cdot 281\cdot 236113465771^{2}$
Root discriminant $252.47$
Ramified primes $5, 7, 41, 281, 236113465771$
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![-2451406229, -5694641308, -1206137365, 6983075268, 7089009467, 1787229816, -807007244, -498874326, -12517755, 42146147, 6192513, -1673273, -392323, 32286, 11507, -245, -168, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 168*x^16 - 245*x^15 + 11507*x^14 + 32286*x^13 - 392323*x^12 - 1673273*x^11 + 6192513*x^10 + 42146147*x^9 - 12517755*x^8 - 498874326*x^7 - 807007244*x^6 + 1787229816*x^5 + 7089009467*x^4 + 6983075268*x^3 - 1206137365*x^2 - 5694641308*x - 2451406229)
 
gp: K = bnfinit(x^18 - 168*x^16 - 245*x^15 + 11507*x^14 + 32286*x^13 - 392323*x^12 - 1673273*x^11 + 6192513*x^10 + 42146147*x^9 - 12517755*x^8 - 498874326*x^7 - 807007244*x^6 + 1787229816*x^5 + 7089009467*x^4 + 6983075268*x^3 - 1206137365*x^2 - 5694641308*x - 2451406229, 1)
 

Normalized defining polynomial

\( x^{18} - 168 x^{16} - 245 x^{15} + 11507 x^{14} + 32286 x^{13} - 392323 x^{12} - 1673273 x^{11} + 6192513 x^{10} + 42146147 x^{9} - 12517755 x^{8} - 498874326 x^{7} - 807007244 x^{6} + 1787229816 x^{5} + 7089009467 x^{4} + 6983075268 x^{3} - 1206137365 x^{2} - 5694641308 x - 2451406229 \)

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:  \(17363537439858654395185498512903406173828125=5^{9}\cdot 7^{12}\cdot 41\cdot 281\cdot 236113465771^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $252.47$
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:  $5, 7, 41, 281, 236113465771$
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}{4043434655153003046056925457711133023683228768270554389} a^{17} - \frac{1521613699069047619045792410626642948619830679091368514}{4043434655153003046056925457711133023683228768270554389} a^{16} + \frac{877375147602712072940871198655083759823628557959915927}{4043434655153003046056925457711133023683228768270554389} a^{15} - \frac{1588621777598285670250882956850534142461911139695094336}{4043434655153003046056925457711133023683228768270554389} a^{14} + \frac{1192532173459214934270087468098672691061862756145160737}{4043434655153003046056925457711133023683228768270554389} a^{13} + \frac{383720457881794747413424869676638901991182632705698688}{4043434655153003046056925457711133023683228768270554389} a^{12} + \frac{307466658688381260468910870908620089867529851184493310}{4043434655153003046056925457711133023683228768270554389} a^{11} - \frac{212611339439550754282077855767149264347879388172021581}{4043434655153003046056925457711133023683228768270554389} a^{10} + \frac{1248318385880460685401867568189733862869325623438160045}{4043434655153003046056925457711133023683228768270554389} a^{9} - \frac{612137707607070718167199423814597423301245703091693772}{4043434655153003046056925457711133023683228768270554389} a^{8} - \frac{151613187670503577856498995775987191993185552567920534}{311033435011769465081301958285471771052556059097734953} a^{7} + \frac{549435526596089658608902843502557084029178349145647942}{4043434655153003046056925457711133023683228768270554389} a^{6} + \frac{213845959187914621658415854326440627556520085589526539}{4043434655153003046056925457711133023683228768270554389} a^{5} - \frac{3174435257891386887926838235673118488680714226285553}{98620357442756171855046962383198366431298262640745229} a^{4} + \frac{915669219382524522135964434580707442838170655413671435}{4043434655153003046056925457711133023683228768270554389} a^{3} - \frac{382720006248552771625434497173500894913056835755203234}{4043434655153003046056925457711133023683228768270554389} a^{2} - \frac{583466979967464808040856975942190905699877563325575782}{4043434655153003046056925457711133023683228768270554389} a + \frac{7225312018488058183785722342764918965843191555779490}{19163197417786744294108651458346602007977387527348599}$

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:  $17$
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:  \( 1949616119630000 \) (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 $18$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $18$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.3.0.1$x^{3} - x + 13$$1$$3$$0$$C_3$$[\ ]^{3}$
281Data not computed
236113465771Data not computed