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Properties

Label 18.10.3356683345...0329.1
Degree $18$
Signature $[10, 4]$
Discriminant $7^{12}\cdot 13\cdot 57138677^{3}$
Root discriminant $82.81$
Ramified primes $7, 13, 57138677$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T926

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10933, 0, -51973, 23494, 41523, -4076, -10007, -32454, 27064, -9962, 10241, -6019, 1676, -864, 399, -81, 22, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 22*x^16 - 81*x^15 + 399*x^14 - 864*x^13 + 1676*x^12 - 6019*x^11 + 10241*x^10 - 9962*x^9 + 27064*x^8 - 32454*x^7 - 10007*x^6 - 4076*x^5 + 41523*x^4 + 23494*x^3 - 51973*x^2 + 10933)
 
gp: K = bnfinit(x^18 - 8*x^17 + 22*x^16 - 81*x^15 + 399*x^14 - 864*x^13 + 1676*x^12 - 6019*x^11 + 10241*x^10 - 9962*x^9 + 27064*x^8 - 32454*x^7 - 10007*x^6 - 4076*x^5 + 41523*x^4 + 23494*x^3 - 51973*x^2 + 10933, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 22 x^{16} - 81 x^{15} + 399 x^{14} - 864 x^{13} + 1676 x^{12} - 6019 x^{11} + 10241 x^{10} - 9962 x^{9} + 27064 x^{8} - 32454 x^{7} - 10007 x^{6} - 4076 x^{5} + 41523 x^{4} + 23494 x^{3} - 51973 x^{2} + 10933 \)

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:  \(33566833452761315517557908483560329=7^{12}\cdot 13\cdot 57138677^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.81$
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:  $7, 13, 57138677$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{592395273933019440064474466475726313} a^{17} - \frac{100062898079295851786031429835363848}{592395273933019440064474466475726313} a^{16} - \frac{231487129319207042643560151405862272}{592395273933019440064474466475726313} a^{15} - \frac{43195101453378717419361453439558455}{592395273933019440064474466475726313} a^{14} + \frac{183781608623137667025497711532468798}{592395273933019440064474466475726313} a^{13} - \frac{129834849901007812050488761500978114}{592395273933019440064474466475726313} a^{12} - \frac{74746279570142608011056807993468113}{592395273933019440064474466475726313} a^{11} - \frac{241414606096074682605060104109299293}{592395273933019440064474466475726313} a^{10} - \frac{170319805845942650430269857146462948}{592395273933019440064474466475726313} a^{9} + \frac{252419426914219434497369494559904711}{592395273933019440064474466475726313} a^{8} - \frac{266002639775398874339049574342471880}{592395273933019440064474466475726313} a^{7} - \frac{95971580972408831131229596281552240}{592395273933019440064474466475726313} a^{6} - \frac{280196361952534461835305186693316899}{592395273933019440064474466475726313} a^{5} + \frac{23545480910750857851627103235651518}{592395273933019440064474466475726313} a^{4} - \frac{193157927671114574404161913947766920}{592395273933019440064474466475726313} a^{3} + \frac{274803130473452041504374627203873236}{592395273933019440064474466475726313} a^{2} + \frac{17822345487628106467382328145214559}{592395273933019440064474466475726313} a - \frac{1082991226070873115056370579789425}{20427423239069635864292222981921597}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 58990119670.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T926:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1119744
The 267 conjugacy class representatives for t18n926 are not computed
Character table for t18n926 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.6.137189963477.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ R $18$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ $18$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
57138677Data not computed