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Properties

Label 18.18.7963437297...5053.1
Degree $18$
Signature $[18, 0]$
Discriminant $7^{12}\cdot 13\cdot 29\cdot 223^{3}\cdot 1051^{3}\cdot 10583273^{3}$
Root discriminant $591.95$
Ramified primes $7, 13, 29, 223, 1051, 10583273$
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![-25675621091, 50872628097, 38964745521, -39404619099, -21792576465, 9562968054, 5143953720, -947632203, -565874667, 46678194, 33557994, -1211598, -1144536, 15795, 22437, -81, -234, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 234*x^16 - 81*x^15 + 22437*x^14 + 15795*x^13 - 1144536*x^12 - 1211598*x^11 + 33557994*x^10 + 46678194*x^9 - 565874667*x^8 - 947632203*x^7 + 5143953720*x^6 + 9562968054*x^5 - 21792576465*x^4 - 39404619099*x^3 + 38964745521*x^2 + 50872628097*x - 25675621091)
 
gp: K = bnfinit(x^18 - 234*x^16 - 81*x^15 + 22437*x^14 + 15795*x^13 - 1144536*x^12 - 1211598*x^11 + 33557994*x^10 + 46678194*x^9 - 565874667*x^8 - 947632203*x^7 + 5143953720*x^6 + 9562968054*x^5 - 21792576465*x^4 - 39404619099*x^3 + 38964745521*x^2 + 50872628097*x - 25675621091, 1)
 

Normalized defining polynomial

\( x^{18} - 234 x^{16} - 81 x^{15} + 22437 x^{14} + 15795 x^{13} - 1144536 x^{12} - 1211598 x^{11} + 33557994 x^{10} + 46678194 x^{9} - 565874667 x^{8} - 947632203 x^{7} + 5143953720 x^{6} + 9562968054 x^{5} - 21792576465 x^{4} - 39404619099 x^{3} + 38964745521 x^{2} + 50872628097 x - 25675621091 \)

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:  \(79634372972300794853892771897752194830441843575053=7^{12}\cdot 13\cdot 29\cdot 223^{3}\cdot 1051^{3}\cdot 10583273^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $591.95$
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:  $7, 13, 29, 223, 1051, 10583273$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{4} - \frac{1}{9} a^{2} - \frac{2}{9}$, $\frac{1}{9} a^{5} - \frac{1}{9} a^{3} - \frac{2}{9} a$, $\frac{1}{27} a^{6} - \frac{1}{9} a^{2} - \frac{2}{27}$, $\frac{1}{27} a^{7} - \frac{1}{9} a^{3} - \frac{2}{27} a$, $\frac{1}{81} a^{8} + \frac{1}{81} a^{6} - \frac{1}{27} a^{4} - \frac{5}{81} a^{2} - \frac{2}{81}$, $\frac{1}{81} a^{9} + \frac{1}{81} a^{7} - \frac{1}{27} a^{5} - \frac{5}{81} a^{3} - \frac{2}{81} a$, $\frac{1}{243} a^{10} - \frac{1}{243} a^{8} + \frac{4}{243} a^{6} + \frac{1}{243} a^{4} - \frac{19}{243} a^{2} - \frac{14}{243}$, $\frac{1}{243} a^{11} - \frac{1}{243} a^{9} + \frac{4}{243} a^{7} + \frac{1}{243} a^{5} - \frac{19}{243} a^{3} - \frac{14}{243} a$, $\frac{1}{729} a^{12} + \frac{1}{243} a^{8} + \frac{5}{729} a^{6} - \frac{2}{81} a^{4} - \frac{11}{243} a^{2} - \frac{14}{729}$, $\frac{1}{729} a^{13} + \frac{1}{243} a^{9} + \frac{5}{729} a^{7} - \frac{2}{81} a^{5} - \frac{11}{243} a^{3} - \frac{14}{729} a$, $\frac{1}{2187} a^{14} + \frac{1}{2187} a^{12} + \frac{1}{729} a^{10} + \frac{8}{2187} a^{8} - \frac{13}{2187} a^{6} - \frac{17}{729} a^{4} - \frac{47}{2187} a^{2} - \frac{14}{2187}$, $\frac{1}{2187} a^{15} + \frac{1}{2187} a^{13} + \frac{1}{729} a^{11} + \frac{8}{2187} a^{9} - \frac{13}{2187} a^{7} - \frac{17}{729} a^{5} - \frac{47}{2187} a^{3} - \frac{14}{2187} a$, $\frac{1}{6561} a^{16} - \frac{1}{6561} a^{14} + \frac{1}{6561} a^{12} + \frac{2}{6561} a^{10} - \frac{29}{6561} a^{8} - \frac{25}{6561} a^{6} + \frac{55}{6561} a^{4} + \frac{80}{6561} a^{2} + \frac{28}{6561}$, $\frac{1}{203805696686657466691568481369492008777607} a^{17} - \frac{4905621830772183330855936860270294911}{67935232228885822230522827123164002925869} a^{16} - \frac{27047857653083850996712861936318086697}{203805696686657466691568481369492008777607} a^{15} - \frac{13525953026978026917001508233843213213}{67935232228885822230522827123164002925869} a^{14} - \frac{4979663482567620592572780764369819513}{203805696686657466691568481369492008777607} a^{13} + \frac{7535637269847940518887958311039522819}{22645077409628607410174275707721334308623} a^{12} + \frac{374338173426481870343069790071057902499}{203805696686657466691568481369492008777607} a^{11} + \frac{8810065793091276203154518189878784707}{67935232228885822230522827123164002925869} a^{10} - \frac{404625363562072051424490087672290768249}{203805696686657466691568481369492008777607} a^{9} - \frac{211177503703049571172560145469346942815}{67935232228885822230522827123164002925869} a^{8} + \frac{3697299255485595054417229879700287877771}{203805696686657466691568481369492008777607} a^{7} - \frac{410929879113685394846489160863875863166}{22645077409628607410174275707721334308623} a^{6} - \frac{7094007992888620926373600784960968198664}{203805696686657466691568481369492008777607} a^{5} - \frac{2912218329841093521972574348416546722293}{67935232228885822230522827123164002925869} a^{4} - \frac{4174898338493226025638381134501723151592}{203805696686657466691568481369492008777607} a^{3} + \frac{324423834406593065129914659148145222161}{2342594214789166283811131969764275962961} a^{2} + \frac{56933026624127631855277269181153810294315}{203805696686657466691568481369492008777607} a + \frac{1989062079924139333150824580806998383358}{7548359136542869136724758569240444769541}$

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:  $17$
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:  \( 18124462468600000000 \) (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

\(\Q(\zeta_{7})^+\), 6.6.5955520696232429.1

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 $18$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.9.0.1}{9} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ $18$ $18$ R $18$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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$7.6.4.3$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}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$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$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
223Data not computed
1051Data not computed
10583273Data not computed