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Properties

Label 18.12.7575432877...0567.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,7^{13}\cdot 52879^{4}$
Root discriminant $45.71$
Ramified primes $7, 52879$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T767

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-435877, -33940, 958638, -542146, -604143, 931975, -26460, -452057, 75962, 103449, -14019, -13126, 528, 590, 202, 27, -32, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 32*x^16 + 27*x^15 + 202*x^14 + 590*x^13 + 528*x^12 - 13126*x^11 - 14019*x^10 + 103449*x^9 + 75962*x^8 - 452057*x^7 - 26460*x^6 + 931975*x^5 - 604143*x^4 - 542146*x^3 + 958638*x^2 - 33940*x - 435877)
 
gp: K = bnfinit(x^18 - x^17 - 32*x^16 + 27*x^15 + 202*x^14 + 590*x^13 + 528*x^12 - 13126*x^11 - 14019*x^10 + 103449*x^9 + 75962*x^8 - 452057*x^7 - 26460*x^6 + 931975*x^5 - 604143*x^4 - 542146*x^3 + 958638*x^2 - 33940*x - 435877, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 32 x^{16} + 27 x^{15} + 202 x^{14} + 590 x^{13} + 528 x^{12} - 13126 x^{11} - 14019 x^{10} + 103449 x^{9} + 75962 x^{8} - 452057 x^{7} - 26460 x^{6} + 931975 x^{5} - 604143 x^{4} - 542146 x^{3} + 958638 x^{2} - 33940 x - 435877 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-757543287754796256482850550567=-\,7^{13}\cdot 52879^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.71$
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:  $7, 52879$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{76680512660818071787838574879678765285526928482401} a^{17} + \frac{7073551850017815724556821200017395484902973818441}{76680512660818071787838574879678765285526928482401} a^{16} - \frac{281808188663729458991124773903820395643256291226}{76680512660818071787838574879678765285526928482401} a^{15} + \frac{4219430630471122931089026641028941529088812254534}{76680512660818071787838574879678765285526928482401} a^{14} - \frac{36979705570325230794411497387196208573854407695559}{76680512660818071787838574879678765285526928482401} a^{13} - \frac{3416360986206418655481787741617883398787531518367}{76680512660818071787838574879678765285526928482401} a^{12} - \frac{35538594332702879687382104992984540341100870714637}{76680512660818071787838574879678765285526928482401} a^{11} + \frac{33989211490636229217384336290605242859278270859615}{76680512660818071787838574879678765285526928482401} a^{10} - \frac{1426181913786044247673690067627179569929336981822}{76680512660818071787838574879678765285526928482401} a^{9} - \frac{4071962494529984908946746413452972592261811933876}{76680512660818071787838574879678765285526928482401} a^{8} + \frac{27774220926741289961224049850243391243425274037284}{76680512660818071787838574879678765285526928482401} a^{7} + \frac{18662668472199294529556832907432888933051390132753}{76680512660818071787838574879678765285526928482401} a^{6} - \frac{420291438585918079201405367957175453241740985429}{25560170886939357262612858293226255095175642827467} a^{5} - \frac{9281716452962275734031021696336505667564593781834}{25560170886939357262612858293226255095175642827467} a^{4} + \frac{19447483139549887800474665292057875737937088545863}{76680512660818071787838574879678765285526928482401} a^{3} + \frac{18248351332858489533908039005435917965548022247476}{76680512660818071787838574879678765285526928482401} a^{2} - \frac{31983703905082177472830774145108155045041292949285}{76680512660818071787838574879678765285526928482401} a + \frac{1724859411461526236542454605972380522134372074366}{76680512660818071787838574879678765285526928482401}$

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:  $14$
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:  \( 361973002.521 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T767:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n767 are not computed
Character table for t18n767 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.7.6221161471.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
52879Data not computed