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Properties

Label 18.16.1232291872...3088.2
Degree $18$
Signature $[16, 1]$
Discriminant $-\,2^{6}\cdot 3^{9}\cdot 7^{12}\cdot 643^{6}$
Root discriminant $68.92$
Ramified primes $2, 3, 7, 643$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-56953, -214976, -104560, 503735, 596828, -289443, -643655, -22604, 257071, 23367, -55094, 1448, 6842, -1588, -360, 208, -8, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 8*x^16 + 208*x^15 - 360*x^14 - 1588*x^13 + 6842*x^12 + 1448*x^11 - 55094*x^10 + 23367*x^9 + 257071*x^8 - 22604*x^7 - 643655*x^6 - 289443*x^5 + 596828*x^4 + 503735*x^3 - 104560*x^2 - 214976*x - 56953)
 
gp: K = bnfinit(x^18 - 8*x^17 - 8*x^16 + 208*x^15 - 360*x^14 - 1588*x^13 + 6842*x^12 + 1448*x^11 - 55094*x^10 + 23367*x^9 + 257071*x^8 - 22604*x^7 - 643655*x^6 - 289443*x^5 + 596828*x^4 + 503735*x^3 - 104560*x^2 - 214976*x - 56953, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} - 8 x^{16} + 208 x^{15} - 360 x^{14} - 1588 x^{13} + 6842 x^{12} + 1448 x^{11} - 55094 x^{10} + 23367 x^{9} + 257071 x^{8} - 22604 x^{7} - 643655 x^{6} - 289443 x^{5} + 596828 x^{4} + 503735 x^{3} - 104560 x^{2} - 214976 x - 56953 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1232291872634528753686122825233088=-\,2^{6}\cdot 3^{9}\cdot 7^{12}\cdot 643^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.92$
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:  $2, 3, 7, 643$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{4}{9} a^{9} - \frac{2}{9} a^{8} + \frac{2}{9} a^{7} - \frac{2}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{9} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{90} a^{15} - \frac{1}{18} a^{13} + \frac{11}{90} a^{12} + \frac{1}{30} a^{11} + \frac{2}{45} a^{10} + \frac{8}{45} a^{9} + \frac{17}{90} a^{8} + \frac{11}{45} a^{7} + \frac{2}{15} a^{6} + \frac{17}{45} a^{5} + \frac{1}{30} a^{3} - \frac{7}{90} a^{2} + \frac{41}{90} a - \frac{13}{30}$, $\frac{1}{90} a^{16} - \frac{1}{18} a^{14} + \frac{11}{90} a^{13} + \frac{1}{30} a^{12} + \frac{2}{45} a^{11} - \frac{7}{45} a^{10} - \frac{43}{90} a^{9} - \frac{19}{45} a^{8} + \frac{7}{15} a^{7} + \frac{2}{45} a^{6} + \frac{1}{3} a^{5} - \frac{3}{10} a^{4} - \frac{37}{90} a^{3} + \frac{41}{90} a^{2} - \frac{13}{30} a + \frac{1}{3}$, $\frac{1}{144592109040147342364276945290} a^{17} - \frac{3953884884473896314265781}{9639473936009822824285129686} a^{16} + \frac{9135770347772400583191487}{3213157978669940941428376562} a^{15} + \frac{2723029268039021741217670453}{72296054520073671182138472645} a^{14} + \frac{9024100216651104021541507924}{72296054520073671182138472645} a^{13} - \frac{1254702265575866357026694851}{144592109040147342364276945290} a^{12} + \frac{11229046392258516310408229288}{72296054520073671182138472645} a^{11} - \frac{19662267610187184066884740673}{144592109040147342364276945290} a^{10} + \frac{7371254239204270997532986213}{16065789893349704707141882810} a^{9} + \frac{11689463073422192870189300591}{72296054520073671182138472645} a^{8} - \frac{445869127735919085548857066}{24098684840024557060712824215} a^{7} + \frac{1153326609511971611587136969}{4819736968004911412142564843} a^{6} - \frac{28703068885772163262395616657}{144592109040147342364276945290} a^{5} - \frac{26727384462409213868513480411}{72296054520073671182138472645} a^{4} + \frac{8836851654293594661784989538}{72296054520073671182138472645} a^{3} - \frac{20338088009334168118135115482}{72296054520073671182138472645} a^{2} + \frac{8933799998982452084374101575}{28918421808029468472855389058} a + \frac{54312123968120429090324039}{123582999179613113131860637}$

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:  $16$
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:  \( 24651944165.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.844471355782761.3

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.12.9.1$x^{12} - 6 x^{8} + 9 x^{4} - 27$$4$$3$$9$$D_4 \times C_3$$[\ ]_{4}^{6}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
643Data not computed