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Properties

Label 18.18.1914722425...7664.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{9}\cdot 7^{15}\cdot 29^{8}$
Root discriminant $62.15$
Ramified primes $2, 3, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $He_3:C_2$ (as 18T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![100591, 824787, 1953503, 183745, -3972458, -2007012, 3046555, 1654492, -1204383, -554078, 267586, 90384, -33669, -7399, 2310, 284, -78, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 78*x^16 + 284*x^15 + 2310*x^14 - 7399*x^13 - 33669*x^12 + 90384*x^11 + 267586*x^10 - 554078*x^9 - 1204383*x^8 + 1654492*x^7 + 3046555*x^6 - 2007012*x^5 - 3972458*x^4 + 183745*x^3 + 1953503*x^2 + 824787*x + 100591)
 
gp: K = bnfinit(x^18 - 4*x^17 - 78*x^16 + 284*x^15 + 2310*x^14 - 7399*x^13 - 33669*x^12 + 90384*x^11 + 267586*x^10 - 554078*x^9 - 1204383*x^8 + 1654492*x^7 + 3046555*x^6 - 2007012*x^5 - 3972458*x^4 + 183745*x^3 + 1953503*x^2 + 824787*x + 100591, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 78 x^{16} + 284 x^{15} + 2310 x^{14} - 7399 x^{13} - 33669 x^{12} + 90384 x^{11} + 267586 x^{10} - 554078 x^{9} - 1204383 x^{8} + 1654492 x^{7} + 3046555 x^{6} - 2007012 x^{5} - 3972458 x^{4} + 183745 x^{3} + 1953503 x^{2} + 824787 x + 100591 \)

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:  \(191472242557130561520018973937664=2^{12}\cdot 3^{9}\cdot 7^{15}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.15$
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:  $2, 3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{13} a^{15} + \frac{3}{13} a^{14} + \frac{1}{13} a^{13} - \frac{1}{13} a^{12} + \frac{1}{13} a^{11} + \frac{1}{13} a^{10} - \frac{1}{13} a^{9} - \frac{4}{13} a^{8} + \frac{1}{13} a^{7} + \frac{2}{13} a^{6} + \frac{1}{13} a^{5} + \frac{3}{13} a^{4} - \frac{3}{13} a^{3} + \frac{5}{13} a^{2} + \frac{1}{13} a - \frac{5}{13}$, $\frac{1}{65} a^{16} + \frac{2}{65} a^{15} - \frac{3}{13} a^{14} + \frac{11}{65} a^{13} - \frac{24}{65} a^{12} + \frac{1}{5} a^{11} - \frac{3}{13} a^{10} - \frac{3}{65} a^{9} + \frac{18}{65} a^{8} + \frac{27}{65} a^{7} + \frac{12}{65} a^{6} - \frac{24}{65} a^{5} - \frac{32}{65} a^{4} - \frac{18}{65} a^{3} + \frac{22}{65} a^{2} - \frac{6}{65} a - \frac{21}{65}$, $\frac{1}{6228235207132027323989443969748795791960787525} a^{17} - \frac{39691375069746196977860107354495389528923386}{6228235207132027323989443969748795791960787525} a^{16} - \frac{35341817380907791408200142274082001922764451}{6228235207132027323989443969748795791960787525} a^{15} + \frac{219316618291967324416054229091646488972356032}{479095015933232871076111074596061214766214425} a^{14} - \frac{1187905586221996606523090441169574124997981527}{6228235207132027323989443969748795791960787525} a^{13} + \frac{567122066753403391010781078677330897248199653}{1245647041426405464797888793949759158392157505} a^{12} + \frac{1440789384306212038656293841936588079424137226}{6228235207132027323989443969748795791960787525} a^{11} + \frac{1717845892527832298168203131774356040768122652}{6228235207132027323989443969748795791960787525} a^{10} + \frac{357183921078660728216502499296462239550516547}{6228235207132027323989443969748795791960787525} a^{9} - \frac{31033632420191456441911309812841444801811049}{144842679235628542418359162087181297487460175} a^{8} - \frac{750304091004899787098558082992085761033990584}{6228235207132027323989443969748795791960787525} a^{7} + \frac{473672485641649565823353334457654141766584051}{1245647041426405464797888793949759158392157505} a^{6} + \frac{174182637228198563853654211135137003434159894}{1245647041426405464797888793949759158392157505} a^{5} - \frac{37178962543650236232305674069936378748041454}{479095015933232871076111074596061214766214425} a^{4} - \frac{1846214579176706441607294099553857277824484669}{6228235207132027323989443969748795791960787525} a^{3} + \frac{110352911361054622344943180289224158478332831}{479095015933232871076111074596061214766214425} a^{2} - \frac{637256359078526845839212675107136638958423443}{6228235207132027323989443969748795791960787525} a + \frac{1091488137070175267534920262420984570892359738}{6228235207132027323989443969748795791960787525}$

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:  \( 32570423525.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$He_3:C_2$ (as 18T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $He_3:C_2$
Character table for $He_3:C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{21})^+\), 9.9.1006519075055424.1 x3

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

Sibling fields

Degree 9 siblings: data not computed
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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$