Properties

Label 18.6.110...000.1
Degree $18$
Signature $[6, 6]$
Discriminant $1.107\times 10^{20}$
Root discriminant $12.99$
Ramified primes $2, 5, 7$
Class number $1$
Class group trivial
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 5*x^16 + 10*x^15 - 19*x^14 + 7*x^13 - 18*x^12 + 34*x^11 + 21*x^10 - 70*x^9 + 22*x^8 + 14*x^7 + 11*x^6 - 7*x^5 - 20*x^4 + 13*x^3 - x^2 + 4*x - 1)
 
gp: K = bnfinit(x^18 - 5*x^17 + 5*x^16 + 10*x^15 - 19*x^14 + 7*x^13 - 18*x^12 + 34*x^11 + 21*x^10 - 70*x^9 + 22*x^8 + 14*x^7 + 11*x^6 - 7*x^5 - 20*x^4 + 13*x^3 - x^2 + 4*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, -1, 13, -20, -7, 11, 14, 22, -70, 21, 34, -18, 7, -19, 10, 5, -5, 1]);
 

\(x^{18} - 5 x^{17} + 5 x^{16} + 10 x^{15} - 19 x^{14} + 7 x^{13} - 18 x^{12} + 34 x^{11} + 21 x^{10} - 70 x^{9} + 22 x^{8} + 14 x^{7} + 11 x^{6} - 7 x^{5} - 20 x^{4} + 13 x^{3} - x^{2} + 4 x - 1\)  Toggle raw display

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[6, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(110730297608000000000\)\(\medspace = 2^{12}\cdot 5^{9}\cdot 7^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $12.99$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 5, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $6$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{61668538231} a^{17} + \frac{17625470048}{61668538231} a^{16} - \frac{22926492839}{61668538231} a^{15} - \frac{16185322443}{61668538231} a^{14} + \frac{22119042450}{61668538231} a^{13} + \frac{8294293274}{61668538231} a^{12} + \frac{10064486282}{61668538231} a^{11} + \frac{15931590811}{61668538231} a^{10} + \frac{2163650743}{61668538231} a^{9} - \frac{28778891729}{61668538231} a^{8} + \frac{6174380026}{61668538231} a^{7} - \frac{28525476316}{61668538231} a^{6} + \frac{14193776766}{61668538231} a^{5} - \frac{27541996467}{61668538231} a^{4} - \frac{5729903377}{61668538231} a^{3} + \frac{25794475256}{61668538231} a^{2} - \frac{10134542531}{61668538231} a + \frac{5843535590}{61668538231}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 742.586512103 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{6}\cdot(2\pi)^{6}\cdot 742.586512103 \cdot 1}{2\sqrt{110730297608000000000}}\approx 0.138945028307$

Galois group

$C_6\times S_3$ (as 18T6):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.980.1, \(\Q(\zeta_{7})^+\), 6.2.4802000.1, 6.6.300125.1, 9.3.941192000.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: 12.0.153664000000.1
Degree 18 sibling: 18.0.56693912375296000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{3}$ R R ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{9}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed