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Properties

Label 18.0.24325752463...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{30}\cdot 3^{24}\cdot 5^{6}\cdot 7^{14}\cdot 31^{14}$
Root discriminant $1542.11$
Ramified primes $2, 3, 5, 7, 31$
Class number $14960973312$ (GRH)
Class group $[2, 2, 4, 12, 77921736]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![194277433462336, 0, 451572633759492, 0, 92453456727660, 0, 7475073472013, 0, 308465253054, 0, 7014895419, 0, 87091648, 0, 537075, 0, 1302, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 1302*x^16 + 537075*x^14 + 87091648*x^12 + 7014895419*x^10 + 308465253054*x^8 + 7475073472013*x^6 + 92453456727660*x^4 + 451572633759492*x^2 + 194277433462336)
 
gp: K = bnfinit(x^18 + 1302*x^16 + 537075*x^14 + 87091648*x^12 + 7014895419*x^10 + 308465253054*x^8 + 7475073472013*x^6 + 92453456727660*x^4 + 451572633759492*x^2 + 194277433462336, 1)
 

Normalized defining polynomial

\( x^{18} + 1302 x^{16} + 537075 x^{14} + 87091648 x^{12} + 7014895419 x^{10} + 308465253054 x^{8} + 7475073472013 x^{6} + 92453456727660 x^{4} + 451572633759492 x^{2} + 194277433462336 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2432575246331888759197234289996630135894514528681984000000=-\,2^{30}\cdot 3^{24}\cdot 5^{6}\cdot 7^{14}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1542.11$
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:  $2, 3, 5, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{434} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{868} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8680} a^{8} - \frac{1}{1736} a^{7} - \frac{1}{868} a^{6} - \frac{1}{4} a^{5} - \frac{1}{40} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{5}$, $\frac{1}{8680} a^{9} - \frac{1}{1736} a^{7} + \frac{9}{40} a^{5} + \frac{1}{8} a^{3} - \frac{1}{20} a$, $\frac{1}{34720} a^{10} + \frac{1}{34720} a^{8} + \frac{13}{34720} a^{6} - \frac{1}{160} a^{4} - \frac{1}{2} a^{3} - \frac{11}{80} a^{2} - \frac{1}{2} a - \frac{1}{5}$, $\frac{1}{7534240} a^{11} + \frac{1}{34720} a^{9} - \frac{11}{34720} a^{7} + \frac{6023}{34720} a^{5} - \frac{3}{80} a^{3} - \frac{1}{5} a$, $\frac{1}{37671200} a^{12} - \frac{1}{43400} a^{8} + \frac{9}{17360} a^{6} + \frac{147}{800} a^{4} - \frac{1}{2} a^{3} - \frac{17}{80} a^{2} - \frac{1}{2} a + \frac{8}{25}$, $\frac{1}{1393834400} a^{13} - \frac{1}{39823840} a^{11} + \frac{121}{6423200} a^{9} + \frac{11}{256928} a^{7} + \frac{90141}{458800} a^{5} - \frac{147}{370} a^{3} - \frac{309}{1850} a$, $\frac{1}{278766880000} a^{14} + \frac{611}{69691720000} a^{12} - \frac{1327}{642320000} a^{10} - \frac{573}{40145000} a^{8} - \frac{1}{1736} a^{7} + \frac{1167809}{1284640000} a^{6} - \frac{1}{4} a^{5} + \frac{201337}{1480000} a^{4} + \frac{1}{8} a^{3} + \frac{23369}{1480000} a^{2} + \frac{1}{4} a - \frac{127}{2500}$, $\frac{1}{120984825920000} a^{15} - \frac{1}{557533760000} a^{14} + \frac{33}{139383440000} a^{13} + \frac{177}{19911920000} a^{12} - \frac{5827}{278766880000} a^{11} - \frac{17173}{1284640000} a^{10} - \frac{1752017}{69691720000} a^{9} + \frac{1781}{45880000} a^{8} - \frac{488623}{2569280000} a^{7} - \frac{2462809}{2569280000} a^{6} + \frac{14322491}{91760000} a^{5} + \frac{42863}{2960000} a^{4} - \frac{23280631}{642320000} a^{3} + \frac{235631}{2960000} a^{2} - \frac{55897}{185000} a - \frac{573}{5000}$, $\frac{1}{1671317704479938368000000} a^{16} - \frac{5391687731}{7701924905437504000000} a^{14} + \frac{37091197421287}{3850962452718752000000} a^{12} - \frac{39742067563760951}{3850962452718752000000} a^{10} + \frac{18112218364949}{7098548299942400000} a^{8} - \frac{36821631882507419}{35492741499712000000} a^{6} + \frac{977911929413512221}{4436592687464000000} a^{4} + \frac{1180775225514201}{40890255184000000} a^{2} - \frac{29603667802099}{69071377000000}$, $\frac{1}{1671317704479938368000000} a^{17} + \frac{4217171373}{1671317704479938368000000} a^{15} - \frac{1}{557533760000} a^{14} + \frac{1312224135287}{3850962452718752000000} a^{13} + \frac{177}{19911920000} a^{12} - \frac{195872304803951}{3850962452718752000000} a^{11} - \frac{17173}{1284640000} a^{10} - \frac{85156142442154867}{1540384981087500800000} a^{9} + \frac{1781}{45880000} a^{8} - \frac{11471814881114419}{35492741499712000000} a^{7} - \frac{2462809}{2569280000} a^{6} + \frac{1023728666026348721}{4436592687464000000} a^{5} + \frac{42863}{2960000} a^{4} + \frac{3325447531692902617}{8873185374928000000} a^{3} + \frac{235631}{2960000} a^{2} + \frac{1117089568009337}{2555640949000000} a - \frac{573}{5000}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{12}\times C_{77921736}$, which has order $14960973312$ (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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{13383675}{53482166543358027776} a^{17} - \frac{16864227559}{53482166543358027776} a^{15} - \frac{14933165085}{123230798487000064} a^{13} - \frac{2060151284115}{123230798487000064} a^{11} - \frac{260403046743675}{246461596974000128} a^{9} - \frac{5386053065025}{162252532570112} a^{7} - \frac{326177059887}{654244082944} a^{5} - \frac{814866213005035}{283941931997696} a^{3} - \frac{221199139395}{81780510368} a \) (order $4$)
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:  \( 162297974877455.78 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{-1}) \), 3.3.8680.1, 3.3.3814209.2, 6.0.1205478400.2, 6.0.931084178923584.3, 9.9.770642875136723935897152000.1

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: data not computed
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 R R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
3Data not computed
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.3$x^{12} - 49 x^{6} + 3969$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$31$31.6.4.1$x^{6} + 1085 x^{3} + 1660608$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.12.10.1$x^{12} + 69161 x^{6} + 2869530624$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$