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Properties

Label 18.6.18147624991...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 5^{9}\cdot 197^{8}$
Root discriminant $37.15$
Ramified primes $2, 5, 197$
Class number $2$
Class group $[2]$
Galois group $S_3\times S_4$ (as 18T69)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25, -35, 754, -3101, 7334, -12128, 14814, -13607, 9385, -4811, 1550, 205, -708, 495, -161, 1, 19, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 19*x^16 + x^15 - 161*x^14 + 495*x^13 - 708*x^12 + 205*x^11 + 1550*x^10 - 4811*x^9 + 9385*x^8 - 13607*x^7 + 14814*x^6 - 12128*x^5 + 7334*x^4 - 3101*x^3 + 754*x^2 - 35*x - 25)
 
gp: K = bnfinit(x^18 - 7*x^17 + 19*x^16 + x^15 - 161*x^14 + 495*x^13 - 708*x^12 + 205*x^11 + 1550*x^10 - 4811*x^9 + 9385*x^8 - 13607*x^7 + 14814*x^6 - 12128*x^5 + 7334*x^4 - 3101*x^3 + 754*x^2 - 35*x - 25, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 19 x^{16} + x^{15} - 161 x^{14} + 495 x^{13} - 708 x^{12} + 205 x^{11} + 1550 x^{10} - 4811 x^{9} + 9385 x^{8} - 13607 x^{7} + 14814 x^{6} - 12128 x^{5} + 7334 x^{4} - 3101 x^{3} + 754 x^{2} - 35 x - 25 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(18147624991591898888000000000=2^{12}\cdot 5^{9}\cdot 197^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.15$
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:  $2, 5, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{155} a^{16} + \frac{4}{155} a^{15} - \frac{27}{155} a^{14} - \frac{36}{155} a^{13} + \frac{13}{155} a^{12} + \frac{3}{155} a^{11} + \frac{3}{31} a^{10} - \frac{11}{31} a^{9} + \frac{12}{31} a^{8} + \frac{24}{155} a^{7} + \frac{64}{155} a^{6} - \frac{28}{155} a^{5} + \frac{66}{155} a^{4} - \frac{47}{155} a^{3} - \frac{53}{155} a^{2} - \frac{74}{155} a + \frac{12}{31}$, $\frac{1}{1614268454605379504635} a^{17} + \frac{7592916984113313}{322853690921075900927} a^{16} + \frac{718606157097853422592}{1614268454605379504635} a^{15} + \frac{680817568353439529767}{1614268454605379504635} a^{14} - \frac{458545491987544469883}{1614268454605379504635} a^{13} + \frac{648496591529690836621}{1614268454605379504635} a^{12} - \frac{1215552140268499317}{1614268454605379504635} a^{11} + \frac{86043452847649186442}{322853690921075900927} a^{10} - \frac{31054664421026019155}{322853690921075900927} a^{9} - \frac{113961667770374591991}{1614268454605379504635} a^{8} + \frac{638322344748055469508}{1614268454605379504635} a^{7} + \frac{465865460825204716531}{1614268454605379504635} a^{6} + \frac{441179995207365727208}{1614268454605379504635} a^{5} + \frac{504368920725380742859}{1614268454605379504635} a^{4} + \frac{153677865575296653573}{322853690921075900927} a^{3} - \frac{403127982940604787642}{1614268454605379504635} a^{2} - \frac{49673131609393705729}{146751677691398136785} a - \frac{42296773445419970671}{322853690921075900927}$

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

Class group and class number

$C_{2}$, which has order $2$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 10939516.986 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times S_4$ (as 18T69):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 15 conjugacy class representatives for $S_3\times S_4$
Character table for $S_3\times S_4$

Intermediate fields

3.3.985.1, 3.3.788.1, 6.2.77618000.2, 9.9.12049107848000.1

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

Sibling fields

Degree 12 sibling: 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
197Data not computed