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Properties

Label 18.0.49418948591...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 5^{9}\cdot 11^{12}$
Root discriminant $30.41$
Ramified primes $2, 3, 5, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8115, 1335, 6796, -592, -649, -1275, -2022, -487, 226, 317, 304, -43, -68, -13, 21, 0, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 4*x^16 + 21*x^14 - 13*x^13 - 68*x^12 - 43*x^11 + 304*x^10 + 317*x^9 + 226*x^8 - 487*x^7 - 2022*x^6 - 1275*x^5 - 649*x^4 - 592*x^3 + 6796*x^2 + 1335*x + 8115)
 
gp: K = bnfinit(x^18 - x^17 - 4*x^16 + 21*x^14 - 13*x^13 - 68*x^12 - 43*x^11 + 304*x^10 + 317*x^9 + 226*x^8 - 487*x^7 - 2022*x^6 - 1275*x^5 - 649*x^4 - 592*x^3 + 6796*x^2 + 1335*x + 8115, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 4 x^{16} + 21 x^{14} - 13 x^{13} - 68 x^{12} - 43 x^{11} + 304 x^{10} + 317 x^{9} + 226 x^{8} - 487 x^{7} - 2022 x^{6} - 1275 x^{5} - 649 x^{4} - 592 x^{3} + 6796 x^{2} + 1335 x + 8115 \)

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:  \(-494189485911995544000000000=-\,2^{12}\cdot 3^{9}\cdot 5^{9}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.41$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{8} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{2}{9} a^{9} + \frac{4}{9} a^{8} - \frac{1}{9} a^{7} + \frac{4}{9} a^{6} - \frac{4}{9} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{401190077917814434110433646685} a^{17} + \frac{11519612523983575459765864057}{401190077917814434110433646685} a^{16} + \frac{3996373509909811689889270153}{44576675324201603790048182965} a^{15} - \frac{26434154100993652029107656459}{401190077917814434110433646685} a^{14} - \frac{52306123760480092776337416191}{401190077917814434110433646685} a^{13} - \frac{3595740482739974327075356066}{401190077917814434110433646685} a^{12} + \frac{3410431573802754389403886129}{401190077917814434110433646685} a^{11} + \frac{45509560436879210273520012089}{401190077917814434110433646685} a^{10} - \frac{176722585652341505620762365584}{401190077917814434110433646685} a^{9} - \frac{14435639387309483039758510208}{80238015583562886822086729337} a^{8} - \frac{1833755585368218795279509893}{4833615396600173904944983695} a^{7} - \frac{191490414689284148919880636084}{401190077917814434110433646685} a^{6} + \frac{13294195704186794113651396667}{133730025972604811370144548895} a^{5} + \frac{85526652504976625486647477783}{401190077917814434110433646685} a^{4} - \frac{245402571945687687017466082}{80238015583562886822086729337} a^{3} + \frac{384666003822181394494447307}{44576675324201603790048182965} a^{2} + \frac{8404889617118649838930784269}{26746005194520962274028909779} a - \frac{1807313252149383399592529479}{8915335064840320758009636593}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -1 \) (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:  \( 2518051.214463339 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.1452.1, 3.1.1815.1 x3, 6.0.790614000.1, 6.0.49413375.1, 9.1.1147971528000.1 x3

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 6 sibling: data not computed
Degree 9 sibling: data not computed
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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$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$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$