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Properties

Label 18.18.6191211349...5497.1
Degree $18$
Signature $[18, 0]$
Discriminant $11^{7}\cdot 43^{15}$
Root discriminant $58.37$
Ramified primes $11, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times A_4$ (as 18T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, -151, -229, 6730, -14542, -19306, 72195, -39622, -32926, 29599, 5632, -7829, -493, 965, 34, -52, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 52*x^16 + 34*x^15 + 965*x^14 - 493*x^13 - 7829*x^12 + 5632*x^11 + 29599*x^10 - 32926*x^9 - 39622*x^8 + 72195*x^7 - 19306*x^6 - 14542*x^5 + 6730*x^4 - 229*x^3 - 151*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^18 - x^17 - 52*x^16 + 34*x^15 + 965*x^14 - 493*x^13 - 7829*x^12 + 5632*x^11 + 29599*x^10 - 32926*x^9 - 39622*x^8 + 72195*x^7 - 19306*x^6 - 14542*x^5 + 6730*x^4 - 229*x^3 - 151*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 52 x^{16} + 34 x^{15} + 965 x^{14} - 493 x^{13} - 7829 x^{12} + 5632 x^{11} + 29599 x^{10} - 32926 x^{9} - 39622 x^{8} + 72195 x^{7} - 19306 x^{6} - 14542 x^{5} + 6730 x^{4} - 229 x^{3} - 151 x^{2} + 5 x + 1 \)

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:  \(61912113497337313434142287855497=11^{7}\cdot 43^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.37$
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:  $11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{3542} a^{16} + \frac{436}{1771} a^{15} + \frac{225}{1771} a^{14} - \frac{64}{1771} a^{13} - \frac{332}{1771} a^{12} - \frac{600}{1771} a^{11} - \frac{69}{154} a^{10} - \frac{1635}{3542} a^{9} - \frac{24}{77} a^{8} - \frac{369}{3542} a^{7} + \frac{878}{1771} a^{6} - \frac{278}{1771} a^{5} + \frac{57}{253} a^{4} + \frac{135}{506} a^{3} + \frac{257}{1771} a^{2} + \frac{278}{1771} a - \frac{1399}{3542}$, $\frac{1}{5825820914848469458} a^{17} - \frac{275419115702691}{5825820914848469458} a^{16} + \frac{72642113491614539}{529620083168042678} a^{15} + \frac{6463772857393387}{75660011881148954} a^{14} - \frac{608106788627353597}{5825820914848469458} a^{13} - \frac{666276639743682809}{5825820914848469458} a^{12} - \frac{1344884860524943637}{2912910457424234729} a^{11} - \frac{984159774208085813}{5825820914848469458} a^{10} - \frac{585627398207335395}{5825820914848469458} a^{9} - \frac{63002731419398410}{264810041584021339} a^{8} - \frac{376560637652828709}{2912910457424234729} a^{7} - \frac{2570623652972678}{11513480068870493} a^{6} + \frac{615174583733264249}{2912910457424234729} a^{5} - \frac{51033520984338929}{832260130692638494} a^{4} + \frac{776451290038429449}{5825820914848469458} a^{3} - \frac{2115949694830147303}{5825820914848469458} a^{2} + \frac{23079934321039495}{264810041584021339} a + \frac{1453447487879386325}{2912910457424234729}$

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:  \( 19367773986.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times A_4$ (as 18T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 12 conjugacy class representatives for $S_3\times A_4$
Character table for $S_3\times A_4$

Intermediate fields

3.3.1849.1, 3.3.473.1, 6.6.1617092873.1, 9.9.361790571383417.1

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 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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

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
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$43$43.6.5.1$x^{6} - 43$$6$$1$$5$$C_6$$[\ ]_{6}$
43.6.5.1$x^{6} - 43$$6$$1$$5$$C_6$$[\ ]_{6}$
43.6.5.1$x^{6} - 43$$6$$1$$5$$C_6$$[\ ]_{6}$