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Properties

Label 18.0.72621606640...9851.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{14}\cdot 67^{6}\cdot 491^{3}$
Root discriminant $51.82$
Ramified primes $7, 67, 491$
Class number $596$ (GRH)
Class group $[2, 298]$ (GRH)
Galois group $C_2\times S_3\times A_4$ (as 18T60)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![692091, -520209, 979440, -476930, 281649, -114144, 62185, -34802, 24013, -8213, 1810, -1433, 1228, -380, 40, -23, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 24*x^16 - 23*x^15 + 40*x^14 - 380*x^13 + 1228*x^12 - 1433*x^11 + 1810*x^10 - 8213*x^9 + 24013*x^8 - 34802*x^7 + 62185*x^6 - 114144*x^5 + 281649*x^4 - 476930*x^3 + 979440*x^2 - 520209*x + 692091)
 
gp: K = bnfinit(x^18 - 8*x^17 + 24*x^16 - 23*x^15 + 40*x^14 - 380*x^13 + 1228*x^12 - 1433*x^11 + 1810*x^10 - 8213*x^9 + 24013*x^8 - 34802*x^7 + 62185*x^6 - 114144*x^5 + 281649*x^4 - 476930*x^3 + 979440*x^2 - 520209*x + 692091, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 24 x^{16} - 23 x^{15} + 40 x^{14} - 380 x^{13} + 1228 x^{12} - 1433 x^{11} + 1810 x^{10} - 8213 x^{9} + 24013 x^{8} - 34802 x^{7} + 62185 x^{6} - 114144 x^{5} + 281649 x^{4} - 476930 x^{3} + 979440 x^{2} - 520209 x + 692091 \)

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:  \(-7262160664015682731094650299851=-\,7^{14}\cdot 67^{6}\cdot 491^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.82$
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:  $7, 67, 491$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{15} a^{16} + \frac{1}{15} a^{15} + \frac{2}{5} a^{14} + \frac{7}{15} a^{13} - \frac{2}{15} a^{12} + \frac{7}{15} a^{11} - \frac{2}{15} a^{10} - \frac{2}{15} a^{9} + \frac{1}{15} a^{8} + \frac{4}{15} a^{7} - \frac{2}{15} a^{6} - \frac{2}{15} a^{5} + \frac{1}{15} a^{4} + \frac{1}{5} a^{3} - \frac{1}{3} a - \frac{1}{5}$, $\frac{1}{55118909347946594930367461727691109054813485935} a^{17} - \frac{154210937358493010669866206446623584852324173}{55118909347946594930367461727691109054813485935} a^{16} + \frac{1295271068597940977032807978809007456931190799}{18372969782648864976789153909230369684937828645} a^{15} + \frac{7724043368958001739582653224322792794050320933}{55118909347946594930367461727691109054813485935} a^{14} + \frac{3481119813035638446546356318884561145288699753}{11023781869589318986073492345538221810962697187} a^{13} - \frac{4341818053564672778105950884252801571612907203}{11023781869589318986073492345538221810962697187} a^{12} - \frac{338092969406137935699888938499180732268477810}{11023781869589318986073492345538221810962697187} a^{11} - \frac{998251896937706810449261159529718439711846844}{55118909347946594930367461727691109054813485935} a^{10} + \frac{184152995517188219989708323739299774060986704}{55118909347946594930367461727691109054813485935} a^{9} + \frac{844848910320021279860893031191229098067647370}{11023781869589318986073492345538221810962697187} a^{8} - \frac{7378929668648955285416958129452069600183789753}{55118909347946594930367461727691109054813485935} a^{7} + \frac{26035799511042148316989769423063723962724335756}{55118909347946594930367461727691109054813485935} a^{6} + \frac{8857084998738939812093716532936454986491397149}{55118909347946594930367461727691109054813485935} a^{5} - \frac{2503698953516591254246500860735730331711643139}{6124323260882954992263051303076789894979276215} a^{4} + \frac{7239669386420663470363510789884314201828385676}{18372969782648864976789153909230369684937828645} a^{3} + \frac{1402328055281184043660480320352770273217729949}{11023781869589318986073492345538221810962697187} a^{2} - \frac{6616412802962199624070081395139859782851393731}{18372969782648864976789153909230369684937828645} a + \frac{1169054893270528972567038061138834231528248153}{6124323260882954992263051303076789894979276215}$

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

Class group and class number

$C_{2}\times C_{298}$, which has order $596$ (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:  \( 122519.134308 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\times A_4$ (as 18T60):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 24 conjugacy class representatives for $C_2\times S_3\times A_4$
Character table for $C_2\times S_3\times A_4$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.469.1, 6.0.1178891.1, 9.9.247691263309.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$67$67.6.0.1$x^{6} + x^{2} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
67.12.6.1$x^{12} + 8978 x^{8} + 7218312 x^{6} + 20151121 x^{4} + 31052877461 x^{2} + 13026007032336$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
491Data not computed