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Properties

Label 18.0.73997032984...3984.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 17^{9}\cdot 163^{15}$
Root discriminant $456.46$
Ramified primes $2, 17, 163$
Class number $1283680736$ (GRH)
Class group $[26, 1612, 30628]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3731618953, -16638384612, 33626242626, -41292548918, 35004517519, -22066150836, 10825160241, -4240439604, 1342940679, -344420830, 70706700, -11262660, 1287658, -80238, -2841, 1202, -87, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 87*x^16 + 1202*x^15 - 2841*x^14 - 80238*x^13 + 1287658*x^12 - 11262660*x^11 + 70706700*x^10 - 344420830*x^9 + 1342940679*x^8 - 4240439604*x^7 + 10825160241*x^6 - 22066150836*x^5 + 35004517519*x^4 - 41292548918*x^3 + 33626242626*x^2 - 16638384612*x + 3731618953)
 
gp: K = bnfinit(x^18 - 2*x^17 - 87*x^16 + 1202*x^15 - 2841*x^14 - 80238*x^13 + 1287658*x^12 - 11262660*x^11 + 70706700*x^10 - 344420830*x^9 + 1342940679*x^8 - 4240439604*x^7 + 10825160241*x^6 - 22066150836*x^5 + 35004517519*x^4 - 41292548918*x^3 + 33626242626*x^2 - 16638384612*x + 3731618953, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 87 x^{16} + 1202 x^{15} - 2841 x^{14} - 80238 x^{13} + 1287658 x^{12} - 11262660 x^{11} + 70706700 x^{10} - 344420830 x^{9} + 1342940679 x^{8} - 4240439604 x^{7} + 10825160241 x^{6} - 22066150836 x^{5} + 35004517519 x^{4} - 41292548918 x^{3} + 33626242626 x^{2} - 16638384612 x + 3731618953 \)

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:  \(-739970329848199945276026357888092034566758313984=-\,2^{12}\cdot 17^{9}\cdot 163^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $456.46$
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:  $2, 17, 163$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{8} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a$, $\frac{1}{102} a^{12} + \frac{2}{51} a^{11} - \frac{2}{51} a^{10} + \frac{1}{102} a^{9} + \frac{2}{51} a^{8} + \frac{49}{102} a^{7} - \frac{8}{17} a^{6} - \frac{14}{51} a^{5} - \frac{11}{34} a^{4} - \frac{13}{102} a^{3} + \frac{43}{102} a^{2} + \frac{4}{17} a - \frac{5}{17}$, $\frac{1}{102} a^{13} - \frac{1}{34} a^{11} - \frac{3}{17} a^{8} + \frac{15}{34} a^{7} + \frac{15}{34} a^{6} - \frac{1}{17} a^{5} - \frac{1}{2} a^{4} - \frac{4}{17} a^{3} - \frac{2}{17} a^{2} - \frac{4}{17} a + \frac{35}{102}$, $\frac{1}{102} a^{14} - \frac{5}{102} a^{11} + \frac{5}{102} a^{10} + \frac{1}{51} a^{9} - \frac{11}{102} a^{8} + \frac{11}{51} a^{7} + \frac{1}{34} a^{6} - \frac{25}{51} a^{5} - \frac{19}{51} a^{4} - \frac{1}{6} a^{3} + \frac{10}{51} a^{2} + \frac{5}{102} a - \frac{11}{51}$, $\frac{1}{16626} a^{15} + \frac{2}{8313} a^{14} - \frac{25}{5542} a^{13} + \frac{10}{2771} a^{12} - \frac{530}{8313} a^{11} + \frac{1}{163} a^{10} - \frac{737}{16626} a^{9} + \frac{1673}{16626} a^{8} + \frac{2305}{8313} a^{7} + \frac{4585}{16626} a^{6} + \frac{845}{8313} a^{5} - \frac{108}{2771} a^{4} + \frac{2264}{8313} a^{3} - \frac{1205}{5542} a^{2} + \frac{1}{17} a + \frac{2233}{8313}$, $\frac{1}{32902854} a^{16} + \frac{1}{16451427} a^{15} - \frac{16178}{5483809} a^{14} + \frac{32053}{16451427} a^{13} + \frac{49513}{32902854} a^{12} + \frac{912770}{16451427} a^{11} + \frac{417706}{16451427} a^{10} + \frac{843575}{32902854} a^{9} + \frac{3001523}{10967618} a^{8} + \frac{15754205}{32902854} a^{7} - \frac{3373928}{16451427} a^{6} + \frac{8074636}{16451427} a^{5} + \frac{2379452}{16451427} a^{4} - \frac{5837965}{32902854} a^{3} + \frac{11779253}{32902854} a^{2} - \frac{2282125}{5483809} a - \frac{2471721}{10967618}$, $\frac{1}{59925551439075890566782331720316398919982} a^{17} + \frac{263358570710284100736957014651369}{29962775719537945283391165860158199459991} a^{16} + \frac{876377738550140047146478791239708311}{59925551439075890566782331720316398919982} a^{15} + \frac{127295733884165465057610293592091628423}{29962775719537945283391165860158199459991} a^{14} - \frac{204431256731871265384545942763911575629}{59925551439075890566782331720316398919982} a^{13} - \frac{175959064519022684436941199477942311785}{59925551439075890566782331720316398919982} a^{12} + \frac{4079212950307527172362381077120514711089}{59925551439075890566782331720316398919982} a^{11} + \frac{1653562952633966400926525351327546659726}{29962775719537945283391165860158199459991} a^{10} - \frac{4194972871714521434563410760166426825789}{59925551439075890566782331720316398919982} a^{9} + \frac{2601139261380973583775455376838718637941}{9987591906512648427797055286719399819997} a^{8} - \frac{2234781648774485510235082467727078459391}{59925551439075890566782331720316398919982} a^{7} - \frac{336072015040234525669548293280809822046}{9987591906512648427797055286719399819997} a^{6} - \frac{5996212996600255399019510243905655099695}{59925551439075890566782331720316398919982} a^{5} + \frac{5086252907388191689930922819968091264213}{19975183813025296855594110573438799639994} a^{4} + \frac{673943454916104417595204918736571947962}{9987591906512648427797055286719399819997} a^{3} - \frac{28570511944827435944932584984716175489587}{59925551439075890566782331720316398919982} a^{2} - \frac{13587576355648772671769977992016682913778}{29962775719537945283391165860158199459991} a - \frac{6318519589136950185843023769848941838704}{29962775719537945283391165860158199459991}$

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

Class group and class number

$C_{26}\times C_{1612}\times C_{30628}$, which has order $1283680736$ (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:  \( 24241802.06091236 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-2771}) \), 3.1.11084.1 x3, 3.3.26569.1, 6.0.340431360176.1, Deg 6 x2, Deg 6, 9.3.961258003845861719744.1 x3

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

Sibling fields

Degree 6 sibling: data not computed
Degree 9 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
$163$163.6.5.1$x^{6} - 163$$6$$1$$5$$C_6$$[\ ]_{6}$
163.6.5.1$x^{6} - 163$$6$$1$$5$$C_6$$[\ ]_{6}$
163.6.5.1$x^{6} - 163$$6$$1$$5$$C_6$$[\ ]_{6}$