↑

Properties

Label 18.0.20498592231...2368.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 7^{6}\cdot 43^{10}$
Root discriminant $42.50$
Ramified primes $2, 3, 7, 43$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![183, 165, 1558, 381, 5358, 858, 10340, 1305, 10875, 2007, 5724, 1233, 1674, -3, 143, -57, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^16 - 57*x^15 + 143*x^14 - 3*x^13 + 1674*x^12 + 1233*x^11 + 5724*x^10 + 2007*x^9 + 10875*x^8 + 1305*x^7 + 10340*x^6 + 858*x^5 + 5358*x^4 + 381*x^3 + 1558*x^2 + 165*x + 183)
 
gp: K = bnfinit(x^18 - 3*x^17 + 9*x^16 - 57*x^15 + 143*x^14 - 3*x^13 + 1674*x^12 + 1233*x^11 + 5724*x^10 + 2007*x^9 + 10875*x^8 + 1305*x^7 + 10340*x^6 + 858*x^5 + 5358*x^4 + 381*x^3 + 1558*x^2 + 165*x + 183, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 9 x^{16} - 57 x^{15} + 143 x^{14} - 3 x^{13} + 1674 x^{12} + 1233 x^{11} + 5724 x^{10} + 2007 x^{9} + 10875 x^{8} + 1305 x^{7} + 10340 x^{6} + 858 x^{5} + 5358 x^{4} + 381 x^{3} + 1558 x^{2} + 165 x + 183 \)

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:  \(-204985922317940382893914042368=-\,2^{12}\cdot 3^{9}\cdot 7^{6}\cdot 43^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.50$
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:  $2, 3, 7, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{8} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{14} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{4}{9} a^{4} - \frac{1}{9} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{364692856776462364811552581743} a^{17} - \frac{13575160268273582354492872753}{364692856776462364811552581743} a^{16} + \frac{20305242015449263631035813969}{364692856776462364811552581743} a^{15} + \frac{31858866107199530139581797610}{364692856776462364811552581743} a^{14} + \frac{10645213317464456761949034571}{121564285592154121603850860581} a^{13} - \frac{12868637883517823621042284504}{121564285592154121603850860581} a^{12} + \frac{1730512843392356132330321477}{40521428530718040534616953527} a^{11} - \frac{12186118551869477245289098411}{121564285592154121603850860581} a^{10} - \frac{39600506003417403538792949330}{121564285592154121603850860581} a^{9} + \frac{12593099158302900300328527585}{40521428530718040534616953527} a^{8} - \frac{12853480360369955907101277946}{40521428530718040534616953527} a^{7} + \frac{53679779494405850342329706033}{121564285592154121603850860581} a^{6} + \frac{148721160306842897027737448342}{364692856776462364811552581743} a^{5} + \frac{57881059049887005717205585132}{364692856776462364811552581743} a^{4} - \frac{83142649488269327876039637475}{364692856776462364811552581743} a^{3} - \frac{11607697022115200880885589463}{364692856776462364811552581743} a^{2} + \frac{16931510097820969058138161169}{40521428530718040534616953527} a - \frac{32745803918666298389843594819}{121564285592154121603850860581}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{149131692413739326}{37703520132935729883} a^{17} + \frac{473976989037599438}{37703520132935729883} a^{16} - \frac{1461017701596462538}{37703520132935729883} a^{15} + \frac{2960482522889615091}{12567840044311909961} a^{14} - \frac{7736448795665348035}{12567840044311909961} a^{13} + \frac{2169783114008740096}{12567840044311909961} a^{12} - \frac{85392600558717187811}{12567840044311909961} a^{11} - \frac{46351121725856008764}{12567840044311909961} a^{10} - \frac{292530515696235301892}{12567840044311909961} a^{9} - \frac{51723144029576407028}{12567840044311909961} a^{8} - \frac{557309903511451615850}{12567840044311909961} a^{7} + \frac{66395116999951970188}{12567840044311909961} a^{6} - \frac{1629414233231806567579}{37703520132935729883} a^{5} + \frac{355354514741260369567}{37703520132935729883} a^{4} - \frac{721259331143792675231}{37703520132935729883} a^{3} + \frac{56479911217287596864}{12567840044311909961} a^{2} - \frac{45730901114817469945}{12567840044311909961} a + \frac{17103887341629606118}{12567840044311909961} \) (order $6$)
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:  \( 55890127.22994664 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.172.1, 6.0.2446227.1, 6.0.798768.3

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$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}$
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
$43$43.3.2.2$x^{3} + 387$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.6.3.2$x^{6} - 1849 x^{2} + 795070$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
43.6.5.4$x^{6} + 387$$6$$1$$5$$C_6$$[\ ]_{6}$