↑

Properties

Label 18.0.82390302717...1051.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 53^{6}\cdot 139^{3}$
Root discriminant $31.28$
Ramified primes $7, 53, 139$
Class number $40$ (GRH)
Class group $[2, 2, 10]$ (GRH)
Galois group $C_6\times S_4$ (as 18T61)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![216, -2412, 14724, -20759, 47915, -44688, 58530, -25559, 25026, -6859, 6618, -1068, 1276, -142, 174, -7, 17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 17*x^16 - 7*x^15 + 174*x^14 - 142*x^13 + 1276*x^12 - 1068*x^11 + 6618*x^10 - 6859*x^9 + 25026*x^8 - 25559*x^7 + 58530*x^6 - 44688*x^5 + 47915*x^4 - 20759*x^3 + 14724*x^2 - 2412*x + 216)
 
gp: K = bnfinit(x^18 + 17*x^16 - 7*x^15 + 174*x^14 - 142*x^13 + 1276*x^12 - 1068*x^11 + 6618*x^10 - 6859*x^9 + 25026*x^8 - 25559*x^7 + 58530*x^6 - 44688*x^5 + 47915*x^4 - 20759*x^3 + 14724*x^2 - 2412*x + 216, 1)
 

Normalized defining polynomial

\( x^{18} + 17 x^{16} - 7 x^{15} + 174 x^{14} - 142 x^{13} + 1276 x^{12} - 1068 x^{11} + 6618 x^{10} - 6859 x^{9} + 25026 x^{8} - 25559 x^{7} + 58530 x^{6} - 44688 x^{5} + 47915 x^{4} - 20759 x^{3} + 14724 x^{2} - 2412 x + 216 \)

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:  \(-823903027170640554647911051=-\,7^{12}\cdot 53^{6}\cdot 139^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.28$
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:  $7, 53, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{4}{9} a^{7} - \frac{4}{9} a^{6} - \frac{4}{9} a^{5} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{2}{9} a^{7} + \frac{4}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{3} a^{4} + \frac{1}{9} a^{3} + \frac{4}{9} a^{2}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{9} + \frac{1}{9} a^{6} - \frac{4}{9} a^{5} + \frac{1}{9} a^{4} - \frac{2}{9} a^{3} - \frac{2}{9} a^{2}$, $\frac{1}{18} a^{16} - \frac{1}{18} a^{14} - \frac{1}{18} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{6} a^{7} - \frac{1}{9} a^{6} - \frac{1}{18} a^{5} - \frac{4}{9} a^{4} - \frac{1}{18} a^{2} + \frac{1}{6} a$, $\frac{1}{14338525657228875905375270056196412} a^{17} + \frac{170674780076076938410625397326561}{7169262828614437952687635028098206} a^{16} + \frac{525704991702574854877977854008309}{14338525657228875905375270056196412} a^{15} + \frac{331453650723231143752298607226787}{14338525657228875905375270056196412} a^{14} + \frac{28449565966659214313456380074613}{3584631414307218976343817514049103} a^{13} - \frac{1051409778552698046056451785891759}{7169262828614437952687635028098206} a^{12} - \frac{44728979770108738957898743610457}{398292379367468775149313057116567} a^{11} - \frac{533885496474933539464238372096524}{3584631414307218976343817514049103} a^{10} - \frac{352614540304480084906952557298525}{2389754276204812650895878342699402} a^{9} - \frac{214938233099621383980739166911547}{1593169517469875100597252228466268} a^{8} - \frac{751884290349743685370138476685672}{3584631414307218976343817514049103} a^{7} + \frac{1307444295059445132224034126429671}{4779508552409625301791756685398804} a^{6} + \frac{585138781592644878714957907895348}{1194877138102406325447939171349701} a^{5} + \frac{581694470051582228308221811959584}{1194877138102406325447939171349701} a^{4} + \frac{4595860751414245958719321459884395}{14338525657228875905375270056196412} a^{3} + \frac{2286203845796283085554953253955799}{14338525657228875905375270056196412} a^{2} - \frac{314164840341435123516018354343989}{796584758734937550298626114233134} a - \frac{136367149911581958989927950762298}{398292379367468775149313057116567}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}$, which has order $40$ (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:  \( 23621.7133529 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_4$ (as 18T61):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 30 conjugacy class representatives for $C_6\times S_4$
Character table for $C_6\times S_4$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 6.0.937472851.1, 9.9.17515230173.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$53$53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
53.6.0.1$x^{6} - x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$139$139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$