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Properties

Label 18.18.9199789121...0621.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{27}\cdot 7^{15}\cdot 71^{4}$
Root discriminant $67.81$
Ramified primes $3, 7, 71$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times A_4^2$ (as 18T109)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45928, 274272, -4102746, 2738371, 6904779, -5476125, -3597869, 3250152, 833481, -913141, -85281, 138711, 1030, -11670, 531, 510, -42, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 42*x^16 + 510*x^15 + 531*x^14 - 11670*x^13 + 1030*x^12 + 138711*x^11 - 85281*x^10 - 913141*x^9 + 833481*x^8 + 3250152*x^7 - 3597869*x^6 - 5476125*x^5 + 6904779*x^4 + 2738371*x^3 - 4102746*x^2 + 274272*x + 45928)
 
gp: K = bnfinit(x^18 - 9*x^17 - 42*x^16 + 510*x^15 + 531*x^14 - 11670*x^13 + 1030*x^12 + 138711*x^11 - 85281*x^10 - 913141*x^9 + 833481*x^8 + 3250152*x^7 - 3597869*x^6 - 5476125*x^5 + 6904779*x^4 + 2738371*x^3 - 4102746*x^2 + 274272*x + 45928, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 42 x^{16} + 510 x^{15} + 531 x^{14} - 11670 x^{13} + 1030 x^{12} + 138711 x^{11} - 85281 x^{10} - 913141 x^{9} + 833481 x^{8} + 3250152 x^{7} - 3597869 x^{6} - 5476125 x^{5} + 6904779 x^{4} + 2738371 x^{3} - 4102746 x^{2} + 274272 x + 45928 \)

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:  \(919978912157596000406985531780621=3^{27}\cdot 7^{15}\cdot 71^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.81$
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:  $3, 7, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{12} + \frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{15} + \frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{13} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{97504537066459017963021844812359912231261212} a^{17} - \frac{2064691828041859626813783355754945242839387}{48752268533229508981510922406179956115630606} a^{16} - \frac{832278092600460119199983075293610169807123}{97504537066459017963021844812359912231261212} a^{15} + \frac{5653485079357376332927145654252412480197823}{97504537066459017963021844812359912231261212} a^{14} + \frac{2118725804131135890160106064730660913491043}{24376134266614754490755461203089978057815303} a^{13} - \frac{35119437077133256884770763273536951055132969}{97504537066459017963021844812359912231261212} a^{12} + \frac{43785801882557637792688613363601954229344469}{97504537066459017963021844812359912231261212} a^{11} + \frac{9270349200003566099882960780691186196155984}{24376134266614754490755461203089978057815303} a^{10} + \frac{13622234069416114521333585054726203611010477}{48752268533229508981510922406179956115630606} a^{9} - \frac{29947757045838251360526824761946071059541787}{97504537066459017963021844812359912231261212} a^{8} + \frac{727913984622491455205083157028877639024939}{97504537066459017963021844812359912231261212} a^{7} + \frac{14962052599403729376609297776469598684573443}{48752268533229508981510922406179956115630606} a^{6} + \frac{4951560172771748585367551911774950036121111}{97504537066459017963021844812359912231261212} a^{5} + \frac{32056735396757871544121784825854942081778181}{97504537066459017963021844812359912231261212} a^{4} + \frac{26141511241209528681266911955787069484120845}{97504537066459017963021844812359912231261212} a^{3} + \frac{12167332699366931268155360763466279866835432}{24376134266614754490755461203089978057815303} a^{2} - \frac{1816696648233329810778301665737539622896314}{24376134266614754490755461203089978057815303} a - \frac{3321076190534505610240273936382265543699107}{24376134266614754490755461203089978057815303}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 30703043328.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_4^2$ (as 18T109):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 288
The 32 conjugacy class representatives for $C_2\times A_4^2$
Character table for $C_2\times A_4^2$ is not computed

Intermediate fields

3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{9})^+\), 9.9.62523502209.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
7Data not computed
71Data not computed