↑

Properties

Label 18.0.12767980896...2231.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 13^{7}\cdot 43^{5}$
Root discriminant $28.21$
Ramified primes $7, 13, 43$
Class number $6$
Class group $[6]$
Galois group 18T188

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2801, -10126, 10797, -3510, 4154, -5818, 4699, -1377, 1231, -1015, 573, -172, 134, -115, 71, -25, 11, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 11*x^16 - 25*x^15 + 71*x^14 - 115*x^13 + 134*x^12 - 172*x^11 + 573*x^10 - 1015*x^9 + 1231*x^8 - 1377*x^7 + 4699*x^6 - 5818*x^5 + 4154*x^4 - 3510*x^3 + 10797*x^2 - 10126*x + 2801)
 
gp: K = bnfinit(x^18 - 3*x^17 + 11*x^16 - 25*x^15 + 71*x^14 - 115*x^13 + 134*x^12 - 172*x^11 + 573*x^10 - 1015*x^9 + 1231*x^8 - 1377*x^7 + 4699*x^6 - 5818*x^5 + 4154*x^4 - 3510*x^3 + 10797*x^2 - 10126*x + 2801, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 11 x^{16} - 25 x^{15} + 71 x^{14} - 115 x^{13} + 134 x^{12} - 172 x^{11} + 573 x^{10} - 1015 x^{9} + 1231 x^{8} - 1377 x^{7} + 4699 x^{6} - 5818 x^{5} + 4154 x^{4} - 3510 x^{3} + 10797 x^{2} - 10126 x + 2801 \)

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:  \(-127679808965803654033432231=-\,7^{12}\cdot 13^{7}\cdot 43^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.21$
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:  $7, 13, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{275734138311526935952164327210395993} a^{17} + \frac{132096475528002549972496592773702349}{275734138311526935952164327210395993} a^{16} - \frac{133140007676513081650959525158125674}{275734138311526935952164327210395993} a^{15} - \frac{115885272326498846513049976511122423}{275734138311526935952164327210395993} a^{14} + \frac{35245131691595575665071430518414186}{275734138311526935952164327210395993} a^{13} - \frac{29950172407765757409848042742220494}{275734138311526935952164327210395993} a^{12} - \frac{122952731088946249325651739080493072}{275734138311526935952164327210395993} a^{11} - \frac{97508119081705779204400748931562431}{275734138311526935952164327210395993} a^{10} - \frac{34613366433217029987378052790310206}{275734138311526935952164327210395993} a^{9} - \frac{90617734899197866313258887391689405}{275734138311526935952164327210395993} a^{8} + \frac{132126197823098880716433406934398238}{275734138311526935952164327210395993} a^{7} + \frac{25530848525434418990552309312561882}{275734138311526935952164327210395993} a^{6} - \frac{51355803972360426232120443947432215}{275734138311526935952164327210395993} a^{5} + \frac{14055393992772319451843048715511775}{275734138311526935952164327210395993} a^{4} + \frac{4260584520621156342036585981734677}{275734138311526935952164327210395993} a^{3} - \frac{30748527054138531542162687844923560}{275734138311526935952164327210395993} a^{2} + \frac{88014704753888173792451178186288305}{275734138311526935952164327210395993} a + \frac{108465192470875357472615082946573347}{275734138311526935952164327210395993}$

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

Class group and class number

$C_{6}$, which has order $6$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 35085.5946511 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T188:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 88 conjugacy class representatives for t18n188 are not computed
Character table for t18n188 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.0.1342159.1, 9.9.36763077169.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R $18$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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}$
$13$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
43Data not computed