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Properties

Label 18.0.13969228803...8551.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 13^{5}\cdot 43^{7}$
Root discriminant $32.22$
Ramified primes $7, 13, 43$
Class number $24$
Class group $[2, 2, 6]$
Galois group 18T188

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27971, -3691, -33816, 6876, 32584, -25670, 6401, -6585, 11716, -9712, 5300, -2720, 1591, -878, 398, -132, 33, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 33*x^16 - 132*x^15 + 398*x^14 - 878*x^13 + 1591*x^12 - 2720*x^11 + 5300*x^10 - 9712*x^9 + 11716*x^8 - 6585*x^7 + 6401*x^6 - 25670*x^5 + 32584*x^4 + 6876*x^3 - 33816*x^2 - 3691*x + 27971)
 
gp: K = bnfinit(x^18 - 6*x^17 + 33*x^16 - 132*x^15 + 398*x^14 - 878*x^13 + 1591*x^12 - 2720*x^11 + 5300*x^10 - 9712*x^9 + 11716*x^8 - 6585*x^7 + 6401*x^6 - 25670*x^5 + 32584*x^4 + 6876*x^3 - 33816*x^2 - 3691*x + 27971, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 33 x^{16} - 132 x^{15} + 398 x^{14} - 878 x^{13} + 1591 x^{12} - 2720 x^{11} + 5300 x^{10} - 9712 x^{9} + 11716 x^{8} - 6585 x^{7} + 6401 x^{6} - 25670 x^{5} + 32584 x^{4} + 6876 x^{3} - 33816 x^{2} - 3691 x + 27971 \)

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:  \(-1396922880341839978152758551=-\,7^{12}\cdot 13^{5}\cdot 43^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.22$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{9652559071478923559442723306492692337373} a^{17} + \frac{3506074096809396744972859550760131601000}{9652559071478923559442723306492692337373} a^{16} + \frac{565116351193727350304196185633849234996}{9652559071478923559442723306492692337373} a^{15} + \frac{3825988126600677024094061437429519149320}{9652559071478923559442723306492692337373} a^{14} + \frac{1433570933427714768176345293266138087981}{9652559071478923559442723306492692337373} a^{13} + \frac{287457959028175358901269704270371185002}{9652559071478923559442723306492692337373} a^{12} + \frac{1681046050997790024399690406928230924120}{9652559071478923559442723306492692337373} a^{11} - \frac{48072601519944637461425215057099768069}{9652559071478923559442723306492692337373} a^{10} - \frac{348670508487132619115286698156487591462}{9652559071478923559442723306492692337373} a^{9} + \frac{3555475548237026471565907426508701429414}{9652559071478923559442723306492692337373} a^{8} + \frac{3662558091981944505886429917771660501580}{9652559071478923559442723306492692337373} a^{7} - \frac{4665102533194470514368104835542477784589}{9652559071478923559442723306492692337373} a^{6} + \frac{4056760915254572443226967124141169078770}{9652559071478923559442723306492692337373} a^{5} - \frac{2164803155770846777425887318331721869360}{9652559071478923559442723306492692337373} a^{4} + \frac{3941084744532743721006930445411273769845}{9652559071478923559442723306492692337373} a^{3} + \frac{1438598443486285840036504003535112336397}{9652559071478923559442723306492692337373} a^{2} - \frac{3118820060530488182255459687464984667976}{9652559071478923559442723306492692337373} a - \frac{272397025404886802497969022113320156117}{9652559071478923559442723306492692337373}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}$, which has order $24$

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 ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ 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$13.3.0.1$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}$
13.3.0.1$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}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
43Data not computed