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Properties

Label 18.4.89326386110...6947.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,7^{13}\cdot 83^{4}\cdot 181^{5}$
Root discriminant $46.13$
Ramified primes $7, 83, 181$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T705

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-71, 249, 858, -2435, -3548, 3316, 1023, 10750, -5633, 7253, -3644, 368, -196, -448, 171, -99, 28, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 28*x^16 - 99*x^15 + 171*x^14 - 448*x^13 - 196*x^12 + 368*x^11 - 3644*x^10 + 7253*x^9 - 5633*x^8 + 10750*x^7 + 1023*x^6 + 3316*x^5 - 3548*x^4 - 2435*x^3 + 858*x^2 + 249*x - 71)
 
gp: K = bnfinit(x^18 - 6*x^17 + 28*x^16 - 99*x^15 + 171*x^14 - 448*x^13 - 196*x^12 + 368*x^11 - 3644*x^10 + 7253*x^9 - 5633*x^8 + 10750*x^7 + 1023*x^6 + 3316*x^5 - 3548*x^4 - 2435*x^3 + 858*x^2 + 249*x - 71, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 28 x^{16} - 99 x^{15} + 171 x^{14} - 448 x^{13} - 196 x^{12} + 368 x^{11} - 3644 x^{10} + 7253 x^{9} - 5633 x^{8} + 10750 x^{7} + 1023 x^{6} + 3316 x^{5} - 3548 x^{4} - 2435 x^{3} + 858 x^{2} + 249 x - 71 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-893263861107131279261229596947=-\,7^{13}\cdot 83^{4}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.13$
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:  $7, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{113} a^{16} + \frac{4}{113} a^{15} + \frac{3}{113} a^{14} + \frac{10}{113} a^{13} - \frac{37}{113} a^{12} + \frac{1}{113} a^{11} - \frac{41}{113} a^{10} + \frac{6}{113} a^{9} - \frac{15}{113} a^{8} + \frac{46}{113} a^{7} - \frac{17}{113} a^{6} + \frac{19}{113} a^{5} - \frac{55}{113} a^{4} - \frac{51}{113} a^{3} - \frac{31}{113} a^{2} + \frac{5}{113} a - \frac{15}{113}$, $\frac{1}{27820872989484925373512420486687273} a^{17} + \frac{57847972473350513989098882133373}{27820872989484925373512420486687273} a^{16} + \frac{4598853681890338533820422985006426}{27820872989484925373512420486687273} a^{15} + \frac{7935099221196991975443686546822425}{27820872989484925373512420486687273} a^{14} + \frac{5392774310230856228022630885374941}{27820872989484925373512420486687273} a^{13} + \frac{4084383280393183815467767550073163}{27820872989484925373512420486687273} a^{12} - \frac{10090380999690936021001410969831442}{27820872989484925373512420486687273} a^{11} - \frac{3256542951324445368601590513766574}{27820872989484925373512420486687273} a^{10} - \frac{11235835130626369921669916236998299}{27820872989484925373512420486687273} a^{9} - \frac{10404287540989093807904430276047356}{27820872989484925373512420486687273} a^{8} - \frac{11741978949088659381489408913727662}{27820872989484925373512420486687273} a^{7} + \frac{3805238739452282296848860955436273}{27820872989484925373512420486687273} a^{6} + \frac{5775726434198936127221375517983544}{27820872989484925373512420486687273} a^{5} - \frac{13329411565775440950589572689453385}{27820872989484925373512420486687273} a^{4} - \frac{2876987111399999242607375705389143}{27820872989484925373512420486687273} a^{3} - \frac{5465475658650701830381035962295388}{27820872989484925373512420486687273} a^{2} - \frac{3370919812533065751579743288284428}{27820872989484925373512420486687273} a + \frac{1657695817306495794933428430079459}{27820872989484925373512420486687273}$

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:  $10$
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:  \( 28987004.2919 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T705:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 64 conjugacy class representatives for t18n705 are not computed
Character table for t18n705 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

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 $18$ $18$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ $18$

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$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.4.0.1$x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$
83.6.4.1$x^{6} + 415 x^{3} + 55112$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
181Data not computed