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Properties

Label 18.0.71235766671...4096.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 7^{12}\cdot 53^{6}\cdot 5669$
Root discriminant $35.27$
Ramified primes $2, 7, 53, 5669$
Class number $96$ (GRH)
Class group $[2, 48]$ (GRH)
Galois group 18T544

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5144, 2636, 12432, -961, 13806, -5399, 14017, -7520, 7272, -3133, 2554, -1097, 611, -218, 116, -33, 12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 12*x^16 - 33*x^15 + 116*x^14 - 218*x^13 + 611*x^12 - 1097*x^11 + 2554*x^10 - 3133*x^9 + 7272*x^8 - 7520*x^7 + 14017*x^6 - 5399*x^5 + 13806*x^4 - 961*x^3 + 12432*x^2 + 2636*x + 5144)
 
gp: K = bnfinit(x^18 - 3*x^17 + 12*x^16 - 33*x^15 + 116*x^14 - 218*x^13 + 611*x^12 - 1097*x^11 + 2554*x^10 - 3133*x^9 + 7272*x^8 - 7520*x^7 + 14017*x^6 - 5399*x^5 + 13806*x^4 - 961*x^3 + 12432*x^2 + 2636*x + 5144, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 12 x^{16} - 33 x^{15} + 116 x^{14} - 218 x^{13} + 611 x^{12} - 1097 x^{11} + 2554 x^{10} - 3133 x^{9} + 7272 x^{8} - 7520 x^{7} + 14017 x^{6} - 5399 x^{5} + 13806 x^{4} - 961 x^{3} + 12432 x^{2} + 2636 x + 5144 \)

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:  \(-7123576667122313292544004096=-\,2^{12}\cdot 7^{12}\cdot 53^{6}\cdot 5669\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.27$
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:  $2, 7, 53, 5669$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{124139796447664074064090721869267348} a^{17} - \frac{23094516068555221950589021470456503}{124139796447664074064090721869267348} a^{16} - \frac{8436327624873798014214392288977466}{31034949111916018516022680467316837} a^{15} + \frac{5084563235296939277435658567315939}{124139796447664074064090721869267348} a^{14} + \frac{6587072544259599311911035414396739}{31034949111916018516022680467316837} a^{13} - \frac{5732542296439624040299473114151493}{62069898223832037032045360934633674} a^{12} + \frac{24293477209302506273324545229075079}{124139796447664074064090721869267348} a^{11} - \frac{58809498761239086026918396394720629}{124139796447664074064090721869267348} a^{10} - \frac{10756407246839640192011731307350267}{62069898223832037032045360934633674} a^{9} + \frac{33459627523257861514108203073768767}{124139796447664074064090721869267348} a^{8} - \frac{9489891932154668167853452824407502}{31034949111916018516022680467316837} a^{7} + \frac{4584167345287455955468148651104538}{31034949111916018516022680467316837} a^{6} + \frac{13579748758956432506135560168485673}{124139796447664074064090721869267348} a^{5} - \frac{23564219953266924061040452559260943}{124139796447664074064090721869267348} a^{4} + \frac{26142080418657823876814730971351879}{62069898223832037032045360934633674} a^{3} - \frac{2641622401017460770908864751742461}{124139796447664074064090721869267348} a^{2} + \frac{11526421405125498542755924733329881}{31034949111916018516022680467316837} a + \frac{6980202664294520792611944615386402}{31034949111916018516022680467316837}$

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

Class group and class number

$C_{2}\times C_{48}$, which has order $96$ (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

18T544:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 9216
The 96 conjugacy class representatives for t18n544 are not computed
Character table for t18n544 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 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 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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.12.22$x^{12} - 52 x^{10} - 7 x^{8} + 32 x^{6} + 35 x^{4} - 44 x^{2} - 29$$2$$6$$12$12T134$[2, 2, 2, 2, 2, 2]^{6}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$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$$[\ ]_{3}^{3}$
$53$53.6.0.1$x^{6} - x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
53.12.6.1$x^{12} + 2382032 x^{6} - 418195493 x^{2} + 1418519112256$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5669Data not computed