Properties

Label 18.0.12619972448...8967.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,23^{9}\cdot 2647^{2}$
Root discriminant $11.51$
Ramified primes $23, 2647$
Class number $1$
Class group Trivial
Galois Group 18T319

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 10, -6, -9, 18, -10, -6, 17, -19, 14, -3, -4, 2, 4, -7, 6, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 - 7*x^15 + 4*x^14 + 2*x^13 - 4*x^12 - 3*x^11 + 14*x^10 - 19*x^9 + 17*x^8 - 6*x^7 - 10*x^6 + 18*x^5 - 9*x^4 - 6*x^3 + 10*x^2 - 5*x + 1)
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 - 7*x^15 + 4*x^14 + 2*x^13 - 4*x^12 - 3*x^11 + 14*x^10 - 19*x^9 + 17*x^8 - 6*x^7 - 10*x^6 + 18*x^5 - 9*x^4 - 6*x^3 + 10*x^2 - 5*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 3 x^{17} \) \(\mathstrut +\mathstrut 6 x^{16} \) \(\mathstrut -\mathstrut 7 x^{15} \) \(\mathstrut +\mathstrut 4 x^{14} \) \(\mathstrut +\mathstrut 2 x^{13} \) \(\mathstrut -\mathstrut 4 x^{12} \) \(\mathstrut -\mathstrut 3 x^{11} \) \(\mathstrut +\mathstrut 14 x^{10} \) \(\mathstrut -\mathstrut 19 x^{9} \) \(\mathstrut +\mathstrut 17 x^{8} \) \(\mathstrut -\mathstrut 6 x^{7} \) \(\mathstrut -\mathstrut 10 x^{6} \) \(\mathstrut +\mathstrut 18 x^{5} \) \(\mathstrut -\mathstrut 9 x^{4} \) \(\mathstrut -\mathstrut 6 x^{3} \) \(\mathstrut +\mathstrut 10 x^{2} \) \(\mathstrut -\mathstrut 5 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-12619972448180608967=-\,23^{9}\cdot 2647^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.51$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $23, 2647$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{687259} a^{17} + \frac{211711}{687259} a^{16} - \frac{9533}{40427} a^{15} + \frac{135755}{687259} a^{14} + \frac{62694}{687259} a^{13} + \frac{164451}{687259} a^{12} + \frac{38070}{687259} a^{11} - \frac{221575}{687259} a^{10} - \frac{291973}{687259} a^{9} + \frac{51755}{687259} a^{8} + \frac{287850}{687259} a^{7} - \frac{129672}{687259} a^{6} - \frac{129804}{687259} a^{5} + \frac{101595}{687259} a^{4} - \frac{61102}{687259} a^{3} + \frac{127323}{687259} a^{2} - \frac{298125}{687259} a - \frac{56954}{687259}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{814384}{687259} a^{17} - \frac{2050601}{687259} a^{16} + \frac{240116}{40427} a^{15} - \frac{4029928}{687259} a^{14} + \frac{1893904}{687259} a^{13} + \frac{2163631}{687259} a^{12} - \frac{2090905}{687259} a^{11} - \frac{3160796}{687259} a^{10} + \frac{10038073}{687259} a^{9} - \frac{11459435}{687259} a^{8} + \frac{9447421}{687259} a^{7} - \frac{1333144}{687259} a^{6} - \frac{7804759}{687259} a^{5} + \frac{11289391}{687259} a^{4} - \frac{2939568}{687259} a^{3} - \frac{5885665}{687259} a^{2} + \frac{5942261}{687259} a - \frac{1378203}{687259} \),  \( \frac{1388599}{687259} a^{17} - \frac{3663246}{687259} a^{16} + \frac{387737}{40427} a^{15} - \frac{6573714}{687259} a^{14} + \frac{1728176}{687259} a^{13} + \frac{4382514}{687259} a^{12} - \frac{4121904}{687259} a^{11} - \frac{6714305}{687259} a^{10} + \frac{17093000}{687259} a^{9} - \frac{18288378}{687259} a^{8} + \frac{13507448}{687259} a^{7} - \frac{551528}{687259} a^{6} - \frac{16842659}{687259} a^{5} + \frac{18241950}{687259} a^{4} - \frac{2678030}{687259} a^{3} - \frac{11906971}{687259} a^{2} + \frac{8814214}{687259} a - \frac{1999798}{687259} \),  \( \frac{492663}{687259} a^{17} - \frac{1060260}{687259} a^{16} + \frac{131200}{40427} a^{15} - \frac{1893056}{687259} a^{14} + \frac{907403}{687259} a^{13} + \frac{1395798}{687259} a^{12} - \frac{992218}{687259} a^{11} - \frac{1708219}{687259} a^{10} + \frac{5199932}{687259} a^{9} - \frac{5720666}{687259} a^{8} + \frac{4709749}{687259} a^{7} + \frac{251068}{687259} a^{6} - \frac{4301656}{687259} a^{5} + \frac{5897105}{687259} a^{4} - \frac{1437685}{687259} a^{3} - \frac{2921335}{687259} a^{2} + \frac{2400310}{687259} a - \frac{1092568}{687259} \),  \( \frac{1011374}{687259} a^{17} - \frac{2725867}{687259} a^{16} + \frac{289877}{40427} a^{15} - \frac{4961945}{687259} a^{14} + \frac{1253475}{687259} a^{13} + \frac{3413156}{687259} a^{12} - \frac{3426331}{687259} a^{11} - \frac{4775474}{687259} a^{10} + \frac{13044290}{687259} a^{9} - \frac{13791027}{687259} a^{8} + \frac{10027867}{687259} a^{7} - \frac{3394}{687259} a^{6} - \frac{12547178}{687259} a^{5} + \frac{14255397}{687259} a^{4} - \frac{2768422}{687259} a^{3} - \frac{9481395}{687259} a^{2} + \frac{7329097}{687259} a - \frac{1243488}{687259} \),  \( \frac{2380737}{687259} a^{17} - \frac{6165075}{687259} a^{16} + \frac{677530}{40427} a^{15} - \frac{11334639}{687259} a^{14} + \frac{3826671}{687259} a^{13} + \frac{7182152}{687259} a^{12} - \frac{6557402}{687259} a^{11} - \frac{10966138}{687259} a^{10} + \frac{29562111}{687259} a^{9} - \frac{31800294}{687259} a^{8} + \frac{24385737}{687259} a^{7} - \frac{1621759}{687259} a^{6} - \frac{25855745}{687259} a^{5} + \frac{31406005}{687259} a^{4} - \frac{5301270}{687259} a^{3} - \frac{19808000}{687259} a^{2} + \frac{15012197}{687259} a - \frac{3166988}{687259} \),  \( \frac{1691543}{687259} a^{17} - \frac{3470460}{687259} a^{16} + \frac{365757}{40427} a^{15} - \frac{4615036}{687259} a^{14} + \frac{35070}{687259} a^{13} + \frac{5108607}{687259} a^{12} - \frac{1875326}{687259} a^{11} - \frac{9006352}{687259} a^{10} + \frac{15857788}{687259} a^{9} - \frac{13737671}{687259} a^{8} + \frac{9643338}{687259} a^{7} + \frac{3942098}{687259} a^{6} - \frac{16600173}{687259} a^{5} + \frac{12819761}{687259} a^{4} + \frac{3656919}{687259} a^{3} - \frac{11892175}{687259} a^{2} + \frac{4277627}{687259} a - \frac{173402}{687259} \),  \( \frac{424221}{687259} a^{17} - \frac{815766}{687259} a^{16} + \frac{97006}{40427} a^{15} - \frac{1495086}{687259} a^{14} + \frac{562592}{687259} a^{13} + \frac{593840}{687259} a^{12} - \frac{493030}{687259} a^{11} - \frac{1729163}{687259} a^{10} + \frac{3611537}{687259} a^{9} - \frac{4428972}{687259} a^{8} + \frac{3959284}{687259} a^{7} - \frac{634}{687259} a^{6} - \frac{3078863}{687259} a^{5} + \frac{2782382}{687259} a^{4} - \frac{1465616}{687259} a^{3} - \frac{1343463}{687259} a^{2} + \frac{1951850}{687259} a - \frac{1179948}{687259} \),  \( \frac{159676}{687259} a^{17} - \frac{1104574}{687259} a^{16} + \frac{127804}{40427} a^{15} - \frac{3457034}{687259} a^{14} + \frac{2174327}{687259} a^{13} + \frac{86004}{687259} a^{12} - \frac{2689571}{687259} a^{11} - \frac{116380}{687259} a^{10} + \frac{5318848}{687259} a^{9} - \frac{9192462}{687259} a^{8} + \frac{7789047}{687259} a^{7} - \frac{4577933}{687259} a^{6} - \frac{2975618}{687259} a^{5} + \frac{8468892}{687259} a^{4} - \frac{6379519}{687259} a^{3} - \frac{2817426}{687259} a^{2} + \frac{5772466}{687259} a - \frac{2437593}{687259} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 86.4501902329 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T319:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 1296
The 22 conjugacy class representatives for t18n319
Character table for t18n319 is not computed

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.12167.1, 9.1.32206049.1

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

Sibling fields

Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
2647Data not computed