Properties

Label 18.0.13950087665...5207.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,11^{4}\cdot 23^{11}$
Root discriminant $11.58$
Ramified primes $11, 23$
Class number $1$
Class group Trivial
Galois Group 18T217

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -1, -1, 6, -5, -2, 11, -12, 1, 7, -9, 6, 1, -5, 4, 0, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 4*x^15 - 5*x^14 + x^13 + 6*x^12 - 9*x^11 + 7*x^10 + x^9 - 12*x^8 + 11*x^7 - 2*x^6 - 5*x^5 + 6*x^4 - x^3 - x^2 - x + 1)
gp: K = bnfinit(x^18 - x^17 + 4*x^15 - 5*x^14 + x^13 + 6*x^12 - 9*x^11 + 7*x^10 + x^9 - 12*x^8 + 11*x^7 - 2*x^6 - 5*x^5 + 6*x^4 - x^3 - x^2 - x + 1, 1)

Normalized defining polynomial

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

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:  \(-13950087665617805207=-\,11^{4}\cdot 23^{11}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.58$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{16867} a^{17} - \frac{6775}{16867} a^{16} - \frac{1257}{16867} a^{15} - \frac{2913}{16867} a^{14} - \frac{1733}{16867} a^{13} - \frac{89}{16867} a^{12} - \frac{4320}{16867} a^{11} - \frac{574}{16867} a^{10} - \frac{7994}{16867} a^{9} + \frac{8287}{16867} a^{8} - \frac{2774}{16867} a^{7} + \frac{1249}{16867} a^{6} + \frac{6506}{16867} a^{5} + \frac{1822}{16867} a^{4} + \frac{4422}{16867} a^{3} + \frac{1163}{16867} a^{2} - \frac{1274}{16867} a - \frac{5829}{16867}$

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:  \( \frac{17410}{16867} a^{17} - \frac{1819}{16867} a^{16} + \frac{8996}{16867} a^{15} + \frac{54340}{16867} a^{14} - \frac{13334}{16867} a^{13} + \frac{19141}{16867} a^{12} + \frac{49354}{16867} a^{11} - \frac{24943}{16867} a^{10} + \frac{61545}{16867} a^{9} + \frac{13219}{16867} a^{8} - \frac{55720}{16867} a^{7} - \frac{13340}{16867} a^{6} - \frac{9312}{16867} a^{5} - \frac{5807}{16867} a^{4} - \frac{27702}{16867} a^{3} + \frac{41164}{16867} a^{2} - \frac{235}{16867} a - \frac{11018}{16867} \),  \( \frac{5790}{16867} a^{17} + \frac{5392}{16867} a^{16} - \frac{8353}{16867} a^{15} + \frac{34464}{16867} a^{14} + \frac{1795}{16867} a^{13} - \frac{26167}{16867} a^{12} + \frac{68429}{16867} a^{11} - \frac{34395}{16867} a^{10} - \frac{2212}{16867} a^{9} + \frac{79450}{16867} a^{8} - \frac{105278}{16867} a^{7} + \frac{29501}{16867} a^{6} + \frac{22596}{16867} a^{5} - \frac{76830}{16867} a^{4} + \frac{33008}{16867} a^{3} - \frac{13030}{16867} a^{2} + \frac{11286}{16867} a + \frac{957}{16867} \),  \( \frac{15616}{16867} a^{17} - \frac{8576}{16867} a^{16} + \frac{3876}{16867} a^{15} + \frac{51492}{16867} a^{14} - \frac{41594}{16867} a^{13} + \frac{10137}{16867} a^{12} + \frac{57481}{16867} a^{11} - \frac{74675}{16867} a^{10} + \frac{65831}{16867} a^{9} + \frac{6168}{16867} a^{8} - \frac{122397}{16867} a^{7} + \frac{56733}{16867} a^{6} - \frac{9112}{16867} a^{5} - \frac{52878}{16867} a^{4} + \frac{34188}{16867} a^{3} + \frac{12516}{16867} a^{2} - \frac{8591}{16867} a + \frac{5535}{16867} \),  \( \frac{9857}{16867} a^{17} - \frac{4722}{16867} a^{16} + \frac{6996}{16867} a^{15} + \frac{27927}{16867} a^{14} - \frac{12777}{16867} a^{13} + \frac{16678}{16867} a^{12} + \frac{23802}{16867} a^{11} - \frac{7473}{16867} a^{10} + \frac{39500}{16867} a^{9} - \frac{1922}{16867} a^{8} - \frac{18778}{16867} a^{7} - \frac{1517}{16867} a^{6} + \frac{18175}{16867} a^{5} - \frac{3901}{16867} a^{4} - \frac{30408}{16867} a^{3} + \frac{27865}{16867} a^{2} - \frac{8770}{16867} a - \frac{7451}{16867} \),  \( \frac{2247}{16867} a^{17} + \frac{7476}{16867} a^{16} - \frac{7690}{16867} a^{15} + \frac{15752}{16867} a^{14} + \frac{19093}{16867} a^{13} - \frac{31313}{16867} a^{12} + \frac{42086}{16867} a^{11} + \frac{8981}{16867} a^{10} - \frac{32897}{16867} a^{9} + \frac{84056}{16867} a^{8} - \frac{42989}{16867} a^{7} - \frac{27153}{16867} a^{6} + \frac{62761}{16867} a^{5} - \frac{55248}{16867} a^{4} + \frac{18438}{16867} a^{3} + \frac{32610}{16867} a^{2} - \frac{12155}{16867} a - \frac{8971}{16867} \),  \( \frac{9221}{16867} a^{17} + \frac{3093}{16867} a^{16} - \frac{3168}{16867} a^{15} + \frac{42092}{16867} a^{14} - \frac{6944}{16867} a^{13} - \frac{11053}{16867} a^{12} + \frac{72602}{16867} a^{11} - \frac{47217}{16867} a^{10} + \frac{29850}{16867} a^{9} + \frac{74385}{16867} a^{8} - \frac{109884}{16867} a^{7} + \frac{47469}{16867} a^{6} + \frac{12774}{16867} a^{5} - \frac{66338}{16867} a^{4} + \frac{58324}{16867} a^{3} + \frac{13478}{16867} a^{2} + \frac{8745}{16867} a - \frac{10947}{16867} \),  \( \frac{9801}{16867} a^{17} + \frac{3604}{16867} a^{16} - \frac{6947}{16867} a^{15} + \frac{39252}{16867} a^{14} - \frac{64}{16867} a^{13} - \frac{28939}{16867} a^{12} + \frac{63318}{16867} a^{11} - \frac{25930}{16867} a^{10} - \frac{1979}{16867} a^{9} + \frac{73750}{16867} a^{8} - \frac{99572}{16867} a^{7} - \frac{3993}{16867} a^{6} + \frac{58647}{16867} a^{5} - \frac{55332}{16867} a^{4} + \frac{25566}{16867} a^{3} + \frac{30205}{16867} a^{2} - \frac{21761}{16867} a - \frac{1500}{16867} \),  \( \frac{17410}{16867} a^{17} - \frac{1819}{16867} a^{16} + \frac{8996}{16867} a^{15} + \frac{54340}{16867} a^{14} - \frac{13334}{16867} a^{13} + \frac{19141}{16867} a^{12} + \frac{49354}{16867} a^{11} - \frac{24943}{16867} a^{10} + \frac{61545}{16867} a^{9} + \frac{13219}{16867} a^{8} - \frac{55720}{16867} a^{7} - \frac{13340}{16867} a^{6} - \frac{9312}{16867} a^{5} - \frac{5807}{16867} a^{4} - \frac{27702}{16867} a^{3} + \frac{41164}{16867} a^{2} + \frac{16632}{16867} a - \frac{11018}{16867} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 91.5286994248 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T217:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 648
The 14 conjugacy class representatives for t18n217
Character table for t18n217

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.12167.1, 9.1.33860761.1

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 siblings: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
$23$23.2.1.2$x^{2} + 46$$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$$[\ ]_{2}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$