Properties

Label 18.0.16230968037...8343.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 43^{4}$
Root discriminant $11.67$
Ramified primes $7, 43$
Class number $1$
Class group Trivial
Galois Group $C_3^3:C_6$ (as 18T85)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 5, 0, -15, 18, 27, -97, 83, 31, -100, 40, 37, -32, 2, 9, -5, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 5*x^16 + 9*x^15 + 2*x^14 - 32*x^13 + 37*x^12 + 40*x^11 - 100*x^10 + 31*x^9 + 83*x^8 - 97*x^7 + 27*x^6 + 18*x^5 - 15*x^4 + 5*x^2 - 3*x + 1)
gp: K = bnfinit(x^18 - x^17 - 5*x^16 + 9*x^15 + 2*x^14 - 32*x^13 + 37*x^12 + 40*x^11 - 100*x^10 + 31*x^9 + 83*x^8 - 97*x^7 + 27*x^6 + 18*x^5 - 15*x^4 + 5*x^2 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut x^{17} \) \(\mathstrut -\mathstrut 5 x^{16} \) \(\mathstrut +\mathstrut 9 x^{15} \) \(\mathstrut +\mathstrut 2 x^{14} \) \(\mathstrut -\mathstrut 32 x^{13} \) \(\mathstrut +\mathstrut 37 x^{12} \) \(\mathstrut +\mathstrut 40 x^{11} \) \(\mathstrut -\mathstrut 100 x^{10} \) \(\mathstrut +\mathstrut 31 x^{9} \) \(\mathstrut +\mathstrut 83 x^{8} \) \(\mathstrut -\mathstrut 97 x^{7} \) \(\mathstrut +\mathstrut 27 x^{6} \) \(\mathstrut +\mathstrut 18 x^{5} \) \(\mathstrut -\mathstrut 15 x^{4} \) \(\mathstrut +\mathstrut 5 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-16230968037754638343=-\,7^{15}\cdot 43^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.67$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{6} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7}$, $\frac{1}{7} a^{14} - \frac{1}{7} a^{7} + \frac{1}{7}$, $\frac{1}{7} a^{15} - \frac{1}{7} a^{8} + \frac{1}{7} a$, $\frac{1}{91} a^{16} - \frac{4}{91} a^{15} + \frac{1}{91} a^{14} + \frac{4}{91} a^{13} - \frac{3}{91} a^{12} - \frac{3}{13} a^{11} - \frac{25}{91} a^{10} - \frac{19}{91} a^{9} - \frac{10}{91} a^{8} + \frac{45}{91} a^{7} + \frac{3}{13} a^{6} + \frac{38}{91} a^{5} - \frac{18}{91} a^{4} - \frac{37}{91} a^{2} + \frac{1}{13} a - \frac{41}{91}$, $\frac{1}{45227} a^{17} + \frac{25}{45227} a^{16} - \frac{349}{45227} a^{15} + \frac{28}{923} a^{14} + \frac{2375}{45227} a^{13} - \frac{407}{45227} a^{12} + \frac{18281}{45227} a^{11} - \frac{7738}{45227} a^{10} + \frac{17392}{45227} a^{9} - \frac{7005}{45227} a^{8} + \frac{677}{3479} a^{7} + \frac{22474}{45227} a^{6} + \frac{14292}{45227} a^{5} - \frac{15056}{45227} a^{4} + \frac{7620}{45227} a^{3} - \frac{1658}{3479} a^{2} + \frac{883}{6461} a + \frac{5142}{45227}$

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:  \( \frac{1590}{923} a^{17} - \frac{344}{497} a^{16} - \frac{59245}{6461} a^{15} + \frac{65497}{6461} a^{14} + \frac{5092}{497} a^{13} - \frac{323870}{6461} a^{12} + \frac{215049}{6461} a^{11} + \frac{601779}{6461} a^{10} - \frac{111767}{923} a^{9} - \frac{163140}{6461} a^{8} + \frac{907360}{6461} a^{7} - \frac{552529}{6461} a^{6} - \frac{95568}{6461} a^{5} + \frac{218464}{6461} a^{4} - \frac{59250}{6461} a^{3} - \frac{43962}{6461} a^{2} + \frac{3015}{497} a - \frac{1787}{923} \) (order $14$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{29083}{45227} a^{17} - \frac{5006}{45227} a^{16} - \frac{12787}{3479} a^{15} + \frac{18919}{6461} a^{14} + \frac{260984}{45227} a^{13} - \frac{814812}{45227} a^{12} + \frac{284360}{45227} a^{11} + \frac{1877000}{45227} a^{10} - \frac{1662242}{45227} a^{9} - \frac{1425547}{45227} a^{8} + \frac{2372688}{45227} a^{7} - \frac{374060}{45227} a^{6} - \frac{768148}{45227} a^{5} + \frac{385922}{45227} a^{4} + \frac{51848}{45227} a^{3} - \frac{45043}{45227} a^{2} + \frac{3210}{6461} a - \frac{4005}{45227} \),  \( \frac{3188}{6461} a^{17} + \frac{109}{6461} a^{16} - \frac{1172}{497} a^{15} + \frac{11625}{6461} a^{14} + \frac{14970}{6461} a^{13} - \frac{77594}{6461} a^{12} + \frac{39309}{6461} a^{11} + \frac{20447}{923} a^{10} - \frac{137009}{6461} a^{9} - \frac{1799}{923} a^{8} + \frac{154020}{6461} a^{7} - \frac{138148}{6461} a^{6} + \frac{33365}{6461} a^{5} + \frac{31765}{6461} a^{4} - \frac{21106}{6461} a^{3} - \frac{11073}{6461} a^{2} + \frac{6717}{6461} a - \frac{3050}{6461} \),  \( \frac{33504}{45227} a^{17} + \frac{1149}{45227} a^{16} - \frac{12385}{3479} a^{15} + \frac{17673}{6461} a^{14} + \frac{167304}{45227} a^{13} - \frac{821480}{45227} a^{12} + \frac{382418}{45227} a^{11} + \frac{1528409}{45227} a^{10} - \frac{1426011}{45227} a^{9} - \frac{267081}{45227} a^{8} + \frac{1670015}{45227} a^{7} - \frac{1137653}{45227} a^{6} + \frac{288802}{45227} a^{5} + \frac{72949}{45227} a^{4} - \frac{167460}{45227} a^{3} - \frac{69820}{45227} a^{2} + \frac{8130}{6461} a - \frac{50721}{45227} \),  \( \frac{2213}{45227} a^{17} - \frac{8291}{45227} a^{16} - \frac{20376}{45227} a^{15} + \frac{6541}{6461} a^{14} + \frac{32902}{45227} a^{13} - \frac{108970}{45227} a^{12} + \frac{11091}{3479} a^{11} + \frac{198741}{45227} a^{10} - \frac{477298}{45227} a^{9} - \frac{218818}{45227} a^{8} + \frac{325712}{45227} a^{7} + \frac{969}{3479} a^{6} + \frac{123366}{45227} a^{5} - \frac{120819}{45227} a^{4} - \frac{58299}{45227} a^{3} + \frac{159613}{45227} a^{2} + \frac{5697}{6461} a - \frac{58712}{45227} \),  \( \frac{2881}{45227} a^{17} + \frac{16858}{45227} a^{16} - \frac{15942}{45227} a^{15} - \frac{11773}{6461} a^{14} + \frac{89636}{45227} a^{13} + \frac{65460}{45227} a^{12} - \frac{439871}{45227} a^{11} + \frac{316843}{45227} a^{10} + \frac{920250}{45227} a^{9} - \frac{1002672}{45227} a^{8} - \frac{483349}{45227} a^{7} + \frac{1317969}{45227} a^{6} - \frac{475336}{45227} a^{5} - \frac{399752}{45227} a^{4} + \frac{373480}{45227} a^{3} - \frac{149503}{45227} a^{2} - \frac{7036}{6461} a + \frac{31831}{45227} \),  \( \frac{17788}{45227} a^{17} + \frac{9328}{45227} a^{16} - \frac{105349}{45227} a^{15} + \frac{207}{497} a^{14} + \frac{227138}{45227} a^{13} - \frac{441750}{45227} a^{12} - \frac{173652}{45227} a^{11} + \frac{1394691}{45227} a^{10} - \frac{414683}{45227} a^{9} - \frac{1685409}{45227} a^{8} + \frac{1460356}{45227} a^{7} + \frac{677506}{45227} a^{6} - \frac{1271664}{45227} a^{5} + \frac{237343}{45227} a^{4} + \frac{315830}{45227} a^{3} - \frac{121619}{45227} a^{2} - \frac{3295}{6461} a + \frac{21872}{45227} \),  \( \frac{18604}{45227} a^{17} - \frac{15499}{45227} a^{16} - \frac{86466}{45227} a^{15} + \frac{22331}{6461} a^{14} + \frac{13625}{45227} a^{13} - \frac{573571}{45227} a^{12} + \frac{678047}{45227} a^{11} + \frac{624021}{45227} a^{10} - \frac{1755055}{45227} a^{9} + \frac{887974}{45227} a^{8} + \frac{1366886}{45227} a^{7} - \frac{2059492}{45227} a^{6} + \frac{873555}{45227} a^{5} + \frac{369689}{45227} a^{4} - \frac{612116}{45227} a^{3} + \frac{129138}{45227} a^{2} + \frac{13907}{6461} a - \frac{65899}{45227} \),  \( \frac{183}{497} a^{17} - \frac{662}{6461} a^{16} - \frac{11925}{6461} a^{15} + \frac{13040}{6461} a^{14} + \frac{11873}{6461} a^{13} - \frac{66979}{6461} a^{12} + \frac{43256}{6461} a^{11} + \frac{16538}{923} a^{10} - \frac{21995}{923} a^{9} - \frac{10709}{6461} a^{8} + \frac{175688}{6461} a^{7} - \frac{118409}{6461} a^{6} + \frac{7643}{6461} a^{5} + \frac{27191}{6461} a^{4} - \frac{1613}{497} a^{3} + \frac{10168}{6461} a^{2} + \frac{690}{923} a + \frac{583}{6461} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 847.661854898 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_3^3:C_6$ (as 18T85):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 162
The 13 conjugacy class representatives for $C_3^3:C_6$
Character table for $C_3^3:C_6$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 9.3.1522731007.1 x3

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$43$43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.3.2.3$x^{3} - 3483$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.3.2.3$x^{3} - 3483$$3$$1$$2$$C_3$$[\ ]_{3}$