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Properties

Label 18.0.10405887262...2499.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 13^{12}\cdot 19^{9}$
Root discriminant $88.19$
Ramified primes $7, 13, 19$
Class number $29484$ (GRH)
Class group $[18, 1638]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![152453839, -6470051, 98419102, -3014537, 29142209, -804120, 5328864, -297270, 668982, -59326, 62002, -6396, 4726, -530, 322, -24, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 7*x^16 - 24*x^15 + 322*x^14 - 530*x^13 + 4726*x^12 - 6396*x^11 + 62002*x^10 - 59326*x^9 + 668982*x^8 - 297270*x^7 + 5328864*x^6 - 804120*x^5 + 29142209*x^4 - 3014537*x^3 + 98419102*x^2 - 6470051*x + 152453839)
 
gp: K = bnfinit(x^18 - 3*x^17 + 7*x^16 - 24*x^15 + 322*x^14 - 530*x^13 + 4726*x^12 - 6396*x^11 + 62002*x^10 - 59326*x^9 + 668982*x^8 - 297270*x^7 + 5328864*x^6 - 804120*x^5 + 29142209*x^4 - 3014537*x^3 + 98419102*x^2 - 6470051*x + 152453839, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 7 x^{16} - 24 x^{15} + 322 x^{14} - 530 x^{13} + 4726 x^{12} - 6396 x^{11} + 62002 x^{10} - 59326 x^{9} + 668982 x^{8} - 297270 x^{7} + 5328864 x^{6} - 804120 x^{5} + 29142209 x^{4} - 3014537 x^{3} + 98419102 x^{2} - 6470051 x + 152453839 \)

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:  \(-104058872623657018481772248523222499=-\,7^{12}\cdot 13^{12}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.19$
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:  $7, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1729=7\cdot 13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1729}(1,·)$, $\chi_{1729}(835,·)$, $\chi_{1729}(1348,·)$, $\chi_{1729}(438,·)$, $\chi_{1729}(1101,·)$, $\chi_{1729}(911,·)$, $\chi_{1729}(1236,·)$, $\chi_{1729}(666,·)$, $\chi_{1729}(989,·)$, $\chi_{1729}(417,·)$, $\chi_{1729}(932,·)$, $\chi_{1729}(1576,·)$, $\chi_{1729}(170,·)$, $\chi_{1729}(172,·)$, $\chi_{1729}(113,·)$, $\chi_{1729}(1654,·)$, $\chi_{1729}(1082,·)$, $\chi_{1729}(191,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{30} a^{12} - \frac{1}{15} a^{11} - \frac{1}{5} a^{9} + \frac{7}{30} a^{8} - \frac{1}{3} a^{7} - \frac{2}{15} a^{6} + \frac{1}{6} a^{5} - \frac{11}{30} a^{4} - \frac{1}{15} a^{3} - \frac{2}{15} a^{2} - \frac{1}{2} a + \frac{13}{30}$, $\frac{1}{30} a^{13} - \frac{2}{15} a^{11} - \frac{1}{5} a^{10} - \frac{1}{6} a^{9} + \frac{2}{15} a^{8} + \frac{1}{5} a^{7} - \frac{1}{10} a^{6} - \frac{1}{30} a^{5} + \frac{1}{5} a^{4} - \frac{4}{15} a^{3} + \frac{7}{30} a^{2} + \frac{13}{30} a - \frac{2}{15}$, $\frac{1}{30} a^{14} + \frac{1}{30} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{2}{15} a^{8} - \frac{13}{30} a^{7} + \frac{13}{30} a^{6} - \frac{2}{15} a^{5} - \frac{7}{30} a^{4} + \frac{7}{15} a^{3} + \frac{2}{5} a^{2} + \frac{11}{30} a - \frac{4}{15}$, $\frac{1}{303630150} a^{15} + \frac{834183}{101210050} a^{14} + \frac{28568}{151815075} a^{13} - \frac{1499948}{151815075} a^{12} - \frac{21127451}{303630150} a^{11} + \frac{51971809}{303630150} a^{10} - \frac{4733959}{303630150} a^{9} + \frac{16611043}{303630150} a^{8} + \frac{6226747}{50605025} a^{7} - \frac{123440117}{303630150} a^{6} - \frac{9063014}{50605025} a^{5} + \frac{29635103}{101210050} a^{4} - \frac{38120753}{151815075} a^{3} - \frac{21501503}{151815075} a^{2} - \frac{65824231}{303630150} a - \frac{4787752}{50605025}$, $\frac{1}{109914114300} a^{16} + \frac{17}{18319019050} a^{15} - \frac{289890871}{54957057150} a^{14} + \frac{285074878}{27478528575} a^{13} + \frac{149833108}{9159509525} a^{12} - \frac{1080627716}{27478528575} a^{11} + \frac{6663005867}{27478528575} a^{10} + \frac{1479641088}{9159509525} a^{9} - \frac{1934347736}{27478528575} a^{8} - \frac{12503555449}{27478528575} a^{7} + \frac{388306805}{2198282286} a^{6} + \frac{5815154408}{27478528575} a^{5} - \frac{1727288141}{27478528575} a^{4} - \frac{439501587}{9159509525} a^{3} - \frac{46557822589}{109914114300} a^{2} - \frac{385244909}{5495705715} a + \frac{6450530623}{36638038100}$, $\frac{1}{33919804422025425790109618731136838300} a^{17} + \frac{30236559446389760726908943}{11306601474008475263369872910378946100} a^{16} - \frac{140695255978920796121599423}{339198044220254257901096187311368383} a^{15} + \frac{10136290422255605446428033670927019}{3391980442202542579010961873113683830} a^{14} + \frac{31684556885951761560253823380185944}{2826650368502118815842468227594736525} a^{13} + \frac{10262708725265545063569466792742372}{1695990221101271289505480936556841915} a^{12} - \frac{52229930378787473793924702613866219}{2826650368502118815842468227594736525} a^{11} - \frac{1445400604944690375807309237410957089}{8479951105506356447527404682784209575} a^{10} - \frac{1036269215336121463591226778601787289}{5653300737004237631684936455189473050} a^{9} - \frac{361778293937181211098351424189910536}{1695990221101271289505480936556841915} a^{8} - \frac{5258357414981377171651659277598002793}{16959902211012712895054809365568419150} a^{7} - \frac{884462925006835197047369546802315429}{5653300737004237631684936455189473050} a^{6} - \frac{959116814249149519970995748876983836}{2826650368502118815842468227594736525} a^{5} + \frac{40675781417340894395731639153501855}{113066014740084752633698729103789461} a^{4} + \frac{581956474843780646023456152862342797}{2261320294801695052673974582075789220} a^{3} - \frac{9063825329707145862949183137638469121}{33919804422025425790109618731136838300} a^{2} - \frac{4182443978708583942019147751929965983}{11306601474008475263369872910378946100} a - \frac{2322806205695963773606355319284782691}{11306601474008475263369872910378946100}$

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

Class group and class number

$C_{18}\times C_{1638}$, which has order $29484$ (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:  \( 205236.825908 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-19}) \), 3.3.8281.1, \(\Q(\zeta_{7})^+\), 3.3.8281.2, 3.3.169.1, 6.0.470355657499.3, 6.0.16468459.1, 6.0.470355657499.2, 6.0.195899899.1, 9.9.567869252041.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$7.9.6.1$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}$
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$19$19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$