↑

Properties

Label 18.0.12397041052...9375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 5^{9}\cdot 7^{12}\cdot 13^{12}$
Root discriminant $78.36$
Ramified primes $3, 5, 7, 13$
Class number $22400$ (GRH)
Class group $[2, 2, 2, 20, 140]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![39701416, -1743720, 28645658, -679057, 9403275, -24520, 1914474, -69734, 264952, -23076, 26144, -2712, 2056, -314, 226, 0, -2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 2*x^16 + 226*x^14 - 314*x^13 + 2056*x^12 - 2712*x^11 + 26144*x^10 - 23076*x^9 + 264952*x^8 - 69734*x^7 + 1914474*x^6 - 24520*x^5 + 9403275*x^4 - 679057*x^3 + 28645658*x^2 - 1743720*x + 39701416)
 
gp: K = bnfinit(x^18 - 3*x^17 - 2*x^16 + 226*x^14 - 314*x^13 + 2056*x^12 - 2712*x^11 + 26144*x^10 - 23076*x^9 + 264952*x^8 - 69734*x^7 + 1914474*x^6 - 24520*x^5 + 9403275*x^4 - 679057*x^3 + 28645658*x^2 - 1743720*x + 39701416, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 2 x^{16} + 226 x^{14} - 314 x^{13} + 2056 x^{12} - 2712 x^{11} + 26144 x^{10} - 23076 x^{9} + 264952 x^{8} - 69734 x^{7} + 1914474 x^{6} - 24520 x^{5} + 9403275 x^{4} - 679057 x^{3} + 28645658 x^{2} - 1743720 x + 39701416 \)

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:  \(-12397041052269497924235545162109375=-\,3^{9}\cdot 5^{9}\cdot 7^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.36$
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:  $3, 5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1365=3\cdot 5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1365}(256,·)$, $\chi_{1365}(1,·)$, $\chi_{1365}(646,·)$, $\chi_{1365}(841,·)$, $\chi_{1365}(74,·)$, $\chi_{1365}(781,·)$, $\chi_{1365}(16,·)$, $\chi_{1365}(1171,·)$, $\chi_{1365}(599,·)$, $\chi_{1365}(211,·)$, $\chi_{1365}(989,·)$, $\chi_{1365}(991,·)$, $\chi_{1365}(1184,·)$, $\chi_{1365}(464,·)$, $\chi_{1365}(809,·)$, $\chi_{1365}(29,·)$, $\chi_{1365}(1199,·)$, $\chi_{1365}(659,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{284061960738} a^{15} + \frac{19714875557}{284061960738} a^{14} + \frac{2012792413}{94687320246} a^{13} - \frac{33703819}{142030980369} a^{12} + \frac{34153152878}{142030980369} a^{11} + \frac{30897692513}{142030980369} a^{10} + \frac{863803521}{15781220041} a^{9} + \frac{5108650473}{31562440082} a^{8} - \frac{88578698645}{284061960738} a^{7} + \frac{27743203349}{284061960738} a^{6} - \frac{6781816280}{15781220041} a^{5} - \frac{55071579002}{142030980369} a^{4} + \frac{16013557090}{142030980369} a^{3} + \frac{33366824830}{142030980369} a^{2} - \frac{15257377318}{142030980369} a + \frac{27067014941}{142030980369}$, $\frac{1}{47154285482508} a^{16} - \frac{29}{23577142741254} a^{15} - \frac{32447640475}{1309841263403} a^{14} - \frac{1399063721521}{23577142741254} a^{13} + \frac{199719853426}{11788571370627} a^{12} - \frac{273030208505}{11788571370627} a^{11} + \frac{5551189223}{7859047580418} a^{10} - \frac{32846658285}{2619682526806} a^{9} + \frac{4707002084915}{23577142741254} a^{8} - \frac{4867780997626}{11788571370627} a^{7} - \frac{1646574697327}{7859047580418} a^{6} + \frac{1377592257965}{11788571370627} a^{5} + \frac{11629820360}{78070009077} a^{4} - \frac{9559624998269}{23577142741254} a^{3} - \frac{202552279751}{47154285482508} a^{2} - \frac{8364041947}{142030980369} a + \frac{547594095569}{1309841263403}$, $\frac{1}{24725189722462355900423992840884} a^{17} - \frac{28808558311998197}{2747243302495817322269332537876} a^{16} - \frac{2250330822861340791}{1373621651247908661134666268938} a^{15} + \frac{79217498026437444399856436987}{1373621651247908661134666268938} a^{14} - \frac{83810269498315438345494162007}{6181297430615588975105998210221} a^{13} + \frac{34273837554968667543754007277}{1373621651247908661134666268938} a^{12} - \frac{1315703456709904288127154649367}{12362594861231177950211996420442} a^{11} - \frac{532274725395947126341127392724}{6181297430615588975105998210221} a^{10} - \frac{59460037353512830905258972517}{12362594861231177950211996420442} a^{9} + \frac{360392674799031559624601900291}{4120864953743725983403998806814} a^{8} - \frac{6714167686781599726861176437}{22767209689191856261900545894} a^{7} + \frac{2650547691604120228464919346683}{6181297430615588975105998210221} a^{6} + \frac{4451095475610612074435727939833}{12362594861231177950211996420442} a^{5} - \frac{196785136651496582621929631595}{1373621651247908661134666268938} a^{4} - \frac{135374361230499884241089707763}{297893852077859709643662564348} a^{3} + \frac{6198551788797651173855084978467}{24725189722462355900423992840884} a^{2} - \frac{466387405387457445608497331089}{6181297430615588975105998210221} a - \frac{1885889753862291534521905882562}{6181297430615588975105998210221}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{20}\times C_{140}$, which has order $22400$ (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

\(\Q(\sqrt{-15}) \), 3.3.8281.1, 3.3.8281.2, 3.3.169.1, \(\Q(\zeta_{7})^+\), 6.0.231440493375.7, 6.0.231440493375.6, 6.0.96393375.1, 6.0.8103375.1, 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
$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}$