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Properties

Label 18.0.65207117393...9083.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{15}\cdot 23^{9}$
Root discriminant $126.12$
Ramified primes $3, 7, 23$
Class number $2517424$ (GRH)
Class group $[2, 2, 26, 24206]$ (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![251122682539, -50842837434, 95124963726, -22003449974, 22666724232, -3134543427, 3750828353, -189463887, 399193434, -5091888, 25847919, -60570, 988554, -237, 21303, -1, 234, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 234*x^16 - x^15 + 21303*x^14 - 237*x^13 + 988554*x^12 - 60570*x^11 + 25847919*x^10 - 5091888*x^9 + 399193434*x^8 - 189463887*x^7 + 3750828353*x^6 - 3134543427*x^5 + 22666724232*x^4 - 22003449974*x^3 + 95124963726*x^2 - 50842837434*x + 251122682539)
 
gp: K = bnfinit(x^18 + 234*x^16 - x^15 + 21303*x^14 - 237*x^13 + 988554*x^12 - 60570*x^11 + 25847919*x^10 - 5091888*x^9 + 399193434*x^8 - 189463887*x^7 + 3750828353*x^6 - 3134543427*x^5 + 22666724232*x^4 - 22003449974*x^3 + 95124963726*x^2 - 50842837434*x + 251122682539, 1)
 

Normalized defining polynomial

\( x^{18} + 234 x^{16} - x^{15} + 21303 x^{14} - 237 x^{13} + 988554 x^{12} - 60570 x^{11} + 25847919 x^{10} - 5091888 x^{9} + 399193434 x^{8} - 189463887 x^{7} + 3750828353 x^{6} - 3134543427 x^{5} + 22666724232 x^{4} - 22003449974 x^{3} + 95124963726 x^{2} - 50842837434 x + 251122682539 \)

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:  \(-65207117393079565410291669478441819083=-\,3^{27}\cdot 7^{15}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $126.12$
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:  $3, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1449=3^{2}\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{1449}(1,·)$, $\chi_{1449}(898,·)$, $\chi_{1449}(68,·)$, $\chi_{1449}(965,·)$, $\chi_{1449}(967,·)$, $\chi_{1449}(1034,·)$, $\chi_{1449}(206,·)$, $\chi_{1449}(1172,·)$, $\chi_{1449}(277,·)$, $\chi_{1449}(1243,·)$, $\chi_{1449}(415,·)$, $\chi_{1449}(482,·)$, $\chi_{1449}(484,·)$, $\chi_{1449}(1381,·)$, $\chi_{1449}(551,·)$, $\chi_{1449}(1448,·)$, $\chi_{1449}(689,·)$, $\chi_{1449}(760,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{181} a^{14} - \frac{29}{181} a^{13} - \frac{54}{181} a^{12} - \frac{9}{181} a^{11} - \frac{43}{181} a^{10} - \frac{38}{181} a^{9} + \frac{62}{181} a^{8} + \frac{59}{181} a^{7} + \frac{60}{181} a^{6} + \frac{40}{181} a^{5} - \frac{50}{181} a^{4} + \frac{87}{181} a^{3} - \frac{32}{181} a^{2} - \frac{47}{181} a + \frac{35}{181}$, $\frac{1}{410327} a^{15} + \frac{62817}{410327} a^{13} + \frac{129107}{410327} a^{12} - \frac{45373}{410327} a^{11} - \frac{192059}{410327} a^{10} + \frac{85116}{410327} a^{9} - \frac{169188}{410327} a^{8} + \frac{64397}{410327} a^{7} - \frac{193519}{410327} a^{6} + \frac{104461}{410327} a^{5} + \frac{31217}{410327} a^{4} + \frac{142223}{410327} a^{3} + \frac{154142}{410327} a^{2} - \frac{88208}{410327} a - \frac{200076}{410327}$, $\frac{1}{12527253356129} a^{16} - \frac{1834230}{12527253356129} a^{15} + \frac{13929247592}{12527253356129} a^{14} - \frac{1252586355732}{12527253356129} a^{13} - \frac{4774277509145}{12527253356129} a^{12} + \frac{3645067880969}{12527253356129} a^{11} + \frac{813171027144}{12527253356129} a^{10} - \frac{2993359217480}{12527253356129} a^{9} - \frac{2783834154364}{12527253356129} a^{8} + \frac{2807443583318}{12527253356129} a^{7} - \frac{3515182688622}{12527253356129} a^{6} - \frac{916435033723}{12527253356129} a^{5} - \frac{1933777673737}{12527253356129} a^{4} + \frac{5411948663759}{12527253356129} a^{3} - \frac{796152602934}{12527253356129} a^{2} + \frac{3104356512916}{12527253356129} a + \frac{651285035969}{12527253356129}$, $\frac{1}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{17} - \frac{2660215850714060414157414451468436872193885486792255973}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{16} + \frac{397983609416650915307012543195016516568824659258849136444980998}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{15} - \frac{1841269010368798988283092258842028750935417629171043207526440494461}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{14} - \frac{478650696577164752142577782326075376094095763894701891714223245333466}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{13} + \frac{65918657641086590418430787764135930568100639596393242900630496745145}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{12} - \frac{243474746954388398599431175739170015353401989036492367222126670706236}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{11} - \frac{492218418161123112481223550291545343467872384354203739946383713799431}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{10} - \frac{39301164653695885027167319865387924785611988830073413592438693849419}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{9} + \frac{4335487772449085187810980436972878795151433049518559156293980237091}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{8} - \frac{321053426298330962109824744674569690872729126645938012429077859614714}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{7} + \frac{345361174250632389968101157872159628026638240728926278361234783389800}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{6} + \frac{818422958492091140991837137935207037966759216693769991257656638437507}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{5} + \frac{7854546413376227926917002423591713739342647121535878741775229399072}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{4} + \frac{177950798984400580123390209777616596857394067180357910668403355529732}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{3} + \frac{898662572862693851988306263586104695410901846324104986903946024530510}{1825660826359346397484253975278409470294936140978270426327462622711657} a^{2} - \frac{366795233924930703511596468350853523138182294594811137901070031393}{805320170427589941545767082169567476971740688565624360973737372171} a - \frac{730328724744047815753789905153104718382549174516156328782782000751940}{1825660826359346397484253975278409470294936140978270426327462622711657}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{26}\times C_{24206}$, which has order $2517424$ (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:  \( 54408.48888868202 \) (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

\(\Q(\sqrt{-483}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{7})^+\), 6.0.82142689923.4, 6.0.4024991806227.4, 6.0.4024991806227.3, 6.0.5521250763.1, 9.9.62523502209.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
7Data not computed
$23$23.6.3.1$x^{6} - 46 x^{4} + 529 x^{2} - 194672$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.1$x^{6} - 46 x^{4} + 529 x^{2} - 194672$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.1$x^{6} - 46 x^{4} + 529 x^{2} - 194672$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$