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This is a model for the curve $XY(X+Y)=Z^3$.

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 1, 0, 0])

gp: E = ellinit([0, 0, 1, 0, 0])

magma: E := EllipticCurve([0, 0, 1, 0, 0]);

$$y^2+y=x^3$$ ## Mordell-Weil group structure

$\Z/{3}\Z$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(0, 0\right)$$ ## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(0, 0\right)$$, $$\left(0, -1\right)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$27$$ = $3^{3}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-27$ = $-1 \cdot 3^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$0$$ = $0$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z[(1+\sqrt{-3})/2]$$ (potential complex multiplication) Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $-1.0464643562610104921410578866\dots$ Stable Faltings height: $-1.3211174284280379149898691958\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $5.2999162508563498719410684990\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $1$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $3$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) sage: r = E.rank(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: ar = ellanalyticrank(E); gp: ar/factorial(ar)  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L(E,1)$ ≈ $0.58887958342848331910456316654932546833$

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - 2q^{4} - q^{7} + 5q^{13} + 4q^{16} - 7q^{19} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3 $\Gamma_0(N)$-optimal: no Manin constant: 3

## Local data

This elliptic curve is not semistable. There is only one prime of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $1$ $II$ Additive -1 3 3 0

## Galois representations

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

The mod-$\ell$ Galois representation has maximal image for all primes $\ell < 1000$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image
$3$ 3Cs.1.1

For all other primes $\ell$ the image is the normalizer of a split Cartan subgroup if $\left(\frac{ -3 }{\ell}\right)=+1$ or the normalizer of a nonsplit Cartan subgroup if $\left(\frac{ -3 }{\ell}\right)=-1$.

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

All $p$-adic regulators are identically $1$ since the rank is $0$.

## Iwasawa invariants

$p$ Reduction type 2 3 ss add 0,5 - 0,0 -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3 and 9.
Its isogeny class 27.a consists of 3 curves linked by isogenies of degrees dividing 27.

## Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/3\Z \times \Z/3\Z$$ 2.0.3.1-81.1-CMa1 $3$ 3.1.108.1 $$\Z/6\Z$$ Not in database $3$ $$\Q(\zeta_{9})^+$$ $$\Z/9\Z$$ 3.3.81.1-27.1-a4 $6$ 6.0.34992.1 $$\Z/6\Z \times \Z/6\Z$$ Not in database $6$ $$\Q(\zeta_{9})$$ $$\Z/3\Z \times \Z/9\Z$$ Not in database $9$ 9.3.918330048.1 $$\Z/18\Z$$ Not in database $12$ 12.2.15045919506432.1 $$\Z/12\Z$$ Not in database $12$ 12.0.241162079949.1 $$\Z/3\Z \times \Z/21\Z$$ Not in database $18$ 18.0.4052555153018976267.1 $$\Z/9\Z \times \Z/9\Z$$ Not in database $18$ 18.0.2529990231179046912.1 $$\Z/6\Z \times \Z/18\Z$$ Not in database

We only show fields where the torsion growth is primitive.

An explicit birational isomorphism from $XY(X+Y)=Z^3$ to $y^2+y = x^3$ is $(x,y) = (Y/X,Z/X)$. The $3$-isogeny from the Fermat cubic $X_1^3 + Y_1^3 = Z_1^3$ is simply $(X:Y:Z) = (X_1^3 : Y_1^3 : X_1 Y_1 Z_1)$.