# Properties

 Label 20691.i Number of curves $3$ Conductor $20691$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 20691.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
20691.i1 20691o3 $$[0, 0, 1, -837804, -295162893]$$ $$-50357871050752/19$$ $$-24537891411$$ $$[]$$ $$97200$$ $$1.7817$$
20691.i2 20691o2 $$[0, 0, 1, -10164, -419598]$$ $$-89915392/6859$$ $$-8858178799371$$ $$[]$$ $$32400$$ $$1.2324$$
20691.i3 20691o1 $$[0, 0, 1, 726, -333]$$ $$32768/19$$ $$-24537891411$$ $$[]$$ $$10800$$ $$0.68308$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 20691.i have rank $$1$$.

## Complex multiplication

The elliptic curves in class 20691.i do not have complex multiplication.

## Modular form 20691.2.a.i

sage: E.q_eigenform(10)

$$q - 2q^{4} - 3q^{5} + q^{7} + 4q^{13} + 4q^{16} - 3q^{17} - q^{19} + O(q^{20})$$ ## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels. 