# Properties

 Label 51984.i Number of curves $3$ Conductor $51984$ CM no Rank $0$ Graph # Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 51984.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51984.i1 51984cw3 $$[0, 0, 0, -39993024, 97347427504]$$ $$-50357871050752/19$$ $$-2669086710706176$$ $$[]$$ $$1866240$$ $$2.7481$$
51984.i2 51984cw2 $$[0, 0, 0, -485184, 138387184]$$ $$-89915392/6859$$ $$-963540302564929536$$ $$[]$$ $$622080$$ $$2.1988$$
51984.i3 51984cw1 $$[0, 0, 0, 34656, 109744]$$ $$32768/19$$ $$-2669086710706176$$ $$[]$$ $$207360$$ $$1.6495$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 51984.i have rank $$0$$.

## Complex multiplication

The elliptic curves in class 51984.i do not have complex multiplication.

## Modular form 51984.2.a.i

sage: E.q_eigenform(10)

$$q - 3q^{5} + q^{7} + 3q^{11} + 4q^{13} + 3q^{17} + O(q^{20})$$ ## 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. 