# Properties

 Label 1216.o Number of curves $3$ Conductor $1216$ CM no Rank $1$ Graph # Learn more

Show commands for: SageMath
sage: E = EllipticCurve("o1")

sage: E.isogeny_class()

## Elliptic curves in class 1216.o

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1216.o1 1216d3 $$[0, -1, 0, -3077, -64681]$$ $$-50357871050752/19$$ $$-1216$$ $$[]$$ $$432$$ $$0.38001$$
1216.o2 1216d2 $$[0, -1, 0, -37, -81]$$ $$-89915392/6859$$ $$-438976$$ $$[]$$ $$144$$ $$-0.16929$$
1216.o3 1216d1 $$[0, -1, 0, 3, -1]$$ $$32768/19$$ $$-1216$$ $$[]$$ $$48$$ $$-0.71860$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1216.o have rank $$1$$.

## Complex multiplication

The elliptic curves in class 1216.o do not have complex multiplication.

## Modular form1216.2.a.o

sage: E.q_eigenform(10)

$$q + 2q^{3} - 3q^{5} - q^{7} + q^{9} - 3q^{11} + 4q^{13} - 6q^{15} - 3q^{17} - q^{19} + O(q^{20})$$ ## 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. 