# Properties

 Label 1216.e Number of curves $3$ Conductor $1216$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("e1")

E.isogeny_class()

## Elliptic curves in class 1216.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1216.e1 1216b3 $$[0, -1, 0, -5473, -1251871]$$ $$-69173457625/2550136832$$ $$-668503069687808$$ $$[]$$ $$3456$$ $$1.5249$$
1216.e2 1216b1 $$[0, -1, 0, -993, 12385]$$ $$-413493625/152$$ $$-39845888$$ $$[]$$ $$384$$ $$0.42628$$ $$\Gamma_0(N)$$-optimal
1216.e3 1216b2 $$[0, -1, 0, 607, 45601]$$ $$94196375/3511808$$ $$-920599396352$$ $$[]$$ $$1152$$ $$0.97558$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1216.2.a.e

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 