# Properties

 Label 64715e Number of curves $3$ Conductor $64715$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 64715e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
64715.c2 64715e1 $$[0, -1, 1, -2465, -51447]$$ $$-262144/35$$ $$-221247706715$$ $$[]$$ $$51408$$ $$0.90945$$ $$\Gamma_0(N)$$-optimal
64715.c3 64715e2 $$[0, -1, 1, 16025, 127906]$$ $$71991296/42875$$ $$-271028440725875$$ $$[]$$ $$154224$$ $$1.4588$$
64715.c1 64715e3 $$[0, -1, 1, -242835, 48262923]$$ $$-250523582464/13671875$$ $$-86424885435546875$$ $$[]$$ $$462672$$ $$2.0081$$

## Rank

sage: E.rank()

The elliptic curves in class 64715e have rank $$1$$.

## Complex multiplication

The elliptic curves in class 64715e do not have complex multiplication.

## Modular form 64715.2.a.e

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{4} + q^{5} - q^{7} - 2q^{9} - 3q^{11} + 2q^{12} + 5q^{13} - q^{15} + 4q^{16} + 3q^{17} - 2q^{19} + O(q^{20})$$ ## 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 Cremona 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 Cremona labels. 