# Properties

 Label 630e Number of curves $4$ Conductor $630$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("630.d1")

sage: E.isogeny_class()

## Elliptic curves in class 630e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
630.d4 630e1 [1, -1, 0, 21, 53]  128 $$\Gamma_0(N)$$-optimal
630.d3 630e2 [1, -1, 0, -159, 665] [2, 2] 256
630.d2 630e3 [1, -1, 0, -789, -7777]  512
630.d1 630e4 [1, -1, 0, -2409, 46115]  512

## Rank

sage: E.rank()

The elliptic curves in class 630e have rank $$1$$.

## Modular form630.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - q^{10} - 4q^{11} - 6q^{13} + q^{14} + q^{16} - 2q^{17} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 