# Properties

 Label 49.a Number of curves $4$ Conductor $49$ CM -7 Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("49.a1")

sage: E.isogeny_class()

## Elliptic curves in class 49.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
49.a1 49a4 [1, -1, 0, -1822, 30393]  14
49.a2 49a3 [1, -1, 0, -107, 552]  7
49.a3 49a2 [1, -1, 0, -37, -78]  2
49.a4 49a1 [1, -1, 0, -2, -1]  1 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 49.a have rank $$0$$.

## Modular form49.2.a.a

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 3q^{8} - 3q^{9} + 4q^{11} - q^{16} - 3q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 