# Properties

 Label 121.d Number of curves $3$ Conductor $121$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("121.d1")

sage: E.isogeny_class()

## Elliptic curves in class 121.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
121.d1 121d3 [0, -1, 1, -946260, 354609639] [] 600
121.d2 121d2 [0, -1, 1, -1250, 31239] [] 120
121.d3 121d1 [0, -1, 1, -40, -221] [] 24 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 121.d have rank $$0$$.

## Modular form121.2.a.d

sage: E.q_eigenform(10)

$$q + 2q^{2} - q^{3} + 2q^{4} + q^{5} - 2q^{6} + 2q^{7} - 2q^{9} + 2q^{10} - 2q^{12} - 4q^{13} + 4q^{14} - q^{15} - 4q^{16} + 2q^{17} - 4q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 