# Properties

 Label 225.a Number of curves 2 Conductor 225 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("225.a1")

sage: E.isogeny_class()

## Elliptic curves in class 225.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
225.a1 225e1 [0, 0, 1, -75, 256] [] 48 $$\Gamma_0(N)$$-optimal
225.a2 225e2 [0, 0, 1, 375, -12344] [] 240

## Rank

sage: E.rank()

The elliptic curves in class 225.a have rank $$1$$.

## Modular form225.2.a.a

sage: E.q_eigenform(10)

$$q - 2q^{2} + 2q^{4} - 3q^{7} - 2q^{11} + q^{13} + 6q^{14} - 4q^{16} - 2q^{17} - 5q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 