# Properties

 Label 525a Number of curves 4 Conductor 525 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("525.a1")

sage: E.isogeny_class()

## Elliptic curves in class 525a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
525.a3 525a1 [1, 1, 1, -63, 156]  96 $$\Gamma_0(N)$$-optimal
525.a2 525a2 [1, 1, 1, -188, -844] [2, 2] 192
525.a1 525a3 [1, 1, 1, -2813, -58594]  384
525.a4 525a4 [1, 1, 1, 437, -4594]  384

## Rank

sage: E.rank()

The elliptic curves in class 525a have rank $$1$$.

## Modular form525.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} + q^{6} - q^{7} + 3q^{8} + q^{9} + q^{12} + 6q^{13} + q^{14} - q^{16} - 2q^{17} - q^{18} - 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 