# Properties

 Label 825.a Number of curves 4 Conductor 825 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("825.a1")

sage: E.isogeny_class()

## Elliptic curves in class 825.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
825.a1 825b3 [1, 0, 0, -3663, 84942]  768
825.a2 825b2 [1, 0, 0, -288, 567] [2, 2] 384
825.a3 825b1 [1, 0, 0, -163, -808]  192 $$\Gamma_0(N)$$-optimal
825.a4 825b4 [1, 0, 0, 1087, 4692]  768

## Rank

sage: E.rank()

The elliptic curves in class 825.a have rank $$1$$.

## Modular form825.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} - q^{6} - 4q^{7} + 3q^{8} + q^{9} + q^{11} - q^{12} + 2q^{13} + 4q^{14} - q^{16} + 2q^{17} - q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 