# Properties

 Label 25200.ej Number of curves $4$ Conductor $25200$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("25200.ej1")

sage: E.isogeny_class()

## Elliptic curves in class 25200.ej

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25200.ej1 25200cx3 [0, 0, 0, -378675, 89579250] [2] 165888
25200.ej2 25200cx4 [0, 0, 0, -270675, 141743250] [2] 331776
25200.ej3 25200cx1 [0, 0, 0, -18675, -860750] [2] 55296 $$\Gamma_0(N)$$-optimal
25200.ej4 25200cx2 [0, 0, 0, 29325, -4556750] [2] 110592

## Rank

sage: E.rank()

The elliptic curves in class 25200.ej have rank $$1$$.

## Modular form 25200.2.a.ej

sage: E.q_eigenform(10)

$$q + q^{7} - 2q^{13} - 2q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.