# Properties

 Label 287490.cs Number of curves $4$ Conductor $287490$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("cs1")

sage: E.isogeny_class()

## Elliptic curves in class 287490.cs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
287490.cs1 287490cs3 [1, 1, 1, -511350, 140529405]  3317760
287490.cs2 287490cs2 [1, 1, 1, -32200, 2150885] [2, 2] 1658880
287490.cs3 287490cs1 [1, 1, 1, -4820, -83323]  829440 $$\Gamma_0(N)$$-optimal
287490.cs4 287490cs4 [1, 1, 1, 8870, 7309277]  3317760

## Rank

sage: E.rank()

The elliptic curves in class 287490.cs have rank $$1$$.

## Complex multiplication

The elliptic curves in class 287490.cs do not have complex multiplication.

## Modular form 287490.2.a.cs

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - q^{7} + q^{8} + q^{9} + q^{10} - 4q^{11} - q^{12} + 2q^{13} - q^{14} - q^{15} + q^{16} + 6q^{17} + q^{18} + O(q^{20})$$

## 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. 