# Properties

 Label 4010.e Number of curves 2 Conductor 4010 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("4010.e1")

sage: E.isogeny_class()

## Elliptic curves in class 4010.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4010.e1 4010e2 [1, 1, 1, -23975, -1742523] [] 21000
4010.e2 4010e1 [1, 1, 1, -175, 9717]  4200 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 4010.e have rank $$0$$.

## Modular form4010.2.a.e

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 