# Properties

 Label 40425y Number of curves 4 Conductor 40425 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("40425.p1")

sage: E.isogeny_class()

## Elliptic curves in class 40425y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
40425.p3 40425y1 [1, 1, 1, -7988, 269156]  55296 $$\Gamma_0(N)$$-optimal
40425.p2 40425y2 [1, 1, 1, -14113, -208594] [2, 2] 110592
40425.p4 40425y3 [1, 1, 1, 53262, -1556094]  221184
40425.p1 40425y4 [1, 1, 1, -179488, -29314594]  221184

## Rank

sage: E.rank()

The elliptic curves in class 40425y have rank $$1$$.

## Modular form 40425.2.a.p

sage: E.q_eigenform(10)

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

## 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 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. 