# Properties

 Label 250025.a Number of curves $2$ Conductor $250025$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("250025.a1")

sage: E.isogeny_class()

## Elliptic curves in class 250025.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
250025.a1 250025a2 [1, -1, 1, -1348980, 588739022]  3732480
250025.a2 250025a1 [1, -1, 1, -1339855, 597280022]  1866240 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 250025.a have rank $$1$$.

## Modular form 250025.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + 4q^{7} + 3q^{8} - 3q^{9} + 4q^{11} - 4q^{14} - q^{16} - 2q^{17} + 3q^{18} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 