# Properties

 Label 18240.bd Number of curves $4$ Conductor $18240$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("18240.bd1")

sage: E.isogeny_class()

## Elliptic curves in class 18240.bd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
18240.bd1 18240cb3 [0, -1, 0, -39745, 3058657]  49152
18240.bd2 18240cb2 [0, -1, 0, -3265, 16225] [2, 2] 24576
18240.bd3 18240cb1 [0, -1, 0, -1985, -33183]  12288 $$\Gamma_0(N)$$-optimal
18240.bd4 18240cb4 [0, -1, 0, 12735, 115425]  49152

## Rank

sage: E.rank()

The elliptic curves in class 18240.bd have rank $$1$$.

## Modular form 18240.2.a.bd

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} + q^{9} + 4q^{11} - 2q^{13} - q^{15} + 2q^{17} - q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 