# Properties

 Label 240.d Number of curves $8$ Conductor $240$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("d1")

E.isogeny_class()

## Elliptic curves in class 240.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
240.d1 240d7 $$[0, 1, 0, -34560, 2461428]$$ $$1114544804970241/405$$ $$1658880$$ $$[4]$$ $$256$$ $$0.98402$$
240.d2 240d5 $$[0, 1, 0, -2160, 37908]$$ $$272223782641/164025$$ $$671846400$$ $$[2, 4]$$ $$128$$ $$0.63744$$
240.d3 240d8 $$[0, 1, 0, -1760, 52788]$$ $$-147281603041/215233605$$ $$-881596846080$$ $$[4]$$ $$256$$ $$0.98402$$
240.d4 240d3 $$[0, 1, 0, -1280, -18060]$$ $$56667352321/15$$ $$61440$$ $$[2]$$ $$64$$ $$0.29087$$
240.d5 240d4 $$[0, 1, 0, -160, 308]$$ $$111284641/50625$$ $$207360000$$ $$[2, 4]$$ $$64$$ $$0.29087$$
240.d6 240d2 $$[0, 1, 0, -80, -300]$$ $$13997521/225$$ $$921600$$ $$[2, 2]$$ $$32$$ $$-0.055704$$
240.d7 240d1 $$[0, 1, 0, 0, -12]$$ $$-1/15$$ $$-61440$$ $$[2]$$ $$16$$ $$-0.40228$$ $$\Gamma_0(N)$$-optimal
240.d8 240d6 $$[0, 1, 0, 560, 2900]$$ $$4733169839/3515625$$ $$-14400000000$$ $$[4]$$ $$128$$ $$0.63744$$

## Rank

sage: E.rank()

The elliptic curves in class 240.d have rank $$0$$.

## Complex multiplication

The elliptic curves in class 240.d do not have complex multiplication.

## Modular form240.2.a.d

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.