# Properties

 Label 6240.x Number of curves $4$ Conductor $6240$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("x1")

E.isogeny_class()

## Elliptic curves in class 6240.x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6240.x1 6240k2 $$[0, 1, 0, -1256, -17556]$$ $$428320044872/73125$$ $$37440000$$ $$[2]$$ $$3072$$ $$0.45975$$
6240.x2 6240k3 $$[0, 1, 0, -536, 4440]$$ $$33324076232/1285245$$ $$658045440$$ $$[2]$$ $$3072$$ $$0.45975$$
6240.x3 6240k1 $$[0, 1, 0, -86, -240]$$ $$1111934656/342225$$ $$21902400$$ $$[2, 2]$$ $$1536$$ $$0.11317$$ $$\Gamma_0(N)$$-optimal
6240.x4 6240k4 $$[0, 1, 0, 239, -1345]$$ $$367061696/426465$$ $$-1746800640$$ $$[2]$$ $$3072$$ $$0.45975$$

## Rank

sage: E.rank()

The elliptic curves in class 6240.x have rank $$0$$.

## Complex multiplication

The elliptic curves in class 6240.x do not have complex multiplication.

## Modular form6240.2.a.x

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} + 4 q^{7} + q^{9} + 4 q^{11} - q^{13} - q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.