# Properties

 Label 6240.e Number of curves $4$ Conductor $6240$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("e1")

E.isogeny_class()

## Elliptic curves in class 6240.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6240.e1 6240c2 $$[0, -1, 0, -696, -6840]$$ $$72929847752/5265$$ $$2695680$$ $$$$ $$2048$$ $$0.28534$$
6240.e2 6240c3 $$[0, -1, 0, -241, 1441]$$ $$379503424/24375$$ $$99840000$$ $$$$ $$2048$$ $$0.28534$$
6240.e3 6240c1 $$[0, -1, 0, -46, -80]$$ $$171879616/38025$$ $$2433600$$ $$[2, 2]$$ $$1024$$ $$-0.061236$$ $$\Gamma_0(N)$$-optimal
6240.e4 6240c4 $$[0, -1, 0, 104, -620]$$ $$240641848/428415$$ $$-219348480$$ $$$$ $$2048$$ $$0.28534$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form6240.2.a.e

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + q^{9} + 4 q^{11} + q^{13} + q^{15} + 6 q^{17} + 4 q^{19} + O(q^{20})$$

## 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. 