# Properties

 Label 6240a Number of curves $4$ Conductor $6240$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 6240a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6240.d3 6240a1 $$[0, -1, 0, -66, 216]$$ $$504358336/38025$$ $$2433600$$ $$[2, 2]$$ $$768$$ $$-0.029371$$ $$\Gamma_0(N)$$-optimal
6240.d2 6240a2 $$[0, -1, 0, -216, -924]$$ $$2186875592/428415$$ $$219348480$$ $$$$ $$1536$$ $$0.31720$$
6240.d1 6240a3 $$[0, -1, 0, -1041, 13281]$$ $$30488290624/195$$ $$798720$$ $$$$ $$1536$$ $$0.31720$$
6240.d4 6240a4 $$[0, -1, 0, 64, 840]$$ $$55742968/658125$$ $$-336960000$$ $$$$ $$1536$$ $$0.31720$$

## Rank

sage: E.rank()

The elliptic curves in class 6240a have rank $$1$$.

## Complex multiplication

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

## Modular form6240.2.a.a

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 