# Properties

 Label 2240m Number of curves $3$ Conductor $2240$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("m1")

sage: E.isogeny_class()

## Elliptic curves in class 2240m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2240.k2 2240m1 $$[0, -1, 0, -5, 7]$$ $$-262144/35$$ $$-2240$$ $$[]$$ $$96$$ $$-0.62458$$ $$\Gamma_0(N)$$-optimal
2240.k3 2240m2 $$[0, -1, 0, 35, -25]$$ $$71991296/42875$$ $$-2744000$$ $$[]$$ $$288$$ $$-0.075270$$
2240.k1 2240m3 $$[0, -1, 0, -525, -4673]$$ $$-250523582464/13671875$$ $$-875000000$$ $$[]$$ $$864$$ $$0.47404$$

## Rank

sage: E.rank()

The elliptic curves in class 2240m have rank $$1$$.

## Complex multiplication

The elliptic curves in class 2240m do not have complex multiplication.

## Modular form2240.2.a.m

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 