# Properties

 Label 5040.bm Number of curves $4$ Conductor $5040$ CM no Rank $1$ Graph # Learn more

Show commands: SageMath
sage: E = EllipticCurve("bm1")

sage: E.isogeny_class()

## Elliptic curves in class 5040.bm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5040.bm1 5040bn3 $$[0, 0, 0, -38547, -2912814]$$ $$2121328796049/120050$$ $$358467379200$$ $$$$ $$12288$$ $$1.2814$$
5040.bm2 5040bn4 $$[0, 0, 0, -12627, 510354]$$ $$74565301329/5468750$$ $$16329600000000$$ $$$$ $$12288$$ $$1.2814$$
5040.bm3 5040bn2 $$[0, 0, 0, -2547, -40014]$$ $$611960049/122500$$ $$365783040000$$ $$[2, 2]$$ $$6144$$ $$0.93487$$
5040.bm4 5040bn1 $$[0, 0, 0, 333, -3726]$$ $$1367631/2800$$ $$-8360755200$$ $$$$ $$3072$$ $$0.58829$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 5040.bm have rank $$1$$.

## Complex multiplication

The elliptic curves in class 5040.bm do not have complex multiplication.

## Modular form5040.2.a.bm

sage: E.q_eigenform(10)

$$q + q^{5} + q^{7} + 4q^{11} - 6q^{13} - 2q^{17} + O(q^{20})$$ ## 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. 