# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("5040.bm1")

sage: E.isogeny_class()

## Elliptic curves in class 5040.bm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5040.bm1 5040bn3 [0, 0, 0, -38547, -2912814]  12288
5040.bm2 5040bn4 [0, 0, 0, -12627, 510354]  12288
5040.bm3 5040bn2 [0, 0, 0, -2547, -40014] [2, 2] 6144
5040.bm4 5040bn1 [0, 0, 0, 333, -3726]  3072 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## 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. 