# Properties

 Label 5040m Number of curves $6$ Conductor $5040$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("5040.l1")

sage: E.isogeny_class()

## Elliptic curves in class 5040m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5040.l4 5040m1 [0, 0, 0, -1578, 24127]  2048 $$\Gamma_0(N)$$-optimal
5040.l3 5040m2 [0, 0, 0, -1623, 22678] [2, 2] 4096
5040.l2 5040m3 [0, 0, 0, -6123, -160022] [2, 2] 8192
5040.l5 5040m4 [0, 0, 0, 2157, 112642]  8192
5040.l1 5040m5 [0, 0, 0, -94323, -11149742]  16384
5040.l6 5040m6 [0, 0, 0, 10077, -863102]  16384

## Rank

sage: E.rank()

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

## Modular form5040.2.a.l

sage: E.q_eigenform(10)

$$q - q^{5} + q^{7} - 4q^{11} - 2q^{13} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 